diff --git a/-NE1T4oBgHgl3EQfoQQY/content/tmp_files/2301.03317v1.pdf.txt b/-NE1T4oBgHgl3EQfoQQY/content/tmp_files/2301.03317v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15a6d40e49d6e68ec3cd9d571e42444924303b4b --- /dev/null +++ b/-NE1T4oBgHgl3EQfoQQY/content/tmp_files/2301.03317v1.pdf.txt @@ -0,0 +1,2445 @@ +1 +ATM-R: An Adaptive Tradeoff Model with +Reference Points for Constrained Multiobjective +Evolutionary Optimization +Bing-Chuan Wang, Yunchuan Qin, Xian-Bing Meng, Zhi-Zhong Liu +Abstract—The goal of constrained multiobjective evolutionary +optimization is to obtain a set of well-converged and well- +distributed feasible solutions. To complete this goal, there should +be a tradeoff among feasibility, diversity, and convergence. +However, it is nontrivial to balance these three elements simulta- +neously by using a single tradeoff model since the importance of +each element varies in different evolutionary phases. As an alter- +native, we adapt different tradeoff models in different phases and +propose a novel algorithm called ATM-R. In the infeasible phase, +ATM-R takes the tradeoff between diversity and feasibility into +account, aiming to move the population toward feasible regions +from diverse search directions. In the semi-feasible phase, ATM-R +promotes the transition from “the tradeoff between feasibility and +diversity” to “the tradeoff between diversity and convergence”, +which can facilitate the discovering of enough feasible regions +and speed up the search for the feasible Pareto optima in +succession. In the feasible phase, the tradeoff between diversity +and convergence is considered to attain a set of well-converged +and well-distributed feasible solutions. It is worth noting that the +merits of reference points are leveraged in ATM-R to accomplish +these tradeoff models. Also, in ATM-R, a multiphase mating +selection strategy is developed to generate promising solutions +beneficial to different evolutionary phases. Systemic experiments +on a wide range of benchmark test functions demonstrate that +ATM-R is effective and competitive, compared against five state- +of-the-art constrained multiobjective optimization evolutionary +algorithms. +Index Terms—Constrained multiobjective evolutionary opti- +mization, adaptive tradeoff model, reference point, multiphase +mating selection +I. INTRODUCTION +M +ANY scientific or engineering problems involve the +optimization of conflicting objectives subject to con- +straints, which can be formulated as constrained multiobjective +optimization problems (CMOPs) [1]: +min +F(x) = (f1(x), f2(x), · · · , fm(x))T ∈ Rm +s.t. +gj(x) < 0, j = 1, · · · , ng +hj(x) = 0, j = ng + 1, · · · , ng + nh +xj ≤ xj ≤ xj, j = 1, · · · , D +, +(1) +B.-C. Wang is with the School of Automation, Central South University, +Changsha 410083, China (email: bingcwang@csu.edu.cn). +Y. Qin and Z.-Z. Liu are with the College of Information Science and +Electronic Engineering, Hunan University, Changsha 410082, China (e-mail: +liuzz@hnu.edu.cn; qinyunchuan@hnu.edu.cn). +X.-B. Meng is with the School of Computer Science and Engineering, +South China University of Technology, Guangzhou 510006, China (e-mail: +axbmeng@scut.edu.cn). +where F(x) denotes the objective vector consisting of m +conflicting objectives (i.e., fi(x), i += +1, · · · , m); x += +(x1, · · · , xD)T is a D-dimensional decision vector/solution; +xj and xj are the lower and upper bounds of xj, respectively; +S = �D +j=1[xj, xj] refers to the decision space; gj(x) and +hj(x) represent the jth inequality and (j − ng)th equality +constraints, respectively; ng and nh are the numbers of the +inequality and equality constraints, respectively. +When solving a CMOP, we always quantify constraint +violation by the degree of constraint violation: +G(x) = +ng+nh +� +j=1 +Gj(x). +(2) +Gj(x) denotes the degree of constraint violation of the jth +constraint [2]: +Gj(x) = +� +max(0, gj(x)), +1 ≤ j ≤ ng +max(0, |hj(x)| − δ), +ng + 1 ≤ j ≤ ng + nh +(3) +where δ is a small positive value used to relax an equality +constraint to some degree. A solution x is called a feasible +solution, if and only if G(x) = 0. All feasible solutions +constitute the feasible region: Ω = {x ∈ RD|G(x) = 0}. For +two solutions xu, xv ∈ Ω, xu is said to Pareto dominate xv, +denoted as xu ≺ xv, if and only if ∀j ∈ {1, · · · , m}, fj(xu) ≤ +fj(xv) � ∃j ∈ {1, · · · , m}, fj(xu) < fj(xv). A solution +xp ∈ Ω is considered as a Pareto optimum if and only if +¬∃xv ∈ Ω, xv ≺ xp. The set of all Pareto optima is called +the constrained Pareto set, and its image in the objective +space is called the constrained Pareto front (CPF). The goal +of constrained multiobjective evolutionary optimization is to +pursue a set of well-converged and well-distributed feasible +solutions to approximate the CPF. +To complete this goal, a consensus has been reached in the +community of constrained multiobjective optimization that a +good tradeoff among feasibility, diversity, and convergence +should be achieved [3]. It is worth noting that the impor- +tance of these three elements varies in different evolutionary +phases. Let us take the element of feasibility for example. In +the infeasible phase, this element is very important because +feasibility information plays an indispensable role in locating +feasible regions, which is crucial for constrained multiobjec- +tive optimization. However, in the feasible phase, this element +is negligible as all solutions become feasible. We only need to +consider the tradeoff between diversity and convergence. Due +arXiv:2301.03317v1 [cs.NE] 9 Jan 2023 + +2 +Convergence +Convergence +tradeoff +tradeoff +transition +Population is +infeasible +Population is +feasible +Population is +semi-feasible +Diversity +Diversity +Feasibility +Feasibility +Fig. 1. Task decomposition of achieving a tradeoff among feasibility, diversity, +and convergence. +to their varied importance, it is nontrivial to balance these three +elements simultaneously by using a single tradeoff model. +As an alternative, we adapt different tradeoff models in +different evolutionary phases, proposing an adaptive tradeoff +model with reference points (ATM-R) to handle CMOPs. +Fig. 1 depicts the tradeoffs considered in ATM-R: +• achieving a tradeoff between feasibility and diversity in +the infeasible phase: when the population is entirely +infeasible, the primary goal is to find as many feasible +regions as possible since the Pareto optima may be scat- +tered in different feasible regions. To this end, a tradeoff +between feasibility and diversity should be achieved to +move the population toward the feasible regions from +diverse search directions. +• promoting the transition from “the tradeoff between fea- +sibility and diversity” to “the tradeoff between diversity +and convergence” in the semi-feasible phase: when the +population is semi-feasible (i.e., the population contains +both infeasible and feasible solutions), two situations +should be considered. In the early stage, only a few +feasible regions are discovered. In this case, the tradeoff +between feasibility and diversity should still be prioritized +to find more promising feasible regions. Once enough +feasible regions are located, in the later stage, attention +should be paid to drive the population toward the CPF +quickly and make them uniformly spread over the CPF +simultaneously. Thus, the tradeoff between convergence +and diversity should be concentrated on. In summary, in +this phase, we should shift from “the tradeoff between +feasibility and diversity” to “the tradeoff between diver- +sity and convergence” [3]. +• achieving a tradeoff between diversity and convergence +in the feasible phase: when the population is completely +feasible, the final task is to move the feasible solutions +toward the CPF quickly while maintaining good diversity. +Apparently, a tradeoff between diversity and convergence +should be realized [4]. +In summary, the core of a CMOEA is how to accomplish +the above tradeoffs. The tradeoff in the feasible phase has +been well studied in the community of evolutionary multiob- +jective optimization. For convenience, in ATM-R, an off-the- +shelf unconstrained multiobjective optimization evolutionary +algorithm (MOEA) is utilized to achieve this tradeoff directly. +As for the tradeoffs in the other two phases, the related +studies remain relatively scarce. Especially for the tradeoff +in the semi-feasible phase, little research focuses on this +topic. Indeed, to achieve the tradeoffs in these two phases, an +important concern is how to deal with the infeasible solutions. +Past experience in the community of evolutionary constrained +multiobjective optimization has shown that the infeasible so- +lutions can not only facilitate maintaining diversity but also +contribute to speeding up the convergence. In ATM-R, the +merits of reference points are leveraged to select different +kinds of infeasible solutions suitable for different evolutionary +phases. In summary, the main contributions of this paper are +as follows: +• Instead of using a single tradeoff model, we adapt dif- +ferent tradeoff models in different evolutionary phases, +proposing a novel constrained multiobjective optimiza- +tion algorithm (CMOEA) called ATM-R. Although it is +inevitable for an algorithm to experience three phases +during the evolution, few attempts have been made to +develop alternate tradeoff models for different phases to +facilitate a more explicit adaptation. +• By leveraging the merits of reference points, we provide +a new perspective that selects promising infeasible so- +lutions suitable for different evolutionary phases. To the +best of our knowledge, relevant work along this direction +remains scarce. +• A multiphase mating selection strategy is developed in +this paper that adaptively selects suitable mating parents +for different evolutionary phases. +• Systemic experiments have been implemented on three +sets of test suites including 36 benchmark CMOPs to +validate the effectiveness of ATM-R. Comparison against +five state-of-the-art CMOEAs suggests that ATM-R is +significantly superior or comparable to the contender +algorithms on most of the test problems. Additionally, the +advantages of some important algorithmic components in +ATM-R have been verified. +The rest of this paper is organized as follows. Section II +conducts a brief review of related CMOEAs. The details of +ATM-R are described in Section III. The performance of ATM- +R is compared with five representative CMOEAs in Section +IV. Section V presents some further analyses of ATM-R in +depth. The concluding remarks and future work are given in +Section VI. +II. RELATED WORK +Constrained multiobjective optimization has become a hot +topic in the community of evolutionary computation and +numerous CMOEAs have been proposed. Based on whether +infeasible solutions are utilized, these CMOEAs can be clas- +sified into two categories: feasibility-driven CMOEAs and +infeasibility-assisted CMOEAs. +A. Feasibility-Driven CMOEAs +A feasibility-driven CMOEA is driven by feasibility infor- +mation, in which feasible solutions are considered to be better +than infeasible ones. Some feasibility-driven CMOEAs use +the constrained dominance principle (CDP) to compare two +solutions [5]. In the CDP, a solution xu is said to be better + +3 +than another solution xv, if one of the following conditions is +met: +• both xu and xv are infeasible, and G(xu) < G(xv); +• xu is feasible and xv is infeasible; +• both xu and xv are feasible, and xu ≺ xv. +Due to its preference for feasible solutions, the CDP can +motivate the population toward feasible regions quickly. It has +been widely integrated with different kinds of MOEAs [6], +[7] and used in a spectrum of engineering optimization prob- +lems [8], [9]. Liu et al. [6] combined an angle-based selection +strategy, the shift-based density estimation strategy, and the +CDP for constrained many-objective optimization. Jain and +Deb [7] proposed a reference-point-based nondominated sort- +ing approach, which is integrated with the CDP for constrained +many-objective optimization. Jan and Khanum [10] embedded +the CDP into the framework of MOEA/D and compared its +performance with that of the stochastic ranking [11]. CDP- +based CMOEAs are often used as the baseline algorithms +when evaluating the performance of a CMOEA [12]–[14]. +The feasibility rule, which is widely used for constrained +single-objective optimization, has been extended to solve +CMOPs. Liu et al. [15] combined the feasibility rule with an +indicator-based MOEA and compared its performance with +that of some other kinds of CMOEAs. Fan et al. [16] carried +out a comparison study on MOEA/D for constrained multiob- +jective optimization. Different constraint-handling techniques +including the feasibility rule are embedded into the framework +of MOEA/D. +Some CMOEAs put emphasis on constraints when the +population contains no feasible solutions. Woldesenbet and +Yen [17] presented a self-adaptive penalty method to solve +CMOPs, in which an adaptive penalty function and a dis- +tance measure are combined for constraint-handling. In fact, +when the population is entirely infeasible, the self-adaptive +penalty method compares two solutions based on constraints +regardless of objectives. Liu and Wang [18] presented a two- +phase CMOEA to solve CMOPs. When the population is +entirely infeasible, all objectives are combined together and the +feasibility rule is used to tackle constraints. Due to the superior +capability of its search algorithm, the two-phase CMOEA can +handle complex constraints in the decision space. Jimenez et +al. [19] designed a CMOEA for constrained multiobjective +optimization, in which the min-max formulation is used to +tackle constraints. In addition, the feasibility rule is used to +compare two solutions when an offspring is inserted into the +new population. Miyakawa et al. +[20] developed a two- +stage nondominated sorting method to solve CMOPs. The +population is divided into several fronts by the nondominated +sorting according to constraints. The obtained fronts are further +partitioned by the nondominated sorting based on objectives. +In this manner, constraints are prior to objectives in environ- +mental selection. +B. Infeasibility-assisted CMOEAs +An infeasibility-assisted CMOEA takes advantage of in- +feasible solutions for constrained multiobjective optimization. +Most state-of-the-art CMOEAs fall into this category. +Some CMOEAs take advantage of infeasible solutions +implicitly by using a comparison criterion that takes both +constraints and objectives into account. Ma and Wang [3] pro- +posed a shifted-based penalty function, in which an infeasible +solution is penalized based on the information provided by the +feasible solutions nearby. Jiao et al. [21] proposed a modified +objective function method. When the population is entirely +infeasible, the modified objective function is equivalent to +a distance measure in which constraints and objectives are +considered equally important. Fan et al. [?] presented an +angle-based CDP for constrained multiobjective optimization. +Given a feasible solution and an infeasible solution, if the +angle between these two solutions is smaller than a predefined +threshold, they would be nondominated each other. Thus, some +infeasible solutions could enter into the new population instead +of some feasible ones. Young [22] proposed a blended ranking +measure to select solutions. By blending an individual’s rank +in the objective space with its rank in the constraint space, an +infeasible solution may be better than a feasible one. Similarly, +Ma et al. [13] designed a new fitness function with two +rankings, in which one ranking value is obtained based on the +CDP and the other is calculated based on the Pareto dominance +without considering constraints. The ε constrained method can +use infeasibility information by tuning a threshold value ε [2]; +thus, it has been widely used to solve CMOPs [23]. Zapotecas- +Mart´ınez and Ponsich [24] combined MOEA/D with the ε +constrained method to solve CMOPs, in which the ε value +is set according to the degree of constraint violation. Fan et +al. [25] improved the ε constrained method by setting the ε +value dynamically. Zhou et al. [26] extended the ε constrained +method to solve CMOPs. When the degree of constraint +violation of an infeasible solution is larger than the ε value, its +diversity will be carefully maintained. The stochastic ranking +that is popular for constrained single-objective optimization +has also been extended to solve CMOPs [15], [27]. +Some CMOEAs leverage the advantages of infeasible so- +lutions explicitly by archiving or coevolution. Ray et al. [28] +proposed an infeasibility-driven EA, in which a small per- +centage of infeasible solutions close to the constraint bound- +aries are maintained. Li et al. [29] designed a two-archive +EA for constrained multiobjective optimization. An archive +is used to promote convergence, while the other is used +to maintain diversity. The diversity archive evolves without +considering constraints; thus, infeasible solutions with good +objective function values can be fully used. Liu et al. [4] +tried to solve CMOPs through bidirectional coevolution. The +CDP is used to drive the main population toward the CPF +from the feasible side of the search space. In addition, a +nondominated sorting procedure and an angle-based selection +scheme are conducted in sequence to motivate the population +toward the CPF within the infeasible region. Tian et al. [30] +developed a coevolutionary framework for constrained mul- +tiobjective optimization. Similarly, one population is updated +by the CDP, while the other is updated by an unconstrained +MOEA. Additionally, the elites of these two populations are +selected to generate offspring. Ishibuchi et al. [31] designed +a dual-grid model of MOEA/D for constrained multiobjective +optimization. Two populations are maintained and infeasible + +4 +solutions with good objective function values are preferred +in the secondary population. Zhu et al. [32] employed two +types of weight vectors in MOEA/D to solve CMOPs. The +solutions associated with the convergence weight vectors are +updated based on the aggregation function, while the solutions +associated with the diversity weight vectors are renewed +according to both the aggregation function and the degree +of constraint violation. Peng et al. [14] used two kinds of +weight vectors for constrained multiobjective optimization. +Specifically, the degree of constraint violation is considered +as another objective. Subsequently, a set of feasible weight +vectors and a set of infeasible weight vectors are used to +update the population. Additionally, the set of infeasible +weight vectors is dynamically adjusted to maintain a number +of infeasible solutions with good objective function values and +small degrees of constraint violation. +Some CMOEAs divide the evolutionary process into several +phases and put emphasis on objectives in one of the phases. +Yang et al. [33] divided the evolutionary process into a +constrained search mode and an unconstrained search mode. +These two search modes are executed by a dynamic constraint- +handling mechanism. Fan et al. [12] proposed a push and pull +search (PPS) framework to solve CMOPs, in which the evo- +lutionary process is divided into two stages: push and pull. In +the push stage, the population is updated by an unconstrained +MOEA. In the pull stage, an improved ε constrained method is +designed to tackle complex constraints. Since its proposition, +the PPS framework has been used in various fields [34], +[35]. Yu et al. [36] proposed a dynamic selection preference- +assisted constrained multiobjective differential evolutionary +(DE) algorithm. The selection preference for a solution shifts +from infeasibility to feasibility as the optimization progresses. +Tian et al. [37] proposed a two-stage CMOEA to balance +objective optimization and constraint sanctification. These two +stages are executed dynamically according to the percentage of +feasible solutions in the population. Recently, Ming et al. [38] +proposed a simple two-stage EA for constrained multiobjective +optimization. The two-stage EA focuses on approaching the +unconstrained Pareto front in the first stage and the feasible +solutions are archived. In the second stage, the method seeks to +approximate the CPF, where the archived feasible solutions are +adopted as the initial population. Peng et al. [39] proposed a +two-phase EA for constrained multiobjective optimization with +deceptive constraints. In the first phase, two subpopulations are +employed to explore the feasible regions and the entire space, +respectively. The second phase aims to approach the CPF. +Additionally, an infeasibility utilization strategy is designed +to leverage the promising information provided by infeasible +solutions. +III. PROPOSED METHOD +The general flow chart of ATM-R is shown in Fig. 2. As its +name implies, ATM-R makes use of reference points to adap- +tively accomplish different tradeoffs in different evolutionary +phases, those are, the infeasible phase, the semi-feasible phase, +and the feasible phase. The details of the update mechanisms +in these three different phases are described in Section III-A, +Infeasible? +Infeasible +Phase +Semi-feasible +Phase +Feasible +Phase +Reproduction +Initialization +Stop? +Semi-feasible? +Output the +Population +Yes +No +Yes +Yes +No +No +Fig. 2. Flow chart of ATM-R. +Algorithm 1: Update Mechanism in the Infeasible +Phase +Input: Population P, offspring population O +Output: New population P +1 Q ← P ∪ O; +2 Divide Q into k fronts based on ˆF(x): F1, · · · , Fk; +3 P ← ∅; +4 for l = 1 : k do +5 +if |P| + |Fl| ≥ N then +6 +Break; +7 +P ← P ∪ Fl; +8 if |P| + |Fl| > N then +9 +Sample n uniformly distributed reference points +and generate corresponding weight vectors: +w1, · · · , wn; +10 +Assign each solution in Fl to a weight vector +according to (5)-(7); +11 +while |P| + |Fl| > N do +12 +Select the weight vector associated with the +largest number of solutions: wc; +13 +Among the solutions assigned to wc, select the +one with the largest value of G(x): xw; +14 +Fl ← Fl\xw; +15 P ← P ∪ Fl; +Section III-B, and Section III-C, respectively. Aside from the +environmental selection procedure, another critical element +of a CMOEA is the mating selection procedure. In ATM- +R, a multiphase mating selection strategy is developed to +generate promising solutions beneficial to different tradeoffs. +The details of this strategy are illustrated in Section III-D. +Finally, the framework of ATM-R and some discussions are +shown in Section III-E and Section III-F, respectively. +A. Update Mechanism in the Infeasible Phase +In this phase, ATM-R aims to strike a balance between feasi- +bility and diversity. In other words, it motivates the population +toward feasibility from diverse search directions, thus locating +as many feasible regions as possible. Algorithm 1 shows how +ATM-R accomplishes this tradeoff. In general, it involves two +essential elements. +1) Nondominated Sorting in the Transformed Objective +Space: Following the ideas in [40], we consider G(x) as +an additional objective function, and transform (1) into an + +5 +unconstrained MOP: +min ˆF(x) = (f1(x), · · · , fm(x), G(x))T ∈ Rm+1. +(4) +Clearly, this transformation does not introduce any extra +parameters. In addition, both objective functions and con- +straints are considered in (4), which can facilitate maintaining +population diversity and enhance driving forces toward the +feasible regions. Based on ˆF(x), the population will be divided +into several fronts, denoted as F1, · · · , Fk, by implementing a +nondominated sorting procedure in the transformed objective +space. Afterward, the solutions in each front will be selected +in turn until �l−1 +i=1 |Fi| < N ≤ �l +i=1 |Fi| where N denotes +the size of the final solution set. +2) Regular Reference Point-based Selection: If �l +i=1 |Fi| +is larger than N, we should further select (n = N−�l−1 +i=1 |Fi|) +solutions from the last desired front Fl. To complete this task, +in this study, a regular reference point-based selection scheme +is developed by taking advantage of uniformly distributed +reference points. Its implementation is quite simple. +• First, a set of regular (i.e., uniformly distributed) refer- +ence points is sampled in the objective space to generate +weight vectors denoted as {w1, · · · , wn} following the +ideas in [41]. +• Subsequently, a solution (denoted as x) in Fl is assigned +to the weight vector with the smallest angle to its nor- +malized objective vector: +I = arg min +j∈{1,··· ,n} +θj, +(5) +θj = arccos +����� +F′(x)Twj +∥F′(x)∥ · ∥wj∥ +����� , j = 1, · · · , n, +(6) +f ′ +j(x) = fj(x) − zmin +j +zmax +j +− zmin +j +, j = 1, · · · , m, +(7) +where I indicates which weight vector the solution x is +assigned to; θj denotes the angle between wj and the nor- +malized objective vector F′(x) = (f ′ +1(x), · · · , f ′ +m(x))T; +∥ · ∥ represents the function to calculate the 2-norm +of a vector; zmax += +(zmax +1 +, · · · , zmax +m )T and zmin += +(zmin +1 , · · · , zmin +m )T refer to the estimated nadir point and +ideal point, respectively. +• Afterward, (|Fl|−n) inferior solutions are deleted one by +one by employing a “diversity first, feasibility second” +strategy. To be specific, it first identifies the weight +vector associated with the largest number of solutions1. +Intuitively, since these solutions are associated with the +same weight vector, they will share highly similar search +directions. To maintain diverse search directions, it is nec- +essary to delete one of them. The feasibility information +of these solutions is considered for the deletion. The one +with the largest value of G(x) is discarded. These two +steps will continue until (|Fl| − n) solutions are deleted. +A simple example is given in Fig. 3 for better understanding +the regular reference point-based selection scheme. We con- +sider a CMOP with two objectives. Suppose there are seven +1Note that the tie is broken at random +A +B +D +C +E +F +Feasible +region +1 +w +2 +w +3 +w +4 +w +' +2f +' +1f +G +G +Fig. 3. Update mechanism in the infeasible phase. +solutions in the population, and they lie in the same front in +the transformed objective space. According to the values of +G(x), these individuals were ranked as F, C, E, A, G, D, and +B in ascending order. The task is to select four solutions for +the next generation. +1) First, four reference points are sampled uniformly to +generate four weight vectors denoted as {w1, · · · , w4}. +2) Next, each solution in the population is assigned to a +weight vector: w1 ↔ {A}, w2 ↔ {B, C}, w3 ↔ {D}, +and w4 ↔ {E, F, G}. +3) Subsequently, three solutions are deleted one by one. +G is first deleted since w4 is matched with the largest +number of solutions and G is the one with the largest +value of G(x) compared with E and F. According to +this principle, B and E will be also removed. +4) Finally, the solutions (i.e., A, C, D, and F will enter into +the next generation. +Remark 1: Both ATMES2 [40] and IDEA [28] employ non- +dominated sorting in the transformed objective space as ATM- +R does. The main difference lies in how to distinguish the +solutions in the same front. Specifically, in ATMES, solutions +are selected based on G(x) only. A solution with a smaller +value of G(x) will be preferred. In this manner, ATMES will +put too much emphasis on constraints. It will cause perfor- +mance deterioration in terms of the search diversity, which +is essential for finding as many promising feasible regions +as possible. On the contrary, in IDEA, only the diversity in +the transformed objective space is considered to update the +last desired front Fl. Unfortunately, this manner will result in +a limited driving force toward the feasible regions, which in +turn leads to a relatively low convergence speed. Unlike these +two methods, ATM-R takes both diversity and feasibility into +account to update Fl, and a “diversity first, feasibility second” +strategy is thus developed. As illustrated in Fig. 3, ATM-R can +strike a good balance between diversity and feasibility, thereby +motivating the population toward feasible regions from diverse +search directions. +B. Update Mechanism in the Semi-feasible Phase +ATM-R intends to promote the transition from “the tradeoff +between feasibility and diversity” to “the tradeoff between +2Although ATMES is originally designed for constrained single-objective +optimization, it can be directly applied to solve CMOPs. + +6 +Algorithm 2: Update Mechanism in the Semi-feasible +Phase +Input: Population P, offspring population O, FEs, +MaxFEs +Output: New population P +1 Q ← P ∪ O, P ← ∅; +2 Qf ← {x ∈ Q|G(x) = 0}, Qif ← {x ∈ Q|G(x) > 0}; +3 if |Qf| > N then +4 +Qf ← N feasible solutions seleted from Qf by an +unconstrained MOEA; +5 P ← P ∪ Qf; +6 if |Qif| > N then +7 +if +F Es +MaxF Es < 0.5 or |Qf| < N then +8 +Qif ← N infeasible solutions selected from +Qif by using Algorithm 1; +9 +else +10 +Generate |Qf| weight vectors by using the +solutions in Qf according to (8)-(9); +11 +Assign each solution in |Qif| to a weight +vector according to (5)-(7); +12 +while |Qif| > N do +13 +Select the weight vector associated with the +largest number of solutions: wc; +14 +Among the solutions assigned to wc, select +the one furthest from the feasible solution +used to generate wc: xw; +15 +Qif ← Qif\xw; +16 P ← P ∪ Qif; +diversity and convergence” in the semi-feasible phase (i.e., +the population contains both infeasible and feasible solutions). +The reasons for this transition are two-fold. In the early +stage of the semi-feasible phase, ATM-R must locate as many +feasible regions as possible. To this end, it must focus on +the tradeoff between feasibility and diversity. After finding a +sufficient number of feasible regions, in the later stage, ATM- +R should steer the population rapidly toward the CPF and +distribute it uniformly along with the CPF simultaneously. +Thus, the tradeoff between convergence and diversity should +be prioritized. Algorithm 2 shows how ATM-R updates the +solutions in the semi-feasible phase. +From Algorithm 2, it is observed that ATM-R updates +the feasible and infeasible solutions separately. To update +the feasible solutions, an unconstrained MOEA is used to +truncate the feasible population Qf if its size is greater than +N; otherwise, all feasible solutions are reserved. To update the +infeasible solutions, ATM-R considers two situations. In the +early stage, it aims to achieve a tradeoff between feasibility +and diversity, which is the same as in the infeasible phase. +Thus, the update mechanism used in the infeasible phase (i.e., +Algorithm 1) can be directly applied in this stage. While in +the later stage, ATM-R shifts the emphasis to the tradeoff +between diversity and convergence. To realize this tradeoff, +an important task is how to preserve those infeasible solutions +that can contribute to both diversity and convergence. ATM-R +1 +w +2 +w +3 +w +' +1f +' +2f +A +B +C +D +E +F +' +1 +w +' +2 +w +' +3 +w +' +4 +w +A +B +C +D +E +F +4 +w +' +1f +' +2f +G +H +I +J +G +H +I +J +Feasible solution +Infeasible solution +CPF +Feasible solution +Infeasible solution +CPF +Feasible +region +Feasible +region +(a) +1 +w +2 +w +3 +w +' +1f +' +2f +A +B +C +D +E +F +' +1 +w +' +2 +w +' +3 +w +' +4 +w +A +B +C +D +E +F +4 +w +' +1f +' +2f +G +H +I +J +G +H +I +J +Feasible solution +Infeasible solution +CPF +Feasible solution +Infeasible solution +CPF +Feasible +region +Feasible +region +(b) +Fig. 4. +Illustration of difference between the weight vectors in the regular +reference point-based selection and those in the adaptive reference point-based +selection: (a) weight vectors in regular reference point-based selection and (b) +weight vectors in adaptive reference point-based selection. +designs the following two steps to accomplish this task. +1) Discovery of the Nondominated Infeasible Solutions: +Compared with the feasible solutions in the current population, +the nondominated infeasible solutions usually have smaller ob- +jective function values. It is natural to leverage their benefits to +promote convergence. To distinguish these infeasible solutions, +we first employ a nondominated sorting procedure to divide +the union population (i.e., Q in Algorithm 2) into several +fronts based on ˆF(x) in +(4). Subsequently, the infeasible +solutions in the first front are picked out. If the number of these +nondominated infeasible solutions (denoted as M) is smaller +than N, all of them will be kept; otherwise, they will be further +distinguished by the following adaptive reference point-based +selection. +2) Adaptive Reference Point-based Selection: Herein, the +regular reference points are no longer used to assist the +selection. The reason is that the CPF might be disconnected +(see Fig. 4), and some weight vectors (i.e., w1 and w3 in +Fig. 4(a)) generated using the uniformly distributed reference +points cannot point to any parts of the CPF. As a result, the +solutions preserved by making use of such weight vectors +(i.e., C and F) are far away from the CPF and hardly con- +tribute to convergence speed, which is not desirable. Instead, +we use adaptive reference points for solution selection. For +convenience, in our study, the feasible solutions are considered +as adaptive reference points since they can deliver important +clues for the localization of the CPF (see Fig. 4(b)). Based +on these reference points, a set of adaptive weight vectors +can be obtained conveniently. To be specific, for the ith + +7 +feasible solution xi, the corresponding weight vector (denoted +as w +′ +i = (w +′ +i,1, · · · , w +′ +i,m)T) is generated as follows: +w +′ +i,j = +f +′ +j(x) +�m +j=1 f +′ +j(x), j = 1, · · · , m, +(8) +f ′ +j(x) = fj(x) − zmin +j +zmax +j +− zmin +j +, j = 1, · · · , m, +(9) +where (f ′ +1(x), · · · , f ′ +m(x))T is the normalized objective vector, +(zmax +1 +, · · · , zmax +m )T and (zmin +1 , · · · , zmin +m )T denote the estimated +nadir point and ideal point, respectively. Fig. 4 shows the +difference between the weight vectors generated using regular +reference points and those using adaptive reference points. +It is evident that the weight vectors obtained using adaptive +reference points fit better to the characteristics of the CPF. +Once the adaptive weight vectors are prepared, the next +procedures in the adaptive reference point-based selection +scheme are quite simple. First, each nondominated infeasi- +ble solution is assigned to a weight vector following the +ideas in the regular reference point-based selection scheme +(see Eqs. (5)-(7)). Afterward, (M-N) infeasible solutions are +deleted one by one in a two-step manner. The first step +is to identify the weight vector associated with the largest +number of solutions. In the second step, among the solutions +assigned to this weight vector, the one furthest from the +feasible solution corresponding to the weight vector in the +objective space will be deleted. In general, the first step is +similar to many decomposition-based approaches and can help +to maintain population diversity. As for the second step, it can +help to retain those infeasible solutions close to the feasible +solutions and thus offer a driving force toward the CPF from +the infeasible side of the search space. Intuitively, this way +can speed up the convergence. +Remark 2: In the semi-feasible phase, the population size +in ATM-R is larger than or equal to N. The reason is that a +larger population can enhance population diversity, which is +critical to both “the tradeoff between feasibility and diversity” +and “the tradeoff between diversity and convergence”. As for +how to determine whether the algorithm has entered the later +stage of the semi-feasible phase, we considered two simple +conditions which should be satisfied simultaneously. The first +condition is that +F Es +MaxF Es should be larger than 0.5. Note +that FEs and MaxFEs denote the function evaluations and +the maximum function evaluations, respectively. The second +condition relies on the number of feasible solutions which +should be equal to N. The first condition implies that enough +search efforts have been devoted to finding feasible regions, +while the second condition is set to ensure a sufficient number +of reference points. In the later stage of the semi-infeasible +phase, if no nondominated infeasible solutions are discovered, +the algorithm will enter the feasible phase. +C. Update Mechanism in the Feasible Phase +In this phase, all solutions are feasible. Under this condition, +only the tradeoff between diversity and convergence should be +considered, thus motivating the feasible solutions toward the +CPF quickly while maintaining good diversity. Apparently, +Algorithm 3: Multiphase Mating Selection Strategy +Input: Population P, N +Output: Mating population C +1 C ← ∅; +2 for i = 1 : N do +3 +Randomly select two different solutions denoted as +xa and xb from P; +4 +if P is entirely infeasible then +5 +if rand < 0.5 then +6 +xm ← the better one between xa and xb +based on the degree of constraint +violation; +7 +else +8 +xm ← the better one between xa and xb +based on the diversity; +9 +else if P is feasible then +10 +if xa ≺ xb then +11 +xm ← xa; +12 +else if xb ≺ xa then +13 +xm ← xb; +14 +else +15 +xm ← the better one between xa and xb +based on the diversity; +16 +else if P is semi-feasible then +17 +if i < N/2 then +18 +xm ← the better one between xa and xb by +using the method in the infeasible phase; +19 +else +20 +xm ← the better one between xa and xb +by using the method in the feasible phase; +21 +C ← C ∪ xm; +a current effective unconstrained MOEA can be applied to +achieve this balance. Thus, in ATM-R, an off-the-shelf uncon- +strained MOEA is employed in this phase straightforwardly. +D. Multiphase Mating Selection Strategy +In addition to the multi-phase strategy in environmental +selection, ATM-R uses a multi-phase strategy for mating selec- +tion. It selects appropriate mating parents suitable for different +evolutionary phases. The details of this multiphase mating +selection strategy are described in Algorithm 3. Similarly, +three different phases are considered in this strategy. +• In the infeasible phase, population diversity and feasi- +bility should be focused on simultaneously. Thus, in the +tournament selection, solutions are compared based on +the diversity and the degree of constraint violation with +the same probability (i.e., 0.5). Note that the diversity is +quantified by the same way as in [30]. +• In the feasible phase, population diversity and conver- +gence should be taken into account. Following the ideas +in NSGA-II [5], in the tournament selection, solutions are +compared based on the Pareto dominance relationship. + +8 +Algorithm 4: ATM-R +Input: A CMOP, N, MaxFEs +Output: Final population P +1 P ← a population initialized from the decision space; +2 FEs ← N; +3 while FEs < MaxFEs do +4 +C ← a mating population selected from P by using +Algorithm 3; +5 +O ← an offspring population generated by +executing genetic operators on C; +6 +FEs ← FEs + N; +7 +Q ← P ∪ O; +8 +if Q is entirely infeasible then +9 +P ← the solutions seleted from Q by using +Algorithm 1; +10 +else if Q is semi-feasible then +11 +P ← the solutions selected from Q by using +Algorithm 2; +12 +else if Q is feasible then +13 +P ← the solutions selected from Q by using an +unconstrained MOEA; +Also, if two solutions do not dominate each other, they +are compared based on the diversity. +• The semi-feasible phase needs to bridge the gap between +the feasible phase and the infeasible phase. Thus, in this +phase, the first half of the mating population is selected +by using the method in the infeasible phase, while the +other half is selected by using the method in the feasible +phase. +E. ATM-R +In summary, the details of ATM-R are given in Algorithm 4. +At the beginning, a population of N solutions is sampled +uniformly in the decision space (Lines 1-2). Afterward, the +population is employed to search for the CPF until the +maximum number of function evaluations is exhausted (Lines +3-15). In the search process, first, N mating parents are +selected for offspring generation by using the multiphase +mating selection strategy in Algorithm 3 (Line 4). Next, +N offspring are produced by the simulated binary crossover +(SBX) [42] and the polynomial mutation (PM) [43] (Lines +5-6). Afterward, promising solutions are selected based on +population feasibility (Lines 7-14) where Algorithm 1 and +Algorithm 2 are used in the infeasible phase and the semi- +feasible phase, respectively. Note that if FEs ≥ MaxFEs, +the final population P would be output. +F. Discussion +In essence, ATM-R is a multiphase CMOEA. ATM-R in- +tends to achieve a tradeoff between diversity and feasibility in +the infeasible phase, promote the transition from “the tradeoff +between feasibility and diversity” to “the tradeoff between +diversity and convergence” in the semi-feasible phase, and +accomplish the tradeoff between diversity and convergence in +the feasible phase. To the best of our knowledge, ATM-R is +the first algorithm considering these tradeoffs simultaneously +during different evolution phases. Also, ATM-R is interesting +in that it selects promising infeasible solutions suitable for +different evolutionary phases by using two kinds of reference +points. As far as we know, relevant studies in this direction +are almost absent. From our analysis, it is apparent that ATM- +R is a brand-new CMOEA for constrained multiobjective +optimization. +The computational time complexity of ATM-R is mainly +determined by the nondominated sorting and the unconstrained +MOEA. Suppose the fast nondominated sorting and NS- +GAII [5] are adopted in ATM-R. In the worst case of the +infeasible phase, no solutions nondominated another in the +transformed objective space. The time complexity of this +nondominated sorting is O((m+1)·N 2). The time complexity +of assigning each solution to a weight vector is O(m·N 2). The +time complexity of selecting N solutions is O(N 2). Thus, the +time complexity of the infeasible phase is O((m + 1) · N 2) + +O(m · N 2) + O(N 2) = O((m + 1) · N 2). In the semi-feasible +phase, the worst-case time complexity of selecting feasible +solutions is O(m·N 2). In the early stage of the semi-feasible +phase, the worst-case time complexity is the same as that of +the infeasible phase: O((m + 1) · N 2). In the worst case of +the later stage, no infeasible solutions nondominated another. +It is the same as that of the infeasible phase. Thus, its time +complexity is O((m + 1) · N 2). The time complexity of the +semi-feasible phase is O(m·N 2)+O((m+1)·N 2)+O((m+ +1) · N 2) = O((m + 1) · N 2). In the feasible phase, the time +complexity is the same as that of NSGAII: O(m · N 2). In +summary, the computational time complexity of ATM-R is +O((m+1)·N 2)+O((m+1)·N 2)+O(m·N 2) = O(m·N 2), +which is indeed acceptable. +IV. PERFORMANCE COMPARISON +In this section, we assess the performance of ATM-R based +on a wide range of benchmark test functions. Specifically, +ATM-R was used to solve three test suites and its performance +was compared with that of five representative CMOEAs. +Note that all experiments were implemented by the PlatEMO +toolbox [44]. +A. Experimental Settings +1) Test Functions: Three test suites consisting of 36 bench- +mark test functions (e.g., MW [45], CTP [46], and LIRC- +MOP [25]) were adopted in our study. These test functions +own various challenging characteristics; thus, they can assess +the performance of a CMOEA adequately. Most state-of-the- +art CMOEAs adopt these test functions for empirical study. +Note that the number of decision variables in MW and +LIRCMOP was set to 15 and 10, respectively. Please see [25], +[45], [46] for the details of these test functions. +2) Peer Algorithms: +For performance comparison, five +representative +CMOEAs +were +taken +into +consideration: +NSGAII-CDP [5], PPS [12], the constrained two-archive EA +(CTAEA) [29], the coevolutionary constrained multiobjective + +9 +TABLE I +THE IGD VALUES OF NSGAII-CDP, PPS, CTAEA, CCMO, TOP, AND ATM-R ON THREE SETS OF BENCHMARK TEST FUNCTIONS. +Test Functions +NSGAII-CDP +mean IGD (std) +PPS +mean IGD (std) +CTAEA +mean IGD (std) +CCMO +mean IGD (std) +ToP +mean IGD (std) +ATM-R +mean IGD (std) +MW1 +4.0545e-2 (1.02e-1) - +2.3190e-2 (4.01e-2) - +2.1884e-3 (9.96e-4) - +1.8990e-3 (1.42e-3) + +NaN (NaN) - +2.1748e-3 (1.70e-3) +MW2 +2.3926e-2 (7.65e-3) - +4.2401e-2 (3.25e-2) - +1.7953e-2 (6.74e-3) ≈ +2.1515e-2 (8.20e-3) - +2.3108e-1 (1.89e-1) - +1.9130e-2 (9.76e-3) +MW3 +7.4318e-2 (2.31e-1) - +7.5935e-3 (9.94e-4) - +5.4804e-3 (4.86e-4) ≈ +5.2178e-3 (4.41e-4) ≈ +5.9698e-1 (2.78e-1) - +5.3646e-3 (4.08e-4) +MW4 +5.5780e-2 (2.97e-3) - +5.3955e-2 (1.73e-3) - +4.6413e-2 (4.99e-4) - +4.1285e-2 (3.48e-4) ≈ +NaN (NaN) - +4.1255e-2 (3.45e-4) +MW5 +4.2761e-1 (3.35e-1) - +1.4507e-1 (1.97e-1) - +1.5758e-2 (3.38e-3) - +4.6474e-3 (7.30e-3) - +NaN (NaN) - +4.0638e-3 (1.06e-2) +MW6 +8.0099e-2 (1.51e-1) - +1.0037e-1 (1.61e-1) - +1.1188e-2 (6.68e-3) ≈ +5.2473e-2 (1.26e-1) - +1.0872e+0 (1.81e-1) - +1.5369e-2 (8.69e-3) +MW7 +1.0205e-1 (1.93e-1) - +2.5520e-2 (1.88e-2) - +7.2156e-3 (5.22e-4) - +4.8994e-3 (4.80e-4) + +4.7226e-1 (2.39e-1) - +5.2004e-3 (4.73e-4) +MW8 +6.1793e-2 (8.78e-3) - +7.4112e-2 (2.69e-2) - +5.5531e-2 (2.47e-3) - +4.9189e-2 (1.58e-2) ≈ +9.5949e-1 (2.01e-1) - +4.6368e-2 (5.74e-3) +MW9 +2.1737e-1 (3.10e-1) - +7.4032e-2 (1.81e-1) - +8.8691e-3 (9.23e-4) ≈ +5.1927e-2 (1.79e-1) - +NaN (NaN) - +9.9563e-3 (2.88e-3) +MW10 +2.3341e-1 (2.36e-1) - +1.3321e-1 (1.48e-1) - +1.7599e-2 (1.22e-2) ≈ +4.2867e-2 (2.54e-2) - +NaN (NaN) - +2.7242e-2 (2.33e-2) +MW11 +4.7335e-1 (3.24e-1) - +1.3565e-2 (2.15e-2) - +1.6564e-2 (2.80e-3) - +6.3416e-3 (5.30e-4) ≈ +9.2141e-1 (1.37e-1) - +6.1791e-3 (2.34e-4) +MW12 +8.2766e-2 (2.23e-1) - +2.9552e-2 (1.20e-1) + +8.0645e-3 (6.84e-4) + +3.0553e-2 (1.40e-1) ≈ +NaN (NaN) - +7.8769e-2 (2.24e-1) +MW13 +2.0642e-1 (2.77e-1) - +1.3735e-1 (6.32e-2) - +3.8211e-2 (2.66e-2) ≈ +8.2172e-2 (4.41e-2) - +8.3328e-1 (5.33e-1) - +5.2527e-2 (3.23e-2) +MW14 +1.2974e-1 (1.27e-2) - +2.5313e-1 (9.28e-2) - +1.1279e-1 (6.91e-3) + +9.8349e-2 (2.41e-3) + +4.9059e-1 (5.81e-1) - +1.1492e-1 (4.72e-2) +CTP1 +8.1699e-2 (6.62e-2) - +1.9234e-2 (1.80e-2) - +1.8672e-2 (3.64e-2) - +4.4317e-3 (1.05e-3) - +3.9400e-3 (1.48e-4) - +3.2367e-3 (7.38e-5) +CTP2 +2.4408e-3 (1.89e-3) - +3.7003e-3 (6.96e-4) - +4.6860e-2 (1.25e-2) - +1.6836e-3 (1.65e-4) - +4.4453e-3 (1.08e-3) - +1.4735e-3 (5.94e-5) +CTP3 +6.2833e-2 (9.95e-2) - +3.1094e-2 (4.07e-3) - +5.8093e-2 (5.79e-3) - +2.2180e-2 (2.24e-3) - +3.2847e-2 (6.99e-3) - +1.0066e-2 (1.60e-3) +CTP4 +2.4494e-1 (1.29e-1) - +1.4930e-1 (1.84e-2) - +1.5350e-1 (1.92e-2) - +1.3538e-1 (2.18e-2) - +1.8414e-1 (3.29e-2) - +7.9556e-2 (1.13e-2) +CTP5 +7.2574e-3 (2.92e-3) - +1.8168e-2 (6.11e-3) - +1.8209e-2 (4.61e-3) - +7.6639e-3 (1.76e-3) - +1.2167e-2 (3.45e-3) - +3.3142e-3 (4.13e-4) +CTP6 +1.1404e-2 (4.04e-4) - +1.3061e-2 (7.81e-4) - +3.8535e-2 (5.23e-3) - +1.0141e-2 (3.56e-4) - +1.5214e-2 (2.78e-3) - +9.7103e-3 (3.13e-4) +CTP7 +1.6882e-3 (1.39e-3) - +1.6825e-3 (7.14e-5) - +1.6364e-3 (1.31e-4) - +1.1669e-3 (4.52e-5) ≈ +1.5176e-3 (5.54e-5) - +1.1599e-3 (4.62e-5) +CTP8 +1.2019e-1 (1.45e-1) - +1.1932e-2 (5.26e-3) - +3.4505e-2 (4.79e-3) - +5.5516e-3 (6.49e-4) - +8.0925e-2 (1.38e-1) - +4.7357e-3 (2.32e-4) +LIRCMOP1 +2.6010e-1 (8.10e-2) - +1.1024e-1 (3.40e-2) - +3.7900e-1 (1.66e-1) - +2.0503e-1 (6.82e-2) - +1.2547e-1 (1.36e-1) - +3.5295e-2 (1.26e-2) +LIRCMOP2 +1.9890e-1 (7.22e-2) - +7.3024e-2 (2.79e-2) - +1.2324e-1 (6.12e-2) - +1.1419e-1 (3.19e-2) - +6.8227e-2 (5.38e-2) - +3.1146e-2 (9.36e-3) +LIRCMOP3 +2.4894e-1 (8.38e-2) - +1.7697e-1 (5.80e-2) - +3.4751e-1 (1.14e-1) - +2.0960e-1 (7.98e-2) - +3.6351e-1 (5.72e-2) - +2.3380e-2 (1.06e-2) +LIRCMOP4 +2.3080e-1 (6.23e-2) - +1.4996e-1 (5.59e-2) - +2.7661e-1 (1.38e-1) - +1.9069e-1 (7.18e-2) - +3.2442e-1 (5.76e-2) - +2.3928e-2 (1.11e-2) +LIRCMOP5 +7.3176e-1 (4.81e-1) - +8.4362e-2 (2.44e-2) - +1.3918e-1 (4.41e-2) - +1.4046e-2 (8.21e-3) ≈ +1.2091e-1 (3.48e-1) - +1.3635e-2 (6.48e-3) +LIRCMOP6 +5.7447e-1 (4.57e-1) - +9.5258e-2 (6.31e-2) - +1.3633e-1 (1.13e-1) - +1.1357e-2 (7.75e-3) ≈ +6.4593e-3 (3.45e-4) + +5.7790e-2 (1.51e-1) +LIRCMOP7 +1.7441e-2 (1.32e-2) ≈ +5.6488e-2 (5.84e-2) - +2.5246e-2 (9.12e-3) - +1.1404e-2 (6.38e-3) ≈ +8.6357e-3 (2.52e-4) + +1.2400e-2 (5.03e-3) +LIRCMOP8 +3.6946e-2 (4.51e-2) - +6.6479e-2 (7.05e-2) - +3.6096e-2 (6.64e-2) - +9.1531e-3 (5.01e-3) ≈ +8.6820e-3 (4.53e-4) + +9.4267e-3 (3.85e-3) +LIRCMOP9 +5.3564e-1 (1.24e-1) - +1.4063e-1 (8.94e-2) ≈ +1.1622e-1 (5.42e-2) ≈ +3.4398e-2 (3.91e-2) + +2.4115e-1 (1.73e-1) - +1.1216e-1 (7.48e-2) +LIRCMOP10 +3.6496e-1 (9.66e-2) - +8.2848e-3 (1.47e-2) - +6.0919e-2 (6.50e-2) - +5.4399e-3 (3.36e-4) + +5.4878e-3 (2.21e-4) + +6.9018e-3 (6.27e-4) +LIRCMOP11 +2.4114e-1 (1.80e-1) - +8.1119e-3 (7.48e-3) - +1.3778e-1 (3.83e-2) - +2.4538e-3 (8.89e-5) + +1.2447e-1 (6.37e-2) - +5.3691e-3 (1.45e-2) +LIRCMOP12 +1.5180e-1 (8.66e-2) - +1.5216e-2 (2.43e-2) ≈ +3.1152e-2 (1.68e-2) - +4.6113e-3 (2.58e-3) ≈ +2.9104e-2 (5.29e-2) ≈ +7.8014e-3 (7.28e-3) +LIRCMOP13 +2.3757e-1 (3.69e-1) - +1.1968e-1 (3.45e-3) - +1.0834e-1 (3.97e-4) - +9.3972e-2 (1.13e-3) - +1.2450e-1 (3.78e-3) - +9.3120e-2 (9.31e-4) +LIRCMOP14 +2.0248e-1 (2.93e-1) - +1.1859e-1 (3.97e-3) - +1.1126e-1 (7.98e-4) - +9.5773e-2 (7.40e-4) - +1.1883e-1 (4.04e-3) - +9.4848e-2 (7.79e-4) ++/-/≈ +0/35/1 +1/33/2 +2/27/7 +6/19/11 +4/25/1 +optimization (CCMO) [30], and the two-phase EA (ToP) [18]. +NSGAII-CDP is a classic CMOEA that is usually adopted +as a baseline algorithm, while the other four CMOEAs are +state-of-the-art algorithms proposed recently. NSGAII-CDP is +a feasibility-driven CMOEA and the others are infeasibility- +assisted CMOEAs. Among these four infeasibility-assisted +CMOEAs, CTAEA and CCMO are multi-population methods +which take advantage of infeasible solutions explicitly by +an archive or an additional population. PPS and ToP are +multiphase methods which divide the evolutionary process into +several phases and put emphasis on objectives in some phases. +Note that ATM-R is also a multiphase method. +3) Performance Metrics: Two frequently used performance +metrics were adopted to assess the performance of a CMOEA: +inverted generational distance (IGD) and hyper-volume (HV). +Both IGD and HV can measure the convergence and coverage +of a solution set. More details of these two metrics can be +found in [47]. +4) Parameter Settings: The parameters involved in the +experiments are given as follows: +• Size of the final solution set: N = 100 for all comparison +CMOEAs; +• MaxFEs: MaxFEs = 60, 000 for the MW and CTP +test suites, and MaxFEs = 300, 000 for the LIRCMOP +test suite; +4.6 +3.89 +3.68 +2.15 +5 +1.68 +4.65 +3.69 +3.78 +2.15 +4.85 +1.87 +0 +1 +2 +3 +4 +5 +6 +NSGAII-CDP +PPS +CTAEA +CCMO +ToP +ATM-R +IGD +HV +Fig. 5. Average rankings of six CMOEAs on 36 test functions in terms of +the IGD/HV value. A lower ranking value denotes a better performance. +• Number of independent runs: 30. +The SBX and PM were used as genetic operators in all +CMOEAs except ToP. The parameters of SBX and PM are as +follows: +• Crossover probability of SBX: 1; +• Mutation probability of PM: 1/D; +• Distribution index of SBX and PM: 20. +In addition, the algorithm-specific parameters of the five +peer CMOEAs were obtained from their original papers. + +10 +TABLE II +THE HV VALUES OF NSGAII-CDP, PPS, CTAEA, CCMO, TOP, AND ATM-R ON THREE SETS OF BENCHMARK TEST FUNCTIONS. +Test Functions +NSGAII-CDP +mean HV (std) +PPS +mean HV (std) +CTAEA +mean HV (std) +CCMO +mean HV (std) +ToP +mean HV (std) +ATM-R +mean HV (std) +MW1 +4.5445e-1 (8.09e-2) - +4.6529e-1 (3.63e-2) - +4.8849e-1 (2.03e-3) - +4.8927e-1 (3.04e-3) + +NaN (NaN) - +4.8853e-1 (3.60e-3) +MW2 +5.4798e-1 (1.15e-2) - +5.2241e-1 (4.44e-2) - +5.5765e-1 (1.14e-2) ≈ +5.5199e-1 (1.30e-2) - +3.2482e-1 (1.46e-1) - +5.5635e-1 (1.54e-2) +MW3 +5.0168e-1 (1.37e-1) - +5.4398e-1 (4.88e-4) + +5.4413e-1 (6.14e-4) + +5.4368e-1 (7.81e-4) + +1.2745e-1 (1.27e-1) - +5.4292e-1 (7.86e-4) +MW4 +8.2309e-1 (5.63e-3) - +8.2478e-1 (2.49e-3) - +8.3814e-1 (4.04e-4) - +8.4116e-1 (4.35e-4) + +NaN (NaN) - +8.4001e-1 (7.93e-4) +MW5 +1.7725e-1 (9.80e-2) - +2.5212e-1 (6.86e-2) - +3.1449e-1 (2.61e-3) - +3.2205e-1 (5.38e-3) - +NaN (NaN) - +3.2214e-1 (6.45e-3) +MW6 +2.8267e-1 (4.89e-2) - +2.5928e-1 (6.08e-2) - +3.1251e-1 (9.93e-3) ≈ +2.9009e-1 (5.16e-2) - +1.2194e-2 (2.75e-2) - +3.0911e-1 (1.20e-2) +MW7 +3.7706e-1 (6.78e-2) ≈ +4.0647e-1 (2.09e-3) - +4.0868e-1 (1.03e-3) - +4.1205e-1 (5.95e-4) + +1.9015e-1 (7.70e-2) - +4.1019e-1 (9.75e-4) +MW8 +4.9733e-1 (2.20e-2) - +4.7275e-1 (5.64e-2) - +5.2198e-1 (1.16e-2) - +5.2798e-1 (3.48e-2) - +4.6501e-2 (7.81e-2) - +5.3338e-1 (1.72e-2) +MW9 +2.6792e-1 (1.71e-1) ≈ +3.4455e-1 (1.00e-1) - +3.9100e-1 (2.43e-3) + +3.7160e-1 (1.01e-1) - +NaN (NaN) - +3.8287e-1 (4.60e-3) +MW10 +3.1175e-1 (1.18e-1) - +3.5982e-1 (7.57e-2) - +4.3564e-1 (1.30e-2) ≈ +4.1378e-1 (1.88e-2) - +NaN (NaN) - +4.2764e-1 (1.94e-2) +MW11 +3.2816e-1 (8.07e-2) - +4.4157e-1 (9.48e-3) - +4.4127e-1 (1.39e-3) - +4.4609e-1 (2.03e-3) - +2.2321e-1 (4.19e-2) - +4.4746e-1 (2.05e-4) +MW12 +5.4172e-1 (1.81e-1) - +5.8181e-1 (1.06e-1) + +6.0052e-1 (7.80e-4) + +5.8415e-1 (1.10e-1) ≈ +NaN (NaN) - +5.4377e-1 (1.82e-1) +MW13 +4.0153e-1 (5.63e-2) - +4.1137e-1 (4.30e-2) - +4.6130e-1 (1.23e-2) ≈ +4.3974e-1 (2.53e-2) - +2.3054e-1 (1.15e-1) - +4.5371e-1 (1.66e-2) +MW14 +4.5123e-1 (5.66e-3) - +4.2008e-1 (2.54e-2) - +4.6575e-1 (3.90e-3) ≈ +4.7246e-1 (1.53e-3) + +3.4138e-1 (1.53e-1) - +4.6217e-1 (1.49e-2) +CTP1 +3.5920e-1 (1.97e-2) - +3.7510e-1 (5.31e-3) - +3.7588e-1 (1.03e-2) - +3.8065e-1 (3.93e-4) - +3.8036e-1 (1.15e-4) - +3.8106e-1 (1.09e-4) +CTP2 +4.3083e-1 (1.66e-3) - +4.2928e-1 (7.86e-4) - +3.9367e-1 (8.22e-3) - +4.3073e-1 (2.94e-4) - +4.2689e-1 (1.40e-3) - +4.3128e-1 (2.45e-4) +CTP3 +3.7267e-1 (5.45e-2) - +3.8376e-1 (4.03e-3) - +3.5588e-1 (7.25e-3) - +3.9219e-1 (2.18e-3) - +3.8119e-1 (6.97e-3) - +4.0507e-1 (1.75e-3) +CTP4 +2.2838e-1 (6.88e-2) - +2.5406e-1 (1.86e-2) - +2.4611e-1 (2.00e-2) - +2.7265e-1 (2.41e-2) - +2.2246e-1 (2.88e-2) - +3.3430e-1 (1.20e-2) +CTP5 +3.9329e-1 (2.59e-2) - +3.8822e-1 (4.53e-3) - +3.5629e-1 (9.01e-3) - +3.9643e-1 (2.52e-3) - +3.8643e-1 (5.29e-3) - +4.0665e-1 (1.86e-3) +CTP6 +4.6359e-1 (3.75e-4) - +4.6198e-1 (6.48e-4) - +4.4896e-1 (2.68e-3) - +4.6381e-1 (3.03e-4) - +4.6034e-1 (1.83e-3) - +4.6468e-1 (2.22e-4) +CTP7 +5.6701e-1 (2.23e-3) - +5.6676e-1 (9.22e-4) - +5.6637e-1 (4.11e-4) - +5.6745e-1 (1.69e-4) ≈ +5.6692e-1 (1.86e-4) - +5.6721e-1 (1.62e-3) +CTP8 +3.4937e-1 (2.45e-2) - +3.6598e-1 (2.71e-3) - +3.5213e-1 (3.79e-3) - +3.6932e-1 (8.68e-4) - +3.5503e-1 (2.42e-2) - +3.7069e-1 (4.25e-4) +LIRCMOP1 +1.3114e-1 (2.17e-2) - +1.9042e-1 (1.06e-2) - +1.0593e-1 (3.85e-2) - +1.4954e-1 (1.82e-2) - +1.8833e-1 (4.61e-2) - +2.2304e-1 (6.36e-3) +LIRCMOP2 +2.5580e-1 (2.95e-2) - +3.2332e-1 (1.35e-2) - +2.9229e-1 (3.67e-2) - +2.9325e-1 (2.05e-2) - +3.2282e-1 (2.82e-2) - +3.4702e-1 (3.44e-3) +LIRCMOP3 +1.1697e-1 (2.61e-2) - +1.4007e-1 (1.86e-2) - +9.9083e-2 (2.08e-2) - +1.2942e-1 (2.47e-2) - +9.1646e-2 (1.42e-2) - +1.9947e-1 (4.24e-3) +LIRCMOP4 +2.1773e-1 (2.72e-2) - +2.4241e-1 (3.29e-2) - +1.8974e-1 (4.86e-2) - +2.3242e-1 (3.14e-2) - +1.8379e-1 (2.31e-2) - +3.0693e-1 (3.76e-3) +LIRCMOP5 +8.9983e-2 (1.05e-1) - +2.4475e-1 (1.22e-2) - +2.4215e-1 (1.32e-2) - +2.8700e-1 (5.09e-3) ≈ +2.6214e-1 (8.89e-2) - +2.8657e-1 (5.90e-3) +LIRCMOP6 +8.6641e-2 (5.68e-2) - +1.7269e-1 (1.24e-2) - +1.4582e-1 (3.86e-2) - +1.9402e-1 (3.37e-3) ≈ +1.9677e-1 (1.59e-4) + +1.8377e-1 (3.56e-2) +LIRCMOP7 +2.8752e-1 (6.87e-3) ≈ +2.6957e-1 (2.12e-2) - +2.8582e-1 (3.18e-3) - +2.9114e-1 (4.01e-3) ≈ +2.9389e-1 (1.55e-4) + +2.8957e-1 (4.05e-3) +LIRCMOP8 +2.8236e-1 (1.47e-2) - +2.6950e-1 (1.80e-2) - +2.8424e-1 (1.40e-2) - +2.9321e-1 (3.40e-3) ≈ +2.9387e-1 (2.04e-4) ≈ +2.9235e-1 (3.25e-3) +LIRCMOP9 +3.5772e-1 (8.15e-2) - +5.2821e-1 (2.63e-2) ≈ +4.9955e-1 (2.81e-2) - +5.5712e-1 (8.54e-3) + +4.9582e-1 (5.19e-2) - +5.3708e-1 (2.00e-2) +LIRCMOP10 +5.1193e-1 (6.34e-2) - +7.0668e-1 (5.59e-3) + +6.7264e-1 (2.74e-2) - +7.0659e-1 (3.77e-4) + +7.0755e-1 (1.24e-4) + +7.0630e-1 (4.01e-4) +LIRCMOP11 +5.3737e-1 (1.32e-1) - +6.9062e-1 (4.88e-3) - +6.4145e-1 (1.47e-2) - +6.9392e-1 (7.61e-5) ≈ +6.1686e-1 (4.29e-2) - +6.9393e-1 (5.71e-5) +LIRCMOP12 +5.5161e-1 (4.68e-2) - +6.1582e-1 (9.12e-3) ≈ +6.0522e-1 (7.22e-3) - +6.1952e-1 (1.29e-3) ≈ +6.0839e-1 (2.46e-2) - +6.1811e-1 (3.04e-3) +LIRCMOP13 +4.7950e-1 (1.63e-1) - +5.3426e-1 (4.14e-3) - +5.4704e-1 (3.37e-4) - +5.5421e-1 (1.49e-3) - +5.1626e-1 (3.56e-3) - +5.5578e-1 (1.23e-3) +LIRCMOP14 +4.9121e-1 (1.36e-1) - +5.3744e-1 (4.86e-3) - +5.4656e-1 (7.33e-4) - +5.5357e-1 (1.22e-3) - +5.2944e-1 (4.28e-3) - +5.5604e-1 (1.16e-3) ++/-/≈ +0/33/3 +3/31/2 +3/28/5 +7/20/9 +3/32/1 +B. Comparison Results +First, we compared the performance of ATM-R with that of +the other five CMOEAs. The mean IGD values and standard +deviations of 36 test functions over 30 independent runs are +summarized in Table I. The results in terms of the HV value +are collected in Table II. In each table, “std” represents the +standard deviation of the IGD/HV values over 30 independent +runs. “NaN” denotes that a CMOEA cannot find a feasible +solution of a test function over all 30 independent runs. +For a given test function, ATM-R was compared with each +competitor by the Friedman test with Bonferroni correction at +a significance level of 0.05. For convenience, “+”, “-”, and +“≈” are used to represent that a competitor is better than, +worse than, and similar to ATM-R, respectively. In addition, +for each test function, the best result among the six CMOEAs +is highlighted in gray. To visualize the results, we plotted the +CPFs obtained by the six CMOEAs in a typical run on three +representative CMOPs in Figs. 6-8. A typical run denotes the +one producing the median IGD value among all runs. +1) General Performance: In general, as shown in Table I +and Table II, ATM-R obtained the best results of most of the +test functions in terms of both the IGD and the HV values. +Additionally, it performed significantly better than the other +five competitors on most of the test functions. The multi- +problem Friedman’s test [?] was implemented to compare +these six CMOEAs simultaneously. As shown in Fig. 5, ATM- +R achieved the lowest ranking value among six CMOEAs. +Furthermore, the results in Figs. 6-8 show that ATM-R can +obtain a set of well-converged and well-distributed solutions. +A more detailed discussion on different test suites is given +next. +2) Performance on MW Test Suite: In terms of the IGD +value, ATM-R performed better than NSGAII-CDP, PPS, +CTAEA, CCMO, and ToP on 14, 13, six, six, and 14 test +functions, respectively. Inversely, these peer CMOEAs were +better than ATM-R on zero, one, two, three, and zero test +functions, respectively. ATM-R obtained the best results of +four test functions on which it performed better than the other +five competitors. CCMO obtained the best results of four test +functions, on one of which it performed similarly to ATM- +R. Although CTAEA obtained the best results of six test +functions, it performed similarly to ATM-R on five of these +test functions. +In terms of the HV value, ATM-R performed better than +NSGAII-CDP, PPS, CTAEA, CCMO, and ToP on 12, 12, six, +eight, and 14 test functions, respectively. On the contrary, these +peer CMOEAs revealed better results than ATM-R on zero, +two, three, five, and zero test functions, respectively. ATM- +R obtained the best results of three test functions on which +it performed better than the other five competitors. CCMO + +11 +0.5 +1 +1.5 +1 +2 +3 +NSGAII-CDP on MW13 +0.5 +1 +1.5 +1 +2 +3 +PPS on MW13 +0.5 +1 +1.5 +1 +2 +3 +CTAEA on MW13 +0.5 +1 +1.5 +1 +2 +3 +CCMO on MW13 +2 +3 +4 +5 +1 +2 +3 +ToP on MW13 +0.5 +1 +1.5 +1 +2 +3 +ATM-R on MW13 +Fig. 6. The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on MW13. +0 +0.2 +0.4 +0.6 +0.8 +0.6 +0.8 +1 +1.2 +NSGAII-CDP on CTP4 +0 +0.2 +0.4 +0.6 +0.8 +0.6 +0.8 +1 +1.2 +PPS on CTP4 +0 +0.5 +1 +5 +10 +15 +CTAEA on CTP4 +0 +0.2 +0.4 +0.6 +0.8 +0.6 +0.8 +1 +1.2 +CCMO on CTP4 +0 +0.2 +0.4 +0.6 +0.8 +0.6 +0.8 +1 +1.2 +ToP on CTP4 +0 +0.2 +0.4 +0.6 +0.8 +0.4 +0.6 +0.8 +1 +ATM-R on CTP4 +Fig. 7. The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on CTP4. +0 +0 +0.5 +0.5 +0.5 +NSGAII-CDP on LIRCMOP14 +1 +1 +1 +1.5 +1.5 +1.5 +0 +0 +0.5 +0.5 +0.5 +PPS on LIRCMOP14 +1 +1 +1 +1.5 +1.5 +1.5 +0 +0 +0.5 +0.5 +0.5 +CTAEA on LIRCMOP14 +1 +1 +1 +1.5 +1.5 +1.5 +0 +0 +0.5 +0.5 +0.5 +1 +CCMO on LIRCMOP14 +1.5 +1 +1 +1.5 +1.5 +0 +0 +0.5 +0.5 +0.5 +ToP on LIRCMOP14 +1 +1 +1 +1.5 +1.5 +1.5 +0 +0 +0.5 +0.5 +0.5 +ATM-R on LIRCMOP14 +1 +1 +1 +1.5 +1.5 +1.5 +Fig. 8. The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on LIRCMOP14. +obtained the best results of four test functions. Although +CTAEA obtained the best results of seven test functions, it +performed similarly to ATM-R on four of these test functions. +Furthermore, as shown in Fig 6, ATM-R obtained a set of +well-converged and well-distributed feasible solutions that is +close to the CPF. However, ToP failed to converge to the CPF. +NSGAII-CDP, PPS, CTAEA, and CCMO lost some parts of +the CPF. The results reflect that ATM-R performs better than +the other five competitors on the MW test suite. +3) Performance on CTP Test Suite: For the CTP test suite, +ATM-R performed better than the other five competitors on +most of the test functions in terms of both the IGD and HV +values. Additionally, it obtained the best IGD/HV values on +most of the test functions. +For CTP1, some parts of the CPF come from the un- +constrained Pareto front. For CTP6, the objective space has +infeasible holes of differing widths toward the Pareto-optimal +regions. For CTP2-CTP5, CTP7, and CTP8, the CPFs are +divided into several disconnected segments. To solve these +problems effectively, infeasibility information should be used +carefully. Thus, NSGAII-CDP and ToP, which only consider +constraints in the infeasible phase, performed worse than +ATM-R. As stated in [4], due to the complex (i.e., disconnected +and discrete) characteristics of the CPFs, the CMOEAs using +reference points or vectors would have inferior performance. +Therefore, CTAEA performed worse than ATM-R. PPS puts +emphasis on objectives in the early stage, while CCMO adopts +a specific population to make use of infeasibility information. +Compared with the MW test suite, the test functions in the +CTP test suite have larger feasibility ratios. Thus, too much +infeasibility information would impair the performance of a +CMOEA. This may be why PPS and CCMO performed worse +than ATM-R. +Furthermore, as shown in Fig. 7, ATM-R can converge to +the CPF of CTP4 more quickly than the other five competitors. +Additionally, it can cover more parts of the CPF than the other +five competitors. The results reflect that ATM-R performs +better than the other five competitors on the CTP test suite. +4) Performance on LIRCMOP Test Suite: For the LIRC- +MOP test suite, ATM-R obtained the best results on half of the +test functions in terms of both the IGD and HV values. Similar +to the CTP test suite, the test functions in the LIRCMOP +test suite have infeasible holes in the objective space and the +CPFs of some test functions are disconnected. To solve these +test functions effectively, infeasibility information should be +used carefully. NSGAII-CDP and ToP performed worse than +ATM-R on most of the test functions because they ignore +the infeasibility information to a great extent. ToP performed +better than ATM-R on four and three test functions in terms +of the IGD and HV values, respectively. This is attributed to +the powerful genetic operator (i.e., differential evolution) used +in ToP. +Among the infeasibility-assisted CMOEAs, PPS motivates +the population toward the unconstrained Pareto front in the +early stage. In CTAEA and CCMO, an additional popula- +tion is employed to approach the unconstrained Pareto front. +Thus, these three CMOEAs will fail to solve a CMOP (i.e., +LIRCMOP1-LIRCMOP4) in which the CPF is far away from + +12 +the unconstrained Pareto front. Regarding LIRCMOP5 and +LIRCMOP6, the CPFs are the same as the unconstrained +Pareto fronts. Regarding LIRCMOP7 and LIRCMOP8, the +CPFs are near the unconstrained Pareto fronts. For these four +test functions, an infeasibility-assisted CMOEA can approach +the CPF easily; thus, uniformity is the key factor affecting its +performance. ATM-R and CCMO performed better than PPS +and CTAEA on these test functions because they can preserve +diversity more effectively. For LIRCMOP9-LIRCMOP12, the +CPFs are divided into several disconnected segments. To solve +these test functions effectively, diversity should be maintained +carefully. Due to the weak cooperation of two populations, +CCMO can maintain diversity effectively during the evolution- +ary process. Thus, it performed better than ATM-R on these +four test functions. For the two three-objective test functions +(i.e., LIRCMOP13-LIRCMOP14), ATM-R performed better +than the other five competitors in terms of both the IGD +and HV values. It implies that ATM-R can achieve a bet- +ter tradeoff among feasibility, diversity, and convergence for +three-objective CMOPs. +Furthermore, as shown in Fig. 8, all six CMOEAs can +converge to the constrained Pareto front successfully. ATM- +R performed better than the other five competitors in terms of +the uniformity since it can achieve a better tradeoff among fea- +sibility, diversity, and convergence for three-objective CMOPs. +The results reflect that ATM-R performs better than the other +five competitors on the LIRCMOP test suite. +In summary, the extensive experiments on 36 test func- +tions with various challenging characteristics demonstrate that +ATM-R is able to solve complex CMOPs successfully. +V. FURTHER ANALYSES +A. Advantages of ATM-R over ATM +As discussed in Remark 1, it is not effective to extend +ATM [40] to solve CMOPs straightforwardly. In this subsec- +tion, the advantages of ATM-R over ATM were demonstrated +through experiments. The comparison results on 36 test func- +tions in terms of the IGD and the HV values are summarized in +Table III, where “+”, “-”, and “≈” denote that ATM performs +better than, worse than, and similarly to ATM-R in terms of the +IGD/HV value, respectively. As shown in Table III, ATM-R +performed better than ATM on these three test suites in terms +of both the IGD and the HV values. Specifically, in terms of +the IGD value, ATM-R was better than ATM on 9, 5, and 12 +test functions of the MW, the CTP, and the LIRCMOP test +suites, respectively. In contrast, ATM was better than ATM- +R on no more than three test functions of these test suites. +With regard to the HV value, ATM-R performed better than +ATM on 8, 6, and 12 test functions, respectively. Inversely, +ATM outperformed ATM-R on no more than 4 test functions +of these test suites. In summary, the experimental results on +these test functions with various characteristics demonstrate +that ATM-R has an edge over ATM. +B. Effectiveness of the Infeasible Phase +To validate the effectiveness of the update mechanism in the +infeasible phase, we implemented three variants (denoted as +TABLE III +RESULTS OF ATM VS ATM-R ON 36 TEST FUNCTIONS. +Test Functions +IGD ++/-/≈ +HV ++/-/≈ +MW1-MW14 +2/9/3 +4/8/2 +CTP1-CTP8 +3/5/0 +2/6/0 +LIRCMOP1-LIRCMOP14 +1/12/1 +1/12/1 +TABLE IV +RESULTS OF ATM-RICDP VS ATM-R, ATM-RIOBJ VS ATM-R, AND +ATM-RIDIV VS ATM-R ON 36 TEST FUNCTIONS. +Algorithms +IGD ++/-/≈ +HV ++/-/≈ +ATM-RICDP vs ATM-R +2/19/15 +2/23/11 +ATM-RIobj vs ATM-R +5/11/20 +7/11/18 +ATM-RIdiv vs ATM-R +8/14/14 +10/14/12 +ATM-RICDP, ATM-RIobj, and ATM-RIdiv) by using different +update mechanisms in this phase. Specifically, in ATM-RICDP, +the CDP is used for solution selection. In ATM-RIobj, the +solutions are selected based on Pareto dominance regardless +of constraints. In ATM-RIdiv, the diversity is quantified and +used to select promising solutions. By comparing ATM-R +with each of ATM-RICDP, ATM-RIobj, and ATM-RIdiv, the +effectiveness of the update mechanism in the infeasible phase +can be validated. The comparison results on 36 test functions +mentioned above are summarized in Table IV, where “+”, “-”, +and “≈” denote that a competitor performs better than, worse +than, and similarly to ATM-R in terms of the IGD/HV value, +respectively. +As shown in Table IV, ATM-R performed better than ATM- +RICDP, ATM-RIobj, and ATM-RIdiv in terms of both the +IGD and the HV values. Specifically, with regard to the IGD +value, ATM-R was better than ATM-RICDP, ATM-RIobj, and +ATM-RIdiv on 19, 11, and 14 test functions, respectively. +Inversely, ATM-RICDP, ATM-RIobj, and ATM-RIdiv outper- +formed ATM-R on 2, 5, and 8 test functions, respectively. In +terms of the HV value, ATM-R performed better than ATM- +RICDP, ATM-RIobj, and ATM-RIdiv on 23, 11, and 14 test +functions, respectively. In contrast, ATM-RICDP, ATM-RIobj, +and ATM-RIdiv outperformed ATM-R on 2, 7, and 4 test +functions, respectively. The experimental results show that the +update mechanism in the infeasible phase is critical to ATM-R. +C. Effectiveness of the Semi-feasible Phase +To validate the effectiveness of the update mechanism +in the semi-feasible phase, we implemented three variants +(i.e., ATM-RSCDP, ATM-RSobj, and ATM-RSdiv) by using +different update mechanisms in this phase. Specifically, in +ATM-RSCDP, ATM-RSobj, and ATM-RSdiv, the CDP, the +Pareto dominance, and the diversity are used for solution +selection, respectively. By comparing ATM-R with each of +ATM-RICDP, ATM-RIobj, and ATM-RIdiv, the effectiveness +of the update mechanism in the semi-feasible phase can +be validated. Specifically, the comparison results on 36 test +functions mentioned above are summarized in Table V, where + +13 +TABLE V +RESULTS OF ATM-RSCDP VS ATM-R, ATM-RSOBJ VS ATM-R, AND +ATM-RSDIV VS ATM-R ON 36 TEST FUNCTIONS. +Algorithms +IGD ++/-/≈ +HV ++/-/≈ +ATM-RSCDP vs ATM-R +3/24/9 +5/23/8 +ATM-RSobj vs ATM-R +3/32/1 +2/31/3 +ATM-RSdiv vs ATM-R +0/36/0 +0/36/0 +“+”, “-”, and “≈” denote that a competitor performs better +than, worse than, and similarly to ATM-R in terms of the +IGD/HV value, respectively. +As shown in Table V, ATM-R performed better than ATM- +RSCDP, ATM-RSobj, and ATM-RSdiv in terms of both the +IGD and the HV values. With regard to the IGD value, +ATM-R was better than ATM-RSCDP, ATM-RSobj, and ATM- +RSdiv on 24, 32, and 36 test functions, respectively. In +contrast, ATM-RSCDP, ATM-RSobj, and ATM-RSdiv out- +performed ATM-R on no more than three test functions. In +terms of the HV value, ATM-R performed better than ATM- +RSCDP, ATM-RSobj, and ATM-RSdiv on 23, 31, and 36 test +functions, respectively. Inversely, ATM-RSCDP, ATM-RSobj, +and ATM-RSdiv outperformed ATM-R on no more than five +test functions. The experimental results show that the update +mechanism in the semi-feasible phase is critical to ATM-R. +D. Effectiveness of the Multiphase Mating Selection Strategy +In order to verify the effectiveness of the multiphase mating +selection strategy, we implemented three variants (i.e., ATM- +RMCDP, ATM-RMobj, and ATM-RMdiv) by using different +selection methods to select mating solutions. Specifically, in +ATM-RMCDP, ATM-RMobj, and ATM-RMdiv, the CDP, the +Pareto dominance, and the diversity are used for solution selec- +tion, respectively. By comparing ATM-R with each of ATM- +RMCDP, ATM-RMobj, and ATM-RMdiv, the effectiveness of +the multiphase mating selection strategy can be demonstrated. +Specifically, the comparison results on 36 test functions men- +tioned above are summarized in Table VI, where “+”, “-”, +and “≈” denote that a competitor performs better than, worse +than, and similarly to ATM-R in terms of the IGD/HV value, +respectively. +As shown in Table VI, ATM-R performed better than ATM- +RMCDP, ATM-RMobj, and ATM-RMdiv in terms of both +the IGD and the HV values. With regard to the IGD value, +ATM-R was better than ATM-RMCDP, ATM-RMobj, and +ATM-RMdiv on 19, 10, and 9 test functions, respectively. In +contrast, ATM-RMCDP, ATM-RMobj, and ATM-RMdiv out- +performed ATM-R on 5, 7, and 5 test functions, respectively. In +terms of the HV value, ATM-R performed better than ATM- +RMCDP, ATM-RMobj, and ATM-RMdiv on 22, 13, and 8 +test functions, respectively. Inversely, ATM-RMCDP, ATM- +RMobj, and ATM-RMdiv outperformed ATM-R on 3, 7, and +4 test functions, respectively. The experimental results show +that the multiphase selection strategy is critical to ATM-R. +TABLE VI +RESULTS OF ATM-RMCDP VS ATM-R, ATM-RMOBJ VS ATM-R, AND +ATM-RMDIV VS ATM-R ON 36 TEST FUNCTIONS. +Algorithms +IGD ++/-/≈ +HV ++/-/≈ +ATM-RMCDP vs ATM-R +5/19/12 +3/22/11 +ATM-RMobj vs ATM-R +7/10/19 +7/13/16 +ATM-RMdiv vs ATM-R +5/9/22 +4/8/24 +VI. CONCLUSIONS +This paper has analyzed the key task of constrained multi- +objective optimization in depth and decomposed it into three +subtasks explicitly for the first time. To accomplish these +three subtasks in different evolutionary phases, an adaptive +tradeoff model with reference points (ATM-R) was designed. +Specifically, ATM-R takes advantage of infeasible solutions +to achieve different tradeoffs in these three subtasks. In the +infeasible phase, ATM-R distinguishes and uses infeasible +solutions with good diversity to enhance the diversity loss +due to its pursuit of feasibility. Thus, the population can +move toward feasible regions from diverse search directions. In +the semi-feasible phase, ATM-R leverages infeasible solutions +with good diversity/objective function values to promote the +transition from “the tradeoff between feasibility and diversity” +to “the tradeoff between convergence and diversity”. Thus, the +population can locate enough feasible solutions and approach +the CPF quickly. In the feasible phase, ATM-R employs +NSGAII to seek a set of well-converged and well-distributed +solutions close to the constrained Pareto front. Moreover, a +multiphase mating selection strategy is proposed to select +appropriate mating parents adaptively. 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Yao, “Quality evaluation of solution sets in multiobjective +optimisation: A survey,” ACM Computing Surveys (CSUR), vol. 52, +no. 2, pp. 1–38, 2019. + diff --git a/-NE1T4oBgHgl3EQfoQQY/content/tmp_files/load_file.txt b/-NE1T4oBgHgl3EQfoQQY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49aa5b4b55bfa6fa40b689b733e51806edd53d28 --- /dev/null +++ b/-NE1T4oBgHgl3EQfoQQY/content/tmp_files/load_file.txt @@ -0,0 +1,2052 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf,len=2051 +page_content='1 ATM-R: An Adaptive Tradeoff Model with Reference Points for Constrained Multiobjective Evolutionary Optimization Bing-Chuan Wang, Yunchuan Qin, Xian-Bing Meng, Zhi-Zhong Liu Abstract—The goal of constrained multiobjective evolutionary optimization is to obtain a set of well-converged and well- distributed feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To complete this goal, there should be a tradeoff among feasibility, diversity, and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' However, it is nontrivial to balance these three elements simulta- neously by using a single tradeoff model since the importance of each element varies in different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As an alter- native, we adapt different tradeoff models in different phases and propose a novel algorithm called ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the infeasible phase, ATM-R takes the tradeoff between diversity and feasibility into account, aiming to move the population toward feasible regions from diverse search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the semi-feasible phase, ATM-R promotes the transition from “the tradeoff between feasibility and diversity” to “the tradeoff between diversity and convergence”, which can facilitate the discovering of enough feasible regions and speed up the search for the feasible Pareto optima in succession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the feasible phase, the tradeoff between diversity and convergence is considered to attain a set of well-converged and well-distributed feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It is worth noting that the merits of reference points are leveraged in ATM-R to accomplish these tradeoff models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Also, in ATM-R, a multiphase mating selection strategy is developed to generate promising solutions beneficial to different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Systemic experiments on a wide range of benchmark test functions demonstrate that ATM-R is effective and competitive, compared against five state- of-the-art constrained multiobjective optimization evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Index Terms—Constrained multiobjective evolutionary opti- mization, adaptive tradeoff model, reference point, multiphase mating selection I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' INTRODUCTION M ANY scientific or engineering problems involve the optimization of conflicting objectives subject to con- straints, which can be formulated as constrained multiobjective optimization problems (CMOPs) [1]: min F(x) = (f1(x), f2(x), · · · , fm(x))T ∈ Rm s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' gj(x) < 0, j = 1, · · · , ng hj(x) = 0, j = ng + 1, · · · , ng + nh xj ≤ xj ≤ xj, j = 1, · · · , D , (1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Wang is with the School of Automation, Central South University, Changsha 410083, China (email: bingcwang@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Qin and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Liu are with the College of Information Science and Electronic Engineering, Hunan University, Changsha 410082, China (e-mail: liuzz@hnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' qinyunchuan@hnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Meng is with the School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China (e-mail: axbmeng@scut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' where F(x) denotes the objective vector consisting of m conflicting objectives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', fi(x), i = 1, · · · , m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' x = (x1, · · · , xD)T is a D-dimensional decision vector/solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' xj and xj are the lower and upper bounds of xj, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' S = �D j=1[xj, xj] refers to the decision space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' gj(x) and hj(x) represent the jth inequality and (j − ng)th equality constraints, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ng and nh are the numbers of the inequality and equality constraints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' When solving a CMOP, we always quantify constraint violation by the degree of constraint violation: G(x) = ng+nh � j=1 Gj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' (2) Gj(x) denotes the degree of constraint violation of the jth constraint [2]: Gj(x) = � max(0, gj(x)), 1 ≤ j ≤ ng max(0, |hj(x)| − δ), ng + 1 ≤ j ≤ ng + nh (3) where δ is a small positive value used to relax an equality constraint to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A solution x is called a feasible solution, if and only if G(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' All feasible solutions constitute the feasible region: Ω = {x ∈ RD|G(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For two solutions xu, xv ∈ Ω, xu is said to Pareto dominate xv, denoted as xu ≺ xv, if and only if ∀j ∈ {1, · · · , m}, fj(xu) ≤ fj(xv) � ∃j ∈ {1, · · · , m}, fj(xu) < fj(xv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A solution xp ∈ Ω is considered as a Pareto optimum if and only if ¬∃xv ∈ Ω, xv ≺ xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The set of all Pareto optima is called the constrained Pareto set, and its image in the objective space is called the constrained Pareto front (CPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The goal of constrained multiobjective evolutionary optimization is to pursue a set of well-converged and well-distributed feasible solutions to approximate the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To complete this goal, a consensus has been reached in the community of constrained multiobjective optimization that a good tradeoff among feasibility, diversity, and convergence should be achieved [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It is worth noting that the impor- tance of these three elements varies in different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Let us take the element of feasibility for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the infeasible phase, this element is very important because feasibility information plays an indispensable role in locating feasible regions, which is crucial for constrained multiobjec- tive optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' However, in the feasible phase, this element is negligible as all solutions become feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' We only need to consider the tradeoff between diversity and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Due arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='03317v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='NE] 9 Jan 2023 2 Convergence Convergence tradeoff tradeoff transition Population is infeasible Population is feasible Population is semi-feasible Diversity Diversity Feasibility Feasibility Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Task decomposition of achieving a tradeoff among feasibility, diversity, and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' to their varied importance, it is nontrivial to balance these three elements simultaneously by using a single tradeoff model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As an alternative, we adapt different tradeoff models in different evolutionary phases, proposing an adaptive tradeoff model with reference points (ATM-R) to handle CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 1 depicts the tradeoffs considered in ATM-R: achieving a tradeoff between feasibility and diversity in the infeasible phase: when the population is entirely infeasible, the primary goal is to find as many feasible regions as possible since the Pareto optima may be scat- tered in different feasible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To this end, a tradeoff between feasibility and diversity should be achieved to move the population toward the feasible regions from diverse search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' promoting the transition from “the tradeoff between fea- sibility and diversity” to “the tradeoff between diversity and convergence” in the semi-feasible phase: when the population is semi-feasible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', the population contains both infeasible and feasible solutions), two situations should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the early stage, only a few feasible regions are discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In this case, the tradeoff between feasibility and diversity should still be prioritized to find more promising feasible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Once enough feasible regions are located, in the later stage, attention should be paid to drive the population toward the CPF quickly and make them uniformly spread over the CPF simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, the tradeoff between convergence and diversity should be concentrated on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In summary, in this phase, we should shift from “the tradeoff between feasibility and diversity” to “the tradeoff between diver- sity and convergence” [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' achieving a tradeoff between diversity and convergence in the feasible phase: when the population is completely feasible, the final task is to move the feasible solutions toward the CPF quickly while maintaining good diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Apparently, a tradeoff between diversity and convergence should be realized [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In summary, the core of a CMOEA is how to accomplish the above tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The tradeoff in the feasible phase has been well studied in the community of evolutionary multiob- jective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For convenience, in ATM-R, an off-the- shelf unconstrained multiobjective optimization evolutionary algorithm (MOEA) is utilized to achieve this tradeoff directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As for the tradeoffs in the other two phases, the related studies remain relatively scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Especially for the tradeoff in the semi-feasible phase, little research focuses on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Indeed, to achieve the tradeoffs in these two phases, an important concern is how to deal with the infeasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Past experience in the community of evolutionary constrained multiobjective optimization has shown that the infeasible so- lutions can not only facilitate maintaining diversity but also contribute to speeding up the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In ATM-R, the merits of reference points are leveraged to select different kinds of infeasible solutions suitable for different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In summary, the main contributions of this paper are as follows: Instead of using a single tradeoff model, we adapt dif- ferent tradeoff models in different evolutionary phases, proposing a novel constrained multiobjective optimiza- tion algorithm (CMOEA) called ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Although it is inevitable for an algorithm to experience three phases during the evolution, few attempts have been made to develop alternate tradeoff models for different phases to facilitate a more explicit adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' By leveraging the merits of reference points, we provide a new perspective that selects promising infeasible so- lutions suitable for different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To the best of our knowledge, relevant work along this direction remains scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A multiphase mating selection strategy is developed in this paper that adaptively selects suitable mating parents for different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Systemic experiments have been implemented on three sets of test suites including 36 benchmark CMOPs to validate the effectiveness of ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Comparison against five state-of-the-art CMOEAs suggests that ATM-R is significantly superior or comparable to the contender algorithms on most of the test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, the advantages of some important algorithmic components in ATM-R have been verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Section II conducts a brief review of related CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The details of ATM-R are described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The performance of ATM- R is compared with five representative CMOEAs in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Section V presents some further analyses of ATM-R in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The concluding remarks and future work are given in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' RELATED WORK Constrained multiobjective optimization has become a hot topic in the community of evolutionary computation and numerous CMOEAs have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Based on whether infeasible solutions are utilized, these CMOEAs can be clas- sified into two categories: feasibility-driven CMOEAs and infeasibility-assisted CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Feasibility-Driven CMOEAs A feasibility-driven CMOEA is driven by feasibility infor- mation, in which feasible solutions are considered to be better than infeasible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Some feasibility-driven CMOEAs use the constrained dominance principle (CDP) to compare two solutions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the CDP, a solution xu is said to be better 3 than another solution xv, if one of the following conditions is met: both xu and xv are infeasible, and G(xu) < G(xv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' xu is feasible and xv is infeasible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' both xu and xv are feasible, and xu ≺ xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Due to its preference for feasible solutions, the CDP can motivate the population toward feasible regions quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It has been widely integrated with different kinds of MOEAs [6], [7] and used in a spectrum of engineering optimization prob- lems [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [6] combined an angle-based selection strategy, the shift-based density estimation strategy, and the CDP for constrained many-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Jain and Deb [7] proposed a reference-point-based nondominated sort- ing approach, which is integrated with the CDP for constrained many-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Jan and Khanum [10] embedded the CDP into the framework of MOEA/D and compared its performance with that of the stochastic ranking [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' CDP- based CMOEAs are often used as the baseline algorithms when evaluating the performance of a CMOEA [12]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The feasibility rule, which is widely used for constrained single-objective optimization, has been extended to solve CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [15] combined the feasibility rule with an indicator-based MOEA and compared its performance with that of some other kinds of CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [16] carried out a comparison study on MOEA/D for constrained multiob- jective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Different constraint-handling techniques including the feasibility rule are embedded into the framework of MOEA/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Some CMOEAs put emphasis on constraints when the population contains no feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Woldesenbet and Yen [17] presented a self-adaptive penalty method to solve CMOPs, in which an adaptive penalty function and a dis- tance measure are combined for constraint-handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In fact, when the population is entirely infeasible, the self-adaptive penalty method compares two solutions based on constraints regardless of objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Liu and Wang [18] presented a two- phase CMOEA to solve CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' When the population is entirely infeasible, all objectives are combined together and the feasibility rule is used to tackle constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Due to the superior capability of its search algorithm, the two-phase CMOEA can handle complex constraints in the decision space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [19] designed a CMOEA for constrained multiobjective optimization, in which the min-max formulation is used to tackle constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In addition, the feasibility rule is used to compare two solutions when an offspring is inserted into the new population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Miyakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [20] developed a two- stage nondominated sorting method to solve CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The population is divided into several fronts by the nondominated sorting according to constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The obtained fronts are further partitioned by the nondominated sorting based on objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In this manner, constraints are prior to objectives in environ- mental selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Infeasibility-assisted CMOEAs An infeasibility-assisted CMOEA takes advantage of in- feasible solutions for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Most state-of-the-art CMOEAs fall into this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Some CMOEAs take advantage of infeasible solutions implicitly by using a comparison criterion that takes both constraints and objectives into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Ma and Wang [3] pro- posed a shifted-based penalty function, in which an infeasible solution is penalized based on the information provided by the feasible solutions nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Jiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [21] proposed a modified objective function method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' When the population is entirely infeasible, the modified objective function is equivalent to a distance measure in which constraints and objectives are considered equally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='] presented an angle-based CDP for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Given a feasible solution and an infeasible solution, if the angle between these two solutions is smaller than a predefined threshold, they would be nondominated each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, some infeasible solutions could enter into the new population instead of some feasible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Young [22] proposed a blended ranking measure to select solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' By blending an individual’s rank in the objective space with its rank in the constraint space, an infeasible solution may be better than a feasible one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Similarly, Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [13] designed a new fitness function with two rankings, in which one ranking value is obtained based on the CDP and the other is calculated based on the Pareto dominance without considering constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The ε constrained method can use infeasibility information by tuning a threshold value ε [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' thus, it has been widely used to solve CMOPs [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Zapotecas- Mart´ınez and Ponsich [24] combined MOEA/D with the ε constrained method to solve CMOPs, in which the ε value is set according to the degree of constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [25] improved the ε constrained method by setting the ε value dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [26] extended the ε constrained method to solve CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' When the degree of constraint violation of an infeasible solution is larger than the ε value, its diversity will be carefully maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The stochastic ranking that is popular for constrained single-objective optimization has also been extended to solve CMOPs [15], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Some CMOEAs leverage the advantages of infeasible so- lutions explicitly by archiving or coevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Ray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [28] proposed an infeasibility-driven EA, in which a small per- centage of infeasible solutions close to the constraint bound- aries are maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [29] designed a two-archive EA for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' An archive is used to promote convergence, while the other is used to maintain diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The diversity archive evolves without considering constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' thus, infeasible solutions with good objective function values can be fully used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [4] tried to solve CMOPs through bidirectional coevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The CDP is used to drive the main population toward the CPF from the feasible side of the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In addition, a nondominated sorting procedure and an angle-based selection scheme are conducted in sequence to motivate the population toward the CPF within the infeasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [30] developed a coevolutionary framework for constrained mul- tiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Similarly, one population is updated by the CDP, while the other is updated by an unconstrained MOEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, the elites of these two populations are selected to generate offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Ishibuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [31] designed a dual-grid model of MOEA/D for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Two populations are maintained and infeasible 4 solutions with good objective function values are preferred in the secondary population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [32] employed two types of weight vectors in MOEA/D to solve CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The solutions associated with the convergence weight vectors are updated based on the aggregation function, while the solutions associated with the diversity weight vectors are renewed according to both the aggregation function and the degree of constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [14] used two kinds of weight vectors for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, the degree of constraint violation is considered as another objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Subsequently, a set of feasible weight vectors and a set of infeasible weight vectors are used to update the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, the set of infeasible weight vectors is dynamically adjusted to maintain a number of infeasible solutions with good objective function values and small degrees of constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Some CMOEAs divide the evolutionary process into several phases and put emphasis on objectives in one of the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [33] divided the evolutionary process into a constrained search mode and an unconstrained search mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' These two search modes are executed by a dynamic constraint- handling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [12] proposed a push and pull search (PPS) framework to solve CMOPs, in which the evo- lutionary process is divided into two stages: push and pull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the push stage, the population is updated by an unconstrained MOEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the pull stage, an improved ε constrained method is designed to tackle complex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Since its proposition, the PPS framework has been used in various fields [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [36] proposed a dynamic selection preference- assisted constrained multiobjective differential evolutionary (DE) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The selection preference for a solution shifts from infeasibility to feasibility as the optimization progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [37] proposed a two-stage CMOEA to balance objective optimization and constraint sanctification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' These two stages are executed dynamically according to the percentage of feasible solutions in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Recently, Ming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [38] proposed a simple two-stage EA for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The two-stage EA focuses on approaching the unconstrained Pareto front in the first stage and the feasible solutions are archived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the second stage, the method seeks to approximate the CPF, where the archived feasible solutions are adopted as the initial population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' [39] proposed a two-phase EA for constrained multiobjective optimization with deceptive constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the first phase, two subpopulations are employed to explore the feasible regions and the entire space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The second phase aims to approach the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, an infeasibility utilization strategy is designed to leverage the promising information provided by infeasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' PROPOSED METHOD The general flow chart of ATM-R is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As its name implies, ATM-R makes use of reference points to adap- tively accomplish different tradeoffs in different evolutionary phases, those are, the infeasible phase, the semi-feasible phase, and the feasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The details of the update mechanisms in these three different phases are described in Section III-A, Infeasible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Infeasible Phase Semi-feasible Phase Feasible Phase Reproduction Initialization Stop?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Semi-feasible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Output the Population Yes No Yes Yes No No Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Flow chart of ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Algorithm 1: Update Mechanism in the Infeasible Phase Input: Population P, offspring population O Output: New population P 1 Q ← P ∪ O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2 Divide Q into k fronts based on ˆF(x): F1, · · · , Fk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3 P ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4 for l = 1 : k do 5 if |P| + |Fl| ≥ N then 6 Break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 7 P ← P ∪ Fl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 8 if |P| + |Fl| > N then 9 Sample n uniformly distributed reference points and generate corresponding weight vectors: w1, · · · , wn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 10 Assign each solution in Fl to a weight vector according to (5)-(7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 11 while |P| + |Fl| > N do 12 Select the weight vector associated with the largest number of solutions: wc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 13 Among the solutions assigned to wc, select the one with the largest value of G(x): xw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 14 Fl ← Fl\\xw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 15 P ← P ∪ Fl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Section III-B, and Section III-C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Aside from the environmental selection procedure, another critical element of a CMOEA is the mating selection procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In ATM- R, a multiphase mating selection strategy is developed to generate promising solutions beneficial to different tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The details of this strategy are illustrated in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Finally, the framework of ATM-R and some discussions are shown in Section III-E and Section III-F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Update Mechanism in the Infeasible Phase In this phase, ATM-R aims to strike a balance between feasi- bility and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In other words, it motivates the population toward feasibility from diverse search directions, thus locating as many feasible regions as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Algorithm 1 shows how ATM-R accomplishes this tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In general, it involves two essential elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 1) Nondominated Sorting in the Transformed Objective Space: Following the ideas in [40], we consider G(x) as an additional objective function, and transform (1) into an 5 unconstrained MOP: min ˆF(x) = (f1(x), · · · , fm(x), G(x))T ∈ Rm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' (4) Clearly, this transformation does not introduce any extra parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In addition, both objective functions and con- straints are considered in (4), which can facilitate maintaining population diversity and enhance driving forces toward the feasible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Based on ˆF(x), the population will be divided into several fronts, denoted as F1, · · · , Fk, by implementing a nondominated sorting procedure in the transformed objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Afterward, the solutions in each front will be selected in turn until �l−1 i=1 |Fi| < N ≤ �l i=1 |Fi| where N denotes the size of the final solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2) Regular Reference Point-based Selection: If �l i=1 |Fi| is larger than N, we should further select (n = N−�l−1 i=1 |Fi|) solutions from the last desired front Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To complete this task, in this study, a regular reference point-based selection scheme is developed by taking advantage of uniformly distributed reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Its implementation is quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' First, a set of regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', uniformly distributed) refer- ence points is sampled in the objective space to generate weight vectors denoted as {w1, · · · , wn} following the ideas in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Subsequently, a solution (denoted as x) in Fl is assigned to the weight vector with the smallest angle to its nor- malized objective vector: I = arg min j∈{1,··· ,n} θj, (5) θj = arccos ����� F′(x)Twj ∥F′(x)∥ · ∥wj∥ ����� , j = 1, · · · , n, (6) f ′ j(x) = fj(x) − zmin j zmax j − zmin j , j = 1, · · · , m, (7) where I indicates which weight vector the solution x is assigned to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' θj denotes the angle between wj and the nor- malized objective vector F′(x) = (f ′ 1(x), · · · , f ′ m(x))T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ∥ · ∥ represents the function to calculate the 2-norm of a vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' zmax = (zmax 1 , · · · , zmax m )T and zmin = (zmin 1 , · · · , zmin m )T refer to the estimated nadir point and ideal point, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Afterward, (|Fl|−n) inferior solutions are deleted one by one by employing a “diversity first, feasibility second” strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To be specific, it first identifies the weight vector associated with the largest number of solutions1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Intuitively, since these solutions are associated with the same weight vector, they will share highly similar search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To maintain diverse search directions, it is nec- essary to delete one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The feasibility information of these solutions is considered for the deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The one with the largest value of G(x) is discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' These two steps will continue until (|Fl| − n) solutions are deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A simple example is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3 for better understanding the regular reference point-based selection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' We con- sider a CMOP with two objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=" Suppose there are seven 1Note that the tie is broken at random A B D C E F Feasible region 1 w 2 w 3 w 4 w ' 2f ' 1f G G Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Update mechanism in the infeasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' solutions in the population, and they lie in the same front in the transformed objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' According to the values of G(x), these individuals were ranked as F, C, E, A, G, D, and B in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The task is to select four solutions for the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 1) First, four reference points are sampled uniformly to generate four weight vectors denoted as {w1, · · · , w4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2) Next, each solution in the population is assigned to a weight vector: w1 ↔ {A}, w2 ↔ {B, C}, w3 ↔ {D}, and w4 ↔ {E, F, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3) Subsequently, three solutions are deleted one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' G is first deleted since w4 is matched with the largest number of solutions and G is the one with the largest value of G(x) compared with E and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' According to this principle, B and E will be also removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4) Finally, the solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', A, C, D, and F will enter into the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Remark 1: Both ATMES2 [40] and IDEA [28] employ non- dominated sorting in the transformed objective space as ATM- R does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The main difference lies in how to distinguish the solutions in the same front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, in ATMES, solutions are selected based on G(x) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A solution with a smaller value of G(x) will be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In this manner, ATMES will put too much emphasis on constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It will cause perfor- mance deterioration in terms of the search diversity, which is essential for finding as many promising feasible regions as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' On the contrary, in IDEA, only the diversity in the transformed objective space is considered to update the last desired front Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Unfortunately, this manner will result in a limited driving force toward the feasible regions, which in turn leads to a relatively low convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Unlike these two methods, ATM-R takes both diversity and feasibility into account to update Fl, and a “diversity first, feasibility second” strategy is thus developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3, ATM-R can strike a good balance between diversity and feasibility, thereby motivating the population toward feasible regions from diverse search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Update Mechanism in the Semi-feasible Phase ATM-R intends to promote the transition from “the tradeoff between feasibility and diversity” to “the tradeoff between 2Although ATMES is originally designed for constrained single-objective optimization, it can be directly applied to solve CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 6 Algorithm 2: Update Mechanism in the Semi-feasible Phase Input: Population P, offspring population O, FEs, MaxFEs Output: New population P 1 Q ← P ∪ O, P ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2 Qf ← {x ∈ Q|G(x) = 0}, Qif ← {x ∈ Q|G(x) > 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3 if |Qf| > N then 4 Qf ← N feasible solutions seleted from Qf by an unconstrained MOEA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 5 P ← P ∪ Qf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 6 if |Qif| > N then 7 if F Es MaxF Es < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 or |Qf| < N then 8 Qif ← N infeasible solutions selected from Qif by using Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 9 else 10 Generate |Qf| weight vectors by using the solutions in Qf according to (8)-(9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 11 Assign each solution in |Qif| to a weight vector according to (5)-(7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 12 while |Qif| > N do 13 Select the weight vector associated with the largest number of solutions: wc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 14 Among the solutions assigned to wc, select the one furthest from the feasible solution used to generate wc: xw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 15 Qif ← Qif\\xw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 16 P ← P ∪ Qif;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' diversity and convergence” in the semi-feasible phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', the population contains both infeasible and feasible solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The reasons for this transition are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the early stage of the semi-feasible phase, ATM-R must locate as many feasible regions as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To this end, it must focus on the tradeoff between feasibility and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' After finding a sufficient number of feasible regions, in the later stage, ATM- R should steer the population rapidly toward the CPF and distribute it uniformly along with the CPF simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, the tradeoff between convergence and diversity should be prioritized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Algorithm 2 shows how ATM-R updates the solutions in the semi-feasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' From Algorithm 2, it is observed that ATM-R updates the feasible and infeasible solutions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To update the feasible solutions, an unconstrained MOEA is used to truncate the feasible population Qf if its size is greater than N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' otherwise, all feasible solutions are reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To update the infeasible solutions, ATM-R considers two situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the early stage, it aims to achieve a tradeoff between feasibility and diversity, which is the same as in the infeasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, the update mechanism used in the infeasible phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', Algorithm 1) can be directly applied in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' While in the later stage, ATM-R shifts the emphasis to the tradeoff between diversity and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To realize this tradeoff, an important task is how to preserve those infeasible solutions that can contribute to both diversity and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM-R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='w ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content="' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Feasible solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Infeasible solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='CPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Feasible solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Infeasible solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='CPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Feasible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Feasible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Illustration of difference between the weight vectors in the regular reference point-based selection and those in the adaptive reference point-based selection: (a) weight vectors in regular reference point-based selection and (b) weight vectors in adaptive reference point-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' designs the following two steps to accomplish this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 1) Discovery of the Nondominated Infeasible Solutions: Compared with the feasible solutions in the current population, the nondominated infeasible solutions usually have smaller ob- jective function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It is natural to leverage their benefits to promote convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To distinguish these infeasible solutions, we first employ a nondominated sorting procedure to divide the union population (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', Q in Algorithm 2) into several fronts based on ˆF(x) in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Subsequently, the infeasible solutions in the first front are picked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' If the number of these nondominated infeasible solutions (denoted as M) is smaller than N, all of them will be kept;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' otherwise, they will be further distinguished by the following adaptive reference point-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2) Adaptive Reference Point-based Selection: Herein, the regular reference points are no longer used to assist the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The reason is that the CPF might be disconnected (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4), and some weight vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', w1 and w3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4(a)) generated using the uniformly distributed reference points cannot point to any parts of the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As a result, the solutions preserved by making use of such weight vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', C and F) are far away from the CPF and hardly con- tribute to convergence speed, which is not desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Instead, we use adaptive reference points for solution selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For convenience, in our study, the feasible solutions are considered as adaptive reference points since they can deliver important clues for the localization of the CPF (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Based on these reference points, a set of adaptive weight vectors can be obtained conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To be specific,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' for the ith 7 feasible solution xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' the corresponding weight vector (denoted as w ′ i = (w ′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' w ′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='m)T) is generated as follows: w ′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='j = f ′ j(x) �m j=1 f ′ j(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' (8) f ′ j(x) = fj(x) − zmin j zmax j − zmin j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' (9) where (f ′ 1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' f ′ m(x))T is the normalized objective vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' (zmax 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' zmax m )T and (zmin 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' zmin m )T denote the estimated nadir point and ideal point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4 shows the difference between the weight vectors generated using regular reference points and those using adaptive reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It is evident that the weight vectors obtained using adaptive reference points fit better to the characteristics of the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Once the adaptive weight vectors are prepared, the next procedures in the adaptive reference point-based selection scheme are quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' First, each nondominated infeasi- ble solution is assigned to a weight vector following the ideas in the regular reference point-based selection scheme (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' (5)-(7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Afterward, (M-N) infeasible solutions are deleted one by one in a two-step manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The first step is to identify the weight vector associated with the largest number of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the second step, among the solutions assigned to this weight vector, the one furthest from the feasible solution corresponding to the weight vector in the objective space will be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In general, the first step is similar to many decomposition-based approaches and can help to maintain population diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As for the second step, it can help to retain those infeasible solutions close to the feasible solutions and thus offer a driving force toward the CPF from the infeasible side of the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Intuitively, this way can speed up the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Remark 2: In the semi-feasible phase, the population size in ATM-R is larger than or equal to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The reason is that a larger population can enhance population diversity, which is critical to both “the tradeoff between feasibility and diversity” and “the tradeoff between diversity and convergence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As for how to determine whether the algorithm has entered the later stage of the semi-feasible phase, we considered two simple conditions which should be satisfied simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The first condition is that F Es MaxF Es should be larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Note that FEs and MaxFEs denote the function evaluations and the maximum function evaluations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The second condition relies on the number of feasible solutions which should be equal to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The first condition implies that enough search efforts have been devoted to finding feasible regions, while the second condition is set to ensure a sufficient number of reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the later stage of the semi-infeasible phase, if no nondominated infeasible solutions are discovered, the algorithm will enter the feasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Update Mechanism in the Feasible Phase In this phase, all solutions are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Under this condition, only the tradeoff between diversity and convergence should be considered, thus motivating the feasible solutions toward the CPF quickly while maintaining good diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Apparently, Algorithm 3: Multiphase Mating Selection Strategy Input: Population P, N Output: Mating population C 1 C ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2 for i = 1 : N do 3 Randomly select two different solutions denoted as xa and xb from P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4 if P is entirely infeasible then 5 if rand < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 then 6 xm ← the better one between xa and xb based on the degree of constraint violation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 7 else 8 xm ← the better one between xa and xb based on the diversity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 9 else if P is feasible then 10 if xa ≺ xb then 11 xm ← xa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 12 else if xb ≺ xa then 13 xm ← xb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 14 else 15 xm ← the better one between xa and xb based on the diversity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 16 else if P is semi-feasible then 17 if i < N/2 then 18 xm ← the better one between xa and xb by using the method in the infeasible phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 19 else 20 xm ← the better one between xa and xb by using the method in the feasible phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 21 C ← C ∪ xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' a current effective unconstrained MOEA can be applied to achieve this balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, in ATM-R, an off-the-shelf uncon- strained MOEA is employed in this phase straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Multiphase Mating Selection Strategy In addition to the multi-phase strategy in environmental selection, ATM-R uses a multi-phase strategy for mating selec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It selects appropriate mating parents suitable for different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The details of this multiphase mating selection strategy are described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Similarly, three different phases are considered in this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the infeasible phase, population diversity and feasi- bility should be focused on simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, in the tournament selection, solutions are compared based on the diversity and the degree of constraint violation with the same probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Note that the diversity is quantified by the same way as in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the feasible phase, population diversity and conver- gence should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Following the ideas in NSGA-II [5], in the tournament selection, solutions are compared based on the Pareto dominance relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 8 Algorithm 4: ATM-R Input: A CMOP, N, MaxFEs Output: Final population P 1 P ← a population initialized from the decision space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2 FEs ← N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3 while FEs < MaxFEs do 4 C ← a mating population selected from P by using Algorithm 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 5 O ← an offspring population generated by executing genetic operators on C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 6 FEs ← FEs + N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 7 Q ← P ∪ O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 8 if Q is entirely infeasible then 9 P ← the solutions seleted from Q by using Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 10 else if Q is semi-feasible then 11 P ← the solutions selected from Q by using Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 12 else if Q is feasible then 13 P ← the solutions selected from Q by using an unconstrained MOEA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Also, if two solutions do not dominate each other, they are compared based on the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The semi-feasible phase needs to bridge the gap between the feasible phase and the infeasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, in this phase, the first half of the mating population is selected by using the method in the infeasible phase, while the other half is selected by using the method in the feasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM-R In summary, the details of ATM-R are given in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' At the beginning, a population of N solutions is sampled uniformly in the decision space (Lines 1-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Afterward, the population is employed to search for the CPF until the maximum number of function evaluations is exhausted (Lines 3-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the search process, first, N mating parents are selected for offspring generation by using the multiphase mating selection strategy in Algorithm 3 (Line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Next, N offspring are produced by the simulated binary crossover (SBX) [42] and the polynomial mutation (PM) [43] (Lines 5-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Afterward, promising solutions are selected based on population feasibility (Lines 7-14) where Algorithm 1 and Algorithm 2 are used in the infeasible phase and the semi- feasible phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Note that if FEs ≥ MaxFEs, the final population P would be output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Discussion In essence, ATM-R is a multiphase CMOEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM-R in- tends to achieve a tradeoff between diversity and feasibility in the infeasible phase, promote the transition from “the tradeoff between feasibility and diversity” to “the tradeoff between diversity and convergence” in the semi-feasible phase, and accomplish the tradeoff between diversity and convergence in the feasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To the best of our knowledge, ATM-R is the first algorithm considering these tradeoffs simultaneously during different evolution phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Also, ATM-R is interesting in that it selects promising infeasible solutions suitable for different evolutionary phases by using two kinds of reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As far as we know, relevant studies in this direction are almost absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' From our analysis, it is apparent that ATM- R is a brand-new CMOEA for constrained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The computational time complexity of ATM-R is mainly determined by the nondominated sorting and the unconstrained MOEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Suppose the fast nondominated sorting and NS- GAII [5] are adopted in ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the worst case of the infeasible phase, no solutions nondominated another in the transformed objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The time complexity of this nondominated sorting is O((m+1)·N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The time complexity of assigning each solution to a weight vector is O(m·N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The time complexity of selecting N solutions is O(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, the time complexity of the infeasible phase is O((m + 1) · N 2) + O(m · N 2) + O(N 2) = O((m + 1) · N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the semi-feasible phase, the worst-case time complexity of selecting feasible solutions is O(m·N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the early stage of the semi-feasible phase, the worst-case time complexity is the same as that of the infeasible phase: O((m + 1) · N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the worst case of the later stage, no infeasible solutions nondominated another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It is the same as that of the infeasible phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, its time complexity is O((m + 1) · N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The time complexity of the semi-feasible phase is O(m·N 2)+O((m+1)·N 2)+O((m+ 1) · N 2) = O((m + 1) · N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the feasible phase, the time complexity is the same as that of NSGAII: O(m · N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In summary, the computational time complexity of ATM-R is O((m+1)·N 2)+O((m+1)·N 2)+O(m·N 2) = O(m·N 2), which is indeed acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' PERFORMANCE COMPARISON In this section, we assess the performance of ATM-R based on a wide range of benchmark test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, ATM-R was used to solve three test suites and its performance was compared with that of five representative CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Note that all experiments were implemented by the PlatEMO toolbox [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Experimental Settings 1) Test Functions: Three test suites consisting of 36 bench- mark test functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', MW [45], CTP [46], and LIRC- MOP [25]) were adopted in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' These test functions own various challenging characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' thus, they can assess the performance of a CMOEA adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Most state-of-the- art CMOEAs adopt these test functions for empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Note that the number of decision variables in MW and LIRCMOP was set to 15 and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Please see [25], [45], [46] for the details of these test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2) Peer Algorithms: For performance comparison, five representative CMOEAs were taken into consideration: NSGAII-CDP [5], PPS [12], the constrained two-archive EA (CTAEA) [29], the coevolutionary constrained multiobjective 9 TABLE I THE IGD VALUES OF NSGAII-CDP, PPS, CTAEA, CCMO, TOP, AND ATM-R ON THREE SETS OF BENCHMARK TEST FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Test Functions NSGAII-CDP mean IGD (std) PPS mean IGD (std) CTAEA mean IGD (std) CCMO mean IGD (std) ToP mean IGD (std) ATM-R mean IGD (std) MW1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0545e-2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='02e-1) - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3190e-2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='01e-2) - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1884e-3 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='96e-4) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8990e-3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='42e-3) + NaN (NaN) - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1748e-3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='70e-3) MW2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='73e-1) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1216e-1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='48e-2) LIRCMOP10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6496e-1 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='66e-2) - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2848e-3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} 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(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='83e-2) - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4538e-3 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='89e-5) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2447e-1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='37e-2) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3691e-3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='45e-2) LIRCMOP12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5180e-1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='66e-2) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5216e-2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='43e-2) ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1152e-2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='68e-2) - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6113e-3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='58e-3) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='9104e-2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='29e-2) ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8014e-3 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='28e-3) LIRCMOP13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3757e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='69e-1) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1968e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='45e-3) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0834e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='97e-4) - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3972e-2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='13e-3) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2450e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='78e-3) - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3120e-2 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='31e-4) LIRCMOP14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0248e-1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='93e-1) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1859e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='97e-3) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1126e-1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='98e-4) - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5773e-2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='40e-4) - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1883e-1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='04e-3) - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4848e-2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='79e-4) +/-/≈ 0/35/1 1/33/2 2/27/7 6/19/11 4/25/1 optimization (CCMO) [30], and the two-phase EA (ToP) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' NSGAII-CDP is a classic CMOEA that is usually adopted as a baseline algorithm, while the other four CMOEAs are state-of-the-art algorithms proposed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' NSGAII-CDP is a feasibility-driven CMOEA and the others are infeasibility- assisted CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Among these four infeasibility-assisted CMOEAs, CTAEA and CCMO are multi-population methods which take advantage of infeasible solutions explicitly by an archive or an additional population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' PPS and ToP are multiphase methods which divide the evolutionary process into several phases and put emphasis on objectives in some phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Note that ATM-R is also a multiphase method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3) Performance Metrics: Two frequently used performance metrics were adopted to assess the performance of a CMOEA: inverted generational distance (IGD) and hyper-volume (HV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Both IGD and HV can measure the convergence and coverage of a solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' More details of these two metrics can be found in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4) Parameter Settings: The parameters involved in the experiments are given as follows: Size of the final solution set: N = 100 for all comparison CMOEAs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' MaxFEs: MaxFEs = 60, 000 for the MW and CTP test suites, and MaxFEs = 300, 000 for the LIRCMOP test suite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='15 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='87 0 1 2 3 4 5 6 NSGAII-CDP PPS CTAEA CCMO ToP ATM-R IGD HV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Average rankings of six CMOEAs on 36 test functions in terms of the IGD/HV value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A lower ranking value denotes a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Number of independent runs: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The SBX and PM were used as genetic operators in all CMOEAs except ToP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The parameters of SBX and PM are as follows: Crossover probability of SBX: 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Mutation probability of PM: 1/D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Distribution index of SBX and PM: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In addition, the algorithm-specific parameters of the five peer CMOEAs were obtained from their original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 10 TABLE II THE HV VALUES OF NSGAII-CDP, PPS, CTAEA, CCMO, TOP, AND ATM-R ON THREE SETS OF BENCHMARK TEST FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Test Functions 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='04e-4) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='9235e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='25e-3) LIRCMOP9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5772e-1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='15e-2) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2821e-1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} 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(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='74e-2) - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0659e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='77e-4) + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0755e-1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='24e-4) + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0630e-1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} 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+page_content='29e-3) ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='0839e-1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='46e-2) - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='1811e-1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='04e-3) LIRCMOP13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='7950e-1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='63e-1) - 5.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='56e-3) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5578e-1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='23e-3) LIRCMOP14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='9121e-1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='36e-1) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='3744e-1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='86e-3) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4656e-1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='33e-4) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5357e-1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='22e-3) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2944e-1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='28e-3) - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5604e-1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='16e-3) +/-/≈ 0/33/3 3/31/2 3/28/5 7/20/9 3/32/1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Comparison Results First, we compared the performance of ATM-R with that of the other five CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The mean IGD values and standard deviations of 36 test functions over 30 independent runs are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The results in terms of the HV value are collected in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In each table, “std” represents the standard deviation of the IGD/HV values over 30 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' “NaN” denotes that a CMOEA cannot find a feasible solution of a test function over all 30 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For a given test function, ATM-R was compared with each competitor by the Friedman test with Bonferroni correction at a significance level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For convenience, “+”, “-”, and “≈” are used to represent that a competitor is better than, worse than, and similar to ATM-R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In addition, for each test function, the best result among the six CMOEAs is highlighted in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To visualize the results, we plotted the CPFs obtained by the six CMOEAs in a typical run on three representative CMOPs in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 6-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A typical run denotes the one producing the median IGD value among all runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 1) General Performance: In general, as shown in Table I and Table II, ATM-R obtained the best results of most of the test functions in terms of both the IGD and the HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, it performed significantly better than the other five competitors on most of the test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The multi- problem Friedman’s test [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='] was implemented to compare these six CMOEAs simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 5, ATM- R achieved the lowest ranking value among six CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Furthermore, the results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 6-8 show that ATM-R can obtain a set of well-converged and well-distributed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' A more detailed discussion on different test suites is given next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 2) Performance on MW Test Suite: In terms of the IGD value, ATM-R performed better than NSGAII-CDP, PPS, CTAEA, CCMO, and ToP on 14, 13, six, six, and 14 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Inversely, these peer CMOEAs were better than ATM-R on zero, one, two, three, and zero test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM-R obtained the best results of four test functions on which it performed better than the other five competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' CCMO obtained the best results of four test functions, on one of which it performed similarly to ATM- R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Although CTAEA obtained the best results of six test functions, it performed similarly to ATM-R on five of these test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In terms of the HV value, ATM-R performed better than NSGAII-CDP, PPS, CTAEA, CCMO, and ToP on 12, 12, six, eight, and 14 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' On the contrary, these peer CMOEAs revealed better results than ATM-R on zero, two, three, five, and zero test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM- R obtained the best results of three test functions on which it performed better than the other five competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' CCMO 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 2 3 NSGAII-CDP on MW13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 2 3 PPS on MW13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 2 3 CTAEA on MW13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 2 3 CCMO on MW13 2 3 4 5 1 2 3 ToP on MW13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 2 3 ATM-R on MW13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on MW13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 NSGAII-CDP on CTP4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 PPS on CTP4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 5 10 15 CTAEA on CTP4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 CCMO on CTP4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 ToP on CTP4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='8 1 ATM-R on CTP4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on CTP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 NSGAII-CDP on LIRCMOP14 1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 PPS on LIRCMOP14 1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 CTAEA on LIRCMOP14 1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 CCMO on LIRCMOP14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 ToP on LIRCMOP14 1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 ATM-R on LIRCMOP14 1 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The constrained Pareto front with median value among 30 runs obtained by NSGAII-CDP, PPS, CTAEA, CCMO, ToP, and ATM-R on LIRCMOP14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' obtained the best results of four test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Although CTAEA obtained the best results of seven test functions, it performed similarly to ATM-R on four of these test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Furthermore, as shown in Fig 6, ATM-R obtained a set of well-converged and well-distributed feasible solutions that is close to the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' However, ToP failed to converge to the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' NSGAII-CDP, PPS, CTAEA, and CCMO lost some parts of the CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The results reflect that ATM-R performs better than the other five competitors on the MW test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 3) Performance on CTP Test Suite: For the CTP test suite, ATM-R performed better than the other five competitors on most of the test functions in terms of both the IGD and HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, it obtained the best IGD/HV values on most of the test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For CTP1, some parts of the CPF come from the un- constrained Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For CTP6, the objective space has infeasible holes of differing widths toward the Pareto-optimal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For CTP2-CTP5, CTP7, and CTP8, the CPFs are divided into several disconnected segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To solve these problems effectively, infeasibility information should be used carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, NSGAII-CDP and ToP, which only consider constraints in the infeasible phase, performed worse than ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As stated in [4], due to the complex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', disconnected and discrete) characteristics of the CPFs, the CMOEAs using reference points or vectors would have inferior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Therefore, CTAEA performed worse than ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' PPS puts emphasis on objectives in the early stage, while CCMO adopts a specific population to make use of infeasibility information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Compared with the MW test suite, the test functions in the CTP test suite have larger feasibility ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, too much infeasibility information would impair the performance of a CMOEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' This may be why PPS and CCMO performed worse than ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 7, ATM-R can converge to the CPF of CTP4 more quickly than the other five competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Additionally, it can cover more parts of the CPF than the other five competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The results reflect that ATM-R performs better than the other five competitors on the CTP test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 4) Performance on LIRCMOP Test Suite: For the LIRC- MOP test suite, ATM-R obtained the best results on half of the test functions in terms of both the IGD and HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Similar to the CTP test suite, the test functions in the LIRCMOP test suite have infeasible holes in the objective space and the CPFs of some test functions are disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To solve these test functions effectively, infeasibility information should be used carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' NSGAII-CDP and ToP performed worse than ATM-R on most of the test functions because they ignore the infeasibility information to a great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ToP performed better than ATM-R on four and three test functions in terms of the IGD and HV values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' This is attributed to the powerful genetic operator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', differential evolution) used in ToP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Among the infeasibility-assisted CMOEAs, PPS motivates the population toward the unconstrained Pareto front in the early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In CTAEA and CCMO, an additional popula- tion is employed to approach the unconstrained Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, these three CMOEAs will fail to solve a CMOP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', LIRCMOP1-LIRCMOP4) in which the CPF is far away from 12 the unconstrained Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Regarding LIRCMOP5 and LIRCMOP6, the CPFs are the same as the unconstrained Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Regarding LIRCMOP7 and LIRCMOP8, the CPFs are near the unconstrained Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For these four test functions, an infeasibility-assisted CMOEA can approach the CPF easily;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' thus, uniformity is the key factor affecting its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM-R and CCMO performed better than PPS and CTAEA on these test functions because they can preserve diversity more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For LIRCMOP9-LIRCMOP12, the CPFs are divided into several disconnected segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To solve these test functions effectively, diversity should be maintained carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Due to the weak cooperation of two populations, CCMO can maintain diversity effectively during the evolution- ary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, it performed better than ATM-R on these four test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' For the two three-objective test functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', LIRCMOP13-LIRCMOP14), ATM-R performed better than the other five competitors in terms of both the IGD and HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' It implies that ATM-R can achieve a bet- ter tradeoff among feasibility, diversity, and convergence for three-objective CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' 8, all six CMOEAs can converge to the constrained Pareto front successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM- R performed better than the other five competitors in terms of the uniformity since it can achieve a better tradeoff among fea- sibility, diversity, and convergence for three-objective CMOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The results reflect that ATM-R performs better than the other five competitors on the LIRCMOP test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In summary, the extensive experiments on 36 test func- tions with various challenging characteristics demonstrate that ATM-R is able to solve complex CMOPs successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' FURTHER ANALYSES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Advantages of ATM-R over ATM As discussed in Remark 1, it is not effective to extend ATM [40] to solve CMOPs straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In this subsec- tion, the advantages of ATM-R over ATM were demonstrated through experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The comparison results on 36 test func- tions in terms of the IGD and the HV values are summarized in Table III, where “+”, “-”, and “≈” denote that ATM performs better than, worse than, and similarly to ATM-R in terms of the IGD/HV value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As shown in Table III, ATM-R performed better than ATM on these three test suites in terms of both the IGD and the HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, in terms of the IGD value, ATM-R was better than ATM on 9, 5, and 12 test functions of the MW, the CTP, and the LIRCMOP test suites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In contrast, ATM was better than ATM- R on no more than three test functions of these test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' With regard to the HV value, ATM-R performed better than ATM on 8, 6, and 12 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Inversely, ATM outperformed ATM-R on no more than 4 test functions of these test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In summary, the experimental results on these test functions with various characteristics demonstrate that ATM-R has an edge over ATM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Effectiveness of the Infeasible Phase To validate the effectiveness of the update mechanism in the infeasible phase, we implemented three variants (denoted as TABLE III RESULTS OF ATM VS ATM-R ON 36 TEST FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Test Functions IGD +/-/≈ HV +/-/≈ MW1-MW14 2/9/3 4/8/2 CTP1-CTP8 3/5/0 2/6/0 LIRCMOP1-LIRCMOP14 1/12/1 1/12/1 TABLE IV RESULTS OF ATM-RICDP VS ATM-R, ATM-RIOBJ VS ATM-R, AND ATM-RIDIV VS ATM-R ON 36 TEST FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Algorithms IGD +/-/≈ HV +/-/≈ ATM-RICDP vs ATM-R 2/19/15 2/23/11 ATM-RIobj vs ATM-R 5/11/20 7/11/18 ATM-RIdiv vs ATM-R 8/14/14 10/14/12 ATM-RICDP, ATM-RIobj, and ATM-RIdiv) by using different update mechanisms in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, in ATM-RICDP, the CDP is used for solution selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In ATM-RIobj, the solutions are selected based on Pareto dominance regardless of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In ATM-RIdiv, the diversity is quantified and used to select promising solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' By comparing ATM-R with each of ATM-RICDP, ATM-RIobj, and ATM-RIdiv, the effectiveness of the update mechanism in the infeasible phase can be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The comparison results on 36 test functions mentioned above are summarized in Table IV, where “+”, “-”, and “≈” denote that a competitor performs better than, worse than, and similarly to ATM-R in terms of the IGD/HV value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As shown in Table IV, ATM-R performed better than ATM- RICDP, ATM-RIobj, and ATM-RIdiv in terms of both the IGD and the HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, with regard to the IGD value, ATM-R was better than ATM-RICDP, ATM-RIobj, and ATM-RIdiv on 19, 11, and 14 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Inversely, ATM-RICDP, ATM-RIobj, and ATM-RIdiv outper- formed ATM-R on 2, 5, and 8 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In terms of the HV value, ATM-R performed better than ATM- RICDP, ATM-RIobj, and ATM-RIdiv on 23, 11, and 14 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In contrast, ATM-RICDP, ATM-RIobj, and ATM-RIdiv outperformed ATM-R on 2, 7, and 4 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The experimental results show that the update mechanism in the infeasible phase is critical to ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Effectiveness of the Semi-feasible Phase To validate the effectiveness of the update mechanism in the semi-feasible phase, we implemented three variants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', ATM-RSCDP, ATM-RSobj, and ATM-RSdiv) by using different update mechanisms in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, in ATM-RSCDP, ATM-RSobj, and ATM-RSdiv, the CDP, the Pareto dominance, and the diversity are used for solution selection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' By comparing ATM-R with each of ATM-RICDP, ATM-RIobj, and ATM-RIdiv, the effectiveness of the update mechanism in the semi-feasible phase can be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, the comparison results on 36 test functions mentioned above are summarized in Table V, where 13 TABLE V RESULTS OF ATM-RSCDP VS ATM-R, ATM-RSOBJ VS ATM-R, AND ATM-RSDIV VS ATM-R ON 36 TEST FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Algorithms IGD +/-/≈ HV +/-/≈ ATM-RSCDP vs ATM-R 3/24/9 5/23/8 ATM-RSobj vs ATM-R 3/32/1 2/31/3 ATM-RSdiv vs ATM-R 0/36/0 0/36/0 “+”, “-”, and “≈” denote that a competitor performs better than, worse than, and similarly to ATM-R in terms of the IGD/HV value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As shown in Table V, ATM-R performed better than ATM- RSCDP, ATM-RSobj, and ATM-RSdiv in terms of both the IGD and the HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' With regard to the IGD value, ATM-R was better than ATM-RSCDP, ATM-RSobj, and ATM- RSdiv on 24, 32, and 36 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In contrast, ATM-RSCDP, ATM-RSobj, and ATM-RSdiv out- performed ATM-R on no more than three test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In terms of the HV value, ATM-R performed better than ATM- RSCDP, ATM-RSobj, and ATM-RSdiv on 23, 31, and 36 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Inversely, ATM-RSCDP, ATM-RSobj, and ATM-RSdiv outperformed ATM-R on no more than five test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The experimental results show that the update mechanism in the semi-feasible phase is critical to ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Effectiveness of the Multiphase Mating Selection Strategy In order to verify the effectiveness of the multiphase mating selection strategy, we implemented three variants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=', ATM- RMCDP, ATM-RMobj, and ATM-RMdiv) by using different selection methods to select mating solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, in ATM-RMCDP, ATM-RMobj, and ATM-RMdiv, the CDP, the Pareto dominance, and the diversity are used for solution selec- tion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' By comparing ATM-R with each of ATM- RMCDP, ATM-RMobj, and ATM-RMdiv, the effectiveness of the multiphase mating selection strategy can be demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, the comparison results on 36 test functions men- tioned above are summarized in Table VI, where “+”, “-”, and “≈” denote that a competitor performs better than, worse than, and similarly to ATM-R in terms of the IGD/HV value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' As shown in Table VI, ATM-R performed better than ATM- RMCDP, ATM-RMobj, and ATM-RMdiv in terms of both the IGD and the HV values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' With regard to the IGD value, ATM-R was better than ATM-RMCDP, ATM-RMobj, and ATM-RMdiv on 19, 10, and 9 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In contrast, ATM-RMCDP, ATM-RMobj, and ATM-RMdiv out- performed ATM-R on 5, 7, and 5 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In terms of the HV value, ATM-R performed better than ATM- RMCDP, ATM-RMobj, and ATM-RMdiv on 22, 13, and 8 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Inversely, ATM-RMCDP, ATM- RMobj, and ATM-RMdiv outperformed ATM-R on 3, 7, and 4 test functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The experimental results show that the multiphase selection strategy is critical to ATM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' TABLE VI RESULTS OF ATM-RMCDP VS ATM-R, ATM-RMOBJ VS ATM-R, AND ATM-RMDIV VS ATM-R ON 36 TEST FUNCTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Algorithms IGD +/-/≈ HV +/-/≈ ATM-RMCDP vs ATM-R 5/19/12 3/22/11 ATM-RMobj vs ATM-R 7/10/19 7/13/16 ATM-RMdiv vs ATM-R 5/9/22 4/8/24 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' CONCLUSIONS This paper has analyzed the key task of constrained multi- objective optimization in depth and decomposed it into three subtasks explicitly for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' To accomplish these three subtasks in different evolutionary phases, an adaptive tradeoff model with reference points (ATM-R) was designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Specifically, ATM-R takes advantage of infeasible solutions to achieve different tradeoffs in these three subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the infeasible phase, ATM-R distinguishes and uses infeasible solutions with good diversity to enhance the diversity loss due to its pursuit of feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, the population can move toward feasible regions from diverse search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the semi-feasible phase, ATM-R leverages infeasible solutions with good diversity/objective function values to promote the transition from “the tradeoff between feasibility and diversity” to “the tradeoff between convergence and diversity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Thus, the population can locate enough feasible solutions and approach the CPF quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' In the feasible phase, ATM-R employs NSGAII to seek a set of well-converged and well-distributed solutions close to the constrained Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Moreover, a multiphase mating selection strategy is proposed to select appropriate mating parents adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' Experimental studies on a wide range of CMOPs demonstrate that: ATM-R achieves better or at least highly competitive performance against other representative CMOEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' ATM-R has a significant advantage over ATM for con- strained multiobjective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE1T4oBgHgl3EQfoQQY/content/2301.03317v1.pdf'} +page_content=' The update mechanisms in the infeasible phase and the semi-feasible phase are both critical to the performance of ATM-R.' metadata={'source': 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Shabana,∗, A. Siemiginowskab, R.M. Suleimanb, M.S. El-Nawawya, A. Alia +aAstronomy, Space Science and Meteorology Department, Faculty of Science, Cairo University, Giza, EGYPT +bCenter for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA +Abstract +We performed a study of high redshift (z > 2) quasars, looking for the main differences be- +tween Radio Loud Quasars (RLQ) and Radio Quiet Quasars (RQQ) in the X-ray band. Our +sample of 472 RQQ and 81 RLQ was selected by cross-matching the SDSS DR7 quasars +catalog with the Chandra Source Catalog. We computed the X-ray luminosity for the two +samples and confirmed the X-ray luminosity excess of RLQ over RQQ. We fit the X-ray +spectra assuming the absorbed power law model and obtained the photon index (Γ) values +for all the sources in the sample. We excluded quasars with a low number of counts (< 10) +and large uncertainty on the best-fit photon index (Γerr > 1), and obtained the mean values +of ΓRLQ = 1.70 +0.36 +−0.33 and ΓRQQ = 2.19 +0.46 +−0.44 for the RLQ and RQQ samples, respectively, +showing that the RLQ have flatter (harder) X-ray spectra than RQQ. The Kuiper-two test +confirms this result with the significant difference between the RLQ and RQQ photon in- +dex distributions (Dk = 0.37 and P-value = 10−6). We also evaluated the hardness ratio +distributions and confirmed that the spectra of RLQ are flatter than the spectra of the RQQ. +The RLQ’s hard-to-soft ratio distribution is skewed towards the hard X-ray band, while the +RQQ is towards the soft X-ray band. The hard-to-medium and medium-to-soft ratios show +no difference. +Keywords: Radio Loud Quasars, Radio Quiet Quasars, X-ray Astrophysics, X-ray photon +index, Hardness Ratio +1. Introduction +There are two main classes of quasars: the Radio Quiet Quasars (RQQ) and the Radio +Loud Quasars (RLQ). They have been identified based on the orientation and presence of +∗corresponding author +Email address: fshaban@sci.cu.edu.eg (F. Shaban ) +Preprint submitted to JHEAP +January 10, 2023 +arXiv:2301.02866v1 [astro-ph.HE] 7 Jan 2023 + +a radio jet (Antonucci, 1993; Wilson and Colbert, 1995; Urry and Padovani, 1995). RLQ +have optical and X-rays luminosity about three times greater than their RQQ counterparts +(Zamorani et al., 1981; Worrall et al., 1987; Miller et al., 2010; Zhu et al., 2020). The X- +ray radiation could be due to the Compton scattering of UV photons by energetic electrons +or due to synchrotron radiation from highly relativistic electrons. (Mushotzky et al., 1993; +Nowak, 1995; Turner and Miller, 2009; Worrall, 2009; Fabian, 2012). +Majority of quasars are RQQ with the X-ray radiation attributed to a hot corona formed in +the accretion flow (Haardt and Maraschi, 1993; Fabian et al., 2015; Zhu et al., 2020). RLQ +are a small minority, about ≈ 10% of all quasars, and are characterized by their relativis- +tic jets generated by an accreting supermassive black hole (SMBH) (Padovani et al., 2017; +Blandford et al., 2019). The amount of jet radiation contributing to the X-ray spectrum in +RLQ is still not fully understood. However, identifying the main radiation components in the +X-ray spectrum is important to the estimates of the quasar power. +The RLQ have flatter X-rays spectra (lower photon index value) than those of the RQQ +(Reeves et al., 1997; Page et al., 2005; Miller et al., 2010). The quasar’s hardness ratio +is consistent with the spectral slope (Freeman et al., 2001; Evans et al., 2010; Peca et al., +2021). The RLQ are divided into Core Dominant (CD) and Lobe Dominant (LD) (Haardt and +Maraschi, 1993; Wilson and Colbert, 1995). The radio emission of CD quasars is dominated +by the relativistic jet, while the LD quasars show significant radio emission from the large- +scale components in comparison to the core (Falcke et al., 1995; Boroson, 2002). These two +populations might have different X-ray radiation processes, which was noted recently by Zhu +et al. (2020). +During the past decades, quasar data from X-ray surveys have become available, allowing +for statistical studies of relatively large samples. Many recent studies considered the high +redshift quasars for survey (Kelly et al., 2007; Vito et al., 2019; Pons et al., 2020; Li et al., +2021). Some studies were focusing on deducing RLQ properties using correlations between +X-ray, radio, and optical (or UV) luminosities to investigate the quasars physical model, +Miller et al. (2010) investigate the disk-jet model, Zhu et al. (2020) deduced the disk-corona- +jet model. Interestingly, Lusso and Risaliti (2017) were studying RQQ and showed that RQQ +could be used as standard candles at high redshifts (z > 2), which is important for distance +measurement and cosmological tests. +In this research, we investigate the differences in the X-ray spectral properties (photon +index, intrinsic absorption, hardness ratios, and X-ray luminosity) between RQQ and RLQ +samples using the data available in the Chandra Source Catalog (CSC2) (Evans et al., 2019). +We study quasars at a high redshift near the peak of cosmic quasar activity, at z > 2. Our +sample contains the largest number of RLQ at high redshift observed with Chandra and +2 + +include faint sources with [10−15 - 10−13] ergcm−2 s−1 1 for the energy range [0.5 - 7.0] +keV. We calculate the photon index by fitting the faint X-ray spectra, thus expanding the +number of quasars with this parameter. The observed Chandra effective energy range is [0.5 +- 7.0] keV and corresponds to the rest frame energy greater than [1.5 - 21.0] keV at redshift +z > 2, so at the higher redshifts we are able to study the X-ray spectra, which are the most +sensitive to the properties of the corona and relativistic jet. +In section 2, we briefly describe the data catalogs, the sample selection criteria, and our +constraints. In section 3, we show the distributions of RLQ and RQQ as functions of X- +ray parameters and illustrate the photon index calculations and constraints, furthermore, we +analyze extreme cases for the photon index. In section 4, we discuss our results and compare +them with previous studies, and conclude with a discussion and outlook for future work. +2. Sample selection +DR7+ CSC2 +(2561) +Z > 2 +(595) +DR7 +105,783 +quasars +First_flag= 2; +Lobe- +Dominant +First_flag= -1; +Out of First +field +First_flag= 0; +Radio Quiet +First_flag= 1; +Core- +Dominant +Radio Loud +R ≥ 10 +Radio intermediate +R < 10 +19 +71 +472 +10 +23 +Radio Loud +R ≥ 10 +Figure 1: The sample selection is based on the DR7, CSC2, redshift and radio-loudness. The filters in the fourth column are showing the +FIRST flag values (-1, 0, 1, and 2) representing quasars (not in the FIRST field, Radio Quiet, Core dominant, and Lobe dominant). The +next column filters the radio-loudness (R) into RQQ, RLQ, and RIQ. The circles represent the quasar’s number in each category. +1https://cxc.cfa.harvard.edu/csc/char.html +3 + +We study the X-ray properties of quasars using archival data from two quasar catalogs. +We use DR7 quasars catalog (Shen et al., 2011), which contains 105,783 quasars with optical +spectra and redshift measurements. Shen et al. (2011) quasars were selected from the SDSS +DR7 sample compiled by Schneider et al. (2010) and all have spectroscopic redshift mea- +surements. Schneider et al. (2010) rejected the pipeline redshift measurements for the quasar +candidates with images exceeding the PSF size in the r-band. They provide the uncertainty +on the redshift measurement to be + +− 0.004. +We use the X-ray data obtained by the Chandra X-ray Observatory (Chandra) during the +first 15 years of the mission available in the Chandra Source Catalog release 2.0 (CSC22). +There are more than 315,000 unique X-ray sources in the CSC2 (Evans et al., 2019). Chandra +has a high-quality angular resolution (better than 5′′), which is important for detecting faint +sources, at high redshift, with good source positions. We cross-matched the 105,783 DR7 +quasars with sources in CSC2, using TOPCAT (Taylor, 2017), and set a search cone radius of +30′′, consistent with the range of the sources offset uncertainty given by Evans et al. (2019). +We found 2,561 sources corresponding to X-ray sources in CSC2. We study the sources at +high redshift (z > 2). After applying the (z > 2) filter, we obtained 595 out of 2,561 quasars. +The details of our full sample selection are presented in Fig.1. +Shen et al. (2011) matched DR7 optical quasars catalog with Faint Images of the Radio +Sky at Twenty Centimeters (FIRST) catalog (White et al., 1997), and estimated the quasar +radio loudness parameter (R) defining RLQ and RQQ based on the following equation +R = +�f6 cm +f2500 +� +(1) +where f6 cm and f2500 are the fluxes density (fν) at rest-frame 6 cm and 2500 ˚A, respec- +tively. The flux density in DR7 is determined from the FIRST integrated flux density at 20 cm +assuming a power-law slope of αν = − 0.5. The flux density at the rest frame of 2500 ˚A is +determined by fitting the optical spectrum with a power-law continuum (Shen et al., 2011). +Similar to Jiang et al. (2007), Shen et al. (2011) have divided RLQ in DR7 into lobe +dominant (LD) and core dominant (CD) with FIRST cone radius of 30′′ and 5′′, respectively. +Shen et al. (2011) have removed the effects of galactic extinction in the SDSS spectra +using the Schlegel et al. (1998) map, and the Milky Way extinction curve by Cardelli et al. +(1989). Furthermore, Shen et al. (2011) shifts the spectra to the rest frame using the cataloged +redshift as the systematic redshift (Hewett and Wild, 2010). +We select the Radio Intermediate quasars (RIQ) to have R < 10 and RLQ with (R ≥ 10) +(Miller et al., 2010). We applied the above selection categories to our initial sample of 595 +2https://cxc.cfa.harvard.edu/csc/ +4 + +quasars in CSC2 and divided them into different radio-loudness categories as given in Figure +1. Because we focus on strong differences between the RLQ and RQQ, therefore we exclude +the intermediate sample and only include RLQ and RQQ in our analysis. Our final sample +contains 81 RLQ and 472 RQQ. +3. Data Analysis and Results +We study several parameters representing the X-ray properties of the quasars in our sam- +ples. The redshift (z), and the radio loudness (R) are provided from DR7 (Shen et al., 2011), +while the X-ray flux (fX), the hardness ratios (HRh/m), the hydrogen column density (NH), +and the X-ray spectral files are given in CSC2 (Evans et al., 2019). We calculate the X-ray +luminosity (LX) and the X-ray photon index (Γ). +In order to evaluate the difference between RLQ and RQQ samples in all X-ray param- +eters we use the Kuiper-two sample test (Watson, 1961). The Kuiper test is a test for the +difference between two samples based on their observed Cumulative Distribution Functions +(CDF). It is an extension of the Kolmogorov–Smirnov test, but it is more sensitive to the +shift between the two distributions and the difference in the tails of the distributions. The +Kuiper test is non-parametric and does not assume any functional form of the sample’s true +distribution and it is appropriate when true distributions are unknown. The test returns DK +and Fk, which are the maximum difference between the two samples and the probability +p-value of the test, respectively. The Fk < 0.05 rejects the hypothesis that the two samples +are drawn from the same distribution, so the smaller the value the stronger the significance +of the difference between the two samples. All the Kuiper-two test values of this study are +given in (Table 2). +In our figures, we use normalized density histograms because we have different samples +size. The histograms represent the probability density function of the parameter distributions +(Hunter, 2007), (i.e., m/M × b), where m is the number of quasars in each specific bin, M +is the total number of quasars in the sample, and b is the bin bandwidth. So the area under +the bins integrates into one. We apply the same number of bins to RLQ and RQQ. The RQQ +sample appears to have a smaller bin bandwidth than the RLQ sample because the bin band- +width is affected by the sample number in the probability density function. We also apply +the Kernel Density Estimation (KDE) smoothing function to account for the small sample +size and different bin sizes (Rosenblatt, 1956). The small sample size may contribute to the +gaps within the histograms, and different binning could lead to statistical biases (Waskom, +2021). We use the following KDE equation: +P(x) = +1 +M × h +M +� +i=1 +k +�x − xi +h +� +(2) +5 + +Where M is the total number of quasars in the sample, h the Kernel bandwidth, k the +chosen kernel weight function in our estimate (Gaussian), x is the point where to calculate +the function, and xi is the parameter value in bin i. The seaborn package 3 for fitting KDE +has a built-in kernel bandwidth optimal estimation using Silverman methods, which are used +for random normally distributed samples (Silverman, 1981). +Figure 2 shows the redshift distributions of RLQ and RQQ samples. We apply the Kuiper- +two test which returns a small difference between the RLQ and RQQ samples with Dk = +0.19 and Fk = 0.08. This confirms that the RLQ and RQQ samples in our studies have +consistent redshift distributions. +3.1. X-ray Luminosity +We calculated the X-ray luminosity using the equation given by: +LX = 4πdL +2fX +(3) +Where LX is X-ray luminosity, dL is the distance luminosity, and fX is the X-ray flux +in [0.5 - 7.0] keV broadband energy band. The cosmological model used in this study is the +WMAP9 with (Ho = 69.33, Ωo = 0.287, ΩΛ = 0.712) parameters (Hinshaw et al., 2013). +We use the WMAP9 under the astropy.cosmology package to obtain the distance lu- +minosity (dL) (Astropy Collaboration et al., 2018). Using fX and dL, and Eq.3 we calculate +the X-ray luminosity (Harris et al., 2020). +Figure 2 shows the X-ray luminosity distributions of RLQ and RQQ samples. The RQQ +KDE (blue) shows a shape consistent with a Gaussian distribution and the RLQ KDE (green) +is skewed to the higher X-ray luminosities. The X-ray luminosity range, given in log scale, +for RLQ is LXmin. = 44.5 and LXmax. = 47.07, while for RQQ are LXmin. = 43.68 and +LXmax. = 46.30. The differences between minimum and maximum luminosities are similar, +2.57 and 2.62 for RLQ and RQQ, respectively. However, the median of LX is higher in the +RLQ sample by 0.53 compared to the RQQ’s median. This difference in the median between +RLQ and RQQ is significant and indicates a reliable difference between the intrinsic proper- +ties of the two samples, RLQ and RQQ. The Kuiper-two test returns a significant difference +in X-ray luminosity distributions between RLQ and RQQ, Dk = 0.42, and Fk = 2.18×10−9. +The Fk value validates the remarkable difference in the X-ray luminosity between the radio- +quiet and radio-loud quasars (see Table 1). +3https://seaborn.pydata.org/generated/seaborn.kdeplot.html +6 + +Figure 2: The two panels show the redshift (left), and the X-ray luminosity (right) distributions comparison between RLQ and RQQ. The +green histogram represents all of the RLQ as a function of X-ray Luminosity, and the solid green curve represents the KDE for the RLQ. +The blue histogram represents the RQQ as a function of the X-ray Luminosity, and the dashed-blue line represents its KDE. The +histograms are normalized. +3.2. The Hardness ratios +The hardness ratio is defined as the flux ratio between two different Chandra energy +bands. The X-ray energy bands in the CSC2 are divided into several categories 4: +• Broad (0.5-7.0) keV +• Hard (2.0-7.0) keV +• Medium (1.2-2.0) keV +• Soft (0.5-1.2) keV +The hardness ratios for hard to medium (HRh/m), medium to soft (HRm/s), and hard to +soft (HRh/s) 5 are given in CSC2 for each detected source. CSC2 provides f(h), f(m), and +f(s) the X-ray fluxes in the hard, medium, and soft energy bands, respectively. +HRh/m += +f(h) − f(m) +f(h) + f(m) +(4) +The HRm/s and HRh/s are defined similar to equation 4. When the hardness ratio exceeds +zero, the flux of the higher energy band dominates over the flux of the lower energy band. +For a general comparison between RLQ and RQQ samples, we investigate the distributions +for HRh/m, HRm/s and HRh/s shown in Figure 3 and Figure 3. The distribution plots were +normalized and smoothed with KDE. +4https://cxc.cfa.harvard.edu/csc/columns/ebands.html +5https://cxc.cfa.harvard.edu/csc/columns/spectral properties.html +7 + +1.75 +RQQ +RLQ +1.50 +1.25 +#1.00 +Densit +0.75 +0.50 +0.25 +0.00 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +Redshift- +RQQ +1.0 +RLQ +0.8 +L +Density +0.6 +0.4 +0.2 +0.0 +44 +45 +46 +47 +48 +log (x-rays_luminosity)Figure 3: The three panels show the HRh/m, HRm/s, and HRh/s distributions. The green solid line represents the RLQ, while the blue +dashed line represents the RQQ. The red vertical lines indicate the photon index values corresponding to each hardness ratio for Γ equal +to (3, 2, 1, 0) from left to right. The left panel shows no difference between RLQ and RQQ distributions. The middle panel shows a slight +difference. The right panel shows a significant difference between RLQ and RQQ with a higher tendency toward soft energy in the RQQ +sample. +We also mark the evolution of the photon index as a function of the hardness ratios. Using +the fake pha function in Sherpa and the standard ACIS-S response files, we fix the photon +index (Γ = 0, 1, 2, 3) to simulate the spectrum and calculate the corresponding hardness +ratios for each Γ. Figures 3 show that the red marks of the photon index decrease (flat +spectrum) as the hardness ratios increase (towards the hard band). RLQ and RQQ samples +have a similar HRh/m distributions (see Figure 3) confirmed by the Kuiper-two test Dk = +0.16, Fk = 0.37. +The HRm/s distribution shown in Figure 3, shows a slight shift towards the soft energy +band for RQQ in comparison to the RLQ sample, also indicated by the Kuiper-two test +Dk = 0.21 and Fk = 0.05. +Finally, the HRh/s distribution shows the most significant difference between RLQ and +RQQ samples (see Figure 3) with the Kuiper-two test results of Dk = 0.25 and Fk = 0.01, +the test accuracy is 99.8% (see Table 2). The HRh/s distribution indicates that the X-ray +spectra of RQQ quasars are softer than the spectra of RLQ quasars in our samples. +We investigate the X-ray properties of the CD and LD quasars separately in our compar- +ison to RQQ by applying the Kuiper-two test on all the parameters. We find that LD and +CD samples are consistent in all of the X-ray physical parameters except for hardness ratios, +HRh/s and HRh/m with Fk = 0.16 and Fk = 0.10, respectively. However, HRh/s and HRh/m +distributions for LD sample are similar to RQQ with Fk = 0.53 and Fk = 0.58, respectively. +On the other hand, our LD sample is small (10 LD quasars). Future studies of CD and LD +quasars with high-quality X-ray spectra are needed to confirm and investigate these results +further. +8 + +3.0 +RQQ +RLQ +2.5 +2.0 +isity +1.5 +1.0 +0.5 +0.0 +1.0 +0.5 +0.0 +0.5 +1.0 +HRh/m2.5 +RQQ +RLQ +2.0 +1.5 +Density +1.0 +0.5 +0.0 +-1.00 +-0.75 +-0.50 +-0.25 +0.00 +0.25 +0.50 +0.75 +HRml/sRQQ +2.00 +RLQ +1.75 +1.50 +ensity +1.25 + 1.00 +0.75 +0.50 +0.25 +0.00 +-1.0 +0.5 +0.0 +0.5 +1.0 +HRh/sTable 1: The Statistical Analysis for X-ray Parameters +Radio Loud Quasars +Radio Quiet Quasars +Parameter +max +min +mean +median +SD +max +min +mean +median +SD +z +4.7 +2.0 +2.88 +2.67 +0.76 +5.42 +2.08 +2.7 +2.45 +0.75 +fX +10 +0.05 +1.4 +0.2 +−0.2 +0.7 +0.17 +−0.14 +2 +3 +0.004 +0.4 +0.10 +−0.09 +0.2 +0.89 +−0.68 +0.4 +LX +47.07 +44.54 +45.66 +0.08 +−0.07 +45.9 +0.09 +−0.07 +0.56 +46.3 +43.68 +45.16 +0.11 +−0.01 +45.16 +0.10 +0.00 +0.45 +HRh/s +0.90 +-0.99 +-0.10 +0.05 +−0.35 +-0.13 +0.06 +−0.33 +0.3 +0.99 +-0.99 +-0.21 −0.05 +−0.53 +-0.28 −0.06 +−0.52 +0.40 +HRm/s +0.99 +-0.74 +-0.21 −0.02 +−0.39 +-0.21 +0.05 +−0.38 +0.29 +0.99 +-0.99 +-0.29 −0.02 +−0.49 +-0.3 −0.08 +−0.51 +0.3 +HRh/m +0.6 +-0.99 +0.11 +0.23 +−0.16 +0.09 +0.27 +−0.11 +0.24 +0.99 +-0.99 +0.09 +0.30 +−0.26 +0.07 +0.28 +−0.26 +0.39 +Γ +3.4 +-0.39 +1.8 +0.38 +−0.34 +1.76 +0.35 +−0.32 +0.50 +4.8 +-0.88 +2.14 +0.0.5 +−0.44 +2.06 +0.48 +−0.43 +0.65 +fX: The given X-ray flux (ergcm−2 s−1) must be multiplied by factor of 10−13. +LX: The X-ray luminosity (ergs−1) is given in log scale. +9 + +3.3. X-ray Spectral Modeling and Photon Index +CSC2 lists the photon index calculated by fitting a power law model multiplied by the +photoelectric absorption. However, the CSC2 pipeline restricted the model fitting to spectra +with at least 150 net counts (after subtracting the background) and applied the spectral bin- +ning of 20 counts per energy bin to use the χ2 fit statistics (Evans et al., 2019; McCollough +et al., 2020). The CSC2 fitting criteria mean that the majority of quasars in our study do not +have a photon index available in the CSC2 catalog. We only found 13 RLQ and 26 RQQ. +On the other hand, CSC2 provides X-ray spectra and response files for all the sources +in the catalog. We obtained these spectral files and fit the absorbed power law model to all +the quasars in our sample. In order to fit a larger number of quasar’s spectra, we put less +restrictive criteria. We accept measurements with total counts greater than or equal to 10 +and use the wstat-statistics appropriate for low counts data fitting (Freeman et al., 2001). +Furthermore, we reject any calculated photon index with an error greater than or equal to +one. With these criteria, we increased the number of sources with the calculated photon +index, for RQQ from 26 to 455, and RLQ from 13 to 63. +We note that some quasars have multiple observations. In RQQ, there are 85 RQQ +quasars with 243 observations. One of these quasars has 11 observations. In the RLQ sam- +ple, we found 5 quasars with 12 observations. We checked for the variability between the +multiple observations and confirmed that there is no variability as the measured flux is con- +sistent for each quasar. +In Figure 4, the left panel shows the distribution of our calculated photon index for the +RQQ sample containing 445 quasars and the RLQ sample containing 63 quasars. The right +panel shows the CSC2 photon index for the RQQ sample containing 26 quasars and the RLQ +sample containing 13 quasars. The fitted photon index has a similar distribution to that of +CSC2. We note that a range of the photon index values in our fitting is larger than in the +CSC2. We discuss this in Section 3.4. +The photon index distribution in the RQQ sample shows a steeper spectrum, with the +mean value of Γ = 2.14 +0.05 +−0.44, while RLQ shows a flatter spectrum, the mean value of Γ = +1.8 +0.38 +−0.34 (see Table 2). In addition, the Kuiper-two test for our calculated photon index +shows that the difference between RLQ and RQQ samples is significant with Dk = 0.37 and +Fk = 7.30×10−6. However, the Kuiper-two test gives an insignificant difference (Dk = 0.46 +and Fk = 0.18) for the CSC2 photon index, which may be due to the small sample size (only +13 RLQ and 26 RQQ). +3.4. Extreme cases in RLQ and RQQ +Figure 4 shows some extreme values of the photon index in the distributions (13 RQQ and +3 RLQ). The three RLQ quasars belong to CD class but they show extreme soft spectrum of +Γ > 3: i.e. Γ = 3.14 +0.58 +−0.56 , 3.00 +0.58 +−0.56 and 3.4 +0.8 +−0.7 , a total counts = 33, 22 and 13 counts, and +10 + +Figure 4: The left panel shows our best-fit photon index using wstat-statistics. We fit 63 RLQ and 445 RQQ. The right panel shows the +photon index we obtain from CSC2 for 13 RLQ and 26 RQQ. The RLQ distributions are represented by the solid green histogram and +green KDE curve. The RQQ distributions are represented by the dashed-blue histogram and KDE curve. +the background counts of 0.87, 0.25 and 0.86 counts, at z = 2.23, 2.11 and 3.7, respectively. +Due to the high uncertainty of Γ and the low number of counts in their spectra we were not +able to investigate their properties in more detail. These are interesting outliers identified in +our RLQ distribution, which need to be observed in the future. +On the other side, we identify 13 RQQ with extremely hard spectra, Γ < 1 , with a range +of the photon index [-0.08 - 0.97]. The total number of counts for these sources range [16 - +827] counts with background counts in the source region [0.12 - 655.8] counts. We selected +three quasars with a relatively good signal-to-noise for detail modeling, with a total number +of counts 133, 88, and 156 and a small number of background counts 0.48, 0.27, and 0.88, at +z = 2.5, 2.1, and 3.2. These are RQQ with hard spectra potentially indicating a presence of +the intrinsic absorption resulting in ”flattening” of the intrinsically soft spectrum (Zickgraf +et al., 1997; de Kool et al., 2002; Page et al., 2005). +We fit these three spectra of the RQQ assuming a power law model with additional +multiplicative absorption components (Sherpa has built-in models for the intrinsic absorp- +tion at the quasar redshift (xszphabs), and the photoelectric Galactic absorption compo- +nent (xsphabs)). The best-fit Γ changes from (0.80 +0.14 +−0.14, 0.82 +0.16 +−0.16, 0.93 +0.12 +−0.13) to (1.69 +0.31 +−0.31, +1.39 +0.32 +−0.32, 1.16 +0.21 +−0.21), bringing the photon index values closer to the bulk of the distribution +(see Figure 4). Figure 5 shows the confidence contours for the best-fit Γ and the intrinsic +absorption NH showing a high uncertainty in both the NH and Γ values. We need higher +quality spectra for these quasars to confirm that they are intrinsically absorbed. +After eliminating the extreme cases, the RLQ Γmean changes from 1.8 +0.38 +−0.34 to 1.70 +0.36 +−0.33 +and from 2.14 +0.0.5 +−0.44 to 2.19 +0.46 +−0.44 for RQQ. Consequently, the Kuiper-two test value between +RLQ and RQQ increased to Dk = 0.38 and its corresponding probability decreased to Fk = +10−7, which confirms a strong difference between RLQ and RQQ samples. Since these +extreme cases are a small percentage, 4% RLQ and 2% RQQ for our sample sets, they are +not changing the primary trend of RLQ (hard spectrum) and RQQ (soft spectrum). +11 + +RQQ(445) +1.2 +RLQ(63) +1.0 +0.8 +Density +0.6 +0.4 +0.2 +0.0 +-2 +-1 +0 +1 +2 +3 +4 +Photonindex2.00 +RQQ_CSC(26) +RLQ_CSC(13) +1.75 +1.50 +ensity +1.25 +Der +1.00 +0.75 +0.50 +0.25 +0.00 +-2 +-1 +0 +1 +2 +3 +4 +PhotonindexTable 2: The Kuiper-two sample test between RLQ and RQQ for all the parameters of interest +Samples +RLQ, RQQ +CD, RQQ +LD, RQQ +CD, LD +Parameters +Dk +Fk +Dk +Fk +Dk +Fk +Dk +Fk +z +0.19 +0.08 +0.24 +0.02 +0.39 +0.31 +0.50 +0.09 +LX +0.42 +2.18x10−9 +0.42 +2.41x10−8 +0.48 +0.09 +0.22 +0.99 +Γ +0.37 +7.30x10−6 +0.39 +2.56x10−5 +0.50 +0.04 +0.24 +0.98 +HRh/s +0.25 +0.01 +0.31 +9.80x10−4 +0.37 +0.53 +0.49 +0.16 +HRm/s +0.21 +0.05 +0.21 +0.10 +0.52 +0.07 +0.41 +0.37 +HRh/m +0.16 +0.37 +0.18 +0.30 +0.34 +0.58 +0.52 +0.10 +Dk: is the maximum absolute difference between the two cumulative distribution functions. +Fk: is a probability (P-value) of the hypothesis that the two samples come from the same population +and therefore have the same CDF. +Bolded values: are highlighting the highest difference distributions. +4. Discussion +We studied a sample of high redshift (z > 2) quasars selected from CSC2. The samples +have similar redshift distribution, but the RQQ sample has (472) a higher number of quasars +than the RLQ sample (81). We calculate the X-ray luminosity and the X-ray photon index. +All the properties of the two samples are summarized in Table 1. The Kuiper-two test shows +a significant difference between RLQ and RQQ for both LX and Γ indicating that the RLQ +spectra were flatter than the spectra of RQQ. The Kuiper-two test values for all the X-ray +parameters are given in Table 2. +4.1. Comparing our parameterized results with literature +Our studies indicate that the X-ray luminosity of RLQ is significantly higher than the X- +ray luminosity of RQQ (Dk = 0.42, Fk = 2.18 × 10−9), see Table 1) in the sample of z > 2 +quasars in CSC2. This result agrees with the earlier studies (Scott et al., 2011), and suggests +an additional X-ray radiation component present in RLQ (Bechtold et al., 1994; Zhu et al., +2020). +12 + +(a) +(b) +(c) +Figure 5: The confidence regions of Γ and NH for the three quasars: (a)2CXO J011513.1+002013, (b)2CXO J123540.1+123620, +(c)2CXO J095858.6+020139 with a number of counts (88, 156, 133), respectively. We fit Γ and NH with a power law model multiplied +by the intrinsic absorption at a given redshift (and including the Galactic absorption). The cross marks the best fit value and the contours +show 1σ (purple), 2σ (blue) and 3σ (yellow) levels. The NH values is in log scale. +This additional component may also cause RLQ’s X-ray spectra to be flatter than the +spectra of RQQ (Reeves and Turner, 2000; Piconcelli et al., 2005). Our studies cover a +relatively high rest frame energies, exceeding 30 keV, in this high redshift sample. These +energies are less sensitive to the intrinsic absorption, thus the flattening of the RLQ is less +likely related to the absorption (e.g. high absorption columns, NH > [1022 − 1026] cm−2, +are required to modify the high energy spectra), but more likely due to the differences in the +radiation processes between the two classes (i.e. RLQ and RQQ). +For our sample, the column density in the direction of the source ranges within [0.57- +12.58]×1020 cm−2, with a mean of 2.49×1020 cm−2. The nuclear obscuration is parameter- +ized by the hydrogen column density NH and the maximum value of NH in our sample is +1.26 × 1021 cm−2, which does not affect the AGN X-ray continuum (Hickox and Alexander, +2018). The obscuration due to the Compton-thick absorption requires a strong reflection +component at E > 10 keV, and a prominent Fe-Kα emission line at 6.4 keV (Ricci et al., +2015). In our sample spectra, we did not find any Fe-Kα emission line. +In addition to the photon index we studied the X-ray hardness ratios for the quasars in the +two samples. Our analysis shows, no difference in HRh/m between RLQ and RQQ samples, +a small difference in HRm/s, and a moderate difference in HRh/s with the RLQ having a +harder spectra (see Table 1 and Table 2). +The soft X-ray radiation might be produced anywhere in the vicinity of a SMBH in both +RLQ and RQQ (Shen et al., 2006). However, we find that the peaks of the HRh/s and HRm/s +distributions (see Figures 3) are shifted towards the soft energy band in RQQ but not in RLQ. +Our result indicates that for RQQ, the soft X-ray radiation dominates over the radiation in the +hard and medium energy bands. However, for RLQ, the radiation in the hard and medium +energy X-ray bands dominates over the soft energy band. +Page et al. (2005) considered a small sample of 7 RQQ and 16 RLQ at (z > 2) observed +with XMM-Newton. They used the broad energy band [0.3 - 10] keV. They found 9 intrinsi- +13 + +18 +16 +14 +Column density (Nn) +12 +10 +8 +6 +4 +2 +0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Photon index ()35 +30 +Column density (Nn) +25 +20 +15 +10 +5 +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Photon index ()35 +30 +Column density (Nn) +25 +20 +15 +10 +5 +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Photon index ()Figure 6: The comparison between our calculated photon index and the CSC2 photon index for the same set of quasars (13 RLQ and 26 +RQQ). Dark blue dashed lines and dark green solid lines show our photon index, and light blue dashed lines and light green solid lines +show the CSC2 photon index. +cally absorbed quasars with NH between [1 - 2]×1022 cm−2 in the rest frame of the objects. +Using the absorbed power law model, they found that RLQ have flatter spectra than the RQQ +counterparts (RLQ ≈ 1.55 and RQQ ≈ 1.98). Some studies compare RLQ and RQQ in a +specific part of the X-rays (hard band) to specify the corresponding mechanism (Gupta et al., +2018; Zhu et al., 2020). In our study, the RLQ is flatter than RQQ by 0.49 +0.10 +−0.11, which is a +bigger difference than that found by Page et al. (2005), due to our larger sample size. We do +not see any clear intrinsically absorbed quasars, which could be due to lower signal-to-noise +spectra in our sample. Furthermore, the extreme cases in our samples did not show strong +evidence for intrinsic absorption. +4.2. Calibrating our calculated photon index with CSC2 +We compare the photon index calculated by our spectral modeling to the photon index +given in CSC2 for the same quasars (13 RLQ and 26 RQQ). Figure 6 shows the RLQ and +RQQ distributions for the calculated Γ and the one given in CSC2. The distributions show +a rough agreement between the two methods, with our modeled values indicating a slightly +wider range. +CSC2 uses the χ2 statistics with background subtraction and binning, while we use wstat- +statistics and no background subtraction appropriate for low counts spectra. van Dyk et al. +(2001) have explained the χ2 statistical bias at low counts spectra, see also (Protassov et al., +14 + +2.00 +- +RQQ_CSC +RQQ +1.75 +RLQ CSC +RLQ +1.50 +1.25 +Density +1.00 +0.75 +0.50 +0.25 +0.00 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +PhotonindexFigure 7: The two panels show the 100 trials of the Kuiper test DK (left panel) and FK (right panel) for the HRh/s parameter, between +RLQ and RQQ. The red color for HRh/s at 2 < z < 2.5, and the blue color for HRh/s at z > 2.5. The solid vertical lines are for the +mean and the dashed vertical lines are for the median. +2002). Humphrey et al. (2009) found that even high counts give an inherent bias in the +χ2 fitting. These studies show that χ2 methods should not routinely be used for fitting an +arbitrary, parameterized model to Poisson-distributed data, irrespective of the number of +counts (Mighell, 1999), and instead, the Cash statistic should be adopted (Humphrey et al., +2009). We used the wstat, which is based on the Poisson likelihood and accounts for the +background6. +We applied the Kuiper-two test to evaluate the difference between the two photon-index +distributions, Γfit and ΓCSC2. The test returns high values of Fk, for RQQ Fk = 0.33 and +Fk = 0.77 for RLQ, which implies that the distributions of Γ resulting from our modeling +are consistent with the CSC2 distributions for these small sub-samples. +4.3. Redshift Dependence of the Hardness Ratio +Our results on the hardness ratio parameter HRh/s indicate that the RQQ spectra are +softer than the spectra of RLQ (see Sec.3.2). We perform simulations to confirm that the +effect is an inherent physical property of RQQ and is not affected by the redshift. Because +the rest frame energy range is shifted towards the lower energy in the observed frame we +check the distributions of the hardness ratio parameter in the two redshift ranges. There are +261 RQQ and 33 RLQ at 2 < z < 2.5 and 211 RQQ and 48 RLQ at 2.5 < z < 5.5 redshift. +Thus the fraction of RQQ is higher at z < 2.5 than at z > 2.5, which may bias the RQQ’s +HRh/s parameter in the full redshift range. +6https://cxc.harvard.edu/sherpa/ahelp/wstat.html +15 + +Z<2.5 +Z>2.5 +6 +2 +0 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +D_K8 +Z<2.5 +Z>2.5 +6 +5 +ISU +3 +2 +1 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +F_KWe apply the Kuiper test to the HRh/s at 2 < z < 2.5 sample and get the Kuiper param- +eter values of DK = 0.33, and FK = 0.04. Then, we select the same sample size of 33 RQQ +and RLQ by randomly selecting 33 quasars from the 261 RQQ sample and using all 33 RLQ +in this low redshift bin. In the first random selection of the 33 RQQ we get DK = 0.47, and +Fk = 0.02. Afterward, we perform the test for the hardness ratio difference by looking at +the distribution of the Kuiper parameters, Dk and Fk, in 100 random samples (see Fig. 7). +We selected the 100 random samples of 33 RQQ and used the existing 33 RLQ. The median +values for the 100 Kuiper test parameters in this step are Dk = 0.36, and FK = 0.05. +For the z > 2.5 sample, the Kuiper test results for HRh/s difference between RLQ and +RQQ are DK = 0.30, FK = 0.04. We then performed the same simulation steps as described +above for the z > 2.5 sample using the 48 RLQ and a random sample of 48 RQQ selected +from the 211 RQQ. The median value of the Kuiper test distribution was Dk = 0.32, and +FK = 0.06 (see Fig.7). +The simulation results show that the difference in the HRh/s parameter between RLQ +and RQQ samples is slightly more significant at 2 < z < 2.5 than z > 2.5. According to +Peca et al. (2021), our sample selection is the least affected by the absorption dependence +with redshift and the Chandra detector contamination. At low redshift (z < 2), the difference +in the flux between hard and soft bands is larger for quasars with NH < 1022 cm−2 because +the soft X-ray emission is present in the observed energy band. At high redshift (z > 2) +the hardness ratio of the quasars with low NH is not affected by the hard band shift to the +lower observed energies and only the quasars with high absorption, NH > 1023 cm−2, will +show the impact on the HRh/s parameter. We conclude that the observed difference in the +hardness ratio between RQQ and RLQ at z > 2 is not affected by redshift. +5. Summary and Conclusions +We studied the X-ray properties of high redshift quasars observed by Chandra. We found +a total of 2,561 DR7 quasars in the CSC2 database. After applying redshift and radio- +loudness filters we obtained two samples, one with 472 RQQ and the second with 81 RLQ. +The two samples have a similar redshift range, 2 < z < 5, with the RLQ sample being +one of the largest samples of RLQ within that redshift range to date. Our main results are +summarized below. +• We found that an average X-ray luminosity of RLQ at high redshift is higher than the +average X-ray luminosity of RQQ, consistent with the previous studies. +• We calculated the mean photon index of ΓRLQ = 1.70 +0.36 +−0.33 and ΓRQQ = 2.19 +0.46 +−0.44 +for the RLQ and RQQ samples, respectively. This result confirms that RLQ spectra +16 + +are flatter than the spectra of RQQ. We identified a few extremely soft RLQ and ex- +tremely hard RQQ, but these sources have low signal-to-noise data and require further +observations to understand their X-ray properties. +• We found that the LD and RQQ have similar distributions of hardness ratios, HRh/m +and HRh/s. In comparison, LD and CD have similar photon index and X-ray luminos- +ity distributions. However, our sample has only 10 LD quasars and more LD observa- +tions are needed to confirm this result. +• The peaks of HRh/s and HRm/s distributions are shifted towards negative values (soft +energy band) in RQQ compared to RLQ, which confirms that the X-ray luminosity in +the RQQ is dominated by soft X-rays in comparison to RLQ. +Our study shows potential directions for further investigation. The quasars of extreme cases +need longer observation. The CD and LD comparison needs larger samples for statistically +meaningful results. The current samples can be extended to include quasars at higher red- +shifts, z > 5, with the future releases of the Chandra Source Catalog. Additionally, the +available quasar catalogs can be used to study the early universe population of quasars using +high redshift infrared observations which will become available with the JWST (Gardner et +al., 2006). +Software: Sherpa (Freeman et al., 2001), Topcat (Taylor, 2017), Python packages: As- +tropy (Astropy Collaboration et al., 2018), Seaborn (Waskom, 2021), Numpy (Harris et al., +2020), and Matplotlib (Hunter, 2007). +6. Acknowledgement +This research has made use of data obtained from the Chandra Data Archive and the +Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the +application packages CIAO and Sherpa. F.S. thanks CXC Helpdesk and Nick Lee for the +support in the analysis of Chandra data. A.S. was supported by NASA contract NAS8-03060 +(Chandra X-ray Center). 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AP 323, +L21–L24. +23 + diff --git a/-dE1T4oBgHgl3EQfCgLC/content/tmp_files/load_file.txt b/-dE1T4oBgHgl3EQfCgLC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3837a7b97e25f459031266bdd1e0af412247256 --- /dev/null +++ b/-dE1T4oBgHgl3EQfCgLC/content/tmp_files/load_file.txt @@ -0,0 +1,1553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf,len=1552 +page_content='X-ray properties of high-redshift Radio Loud and Radio Quiet Quasars observed by Chandra F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Shabana,∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Siemiginowskab, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Suleimanb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' El-Nawawya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Alia aAstronomy, Space Science and Meteorology Department, Faculty of Science, Cairo University, Giza, EGYPT bCenter for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA Abstract We performed a study of high redshift (z > 2) quasars, looking for the main differences be- tween Radio Loud Quasars (RLQ) and Radio Quiet Quasars (RQQ) in the X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our sample of 472 RQQ and 81 RLQ was selected by cross-matching the SDSS DR7 quasars catalog with the Chandra Source Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We computed the X-ray luminosity for the two samples and confirmed the X-ray luminosity excess of RLQ over RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We fit the X-ray spectra assuming the absorbed power law model and obtained the photon index (Γ) values for all the sources in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We excluded quasars with a low number of counts (< 10) and large uncertainty on the best-fit photon index (Γerr > 1), and obtained the mean values of ΓRLQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='70 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='33 and ΓRQQ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='19 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='44 for the RLQ and RQQ samples, respectively, showing that the RLQ have flatter (harder) X-ray spectra than RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Kuiper-two test confirms this result with the significant difference between the RLQ and RQQ photon in- dex distributions (Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='37 and P-value = 10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We also evaluated the hardness ratio distributions and confirmed that the spectra of RLQ are flatter than the spectra of the RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RLQ’s hard-to-soft ratio distribution is skewed towards the hard X-ray band, while the RQQ is towards the soft X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The hard-to-medium and medium-to-soft ratios show no difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Keywords: Radio Loud Quasars, Radio Quiet Quasars, X-ray Astrophysics, X-ray photon index, Hardness Ratio 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Introduction There are two main classes of quasars: the Radio Quiet Quasars (RQQ) and the Radio Loud Quasars (RLQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' They have been identified based on the orientation and presence of ∗corresponding author Email address: fshaban@sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='eg (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Shaban ) Preprint submitted to JHEAP January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='02866v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='HE] 7 Jan 2023 a radio jet (Antonucci, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Wilson and Colbert, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Urry and Padovani, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' RLQ have optical and X-rays luminosity about three times greater than their RQQ counterparts (Zamorani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Worrall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The X- ray radiation could be due to the Compton scattering of UV photons by energetic electrons or due to synchrotron radiation from highly relativistic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (Mushotzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Nowak, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Turner and Miller, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Worrall, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Fabian, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Majority of quasars are RQQ with the X-ray radiation attributed to a hot corona formed in the accretion flow (Haardt and Maraschi, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' RLQ are a small minority, about ≈ 10% of all quasars, and are characterized by their relativis- tic jets generated by an accreting supermassive black hole (SMBH) (Padovani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Blandford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The amount of jet radiation contributing to the X-ray spectrum in RLQ is still not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, identifying the main radiation components in the X-ray spectrum is important to the estimates of the quasar power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RLQ have flatter X-rays spectra (lower photon index value) than those of the RQQ (Reeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The quasar’s hardness ratio is consistent with the spectral slope (Freeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Peca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RLQ are divided into Core Dominant (CD) and Lobe Dominant (LD) (Haardt and Maraschi, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Wilson and Colbert, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The radio emission of CD quasars is dominated by the relativistic jet, while the LD quasars show significant radio emission from the large- scale components in comparison to the core (Falcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Boroson, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' These two populations might have different X-ray radiation processes, which was noted recently by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' During the past decades, quasar data from X-ray surveys have become available, allowing for statistical studies of relatively large samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Many recent studies considered the high redshift quasars for survey (Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Vito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Pons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Some studies were focusing on deducing RLQ properties using correlations between X-ray, radio, and optical (or UV) luminosities to investigate the quasars physical model, Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2010) investigate the disk-jet model, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2020) deduced the disk-corona- jet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Interestingly, Lusso and Risaliti (2017) were studying RQQ and showed that RQQ could be used as standard candles at high redshifts (z > 2), which is important for distance measurement and cosmological tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In this research, we investigate the differences in the X-ray spectral properties (photon index, intrinsic absorption, hardness ratios, and X-ray luminosity) between RQQ and RLQ samples using the data available in the Chandra Source Catalog (CSC2) (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We study quasars at a high redshift near the peak of cosmic quasar activity, at z > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our sample contains the largest number of RLQ at high redshift observed with Chandra and 2 include faint sources with [10−15 - 10−13] ergcm−2 s−1 1 for the energy range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0] keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We calculate the photon index by fitting the faint X-ray spectra, thus expanding the number of quasars with this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The observed Chandra effective energy range is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0] keV and corresponds to the rest frame energy greater than [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0] keV at redshift z > 2, so at the higher redshifts we are able to study the X-ray spectra, which are the most sensitive to the properties of the corona and relativistic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In section 2, we briefly describe the data catalogs, the sample selection criteria, and our constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In section 3, we show the distributions of RLQ and RQQ as functions of X- ray parameters and illustrate the photon index calculations and constraints, furthermore, we analyze extreme cases for the photon index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In section 4, we discuss our results and compare them with previous studies, and conclude with a discussion and outlook for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Sample selection DR7+ CSC2 (2561) Z > 2 (595) DR7 105,783 quasars First_flag= 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Lobe- Dominant First_flag= -1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Out of First field First_flag= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Radio Quiet First_flag= 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Core- Dominant Radio Loud R ≥ 10 Radio intermediate R < 10 19 71 472 10 23 Radio Loud R ≥ 10 Figure 1: The sample selection is based on the DR7, CSC2, redshift and radio-loudness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The filters in the fourth column are showing the FIRST flag values (-1, 0, 1, and 2) representing quasars (not in the FIRST field, Radio Quiet, Core dominant, and Lobe dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The next column filters the radio-loudness (R) into RQQ, RLQ, and RIQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The circles represent the quasar’s number in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 1https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='edu/csc/char.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='html 3 We study the X-ray properties of quasars using archival data from two quasar catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We use DR7 quasars catalog (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2011), which contains 105,783 quasars with optical spectra and redshift measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2011) quasars were selected from the SDSS DR7 sample compiled by Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2010) and all have spectroscopic redshift mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2010) rejected the pipeline redshift measurements for the quasar candidates with images exceeding the PSF size in the r-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' They provide the uncertainty on the redshift measurement to be + − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We use the X-ray data obtained by the Chandra X-ray Observatory (Chandra) during the first 15 years of the mission available in the Chandra Source Catalog release 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 (CSC22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' There are more than 315,000 unique X-ray sources in the CSC2 (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Chandra has a high-quality angular resolution (better than 5′′), which is important for detecting faint sources, at high redshift, with good source positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We cross-matched the 105,783 DR7 quasars with sources in CSC2, using TOPCAT (Taylor, 2017), and set a search cone radius of 30′′, consistent with the range of the sources offset uncertainty given by Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We found 2,561 sources corresponding to X-ray sources in CSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We study the sources at high redshift (z > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' After applying the (z > 2) filter, we obtained 595 out of 2,561 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The details of our full sample selection are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2011) matched DR7 optical quasars catalog with Faint Images of the Radio Sky at Twenty Centimeters (FIRST) catalog (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1997), and estimated the quasar radio loudness parameter (R) defining RLQ and RQQ based on the following equation R = �f6 cm f2500 � (1) where f6 cm and f2500 are the fluxes density (fν) at rest-frame 6 cm and 2500 ˚A, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The flux density in DR7 is determined from the FIRST integrated flux density at 20 cm assuming a power-law slope of αν = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The flux density at the rest frame of 2500 ˚A is determined by fitting the optical spectrum with a power-law continuum (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Similar to Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2007), Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2011) have divided RLQ in DR7 into lobe dominant (LD) and core dominant (CD) with FIRST cone radius of 30′′ and 5′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2011) have removed the effects of galactic extinction in the SDSS spectra using the Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (1998) map, and the Milky Way extinction curve by Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Furthermore, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2011) shifts the spectra to the rest frame using the cataloged redshift as the systematic redshift (Hewett and Wild, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We select the Radio Intermediate quasars (RIQ) to have R < 10 and RLQ with (R ≥ 10) (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We applied the above selection categories to our initial sample of 595 2https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='edu/csc/ 4 quasars in CSC2 and divided them into different radio-loudness categories as given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Because we focus on strong differences between the RLQ and RQQ, therefore we exclude the intermediate sample and only include RLQ and RQQ in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our final sample contains 81 RLQ and 472 RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Data Analysis and Results We study several parameters representing the X-ray properties of the quasars in our sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The redshift (z), and the radio loudness (R) are provided from DR7 (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2011), while the X-ray flux (fX), the hardness ratios (HRh/m), the hydrogen column density (NH), and the X-ray spectral files are given in CSC2 (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We calculate the X-ray luminosity (LX) and the X-ray photon index (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In order to evaluate the difference between RLQ and RQQ samples in all X-ray param- eters we use the Kuiper-two sample test (Watson, 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Kuiper test is a test for the difference between two samples based on their observed Cumulative Distribution Functions (CDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' It is an extension of the Kolmogorov–Smirnov test, but it is more sensitive to the shift between the two distributions and the difference in the tails of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Kuiper test is non-parametric and does not assume any functional form of the sample’s true distribution and it is appropriate when true distributions are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The test returns DK and Fk, which are the maximum difference between the two samples and the probability p-value of the test, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Fk < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='05 rejects the hypothesis that the two samples are drawn from the same distribution, so the smaller the value the stronger the significance of the difference between the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' All the Kuiper-two test values of this study are given in (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In our figures, we use normalized density histograms because we have different samples size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The histograms represent the probability density function of the parameter distributions (Hunter, 2007), (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', m/M × b), where m is the number of quasars in each specific bin, M is the total number of quasars in the sample, and b is the bin bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' So the area under the bins integrates into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We apply the same number of bins to RLQ and RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RQQ sample appears to have a smaller bin bandwidth than the RLQ sample because the bin band- width is affected by the sample number in the probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We also apply the Kernel Density Estimation (KDE) smoothing function to account for the small sample size and different bin sizes (Rosenblatt, 1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The small sample size may contribute to the gaps within the histograms, and different binning could lead to statistical biases (Waskom, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We use the following KDE equation: P(x) = 1 M × h M � i=1 k �x − xi h � (2) 5 Where M is the total number of quasars in the sample, h the Kernel bandwidth, k the chosen kernel weight function in our estimate (Gaussian), x is the point where to calculate the function, and xi is the parameter value in bin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The seaborn package 3 for fitting KDE has a built-in kernel bandwidth optimal estimation using Silverman methods, which are used for random normally distributed samples (Silverman, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Figure 2 shows the redshift distributions of RLQ and RQQ samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We apply the Kuiper- two test which returns a small difference between the RLQ and RQQ samples with Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='19 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' This confirms that the RLQ and RQQ samples in our studies have consistent redshift distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' X-ray Luminosity We calculated the X-ray luminosity using the equation given by: LX = 4πdL 2fX (3) Where LX is X-ray luminosity, dL is the distance luminosity, and fX is the X-ray flux in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0] keV broadband energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The cosmological model used in this study is the WMAP9 with (Ho = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='33, Ωo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='287, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='712) parameters (Hinshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We use the WMAP9 under the astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='cosmology package to obtain the distance lu- minosity (dL) (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Using fX and dL, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='3 we calculate the X-ray luminosity (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Figure 2 shows the X-ray luminosity distributions of RLQ and RQQ samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RQQ KDE (blue) shows a shape consistent with a Gaussian distribution and the RLQ KDE (green) is skewed to the higher X-ray luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The X-ray luminosity range, given in log scale, for RLQ is LXmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 and LXmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='07, while for RQQ are LXmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='68 and LXmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' = 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The differences between minimum and maximum luminosities are similar, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='57 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='62 for RLQ and RQQ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, the median of LX is higher in the RLQ sample by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='53 compared to the RQQ’s median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' This difference in the median between RLQ and RQQ is significant and indicates a reliable difference between the intrinsic proper- ties of the two samples, RLQ and RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Kuiper-two test returns a significant difference in X-ray luminosity distributions between RLQ and RQQ, Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='42, and Fk = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='18×10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Fk value validates the remarkable difference in the X-ray luminosity between the radio- quiet and radio-loud quasars (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 3https://seaborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='pydata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='org/generated/seaborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='kdeplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='html 6 Figure 2: The two panels show the redshift (left), and the X-ray luminosity (right) distributions comparison between RLQ and RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The green histogram represents all of the RLQ as a function of X-ray Luminosity, and the solid green curve represents the KDE for the RLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The blue histogram represents the RQQ as a function of the X-ray Luminosity, and the dashed-blue line represents its KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The histograms are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Hardness ratios The hardness ratio is defined as the flux ratio between two different Chandra energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The X-ray energy bands in the CSC2 are divided into several categories 4: Broad (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0) keV Hard (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0) keV Medium (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0) keV Soft (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2) keV The hardness ratios for hard to medium (HRh/m), medium to soft (HRm/s), and hard to soft (HRh/s) 5 are given in CSC2 for each detected source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' CSC2 provides f(h), f(m), and f(s) the X-ray fluxes in the hard, medium, and soft energy bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' HRh/m = f(h) − f(m) f(h) + f(m) (4) The HRm/s and HRh/s are defined similar to equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' When the hardness ratio exceeds zero, the flux of the higher energy band dominates over the flux of the lower energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' For a general comparison between RLQ and RQQ samples, we investigate the distributions for HRh/m, HRm/s and HRh/s shown in Figure 3 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The distribution plots were normalized and smoothed with KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 4https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='edu/csc/columns/ebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='html 5https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='edu/csc/columns/spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='html 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 RQQ RLQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 #1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 Densit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 Redshift- RQQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 RLQ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 L Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 44 45 46 47 48 log (x-rays_luminosity)Figure 3: The three panels show the HRh/m, HRm/s, and HRh/s distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The green solid line represents the RLQ, while the blue dashed line represents the RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The red vertical lines indicate the photon index values corresponding to each hardness ratio for Γ equal to (3, 2, 1, 0) from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The left panel shows no difference between RLQ and RQQ distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The middle panel shows a slight difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The right panel shows a significant difference between RLQ and RQQ with a higher tendency toward soft energy in the RQQ sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We also mark the evolution of the photon index as a function of the hardness ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Using the fake pha function in Sherpa and the standard ACIS-S response files, we fix the photon index (Γ = 0, 1, 2, 3) to simulate the spectrum and calculate the corresponding hardness ratios for each Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Figures 3 show that the red marks of the photon index decrease (flat spectrum) as the hardness ratios increase (towards the hard band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' RLQ and RQQ samples have a similar HRh/m distributions (see Figure 3) confirmed by the Kuiper-two test Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16, Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The HRm/s distribution shown in Figure 3, shows a slight shift towards the soft energy band for RQQ in comparison to the RLQ sample, also indicated by the Kuiper-two test Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='21 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Finally, the HRh/s distribution shows the most significant difference between RLQ and RQQ samples (see Figure 3) with the Kuiper-two test results of Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='01, the test accuracy is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8% (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The HRh/s distribution indicates that the X-ray spectra of RQQ quasars are softer than the spectra of RLQ quasars in our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We investigate the X-ray properties of the CD and LD quasars separately in our compar- ison to RQQ by applying the Kuiper-two test on all the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We find that LD and CD samples are consistent in all of the X-ray physical parameters except for hardness ratios, HRh/s and HRh/m with Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, HRh/s and HRh/m distributions for LD sample are similar to RQQ with Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='53 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='58, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' On the other hand, our LD sample is small (10 LD quasars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Future studies of CD and LD quasars with high-quality X-ray spectra are needed to confirm and investigate these results further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 RQQ RLQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 isity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 HRh/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 RQQ RLQ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='14 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='06 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='65 fX: The given X-ray flux (ergcm−2 s−1) must be multiplied by factor of 10−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' LX: The X-ray luminosity (ergs−1) is given in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' X-ray Spectral Modeling and Photon Index CSC2 lists the photon index calculated by fitting a power law model multiplied by the photoelectric absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, the CSC2 pipeline restricted the model fitting to spectra with at least 150 net counts (after subtracting the background) and applied the spectral bin- ning of 20 counts per energy bin to use the χ2 fit statistics (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' McCollough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The CSC2 fitting criteria mean that the majority of quasars in our study do not have a photon index available in the CSC2 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We only found 13 RLQ and 26 RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' On the other hand, CSC2 provides X-ray spectra and response files for all the sources in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We obtained these spectral files and fit the absorbed power law model to all the quasars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In order to fit a larger number of quasar’s spectra, we put less restrictive criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We accept measurements with total counts greater than or equal to 10 and use the wstat-statistics appropriate for low counts data fitting (Freeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Furthermore, we reject any calculated photon index with an error greater than or equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' With these criteria, we increased the number of sources with the calculated photon index, for RQQ from 26 to 455, and RLQ from 13 to 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We note that some quasars have multiple observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In RQQ, there are 85 RQQ quasars with 243 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' One of these quasars has 11 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In the RLQ sam- ple, we found 5 quasars with 12 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We checked for the variability between the multiple observations and confirmed that there is no variability as the measured flux is con- sistent for each quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In Figure 4, the left panel shows the distribution of our calculated photon index for the RQQ sample containing 445 quasars and the RLQ sample containing 63 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The right panel shows the CSC2 photon index for the RQQ sample containing 26 quasars and the RLQ sample containing 13 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The fitted photon index has a similar distribution to that of CSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We note that a range of the photon index values in our fitting is larger than in the CSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We discuss this in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The photon index distribution in the RQQ sample shows a steeper spectrum, with the mean value of Γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='14 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='44, while RLQ shows a flatter spectrum, the mean value of Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='34 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In addition, the Kuiper-two test for our calculated photon index shows that the difference between RLQ and RQQ samples is significant with Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='37 and Fk = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='30×10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, the Kuiper-two test gives an insignificant difference (Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='46 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='18) for the CSC2 photon index, which may be due to the small sample size (only 13 RLQ and 26 RQQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Extreme cases in RLQ and RQQ Figure 4 shows some extreme values of the photon index in the distributions (13 RQQ and 3 RLQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The three RLQ quasars belong to CD class but they show extreme soft spectrum of Γ > 3: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Γ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='14 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='56 , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='56 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='7 , a total counts = 33, 22 and 13 counts, and 10 Figure 4: The left panel shows our best-fit photon index using wstat-statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We fit 63 RLQ and 445 RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The right panel shows the photon index we obtain from CSC2 for 13 RLQ and 26 RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RLQ distributions are represented by the solid green histogram and green KDE curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The RQQ distributions are represented by the dashed-blue histogram and KDE curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' the background counts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='87, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='86 counts, at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='23, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Due to the high uncertainty of Γ and the low number of counts in their spectra we were not able to investigate their properties in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' These are interesting outliers identified in our RLQ distribution, which need to be observed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' On the other side, we identify 13 RQQ with extremely hard spectra, Γ < 1 , with a range of the photon index [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='08 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The total number of counts for these sources range [16 - 827] counts with background counts in the source region [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='12 - 655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8] counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We selected three quasars with a relatively good signal-to-noise for detail modeling, with a total number of counts 133, 88, and 156 and a small number of background counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='48, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='27, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='88, at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='1, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' These are RQQ with hard spectra potentially indicating a presence of the intrinsic absorption resulting in ”flattening” of the intrinsically soft spectrum (Zickgraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' de Kool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We fit these three spectra of the RQQ assuming a power law model with additional multiplicative absorption components (Sherpa has built-in models for the intrinsic absorp- tion at the quasar redshift (xszphabs), and the photoelectric Galactic absorption compo- nent (xsphabs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The best-fit Γ changes from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='80 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='82 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='93 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='13) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='69 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='31, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='39 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='32, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='21), bringing the photon index values closer to the bulk of the distribution (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Figure 5 shows the confidence contours for the best-fit Γ and the intrinsic absorption NH showing a high uncertainty in both the NH and Γ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We need higher quality spectra for these quasars to confirm that they are intrinsically absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' After eliminating the extreme cases, the RLQ Γmean changes from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='34 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='70 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='33 and from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='14 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='44 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='19 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='44 for RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Consequently, the Kuiper-two test value between RLQ and RQQ increased to Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='38 and its corresponding probability decreased to Fk = 10−7, which confirms a strong difference between RLQ and RQQ samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Since these extreme cases are a small percentage, 4% RLQ and 2% RQQ for our sample sets, they are not changing the primary trend of RLQ (hard spectrum) and RQQ (soft spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 11 RQQ(445) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2 RLQ(63) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 2 1 0 1 2 3 4 Photonindex2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 RQQ_CSC(26) RLQ_CSC(13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 ensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 Der 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 2 1 0 1 2 3 4 PhotonindexTable 2: The Kuiper-two sample test between RLQ and RQQ for all the parameters of interest Samples RLQ, RQQ CD, RQQ LD, RQQ CD, LD Parameters Dk Fk Dk Fk Dk Fk Dk Fk z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='09 LX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='18x10−9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='41x10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} 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+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='98 HRh/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='31 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='80x10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16 HRm/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='37 HRh/m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='10 Dk: is the maximum absolute difference between the two cumulative distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Fk: is a probability (P-value) of the hypothesis that the two samples come from the same population and therefore have the same CDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Bolded values: are highlighting the highest difference distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Discussion We studied a sample of high redshift (z > 2) quasars selected from CSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The samples have similar redshift distribution, but the RQQ sample has (472) a higher number of quasars than the RLQ sample (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We calculate the X-ray luminosity and the X-ray photon index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' All the properties of the two samples are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Kuiper-two test shows a significant difference between RLQ and RQQ for both LX and Γ indicating that the RLQ spectra were flatter than the spectra of RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The Kuiper-two test values for all the X-ray parameters are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Comparing our parameterized results with literature Our studies indicate that the X-ray luminosity of RLQ is significantly higher than the X- ray luminosity of RQQ (Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='42, Fk = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='18 × 10−9), see Table 1) in the sample of z > 2 quasars in CSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' This result agrees with the earlier studies (Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2011), and suggests an additional X-ray radiation component present in RLQ (Bechtold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 12 (a) (b) (c) Figure 5: The confidence regions of Γ and NH for the three quasars: (a)2CXO J011513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='1+002013, (b)2CXO J123540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='1+123620, (c)2CXO J095858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='6+020139 with a number of counts (88, 156, 133), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We fit Γ and NH with a power law model multiplied by the intrinsic absorption at a given redshift (and including the Galactic absorption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The cross marks the best fit value and the contours show 1σ (purple), 2σ (blue) and 3σ (yellow) levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The NH values is in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' This additional component may also cause RLQ’s X-ray spectra to be flatter than the spectra of RQQ (Reeves and Turner, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Piconcelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our studies cover a relatively high rest frame energies, exceeding 30 keV, in this high redshift sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' These energies are less sensitive to the intrinsic absorption, thus the flattening of the RLQ is less likely related to the absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' high absorption columns, NH > [1022 − 1026] cm−2, are required to modify the high energy spectra), but more likely due to the differences in the radiation processes between the two classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' RLQ and RQQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' For our sample, the column density in the direction of the source ranges within [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='57- 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='58]×1020 cm−2, with a mean of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='49×1020 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The nuclear obscuration is parameter- ized by the hydrogen column density NH and the maximum value of NH in our sample is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='26 × 1021 cm−2, which does not affect the AGN X-ray continuum (Hickox and Alexander, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The obscuration due to the Compton-thick absorption requires a strong reflection component at E > 10 keV, and a prominent Fe-Kα emission line at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4 keV (Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In our sample spectra, we did not find any Fe-Kα emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In addition to the photon index we studied the X-ray hardness ratios for the quasars in the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our analysis shows, no difference in HRh/m between RLQ and RQQ samples, a small difference in HRm/s, and a moderate difference in HRh/s with the RLQ having a harder spectra (see Table 1 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The soft X-ray radiation might be produced anywhere in the vicinity of a SMBH in both RLQ and RQQ (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, we find that the peaks of the HRh/s and HRm/s distributions (see Figures 3) are shifted towards the soft energy band in RQQ but not in RLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our result indicates that for RQQ, the soft X-ray radiation dominates over the radiation in the hard and medium energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, for RLQ, the radiation in the hard and medium energy X-ray bands dominates over the soft energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2005) considered a small sample of 7 RQQ and 16 RLQ at (z > 2) observed with XMM-Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' They used the broad energy band [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='3 - 10] keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' They found 9 intrinsi- 13 18 16 14 Column density (Nn) 12 10 8 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 Photon index ()35 30 Column density (Nn) 25 20 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 Photon index ()35 30 Column density (Nn) 25 20 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 Photon index ()Figure 6: The comparison between our calculated photon index and the CSC2 photon index for the same set of quasars (13 RLQ and 26 RQQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Dark blue dashed lines and dark green solid lines show our photon index, and light blue dashed lines and light green solid lines show the CSC2 photon index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' cally absorbed quasars with NH between [1 - 2]×1022 cm−2 in the rest frame of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Using the absorbed power law model, they found that RLQ have flatter spectra than the RQQ counterparts (RLQ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='55 and RQQ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Some studies compare RLQ and RQQ in a specific part of the X-rays (hard band) to specify the corresponding mechanism (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In our study, the RLQ is flatter than RQQ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='49 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='11, which is a bigger difference than that found by Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2005), due to our larger sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We do not see any clear intrinsically absorbed quasars, which could be due to lower signal-to-noise spectra in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Furthermore, the extreme cases in our samples did not show strong evidence for intrinsic absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Calibrating our calculated photon index with CSC2 We compare the photon index calculated by our spectral modeling to the photon index given in CSC2 for the same quasars (13 RLQ and 26 RQQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Figure 6 shows the RLQ and RQQ distributions for the calculated Γ and the one given in CSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The distributions show a rough agreement between the two methods, with our modeled values indicating a slightly wider range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' CSC2 uses the χ2 statistics with background subtraction and binning, while we use wstat- statistics and no background subtraction appropriate for low counts spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' van Dyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2001) have explained the χ2 statistical bias at low counts spectra, see also (Protassov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 RQQ_CSC RQQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 RLQ CSC RLQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 PhotonindexFigure 7: The two panels show the 100 trials of the Kuiper test DK (left panel) and FK (right panel) for the HRh/s parameter, between RLQ and RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The red color for HRh/s at 2 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5, and the blue color for HRh/s at z > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The solid vertical lines are for the mean and the dashed vertical lines are for the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Humphrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2009) found that even high counts give an inherent bias in the χ2 fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' These studies show that χ2 methods should not routinely be used for fitting an arbitrary, parameterized model to Poisson-distributed data, irrespective of the number of counts (Mighell, 1999), and instead, the Cash statistic should be adopted (Humphrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We used the wstat, which is based on the Poisson likelihood and accounts for the background6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We applied the Kuiper-two test to evaluate the difference between the two photon-index distributions, Γfit and ΓCSC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The test returns high values of Fk, for RQQ Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='33 and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='77 for RLQ, which implies that the distributions of Γ resulting from our modeling are consistent with the CSC2 distributions for these small sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Redshift Dependence of the Hardness Ratio Our results on the hardness ratio parameter HRh/s indicate that the RQQ spectra are softer than the spectra of RLQ (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We perform simulations to confirm that the effect is an inherent physical property of RQQ and is not affected by the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Because the rest frame energy range is shifted towards the lower energy in the observed frame we check the distributions of the hardness ratio parameter in the two redshift ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' There are 261 RQQ and 33 RLQ at 2 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 and 211 RQQ and 48 RLQ at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 < z < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Thus the fraction of RQQ is higher at z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 than at z > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5, which may bias the RQQ’s HRh/s parameter in the full redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 6https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='edu/sherpa/ahelp/wstat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='html 15 Z<2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='0 F_KWe apply the Kuiper test to the HRh/s at 2 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 sample and get the Kuiper param- eter values of DK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='33, and FK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Then, we select the same sample size of 33 RQQ and RLQ by randomly selecting 33 quasars from the 261 RQQ sample and using all 33 RLQ in this low redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In the first random selection of the 33 RQQ we get DK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='47, and Fk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Afterward, we perform the test for the hardness ratio difference by looking at the distribution of the Kuiper parameters, Dk and Fk, in 100 random samples (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We selected the 100 random samples of 33 RQQ and used the existing 33 RLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The median values for the 100 Kuiper test parameters in this step are Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='36, and FK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' For the z > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 sample, the Kuiper test results for HRh/s difference between RLQ and RQQ are DK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='30, FK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We then performed the same simulation steps as described above for the z > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 sample using the 48 RLQ and a random sample of 48 RQQ selected from the 211 RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The median value of the Kuiper test distribution was Dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='32, and FK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='06 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The simulation results show that the difference in the HRh/s parameter between RLQ and RQQ samples is slightly more significant at 2 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5 than z > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' According to Peca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' (2021), our sample selection is the least affected by the absorption dependence with redshift and the Chandra detector contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' At low redshift (z < 2), the difference in the flux between hard and soft bands is larger for quasars with NH < 1022 cm−2 because the soft X-ray emission is present in the observed energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' At high redshift (z > 2) the hardness ratio of the quasars with low NH is not affected by the hard band shift to the lower observed energies and only the quasars with high absorption, NH > 1023 cm−2, will show the impact on the HRh/s parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We conclude that the observed difference in the hardness ratio between RQQ and RLQ at z > 2 is not affected by redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Summary and Conclusions We studied the X-ray properties of high redshift quasars observed by Chandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We found a total of 2,561 DR7 quasars in the CSC2 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' After applying redshift and radio- loudness filters we obtained two samples, one with 472 RQQ and the second with 81 RLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The two samples have a similar redshift range, 2 < z < 5, with the RLQ sample being one of the largest samples of RLQ within that redshift range to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our main results are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We found that an average X-ray luminosity of RLQ at high redshift is higher than the average X-ray luminosity of RQQ, consistent with the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We calculated the mean photon index of ΓRLQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='70 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='33 and ΓRQQ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='19 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='44 for the RLQ and RQQ samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' This result confirms that RLQ spectra 16 are flatter than the spectra of RQQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We identified a few extremely soft RLQ and ex- tremely hard RQQ, but these sources have low signal-to-noise data and require further observations to understand their X-ray properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' We found that the LD and RQQ have similar distributions of hardness ratios, HRh/m and HRh/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' In comparison, LD and CD have similar photon index and X-ray luminos- ity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' However, our sample has only 10 LD quasars and more LD observa- tions are needed to confirm this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The peaks of HRh/s and HRm/s distributions are shifted towards negative values (soft energy band) in RQQ compared to RLQ, which confirms that the X-ray luminosity in the RQQ is dominated by soft X-rays in comparison to RLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Our study shows potential directions for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The quasars of extreme cases need longer observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The CD and LD comparison needs larger samples for statistically meaningful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' The current samples can be extended to include quasars at higher red- shifts, z > 5, with the future releases of the Chandra Source Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Additionally, the available quasar catalogs can be used to study the early universe population of quasars using high redshift infrared observations which will become available with the JWST (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Software: Sherpa (Freeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2001), Topcat (Taylor, 2017), Python packages: As- tropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2018), Seaborn (Waskom, 2021), Numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 2020), and Matplotlib (Hunter, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Acknowledgement This research has made use of data obtained from the Chandra Data Archive and the Chandra Source Catalog, and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO and Sherpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' thanks CXC Helpdesk and Nick Lee for the support in the analysis of Chandra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' was supported by NASA 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Zickgraf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', Voges, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', Krautter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', Thiering, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', Appenzeller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', Mujica, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', Serrano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=', 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Identification of a complete sample of northern ROSAT All-Sky Survey X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' Discovery of a z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='28 QSO near the RASS source RX J1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content='6-0844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' AP 323, L21–L24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfCgLC/content/2301.02866v1.pdf'} diff --git a/.gitattributes b/.gitattributes index a274acb944d77a67e0d54f1d745dc5157cb26c6e..4dc1cfce5eded1828a64f27999eeaacc9aa5d271 100644 --- a/.gitattributes +++ b/.gitattributes @@ -7893,3 +7893,61 @@ JdE1T4oBgHgl3EQfsAXy/content/2301.03362v1.pdf filter=lfs diff=lfs merge=lfs -tex JtE5T4oBgHgl3EQfXg-b/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf filter=lfs diff=lfs merge=lfs -text 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b/09E1T4oBgHgl3EQfRgOv/content/tmp_files/2301.03054v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4743b6c9016c51dae40c7646ccb3cbdb633c214 --- /dev/null +++ b/09E1T4oBgHgl3EQfRgOv/content/tmp_files/2301.03054v1.pdf.txt @@ -0,0 +1,795 @@ +arXiv:2301.03054v1 [cond-mat.str-el] 8 Jan 2023 +Time-crystalline spin ice and Dirac strings in a driven magnet +Mingxi Yue1 and Zi Cai1, 2, ∗ +1Wilczek Quantum Center and Key Laboratory of Artificial Structures and Quantum Control, +School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China +2Shanghai Research Center for Quantum Sciences, Shanghai 201315, China +Studies on systems far from equilibrium open up new avenues for investigating exotic phases of +matter. A driven-dissipative frustrated spin system is examined in this study, and we suggest an +out-of-equilibrium non-magnetic phase where the spins do not order but adhere to the ice rule in +space and establish a long-range crystalline order in time. It is shown that this time-crystalline +spin ice phase is distinct from the equilibrium spin ice and conventional discrete time crystal. The +dynamics of monopoles, Dirac strings and other space-time topological defects have been examined +in the context of this far-from-equilibrium system, and the possible experimental realization of our +model has been discussed. +Introduction – Spin ice (SI) is an unusual magnet that +does not order even as the temperature tends towards +zero[1]. Here, geometrical frustration results in ground +states with extensive degeneracy yet local constraints +known as ice rule. +For example, in the rare-earth ti- +tanates such as Dy2Ti2O7 and Ho2Ti2O7, the energy is +minimized for those configurations satisfying two spins +pointing in and two out in each tetrahedra of the py- +rochlore lattice[2–4]. Despite its simplicity, the ice rule is +responsible for a wealth of interesting phenomena includ- +ing the zero point entropy[5, 6], fractionalization[7, 8], +and the emergent gauge field[9, 10]. +Locally break- +ing the ice rule produces a pair of point-like defects - +condensed matter analogs of monopoles[11]- that can be +separated to a large distance with a finite energy cost. +Most studies on this topic focused on the equilibrium or +near-equilibrium (relaxation[12, 13] or transport[14–16] +of monopoles) properties, while the spin ice physics in +far-from-equilibrium systems is elusive. Because the ice +rule is rooted in the energy minimization principle, while +non-equilibrium systems, especially driven systems, are +usually far from ground states. +The scope of nonequilibrium physics is considerable. +Nevertheless, far from equilibrium less is known in gen- +eral. However, nonequilibrium systems present fresh op- +portunities for investigating novel phases of matters ab- +sent in thermal equilibrium. A prototypical example is +the time crystal phase, which spontaneously breaks the +temporal translational symmetry[17–26]. Further incor- +porating spatial degrees of freedom might lead to more +complex non-equilibrium phases with intriguing space- +time structures[27–30]. As for magnetic systems, the role +of frustration in a nonequilibrium magnet is still unclear +despite great efforts[31–35]. For example, one may won- +der whether a magnet driven far from the ground state +can host an out-of-equilibrium analog of the SI phase, +which does not order yet still obeys the ice rule that is +typically assumed to be held only for the ground state. +If exists, how does such a non-equilibrium SI differ from +its equilibrium counterpart? Is it possible to define and +characterize “excitations” above such a non-equilibrium +state that has already been highly excited? +In this study, we attempt to answer these questions by +investigating a periodically driven classical spin system in +a checkerboard lattice. Dynamical simulations of classi- +cal spin systems, unlike quantum many-body systems, do +not suffer from the notorious exponential wall problem, +thus allowing us to simulate 2D systems up to very high +system sizes. On the other hand, it has been realized that +certain intriguing features of non-equilibrium physics do +not crucially depend on the quantum or classical nature +of the systems[36], and discrete time crystal (DTC) or +other exotic orders have been investigated in classical pe- +riodically driven systems[35–40]. In terms of SI physics, +typically, a periodical driving will pump energy into the +system thus is detrimental to the SI phase[41]. Here, we +demonstrate that the interplay between periodic driving +and frustration can lead to a non-equilibrium phase that +displays oscillating SI patterns in space, accompanied by +a DTC order in time. Furthermore, the dynamics of the +space-time defects (monopoles, Dirac string and instan- +ton) emerged from such a time-crystalline spin ice (TC- +SI) phase have also been discussed. +FIG. 1: +(Color online)(a)Schematic of a perfect spin ice +configuration (monopole vacuum) in a checkerboard lat- +tice(blue/red dots indicate spin up/down); (b)Flipping one +spin creates two monopole excitations (dark ⊠) above the vac- +uum; (c)Two monopoles are separated in space by flipping all +the spins in the associated Dirac string (the red line). +The model – To examine the spin ice phase, we em- +ploy a classical transverse Ising model in a checkerboard + +(a) +(b) +c2 +-2 +0 +2 +4 +-1 +0 +1 +0.0 +0.2 +0.4 +0.6 +s +x +(t) + + + +s +z +(t) +(b) +t +3 +t +4 +t +2 + +(t) +t +1 +548 +546 +544 +542 + +t +[J +-1 +] +540 +0 +5 +10 +-0.5 +0.0 +0.5 +1.0 +1 +10 +0.1 +1 + + +S(r +i +) +i+1 +(c) + + +|S(r +i +)| +i+1 +|S(r)|~r +- +FIG. 2: (Color online) (a)the snapshot of {sz +i } at three typical +time slices; (b)dynamics of the x and z components of spin +on site i; (c)equal-time correlation function at a AF time slice +t = t1 (left panel), which exhibits an algebraic decay (right +panel). The parameters are chosen as J′ = 4J, ω = 2πJ, +Γ = 1.5J, γ = J, D = 0.01J and L = 30 (a 10 × 10 section is +plotted in (a)). +lattice, whose Hamiltonian reads: +Hice = +� +⊠ +� +ij∈⊠ +Θ(t)sz +i sz +j − Γ +� +i +sx +i , +(1) +where ⊠ indicates the plaquette in the checkerboard +lattice with the next nearest neighboring (NNN) cou- +pling (the grey plaquette in Fig.1 a). +si = [sx +i , sy +i , sz +i ] +is a classical vector with a fixed length |si| = 1. +Γ +is the strength of a time-independent transverse field, +and Θ(t) = J + J′ cos ωt is a periodically varying inter- +action strength, where Θ(t) being positive/negative in- +dicates anti-ferromagnetic(AF)/ferromagnetic(FM) cou- +pling, whereas J′ and ω represent the amplitude and +frequency of the driving. +Throughout this paper, we +fix these Hamiltonian parameters. +However, we shall +demonstrate in the supplementary material(SM)that the +key results of this work do not crucially depend on this +specific choice of parameters[42] . +Typically, periodic driving will heat closed interact- +ing systems towards an infinite temperature state. We +incorporate dissipation into our model by coupling each +spin to a thermal bath, which can be phenomenologically +described using stochastic methods, to avoid this feature- +less asymptotic state. In the presence of a thermal bath, +the dynamics of spin i can be described by a stochastic +Landau-Lifshitz-Gilbert equation[43]: +˙si = hi(t) × si − γsi × (si × hi(t)) +(2) +where γ is the dissipation strength, which is fixed as +γ = J for the numerical convenience. +Although this +value is larger than that in conventional magnet, the +long-time asymptotic state does not importantly depends +on γ[42]. hi(t) = h0 +i (t) + ξi(t), where h0 +i = −∇siHice = +[Γ, 0, −Θ(t)¯sz +i ] is the effective magnetic field on site i +and ¯sz +i = � +j sz +j where the summation is over all the six +neighboring spins of site i. ξi(t) is a 3D zero-mean ran- +dom field representing thermal fluctuations. The local +bath satisfies: ⟨ξα +i (t)ξβ +j (t′)⟩ξ = D2δαβδijδ(t − t′) where +α, β = x, y, z and D is the strength of the noise. If the +bath is in thermal equilibrium, γ and D should satisfy +D2 = 2T γ, where T is the temperature of the bath. The +stochastic differential equation can be numerically solved +by the standard Heun method with a Stratonovich’s dis- +cretization formula[44], in which we select the discrete +time step ∆t = 10−3J−1 (the convergence with smaller +∆t has been verified). The simulation is performed over a +L×L checkerboard lattice with periodic boundary condi- +tion. In our simulations, we choose random initial states +whose effect has also been analyzed in SM[42]. In the fol- +lowing, we will focus on the long-time asymptotic dynam- +ics of this model. The dynamical phase diagram of this +model is extremely rich as shown in the SM[42]. Here, we +only consider the scenario when the system concurrently +displays SI patterns in space and DTC order in time, as +opposed to listing all the dynamical phases. +Time-crystalline spin ice – We consider the case where +Θ(t) oscillates between the AF and FM couplings (this +condition, however, is not necessary for the TC-SI phase +as illustrated in the SM[42]), and the spin configuration +accordingly varies. The snapshots of {sz +i } at three typical +time slices have been plotted in Fig.2 (a). At a time slice +t1 = 541.2T0 with AF coupling (T0 = 1/J is the period +of Θ(t), the magnetization has a 0.2T0 phase lag with +respect to Θ(t)), each sz +i reaches its maximum (|sz +i | = +0.9994), and {sz +i } obeys the ice rule (� +ij∈⊠ sz +i vanishes +for all ⊠). The {sz +i } snapshot at the next time slice t2 = +t1 + 0.5T0 with FM coupling (Θ(t2) < 0) shows neither +spin ice pattern, nor FM long-range order, rather, it is +a paramagnetic phase (PM) with magnetization along +the x-direction (see Fig.2 b). +At the time slices t3 = +t1+T0, the system Hamiltonian return to the original one +(Hice(t3) = Hice(t1)), but {sz +i } does not. Instead, all of +them are simultaneously reversed {sz +i (t3)} = {−sz +i (t1)}, +thus the ice rule is still preserved. +{sz +i } return to its +original values after two periods of driving at t4 = t1+2T0 +({sz +i (t4) = sz +i (t1)}), which indicates a spontaneous Z2 + +(a) +0.8 +0.6 +0.4 +0.2 +0.2 +-0.4 +0.6 +-0.8 +t,=541.2T。 +t=541.7To +t,=542.2To3 +time translational symmetry breaking (TTSB). +The origin of the DTC order can be understood as a +consequence of the periodically driven interaction. For a +pair of adjacent sites ij, if sz +i (t) and sz +j(t) synchronize as +sz +i = sz +j ∼ cos[ω′t + φ], the instantaneous interacting en- +ergy HI(t) ∼ Θ(t) cos[2ω′t + 2φ] with Θ(t) ∼ cos ωt can +be expressed as HI(t) ∼ cos[δωt − 2φ] + cos[Ωt + 2φ] +with δω = ω − 2ω′ and Ω = ω + 2ω′. +HI(t) oscil- +lates around zero except for the period doubling case +(ω′ = ω/2), where H(t) ∼ cos 2φ (the fast oscillating +term cos[2ωt + 2φ] is omitted). Therefore HI(t) becomes +approximately time-independent and takes its minimum +value at two degenerate points φ1 = +π +2 and φ2 = +3π +2 , +which is responsible for the spontaneous Z2 TTSB in the +DTC. This intuitive picture also explains the fact only +{sz +i } exhibit period doubling, while {sx +i } do not, as shown +in Fig.2 (b). +The equilibrium SI supports a Coulomb phase char- +acterized by an algebraic decay of the spatial corre- +lation function, one may query whether this property +holds for the non-equilibrium TC-SI phase. To answer +this question, we select an AF time slices (t = t1), +and calculate the equal-time correlation function S(r) = +1 +L2 +� +i⟨sz +i (t1)sz +i+r(t1)⟩, where the average ⟨⟩ is performed +over the trajectories starting from different random ini- +tial states. +As shown in Fig.2 (c), along the diagonal +direction r = +1 +√ +2(r, r) with r = |r|, S(r) decays al- +gebraically in distance S(r) ∼ r−α, with α = 1.9(2) +agreeing very well with the exponent predicted by the +Coulomb phase[10] (α = d with d = 2 the dimension of +the lattice). +However, this agreement does not indicate that the +asymptotic state in our model adiabatically follows the +ground state of the Hice. First, the ice rule only hold at +the time slices with AF coupling. For example, at a time +slice with FM coupling (e.g. t = t2), the ground state +of Hice(t2) is supposed to be an FM state along the z- +direction, while the system shows a PM state in our case. +Furthermore, the spontaneous TTSB can is forbidden in +thermal equilibrium due to the no-go theorem[45, 46]. +Therefore, the asymptotic state in our model is a gen- +uine non-equilibrium state with alternating SI and PM +configurations in space and DTC order in time. +Dynamics of monopoles after a local spin flip – In a +conventional SI, the elementary excitations can be intro- +duced by flipping one spin in a perfect SI configuration, +which violates the ice rule in the two adjacent ⊠. For a +monopole “vacuum” (a perfect SI configuration), flipping +a spin equals to create of a pair of monopoles, which can +be separated by properly identifying a chain of spins with +alternating spin up and down and flipping them simul- +taneously, as shown in Fig.1 (c). The energy required to +separate two monopoles in a SI model with short-range +coupling is independent of their distance, and the string +composed of the flipped spins is a condensed matter ana- +100 +1000 +0.0 +0.5 +1.0 +4 +5 +6 +7 +8 +9 +10 +100 +< +> +l + + +(t +n +) +t +n + l=3 + l=5 + l=7 +(c) +[J +-1 +] +FIG. 3: (Color online) The spin difference configuration {δsz +i } +at the time slices t = t1 (initial state) and t = tN (final +states). At t = t1, (a) only one spin is flipped and (b) a string +of spins are flipped (Dirac string); (c) stroboscopic dynamics +of the excess energy ∆E(tn) at the AF time slices with tn = +t0 + 2T0(n − 1) starting from different initial states, each of +which contains one Dirac string with different length l. The +inset indicates the average relaxation time ⟨τ⟩ξ as a function +of the length of the Dirac string in the initial state. Other +parameters are chosen the same as in Fig.2. +log of the Dirac string[47]. +In general, the definition of “excitation” above an out- +of-equilibrium state is elusive. Nevertheless, for the TC- +SI phase in our model, we adopt a similar procedure of +perturbing the state by flipping one spin, and monitor- +ing the subsequential dynamics. +For this purpose, we +first choose an AF time slice t0 when all sz +i reach their +maximum and the corresponding spin configuration {s0 +j} +obeys the ice rule. Then we randomly pick a site (say, site +i), flip its spin then study the evolution from such a con- +figuration {s1 +j} (s1 +j = s0 +j except j = i where s1 +i = −s0 +i ). +We only focus on the stroboscopic dynamics at the AF +time slice with tn = t0 + 2T0(n − 1). +At the time slice tn, we defined {δsn +j } (δsn +j = sn +j − s0 +j) +to measure the change of the spin configuration with re- +spect to the initial SI configuration before the spin flip +{s0 +j}. At t = t1, only one spin is flipped, and thus δs1 +j = 0 +except j = i. Due to the dissipative nature of the dynam- +ics, after sufficiently long time (tn > tN), the system will +approach a new SI configuration, which differs from the +initial state as shown in {δsN +j } in Fig.3 (a). By com- +paring the final and initial state, we can find that the +spins which have been flipped during this process form +a closed ring, along which the δsN +j exhibits an alternat- +ing + and − structure. Flipping one spin produces two +monopoles, each of which can propagate from one ⊠ to +another by flipping the spin between them. The motion +of the monopoles resembles a random walk under certain + +(a) +(OsN) +(0s,) +(SsN +14 +constraint. If these two monopoles contact and their tra- +jectories form a closed ring, they could annihilate with +each other, leaving behind a new SI configuration that +differs from the original one by flipping all of the spins +along the closed ring that the monopoles went through. +Topology protected relaxation dynamics – A more in- +triguing dynamics occur if we start from an initial state +with a pair of well-separated monopoles attached by a +Dirac string, as shown in Fig.3 (b). +A monopole is a +topological fractionalized object that can not be created +or annihilated by itself. As an alternative, monopoles can +only be annihilated in pairs when they intersect. There- +fore, for a configuration with only two well-separated +monopoles, despite the dissipative nature of the dynam- +ics, the excess energy ∆En = ⟨Hice(tn)⟩ξ − ⟨Hice(t0)⟩ξ +(⟨⟩ξ indicates the ensemble average over the trajectories +of the thermal noise) can be protected for a sufficiently +long time before these two monopoles collide. +Conse- +quently, the relaxation is supposed to be slower from an +initial state with a pair of monopoles with larger separa- +tion (see Fig.3 c). The inset of Fig.3 (c) shows that the +average relaxation time ⟨τ⟩ξ exponentially diverges with +the length of the Dirac string l. +-4 +0 +4 +95 +100 +105 +110 +-1 +0 +1 + + +(t) +(a) + +s +z +i +t + D=0.01J + D=0.1J +[J +-1 +] +-phase shift +0 +10 +20 +30 +40 +50 +60 +-1 +0 +1 +0 +20 +40 +0.4 +0.6 +0.8 +1 +| +| +t + + + +t +(b) +[J +-1 +] +FIG. 4: (Color online) (a) The dynamics of sz +i on site i in +a single noise trajectory with D = 0.01J and D = 0.1J, the +former demonstrates a perfect DTC order, while in the latter, +thermal fluctuations activate a π-phase shift; (b) The dynam- +ics of the average ⟨sz +i ⟩ξ starting from a perfect SI state after +ensemble average over 103 noise trajectories. The envelope +of ⟨sz +i ⟩ξ exhibits an exponential decay as shown in the inset. +Other parameters except D are chosen the same as in Fig.2. +Instanton activated by the thermal fluctuation – Al- +though the stroboscopic dynamics of monopoles resemble +the relaxation dynamics in the conventional SI phase, the +proposed TC-SI phase is distinct from the equilibrium SI +because of the spontaneous TTSB. A natural question +thus arises: what’s the effect of the monopoles on the +temporal order of the TC-SI phase. The answer to this +question is directly related to the stability of TC-SI phase +against thermal fluctuations, which excite monopole with +a finite density. The Coulomb phase in equilibrium SI +does not breaks any symmetry, and is not robust at finite +temperature. However, the TC-SI phase is characterized +by a spontaneous Z2 TTSB, while a discrete symmetry +breaking phase is typically assumed to be robust against +weak thermal fluctuations in 2D systems. For example, +in a similar model without frustration, the corresponding +DTC phase is indeed stable at low temperature[48]. The +impact of thermal fluctuation on the TC-SI phase will +then be discussed. +Unlike the conventional SI phase, once a spin in our +TC-SI phase is suddenly flipped at a typical AF time +slice, it does not only produce a pair of monopoles in +space, but also results in a π−phase shift on top of the +periodic dynamics of this flipped spin, which corresponds +to tunneling from one “degenerate” DTC phase (φ = π +2 ) +to the other (φ = 3π +2 ). Such a fluctuation-activated tun- +neling between the two Z2 symmetry breaking states (see +Fig.4 a) resembles the instanton excitation in the field +theory[49], and is a topological defect in the temporal do- +main. These instanton excitations, no matter how rare +they are, are detrimental to the DTC long-range order +in the time domain and result in an exponential decay +of the DTC order at any finite temperature, as shown in +Fig.4 (b). However, the life-time of TC-SI phase can be +extraordinarily long at a temperatures much lower than +the activated temperature of monopoles. +Discussion – The proposed model is classical, while +a quantum generalization might provide a new perspec- +tive, although it is extremely challenging, if not impos- +sible, to simulate its real-time evolution. For a quantum +transverse Ising model in a checkerboard lattice, quan- +tum fluctuation lifts the extensive classical degeneracy +and leads to ordered ground states[50]. These magnetic +orderings could be suppressed by increasing temperature, +which leads to non-magnetic phases that resemble the +Coulomb phase. In terms of a quantum generalization of +our driven-dissipative model, we hypothesize that there +may be a regime of intermediate temperature where the +temperature could overwhelm the quantum fluctuation +while remaining significantly lower than the activated +temperature of monopoles, and the quantum system may +exhibit dynamics similar to those in our classical model. +Experimental realizations of dynamically modulated +interactions– One of the primary obstacles to the experi- +mental realization of our model is that it requires a peri- +odical driving imposed on the interaction rather than on +the external field, which seems unrealistic for solid-state +magnets. This dynamical modulated interaction can be +achieved using magnetophononics, in which the electric + +5 +field of a laser is coupled to the optical phonon, and the +consequent periodic atomic displacements could dynami- +cally modulate the magnetic exchange couplings between +the spins[51, 52]. +This proposal has been realized ex- +perimentally in the AF semiconductor α-MnTe[53]. Al- +though the tunable coupling regime is small and it is +impossible to change the sign of the interaction, we show +in the SM[42] that, for a slower driving (e.g. ω = 0.5πJ) +the TC-SI can exist even when the coupling is always +AF (J′ < J). This periodically modulated interaction is +accessible in synthetic quantum systems such as trapped +ions[54] and cavity QED systems[55]. For instance, in the +latter, by applying a periodic driving to the cavity pho- +tons, the magnetic interaction mediated by cavity can be +dynamically controlled. +Conclusion and outlook – In summary, we examined +a driven-dissipative frustrated magnetic system, and our +results demonstrate that the interplay between the pe- +riodic driving and frustration can give rise to a non- +equilibrium TC-SI phase. +Unlike earlier studies, the +aim of this work is to investigate SI physics in the con- +text of far-from-equilibrium systems rather than the out- +of-equilibrium features of a conventional SI phase. +In +frustrated quantum magnetism, a similar phase without +spontaneous symmetry breaking is the quantum spin liq- +uid. +One thus may wonder whether it is possible to +realize similar exotic quantum phases of matter out of +equilibrium[56], which can simultaneously show spatial +topological order and non-trivial temporal (long-range or +quasi-long-range) orders. 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Knap, arXiv e-prints +arXiv:2211.10453 (2022), 2211.10453. + diff --git a/09E1T4oBgHgl3EQfRgOv/content/tmp_files/load_file.txt b/09E1T4oBgHgl3EQfRgOv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7b14bfc5a7b76f89b3dcdf30e96c54a689bf34f --- /dev/null +++ b/09E1T4oBgHgl3EQfRgOv/content/tmp_files/load_file.txt @@ -0,0 +1,632 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf,len=631 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='03054v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='str-el] 8 Jan 2023 Time-crystalline spin ice and Dirac strings in a driven magnet Mingxi Yue1 and Zi Cai1, 2, ∗ 1Wilczek Quantum Center and Key Laboratory of Artificial Structures and Quantum Control, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 2Shanghai Research Center for Quantum Sciences, Shanghai 201315, China Studies on systems far from equilibrium open up new avenues for investigating exotic phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' A driven-dissipative frustrated spin system is examined in this study, and we suggest an out-of-equilibrium non-magnetic phase where the spins do not order but adhere to the ice rule in space and establish a long-range crystalline order in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' It is shown that this time-crystalline spin ice phase is distinct from the equilibrium spin ice and conventional discrete time crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The dynamics of monopoles, Dirac strings and other space-time topological defects have been examined in the context of this far-from-equilibrium system, and the possible experimental realization of our model has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Introduction – Spin ice (SI) is an unusual magnet that does not order even as the temperature tends towards zero[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Here, geometrical frustration results in ground states with extensive degeneracy yet local constraints known as ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For example, in the rare-earth ti- tanates such as Dy2Ti2O7 and Ho2Ti2O7, the energy is minimized for those configurations satisfying two spins pointing in and two out in each tetrahedra of the py- rochlore lattice[2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Despite its simplicity, the ice rule is responsible for a wealth of interesting phenomena includ- ing the zero point entropy[5, 6], fractionalization[7, 8], and the emergent gauge field[9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Locally break- ing the ice rule produces a pair of point-like defects - condensed matter analogs of monopoles[11]- that can be separated to a large distance with a finite energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Most studies on this topic focused on the equilibrium or near-equilibrium (relaxation[12, 13] or transport[14–16] of monopoles) properties, while the spin ice physics in far-from-equilibrium systems is elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Because the ice rule is rooted in the energy minimization principle, while non-equilibrium systems, especially driven systems, are usually far from ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The scope of nonequilibrium physics is considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Nevertheless, far from equilibrium less is known in gen- eral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' However, nonequilibrium systems present fresh op- portunities for investigating novel phases of matters ab- sent in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' A prototypical example is the time crystal phase, which spontaneously breaks the temporal translational symmetry[17–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Further incor- porating spatial degrees of freedom might lead to more complex non-equilibrium phases with intriguing space- time structures[27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' As for magnetic systems, the role of frustration in a nonequilibrium magnet is still unclear despite great efforts[31–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For example, one may won- der whether a magnet driven far from the ground state can host an out-of-equilibrium analog of the SI phase, which does not order yet still obeys the ice rule that is typically assumed to be held only for the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' If exists, how does such a non-equilibrium SI differ from its equilibrium counterpart?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Is it possible to define and characterize “excitations” above such a non-equilibrium state that has already been highly excited?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In this study, we attempt to answer these questions by investigating a periodically driven classical spin system in a checkerboard lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Dynamical simulations of classi- cal spin systems, unlike quantum many-body systems, do not suffer from the notorious exponential wall problem, thus allowing us to simulate 2D systems up to very high system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' On the other hand, it has been realized that certain intriguing features of non-equilibrium physics do not crucially depend on the quantum or classical nature of the systems[36], and discrete time crystal (DTC) or other exotic orders have been investigated in classical pe- riodically driven systems[35–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In terms of SI physics, typically, a periodical driving will pump energy into the system thus is detrimental to the SI phase[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Here, we demonstrate that the interplay between periodic driving and frustration can lead to a non-equilibrium phase that displays oscillating SI patterns in space, accompanied by a DTC order in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Furthermore, the dynamics of the space-time defects (monopoles, Dirac string and instan- ton) emerged from such a time-crystalline spin ice (TC- SI) phase have also been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' 1: (Color online)(a)Schematic of a perfect spin ice configuration (monopole vacuum) in a checkerboard lat- tice(blue/red dots indicate spin up/down);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' (b)Flipping one spin creates two monopole excitations (dark ⊠) above the vac- uum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' (c)Two monopoles are separated in space by flipping all the spins in the associated Dirac string (the red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The model – To examine the spin ice phase, we em- ploy a classical transverse Ising model in a checkerboard (a) (b) c2 2 0 2 4 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='6 s x (t) s z (t) (b) t 3 t 4 t 2 (t) t 1 548 546 544 542 t [J 1 ] 540 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='0 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='1 1 S(r i ) i+1 (c) |S(r i )| i+1 |S(r)|~r FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' 2: (Color online) (a)the snapshot of {sz i } at three typical time slices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' (b)dynamics of the x and z components of spin on site i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' (c)equal-time correlation function at a AF time slice t = t1 (left panel), which exhibits an algebraic decay (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The parameters are chosen as J′ = 4J, ω = 2πJ, Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='5J, γ = J, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='01J and L = 30 (a 10 × 10 section is plotted in (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' lattice, whose Hamiltonian reads: Hice = � ⊠ � ij∈⊠ Θ(t)sz i sz j − Γ � i sx i , (1) where ⊠ indicates the plaquette in the checkerboard lattice with the next nearest neighboring (NNN) cou- pling (the grey plaquette in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' si = [sx i , sy i , sz i ] is a classical vector with a fixed length |si| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Γ is the strength of a time-independent transverse field, and Θ(t) = J + J′ cos ωt is a periodically varying inter- action strength, where Θ(t) being positive/negative in- dicates anti-ferromagnetic(AF)/ferromagnetic(FM) cou- pling, whereas J′ and ω represent the amplitude and frequency of the driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Throughout this paper, we fix these Hamiltonian parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' However, we shall demonstrate in the supplementary material(SM)that the key results of this work do not crucially depend on this specific choice of parameters[42] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Typically, periodic driving will heat closed interact- ing systems towards an infinite temperature state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' We incorporate dissipation into our model by coupling each spin to a thermal bath, which can be phenomenologically described using stochastic methods, to avoid this feature- less asymptotic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In the presence of a thermal bath, the dynamics of spin i can be described by a stochastic Landau-Lifshitz-Gilbert equation[43]: ˙si = hi(t) × si − γsi × (si × hi(t)) (2) where γ is the dissipation strength, which is fixed as γ = J for the numerical convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Although this value is larger than that in conventional magnet, the long-time asymptotic state does not importantly depends on γ[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' hi(t) = h0 i (t) + ξi(t), where h0 i = −∇siHice = [Γ, 0, −Θ(t)¯sz i ] is the effective magnetic field on site i and ¯sz i = � j sz j where the summation is over all the six neighboring spins of site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' ξi(t) is a 3D zero-mean ran- dom field representing thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The local bath satisfies: ⟨ξα i (t)ξβ j (t′)⟩ξ = D2δαβδijδ(t − t′) where α, β = x, y, z and D is the strength of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' If the bath is in thermal equilibrium, γ and D should satisfy D2 = 2T γ, where T is the temperature of the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The stochastic differential equation can be numerically solved by the standard Heun method with a Stratonovich’s dis- cretization formula[44], in which we select the discrete time step ∆t = 10−3J−1 (the convergence with smaller ∆t has been verified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The simulation is performed over a L×L checkerboard lattice with periodic boundary condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In our simulations, we choose random initial states whose effect has also been analyzed in SM[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In the fol- lowing, we will focus on the long-time asymptotic dynam- ics of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The dynamical phase diagram of this model is extremely rich as shown in the SM[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Here, we only consider the scenario when the system concurrently displays SI patterns in space and DTC order in time, as opposed to listing all the dynamical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Time-crystalline spin ice – We consider the case where Θ(t) oscillates between the AF and FM couplings (this condition, however, is not necessary for the TC-SI phase as illustrated in the SM[42]), and the spin configuration accordingly varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The snapshots of {sz i } at three typical time slices have been plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' At a time slice t1 = 541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2T0 with AF coupling (T0 = 1/J is the period of Θ(t), the magnetization has a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2T0 phase lag with respect to Θ(t)), each sz i reaches its maximum (|sz i | = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='9994), and {sz i } obeys the ice rule (� ij∈⊠ sz i vanishes for all ⊠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The {sz i } snapshot at the next time slice t2 = t1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='5T0 with FM coupling (Θ(t2) < 0) shows neither spin ice pattern, nor FM long-range order, rather, it is a paramagnetic phase (PM) with magnetization along the x-direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' At the time slices t3 = t1+T0, the system Hamiltonian return to the original one (Hice(t3) = Hice(t1)), but {sz i } does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Instead, all of them are simultaneously reversed {sz i (t3)} = {−sz i (t1)}, thus the ice rule is still preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' {sz i } return to its original values after two periods of driving at t4 = t1+2T0 ({sz i (t4) = sz i (t1)}), which indicates a spontaneous Z2 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='8 t,=541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' t=541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='7To t,=542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2To3 time translational symmetry breaking (TTSB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The origin of the DTC order can be understood as a consequence of the periodically driven interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For a pair of adjacent sites ij, if sz i (t) and sz j(t) synchronize as sz i = sz j ∼ cos[ω′t + φ], the instantaneous interacting en- ergy HI(t) ∼ Θ(t) cos[2ω′t + 2φ] with Θ(t) ∼ cos ωt can be expressed as HI(t) ∼ cos[δωt − 2φ] + cos[Ωt + 2φ] with δω = ω − 2ω′ and Ω = ω + 2ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' HI(t) oscil- lates around zero except for the period doubling case (ω′ = ω/2), where H(t) ∼ cos 2φ (the fast oscillating term cos[2ωt + 2φ] is omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Therefore HI(t) becomes approximately time-independent and takes its minimum value at two degenerate points φ1 = π 2 and φ2 = 3π 2 , which is responsible for the spontaneous Z2 TTSB in the DTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' This intuitive picture also explains the fact only {sz i } exhibit period doubling, while {sx i } do not, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The equilibrium SI supports a Coulomb phase char- acterized by an algebraic decay of the spatial corre- lation function, one may query whether this property holds for the non-equilibrium TC-SI phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' To answer this question, we select an AF time slices (t = t1), and calculate the equal-time correlation function S(r) = 1 L2 � i⟨sz i (t1)sz i+r(t1)⟩, where the average ⟨⟩ is performed over the trajectories starting from different random ini- tial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2 (c), along the diagonal direction r = 1 √ 2(r, r) with r = |r|, S(r) decays al- gebraically in distance S(r) ∼ r−α, with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='9(2) agreeing very well with the exponent predicted by the Coulomb phase[10] (α = d with d = 2 the dimension of the lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' However, this agreement does not indicate that the asymptotic state in our model adiabatically follows the ground state of the Hice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' First, the ice rule only hold at the time slices with AF coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For example, at a time slice with FM coupling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' t = t2), the ground state of Hice(t2) is supposed to be an FM state along the z- direction, while the system shows a PM state in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Furthermore, the spontaneous TTSB can is forbidden in thermal equilibrium due to the no-go theorem[45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Therefore, the asymptotic state in our model is a gen- uine non-equilibrium state with alternating SI and PM configurations in space and DTC order in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Dynamics of monopoles after a local spin flip – In a conventional SI, the elementary excitations can be intro- duced by flipping one spin in a perfect SI configuration, which violates the ice rule in the two adjacent ⊠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For a monopole “vacuum” (a perfect SI configuration), flipping a spin equals to create of a pair of monopoles, which can be separated by properly identifying a chain of spins with alternating spin up and down and flipping them simul- taneously, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The energy required to separate two monopoles in a SI model with short-range coupling is independent of their distance, and the string composed of the flipped spins is a condensed matter ana- 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='0 4 5 6 7 8 9 10 100 < > l (t n ) t n l=3 l=5 l=7 (c) [J 1 ] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' 3: (Color online) The spin difference configuration {δsz i } at the time slices t = t1 (initial state) and t = tN (final states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' At t = t1, (a) only one spin is flipped and (b) a string of spins are flipped (Dirac string);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' (c) stroboscopic dynamics of the excess energy ∆E(tn) at the AF time slices with tn = t0 + 2T0(n − 1) starting from different initial states, each of which contains one Dirac string with different length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The inset indicates the average relaxation time ⟨τ⟩ξ as a function of the length of the Dirac string in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Other parameters are chosen the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' log of the Dirac string[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In general, the definition of “excitation” above an out- of-equilibrium state is elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Nevertheless, for the TC- SI phase in our model, we adopt a similar procedure of perturbing the state by flipping one spin, and monitor- ing the subsequential dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For this purpose, we first choose an AF time slice t0 when all sz i reach their maximum and the corresponding spin configuration {s0 j} obeys the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Then we randomly pick a site (say, site i), flip its spin then study the evolution from such a con- figuration {s1 j} (s1 j = s0 j except j = i where s1 i = −s0 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' We only focus on the stroboscopic dynamics at the AF time slice with tn = t0 + 2T0(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' At the time slice tn, we defined {δsn j } (δsn j = sn j − s0 j) to measure the change of the spin configuration with re- spect to the initial SI configuration before the spin flip {s0 j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' At t = t1, only one spin is flipped, and thus δs1 j = 0 except j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Due to the dissipative nature of the dynam- ics, after sufficiently long time (tn > tN), the system will approach a new SI configuration, which differs from the initial state as shown in {δsN j } in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' By com- paring the final and initial state, we can find that the spins which have been flipped during this process form a closed ring, along which the δsN j exhibits an alternat- ing + and − structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Flipping one spin produces two monopoles, each of which can propagate from one ⊠ to another by flipping the spin between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The motion of the monopoles resembles a random walk under certain (a) (OsN) (0s,) (SsN 14 constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' If these two monopoles contact and their tra- jectories form a closed ring, they could annihilate with each other, leaving behind a new SI configuration that differs from the original one by flipping all of the spins along the closed ring that the monopoles went through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Topology protected relaxation dynamics – A more in- triguing dynamics occur if we start from an initial state with a pair of well-separated monopoles attached by a Dirac string, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' A monopole is a topological fractionalized object that can not be created or annihilated by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' As an alternative, monopoles can only be annihilated in pairs when they intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' There- fore, for a configuration with only two well-separated monopoles, despite the dissipative nature of the dynam- ics, the excess energy ∆En = ⟨Hice(tn)⟩ξ − ⟨Hice(t0)⟩ξ (⟨⟩ξ indicates the ensemble average over the trajectories of the thermal noise) can be protected for a sufficiently long time before these two monopoles collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Conse- quently, the relaxation is supposed to be slower from an initial state with a pair of monopoles with larger separa- tion (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='3 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='3 (c) shows that the average relaxation time ⟨τ⟩ξ exponentially diverges with the length of the Dirac string l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' 4 0 4 95 100 105 110 1 0 1 (t) (a) s z i t D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='01J D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='1J [J 1 ] phase shift 0 10 20 30 40 50 60 1 0 1 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='8 1 | | t t (b) [J 1 ] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' 4: (Color online) (a) The dynamics of sz i on site i in a single noise trajectory with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='01J and D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='1J, the former demonstrates a perfect DTC order, while in the latter, thermal fluctuations activate a π-phase shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' (b) The dynam- ics of the average ⟨sz i ⟩ξ starting from a perfect SI state after ensemble average over 103 noise trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The envelope of ⟨sz i ⟩ξ exhibits an exponential decay as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Other parameters except D are chosen the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Instanton activated by the thermal fluctuation – Al- though the stroboscopic dynamics of monopoles resemble the relaxation dynamics in the conventional SI phase, the proposed TC-SI phase is distinct from the equilibrium SI because of the spontaneous TTSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' A natural question thus arises: what’s the effect of the monopoles on the temporal order of the TC-SI phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The answer to this question is directly related to the stability of TC-SI phase against thermal fluctuations, which excite monopole with a finite density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The Coulomb phase in equilibrium SI does not breaks any symmetry, and is not robust at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' However, the TC-SI phase is characterized by a spontaneous Z2 TTSB, while a discrete symmetry breaking phase is typically assumed to be robust against weak thermal fluctuations in 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For example, in a similar model without frustration, the corresponding DTC phase is indeed stable at low temperature[48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' The impact of thermal fluctuation on the TC-SI phase will then be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Unlike the conventional SI phase, once a spin in our TC-SI phase is suddenly flipped at a typical AF time slice, it does not only produce a pair of monopoles in space, but also results in a π−phase shift on top of the periodic dynamics of this flipped spin, which corresponds to tunneling from one “degenerate” DTC phase (φ = π 2 ) to the other (φ = 3π 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Such a fluctuation-activated tun- neling between the two Z2 symmetry breaking states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='4 a) resembles the instanton excitation in the field theory[49], and is a topological defect in the temporal do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' These instanton excitations, no matter how rare they are, are detrimental to the DTC long-range order in the time domain and result in an exponential decay of the DTC order at any finite temperature, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' However, the life-time of TC-SI phase can be extraordinarily long at a temperatures much lower than the activated temperature of monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Discussion – The proposed model is classical, while a quantum generalization might provide a new perspec- tive, although it is extremely challenging, if not impos- sible, to simulate its real-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For a quantum transverse Ising model in a checkerboard lattice, quan- tum fluctuation lifts the extensive classical degeneracy and leads to ordered ground states[50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' These magnetic orderings could be suppressed by increasing temperature, which leads to non-magnetic phases that resemble the Coulomb phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In terms of a quantum generalization of our driven-dissipative model, we hypothesize that there may be a regime of intermediate temperature where the temperature could overwhelm the quantum fluctuation while remaining significantly lower than the activated temperature of monopoles, and the quantum system may exhibit dynamics similar to those in our classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Experimental realizations of dynamically modulated interactions– One of the primary obstacles to the experi- mental realization of our model is that it requires a peri- odical driving imposed on the interaction rather than on the external field, which seems unrealistic for solid-state magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' This dynamical modulated interaction can be achieved using magnetophononics, in which the electric 5 field of a laser is coupled to the optical phonon, and the consequent periodic atomic displacements could dynami- cally modulate the magnetic exchange couplings between the spins[51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' This proposal has been realized ex- perimentally in the AF semiconductor α-MnTe[53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Al- though the tunable coupling regime is small and it is impossible to change the sign of the interaction, we show in the SM[42] that, for a slower driving (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='5πJ) the TC-SI can exist even when the coupling is always AF (J′ < J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' This periodically modulated interaction is accessible in synthetic quantum systems such as trapped ions[54] and cavity QED systems[55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' For instance, in the latter, by applying a periodic driving to the cavity pho- tons, the magnetic interaction mediated by cavity can be dynamically controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Conclusion and outlook – In summary, we examined a driven-dissipative frustrated magnetic system, and our results demonstrate that the interplay between the pe- riodic driving and frustration can give rise to a non- equilibrium TC-SI phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Unlike earlier studies, the aim of this work is to investigate SI physics in the con- text of far-from-equilibrium systems rather than the out- of-equilibrium features of a conventional SI phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' In frustrated quantum magnetism, a similar phase without spontaneous symmetry breaking is the quantum spin liq- uid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' One thus may wonder whether it is possible to realize similar exotic quantum phases of matter out of equilibrium[56], which can simultaneously show spatial topological order and non-trivial temporal (long-range or quasi-long-range) orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Another important direction is to classify the topological space-time defects that emerge from non-equilibrium phases of matter and identify their properties, which are directly relevant to the physical ob- servable effect of these nonequilibrium phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='—This work is supported by the National Key Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content=' 2020YFA0309000), NSFC of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} +page_content='12174251), Natural Science Foundation of Shanghai (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQfRgOv/content/2301.03054v1.pdf'} 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in NGC 4839 Falling into Coma: Evidence for the Second Infall? +Seong-A Oh,1 Myung Gyoon Lee,1 and In Sung Jang2 +1Astronomy Program, Department of Physics and Astronomy, SNUARC, Seoul National University, 1 Gwanak-ro, +Gwanak-gu, Seoul 08826, Republic of Korea +2Department of Astronomy & Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA +ABSTRACT +NGC 4839 is the brightest galaxy (cD) of the NGC 4839 group at R ≈ 1 Mpc in the +south-west of the Coma cluster, which is known to be falling into Coma. However, it +has been controversial whether it is in the first phase of infall or in the second phase of +infall after passing the Coma center. We present a wide field study of globular clusters +(GCs) in NGC 4839 and its environment based on Hyper Suprime-Cam gr images in +the Subaru archive. We compare the GC system of NGC 4839 with that of NGC 4816, +which is the brightest member (S0) of the nearby group and lies at a similar distance +in the west from the Coma center. Interestingly the spatial distribution of the GCs +in NGC 4839 is significantly more compact than that of the GCs in NGC 4816. +In +addition, the radial number density profile of the GCs in NGC 4839 shows an abrupt +drop at RN4839 ≈ 80 kpc, while that of the GCs in NGC 4816 shows a continuous slow +decline even in the outer region at 80 < RN4816 < 500 kpc. The effective radius of +the NGC 4839 GC system is about three times smaller than that of the NGC 4816 GC +system. This striking difference can be explained if NGC 4839 lost a significant fraction +of the GCs in its outskirt when it passed through Coma. This supports strongly the +second infall scenario where the NGC 4839 passed the Coma center about 1.6 Gyr ago, +and began the second infall after reaching the apocenter in the south-west recently. +1. INTRODUCTION +1.1. The NGC 4839 Group and the Main +Cluster in Coma +Coma is the most massive galaxy cluster in +the local universe. +It is connected with fil- +aments from neighboring galaxy clusters and +hosts various substructures indicating that it +is a complex merger system (Colless & Dunn +1996; Malavasi et al. 2020; Healy et al. 2021). +Thus, Coma is one of the best targets to study +how large scale substructures are assembled and +Corresponding author: Myung Gyoon Lee +sao@astro.snu.ac.kr,mglee@astro.snu.ac.kr +evolve, and has been a focus of many cluster +studies in various aspects (see Biviano (1998); +Churazov et al. (2021) and references therein). +Two most prominent substructures in Coma are +the main cluster core in the center and the +NGC 4839 group in the south-west, as shown by +galaxy number density maps (Colless & Dunn +1996; Healy et al. 2021), X-ray images of hot +gas (White et al. 1993; Neumann et al. 2001; +Lyskova et al. 2019; Churazov et al. 2021), and +radio images of synchrotron emission (Bonafede +et al. 2021, 2022; Lal et al. 2022). The main +cluster core hosts two giant galaxies (NGC 4874 +(cD) and NGC 4889 (D)), which are merging +now. The NGC 4839 group is at R ≈ 1 Mpc in +arXiv:2301.05269v1 [astro-ph.GA] 12 Jan 2023 + +2 +Oh et al. +the south-west of Coma, and it is much smaller +and less massive than the main cluster core +(Colless & Dunn 1996; Lyskova et al. 2019). The +NGC 4839 group is considered to be falling into +Coma and that the two systems will merge to +form a more massive system in the future (Bi- +viano (1998) and references therein). +1.2. Merger Scenarios for the NGC 4839 +Group: A Pre-merger or a Post-merger? +It is generally accepted that the NGC 4839 +group is merging with the main cluster. How- +ever, whether it is a pre-merger where the +NGC 4839 group is in the first phase of infall +(Briel et al. 1992; White et al. 1993; Colless & +Dunn 1996; Neumann et al. 2001; Akamatsu et +al. 2013) or a post-merger (Burns et al. 1994; +Lyskova et al. 2019; Churazov et al. 2021) has +been controversial (Sanders et al. 2020; Healy +et al. 2021). +We summarize the observational features re- +lated with the merging of the NGC 4839 group +in the previous studies in Table 1. These fea- +tures include several substructures seen in X- +ray and radio images, an excess of E+A galax- +ies in the SW region of the cluster, and sub- +structures found in the spatial distribution and +kinematics of galaxies. Each feature can be ex- +plained with either the pre-merger scenario or +the post-merger scenario. +Recently the post- +merger scenario, which can better explain the +existence of X-ray/radio substructures (in par- +ticular, bridges and streams), appears to be +more supported (Lyskova et al. 2019; Churazov +et al. 2021, 2022; Bonafede et al. 2021). How- +ever, even in the recent discussions of both sce- +narios based on various observations, Healy et +al. (2021) state that Nevertheless, the question +whether the NGC 4839 group is on its first in- +fall or has already passed through the cluster, +remains open. +1.3. Globular Clusters as a Probe +The halos of massive galaxies in galaxy clus- +ters grow via numerous mergers of less massive +galaxies and host a large number of globular +clusters (GCs). +Thus, GCs are an excellent +probe for investigating the structure of the outer +halos in massive galaxies in the local universe, +and they provide a critical clue for revealing the +assembly history of galaxy halos. +In this study, we present a wide field sur- +vey of GCs covering the NGC 4839 group +and its environment, based on the archival +Subaru/Hypersuprime-Cam (HSC) gr images. +The primary goals of this study are to derive +wide field number density maps of GCs and to +use them to constrain the merger scenarios of +the NGC 4839 group. We adopt the distance to +Coma as 100 Mpc (de Grijs & Bono 2020). +1.4. Previous Studies of NGC 4839 GCs +The main host of the NGC 4839 group is +NGC 4839 (MV += −23.1 mag, vh = 7338 +km s−1), which is an elongated cD galaxy +(Schombert 1988; Ali et al. 2014). +There +are only two previous studies of the GCs in +NGC 4839. Mar´ın-Franch & Aparicio (2002) ap- +plied the surface brightness fluctuation (SBF) +method to estimate indirectly the total number +of GCs in several bright Coma galaxies from +r-band images obtained at the 2.5m Issac New- +ton Telescope. They found that NGC 4839 is +the second-most GC-rich in Coma, following +NGC 4874 in their sample. +Later Jord´an et +al. (2004) presented a F450W(B)/F814W(I) +photometry of GCs in NGC 4839 based on +HST/WFPC2 images, deriving the total num- +ber of GCs to be Ntot(GC) = 3060±850, which +is three times smaller than the value given by +Mar´ın-Franch & Aparicio (2002). These previ- +ous studies either covered only the small field +of NGC 4839 or used only one band, so lit- +tle is known about the GCs in the outskirt +of NGC 4839. Other previous HST surveys of +GCs in Coma covered mainly the main cluster + +Globular Clusters in NGC 4839 +3 +core, and did not cover the NGC 4839 region +(Peng et al. 2011; Madrid et al. 2018). +2. DATA +Utilizing the Subaru/Hyper Suprime-Cam +(HSC) archival gr images from the Subaru Mi- +taka Okayama Kiso Archive system (SMOKA) +(Aihara et al. 2019), Oh et al. (2023) provided +a wide field survey of GCs in the entire Coma +cluster. At the distance of Coma (100 Mpc), one +arcsec (arcmin) corresponds to a linear scale of +484.8 pc (29.1 kpc). Thus GCs at the distance +of Coma appear as point sources in the HSC im- +ages. Oh et al. (2023) obtained photometry of +the point sources in the seven HSC fields cover- +ing the entire Coma cluster, using DAOPHOT +(Stetson 1987). We adopt the AB magnitudes in +the SDSS system. The limiting magnitude with +50% completeness of detection derived from ar- +tificial star experiments is r ≈ 27.1 mag. De- +tailed description of the detection and photom- +etry of the point sources is given in Oh et al. +(2023), of which we used the data for NGC 4839 +and its environment in this study. +We apply +the foreground extinction correction using the +extinction maps for Coma given in Schlegel et +al. (1998); Schlafly & Finkbeiner (2011). +3. RESULTS +3.1. NGC 4839 in Comparison with NGC 4816 +In Figure 1 we show a gray scale map of the +r-band SDSS image of the Coma cluster region +including the NGC 4839 group. +The zoom-in +images (10′ × 10′) of NGC 4839 and NGC 4816 +show that the two galaxies are similar in their +luminosity and size. In the following analysis of +NGC 4839 we chose NGC 4816, a nearby bright +S0 galaxy, as a comparison galaxy. +NGC 4839 and NGC 4816 are at similar pro- +jected distances from the Coma center. +The +projected separation between the two galaxies +in the sky is 21.′8 (0.63 Mpc at the distance of +Coma). Healy et al. (2021) found 15 groups us- +ing the catalog of Coma member galaxies, and +provided the number of members and velocity +dispersion of each group. Groups S11 and S14 +in their study correspond to the NGC 4816 and +NGC 4839 groups, respectively. We used these +group data in the following analysis. +NGC 4816 is the brightest member of the S14 +group (with N(member) = 17 and σv = 521 km +s−1) at R = 49′ in the west of Coma (see Fig. 12 +in Healy et al. (2021)). Similarly, NGC 4839 is +the brightest cD/SA0 member of the NGC 4839 +group (the S11 group with N(member) = 24 +and σv = 462 km s−1) at R = 43′, but in the +south-west of Coma. +Thus both galaxies are very bright, and the +V +magnitude of NGC 4816 is only 0.9 mag +fainter than that of NGC 4839. +While the +NGC 4839 group shows a strong X-ray emission, +the NGC 4816 group shows little detected X- +ray emission even in the recent X-ray images +(Lyskova et al. 2019; Sanders et al. 2020; Mi- +rakhor & Walker 2020; Churazov et al. 2021, +2022). +Table 2 lists the basic parameters of the +NGC 4839 and NGC 4816 groups in compari- +son with the main cluster. We calculated the +virial mass from the velocity dispersion of the +two groups (Healy et al. 2021) using the group +virial mass equation: Mvir/M⊙ = 1.5×106h−1σ3 +v +in Tully (2015) (adopting h = 0.7), as listed +in Table 2: +Mvir += 2.1 × 1014M⊙ for the +NGC 4839 group, and Mvir = 3.0 × 1014M⊙ for +the NGC 4816 group. We also list the mass for +the subhalo 2 corresponding to the NGC 4816 +group (Mvir = 1.3 × 1013M⊙), and the sub- +halo 9 corresponding to the NGC 4839 group +(Mvir = 1.7 × 1013M⊙) derived from the weak +lensing analysis in Okabe et al. (2014). +Ok- +abe et al. (2014) derived the mass within the +truncation radius of each subhalo. The trunca- +tion radius of the subhalo 9 is 98 kpc, which is +much smaller than the virial radius of the typi- +cal galaxy groups (the truncation radius of the +subhalo 2 is not given in Okabe et al. (2014)). + +4 +Oh et al. +Thus weak-lensing masses of the two groups are +significantly smaller than the dynamical masses. +These results show that the NGC 4839 and +NGC 4816 groups have comparable high masses. +This indicates that the NGC 4816 group should +show strong X-ray emission like the NGC 4839 +group, but it is not yet detected in any previous +X-ray observations. Not all galaxy groups are +detected in X-ray observations. About a half +of the nearby galaxy groups show X-ray emis- +sion (Mulchaey 2000). It is not clear why the +NGC 4816 group does not show any strong X- +ray emission, unlike the NGC 4839 group. It +may need a study to investigate this issue fur- +ther. +3.2. CMDs of the GCs +In Figure 2 we plot the color-magnitude dia- +grams (CMDs) of the point sources in the cen- +tral regions (Rgal < 3′ (< 87 kpc)) of NGC 4839 +and NGC 4816 as well as a nearby background +region with the same area as the galaxy region. +We also plot the color histograms of the bright +sources with r0 < 26.5 mag of each region. To +show the net color histograms we display the +contribution of the background sources in the +galaxy regions using the background histogram. +The color histograms of the sources in the +two galaxies clearly show an excess (open his- +tograms) with respect to the background re- +gion (hatched histograms) in the color range of +(0.0 < (g − r)0 < 1.3). In the CMDs the verti- +cal structure seen inside the red box represents +mainly GCs in NGC 4839 and NGC 4816. We +select GC candidates from the entire field using +the color-magnitude criteria marked by the red +box (0.35 < (g − r)0 < 1.0, and 22.5 < r0 < +26.5 mag) in the CMDs for the following anal- +ysis. +3.3. Spatial Distribution of the GCs +In Figure 3 we display the spatial number +density contour map of the selected GC can- +didates in NGC 4839 and its environment. The +region covers out to the virial radius of Coma +(96′ = 2.8 Mpc), and NGC 4839 and NGC 4816 +are located approximately at the half virial ra- +dius. +The strongest peak of the GC number +density is seen at the position of NGC 4874, +which is adopted as the Coma center in this +study. Two other strong peaks in the main clus- +ter core are visible at the position of NGC 4889 +and IC 4051. The main cluster core shows also a +large extended distribution of intracluster GCs, +the details of which are presented in Oh et al. +(2023); Lee et al. (2022). Note that in the south- +west outskirt two more strong peaks are found +at the positions of NGC 4839 and NGC 4816, +similar to those of NGC 4889 and IC 4051 in the +main cluster core. +One striking feature seen in this figure is a +clear difference in the spatial distribution of +GCs between NGC 4839 and NGC 4816: +the +spatial extent of the GC system in NGC 4839 +is very compact and that of the GC system in +NGC 4816 is much more extended, despite both +galaxies showing a similarly strong peak at their +centers. +NGC 4839 shows a weak excess tail +of GCs in the east. There is no corresponding +galaxies around the center of this excess. This +may be due to stripped GCs from NGC 4839. +Sasaki et al. (2016) noted that the center of the +massive subhalo 9 in Okabe et al. (2014) is 1′ +east of NGC 4839 that is located at the X-ray +peak in the XMM-Newton and Suzaku images. +This offset of the subhalo 9 might have also pro- +duced the east tail of the GCs in NGC 4839. +No bright galaxies are found in the outer +region of NGC 4816, which might have con- +tributed to the extended distribution of GCs. +The strong central concentration of the GCs +around NGC 4816 (R < 1200′′) in the radial +number density profiles, as shown in the fol- +lowing section, indicates that a majority of the +GCs in this region are bound to the NGC 4816 +group. In the GC number density map of Fig- +ure 3 there are several weak GC clumps in the + +Globular Clusters in NGC 4839 +5 +outskirts of the NGC 4816 group, some of which +can be due to some non-group member galaxies, +but their contribution to the group GC system +is negligible. +3.4. Radial Number Density Profiles of the +GCs +We derive the radial number density profiles of +the GCs in NGC 4839 and NGC 4816. We esti- +mate the background levels from the surround- +ing regions (at Rgal = 9.2′ for NGC 4839 and +Rgal = 26.4′ for NGC 4816), and subtract them +from the original counts for the galaxy regions. +GC colors such as (g−i) are a useful proxy for +metallicity. The (g−r) color in this study is less +sensitive than the (g − i) color, but is still use- +ful. We divide the GC sample into two subsam- +ples according to their color: blue (metal-poor) +GCs with 0.35 < (g − i) < 0.655, and the red +(metal-rich) GCs with 0.655 < (g − r) < 1.0, as +described in Oh et al. (2023). We derive the ra- +dial number density profiles of the blue GCs and +red GCs in NGC 4839 and NGC ,4816, display- +ing them as well as that of all GCs in Figure 4. +This figure shows that the blue GC system is +slightly more extended than the red GC system +in both NGC 4839 and NGC 4816. +In Figure 5, we compare the radial profiles of +the GC number density and surface brightness +of galaxy light in NGC 4839 and NG 4816 For +comparison with galaxy light, we derive the ra- +dial surface brightness profiles of the two galax- +ies from the HSC r-band images. First we mask +out several bright sources except for the two +galaxies in the images. Then we obtain surface +brightness profiles of the galaxies using annu- +lar aperture photometry, and plot them in the +same figure. +Several interesting features are noted in Fig- +ure 5. First, the radial number density profiles +of the GCs in the two galaxies show a striking +difference in the outer region, while they are +similar in the inner region. The decline in the +central region at Rgal < 20′′ (< 10 kpc) is due +to incompleteness of our photometry, so we use +only the data for the outer region at Rgal > 20′′. +We note only the difference in the outer regions +between the two galaxies. The radial number +density profile of the NGC 4839 GCs shows a +sudden drop at RN4839 ≈ 80 kpc, and few GCs +are found at RN4839 > 100 kpc. On the other +hand, the radial number density profile of the +NGC 4816 GCs shows a slow decline even in the +outer region at RN4816 > 100 kpc, and some +GCs are found even out to RN4816 ≈ 500 kpc. +Second, the surface brightness profiles of the +two galaxies are similar in the inner region at +1 < Rgal < 20 kpc, and show a slight difference +in the outer region at 20 < Rgal < 50 kpc. The +shapes of these profiles are also similar to that +of the GC number density profile of NGC 4816, +but showing a clear difference against that of +the GC number density profile of NGC 4839. +Third, we fit the surface brightness profiles of +the galaxies (3′′ < Rgal < 30′′) with a S´ersic +law for n = 4 (i.e., a de Vaucouleurs law), as +shown by the dot-dashed lines. +The surface +brightness profiles of the galaxy light in the in- +ner regions of the two galaxies are reasonably +fit by the S´ersic law. +The effective radius of +the NGC 4839 galaxy light, Reff,N4839 = 23.′′5 ± +0.′′7 = 11.4 ± 0.3 kpc, is similar to that of the +NGC 4816 galaxy light, Reff,N4816 = 23.′′9±1.′′1 = +11.6 ± 0.5 kpc. The surface brightness profile of +NGC 4839 shows a slight excess over the fitting +line at R > 1′, which is a cD envelope, con- +sistent with the previous results in Schombert +(1988); Ali et al. (2014). On the other hand, this +excess is much weaker in the case of NGC 4816. +Fourth, we fit the radial number density pro- +files of GCs at 50′′ < Rgal < 1260′′ in NGC 4816 +with a S´ersic law for n = 4, as shown by the +dotted line in the figure. +The radial number +density profile of the NGC 4816 GCs is approxi- +mately fit by the S´ersic law. The effective radius +of the NGC 4816 GC system derived from this +fitting, is Reff,GCS = 124±37 kpc. In the case of + +6 +Oh et al. +NGC 4839, the radial number density profile of +the GCs in the inner region (50′′ < Rgal < 200′′) +is roughly fit by the S´ersic law, but the number +density is significantly lower than the fitted line +in the outer region at Rgal > 200′′. We derive +the GC system effective radius of the two galax- +ies, from the cumulative radial distribution of +GCs. We assume that the number density pro- +file is flat in the central region (Rgal < 25′′) +where our data is incomplete (see Lee et al. +(2008) for the radial number density profile of +M60 GCs)). The effective radius of the GC sys- +tem derived from this, is Reff,GCS = 101.′′5 ± 3.′′1 += 49.1 ± 1.5 kpc for NGC 4839. We resample +the radial density profiles from the data 1000 +times, and repeat the same procedure to de- +rive an effective radius from each profile. From +this we obtain a standard deviation of resam- +pled Reff,GCS as a measuring error. Note that +the true error must be larger than this error. +Similarly we obtain Reff,GCS = 331.′′2 ± 10.′′8 = +160.1 ± 5.2 kpc for NGC 4816, which is larger +than, but consistent, within the error, with the +value based on the fitting. Thus the effective ra- +dius of the NGC 4839 GC system is about three +times smaller than that of the NGC 4816 GC +system. +4. DISCUSSION AND CONCLUSION +Coma is an ideal target for investigating not +only the general assembly process of galaxy clus- +ters but also the details of the merging pro- +cess including the infall phase of substructures. +Various substructures related with the merging +process in Coma were discovered in previous X- +ray images (see Briel et al. (1992); White et al. +(1993); Neumann et al. (2001); Sanders et al. +(2020); Mirakhor & Walker (2020); Churazov et +al. (2021) and references therein). Early stud- +ies based on X-ray observations suggested that +the NGC 4839 group is in the first phase of in- +fall (Briel et al. 1992; White et al. 1993). Then +Burns et al. (1994) presented a new scenario, +based on hydro/N-body simulations, that Coma +already had a lunch (the NGC 4839 group) and +the NGC 4839 group is in the second infall, +which can explain the optical, radio, and X- +ray properties of Coma. Later Colless & Dunn +(1996) pointed out the shortcomings of the ar- +guments in Burns et al. (1994), and argued that +NGC 4839 is in the first phase of infall, based on +dynamics of a large number of Coma galaxies. +Most of these substructures could be explained +either in pre-merger scenarios or in post-merger +scenarios, as summarized in Table 1. +Later Lyskova et al. (2019) noted two promi- +nent features seen in the XMM-Newton and +Chandra images of the NGC 4839 group: a long +(600 kpc) bent tail of cool gas of NGC 4839, and +a sheath of enhanced X-ray surface brightness +due to hotter gas in the southwest, and tried +SPH simulations to test both pre-merger and +post-merger scenarios. They concluded that the +post-merger scenario can explain better the ob- +servational results (X-ray brightness and tem- +peratures) than the pre-merger scenario. +Ac- +cording to this scenario (see their Fig. 8), the +NGC 4839 group began falling to the main clus- +ter from the northeast about 2 Gyr ago, passed +the center about 1.6 Gyr ago, and began the +second infall after reaching the apocenter in the +southwest recently. +Recently from the X-ray images obtained +with the SRG/eROSITA, Churazov et al. (2021, +2022) found a faint X-ray bridge connecting the +NGC 4839 group with the main cluster. This +bridge may be a remnant of stripped gas while +NGC 4839 moves outward from the main clus- +ter to the current position, showing that it is +strong evidence that NGC 4839 already passed +the main cluster core (see their Fig. 11). Chura- +zov et al. (2021, 2022) also pointed out that the +existence of the bow shock at R ≈ 33′ (960 kpc) +in the west and the radio relic at R ≈ 2.1 Mpc +in the southwest (Bonafede et al. 2021) may cor- +respond, respectively, to the secondary shock +(produced when crossing the apocenter) and + +Globular Clusters in NGC 4839 +7 +the primary shock (produced when crossing the +main cluster core) caused by the merging event +with NGC 4839. +In Figure 6 we compare the GC number den- +sity map (pseudocolor map) with the XMM- +Newton X-ray contour map of hot gas ob- +tained after β model subtraction (showing sub- +structures better, Neumann et al. (2001, 2003)) +(based on Fig. 3 in Adami et al. (2005)). +In +this figure, the X-ray contours around the NGC +4839 region show a slight offset from the cen- +ter of the NGC 4839 GC clump. This offset is +not seen in the recent X-ray data (Lyskova et +al. 2019; Churazov et al. 2021). This offset is +due to the outdated X-ray data (Neumann et +al. 2003) used in Adami et al. (2005). The X- +ray map shows three prominent substructures: +(a) the NGC 4839 group where a strong con- +centration of GCs is seen only at the position of +NGC 4839, (b) a large arc-like western substruc- +ture where few GCs are found, and (c) a smaller +substructure associated with NGC 4911/4921 in +the southeast where only a small population of +GCs are seen. +Note that the X-ray emission +substructure is seen in the NGC 4839 group, but +not in the NGC 4816 group. +The center of the NGC 4839 group (G2 in +Adami et al. (2005)) was close to NGC 4839 in +the old study by Adami et al. (2005). However, +the recent study based on a much larger sam- +ple of Coma members by Healy et al. (2021) +shows that the center of the NGC 4839 group +(S11) is significantly offset to the southwest +from NGC 4839 (see their Fig. 11). +On the +other hand, the recent SRG/eROSITA X-ray +data with higher spatial resolution (Churazov +et al. 2021) (as well as XMM-Newton data) +shows clearly an X-ray peak at the position of +NGC 4839 which is embedded in a much more +diffuse X-ray emission. This diffuse component +is significantly overlapped with the galaxy dis- +tribution of the S11 group (see Fig. 11 in Healy +et al. (2021)). +In the figure we also add the trajectory (red +dashed line) of the NGC 4839 group suggested +for the second-infall scenario(Lyskova et al. +2019; Churazov et al. 2021) (from Figure 11 in +Churazov et al. (2021)), as well as other known +substructures. The very compact spatial extent +of the GC system in NGC 4839, much smaller +than the GC system in NGC 4816, can be ex- +plained if NGC 4839 lost a significant number of +GCs in the outskirt of NGC 4839 when it passed +the main cluster. +On the other hand, the more extended GC +system in NGC 4816 indicates that it may be in +the first phase of infall, as described below. The +radial velocity of the NGC 4839 group is 768 km +s−1 +larger than that of the main cluster (6853 +km s−1). Colless & Dunn (1996) suggested that +the angle between the observer and the velocity +vector of the NGC 4839 group is about 74 deg +so the merger is happening with ∆v = 1700 km +s−1 almost in the projected sky plane. Which of +the main cluster and NGC 4839 is closer to us is +not yet known. On the other hand, the relative +velocity of the NGC 4816 group with respect to +the main cluster is only +35 km s−1 +and the +NGC 4816 group is located along the large scale +filament connecting with Abell 1367. Consider- +ing these we infer that the NGC 4816 group is +infalling to the cluster center in the sky plane. +In addition, the GC system of the NGC 4816 +shows an extended structure with a continu- +ously declining radial number density profile. +These results indicate that the NGC 4816 is in +its first infall. If it is in its second infall, its ra- +dial profile of the GC system would have shown +a significant drop in the outer region like the +one in the NGC 4839 group. +If NGC 4839 is in the first phase of infall, +it should show a similar distribution to that +of NGC 4816, and it would be difficult to ex- +plain the observed difference between NGC 4839 +and NGC 4816. +When NGC 4839 crosses the + +8 +Oh et al. +main cluster core again, it would lose more GCs, +which will become part of the intracluster GCs. +In conclusion, the spatial distribution of GCs +in NGC 4839 and its environment supports +the second infall scenario where the NGC 4839 +passed the Coma center about 1.6 Gyr ago, and +began the second infall after reaching the apoc- +enter in the southwest. Previous simulations on +GCs in galaxy clusters (e.g., Ramos-Almendares +et al. (2018, 2020)) are useful to understand +the spatial distribution and kinematics of the +GCs in large scales. However none of them pro- +vide any results on how the motion of individual +groups in galaxy clusters affects the size of the +GC systems in individual galaxies, which could +be compared with the results in this study. We +expect that our results motivate future simula- +tions to address this issue. +ACKNOWLEDGMENTS +This work was supported by the National Re- +search Foundation grant funded by the Korean +Government (NRF-2019R1A2C2084019). +We +thank Brian S. Cho for his help in improving +the English in the manuscript. +The authors +are grateful to the anonymous referee for use- +ful comments. +Facilities: Subaru(Hyper Suprime-Cam) + +Globular Clusters in NGC 4839 +9 +REFERENCES +Adami, C., Biviano, A., Durret, F., et al. 2005, +A&A, 443, 17. doi:10.1051/0004-6361:20053504 +Aihara, H., AlSayyad, Y., Ando, M., et al. 2019, +PASJ, 71, 114 +Akamatsu, H., Inoue, S., Sato, T., et al. 2013, +PASJ, 65, 89. doi:10.1093/pasj/65.4.89 +Ali, G. B., Shaban, E. A., Amin, M. 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Merging-related Features for the NGC 4839 Group +Band +Features +Infall Phasea Referenceb +X-ray +NGC 4839 inner tail (SW), SW main tail (sheath) +1,2 +1,2 +SW bridge connecting NGC 4839 and the main cluster 2 +W sharp edge, E contact discontinuity +1,2 +Radio (cont) Coma radio halo, W halo front +2 +3,4 +SW bridge, SW streams, SW relic (R = 2.1 Mpc) +2 +Radio (HI) +HI deficiency and old galaxies in NGC 4839 group +1,2 +5 +Optical +E+A galaxies in SWc +2 +6 +galaxy distribution and kinematics +1,2 +7 +Compact globular cluster system in NGC 4839 +2 +This study +a1 for the first infall phase (a pre-merger scenario), and 2 for the second infall phase (a post-merger +scenario). +b1.Lyskova et al. (2019);2.Churazov et al. (2021, 2022); 3. Kim et al. (1989); 4.Bonafede et al. +(2021, 2022);5.Healy et al. (2021);6.Caldwell et al. (1993); 7.Colless & Dunn (1996). +cColless & Dunn (1996) pointed out that a few of the E+A galaxies that are the members of the +NGC 4839 group, and that these E+A galaxies may be falling recently into Coma like NGC 4839. + +12 +Oh et al. +Table 2. Basic Parameters for the Main Cluster, NGC 4839 and NGC 4816 Group in Coma +Parameter +Main cluster +NGC 4839 group NGC 4816 group Referencea +Heliocentric galaxy velocity, vh +7167 km s−1 +7338 km s−1 +6915 km s−1 +1,2,3 +Heliocentric group velocity, vh +6853 km s−1 +7621 km s−1 +6898 km s−1 +1,2,3 +Velocity dispersion, σv +1082 km s−1 +462 km s−1 +521 km s−1 +2,3 +Virial Mass (dynamics)b, Mvir +2.7 × 1015M⊙ +2.1 × 1014M⊙ +3.0 × 1014M⊙ +4 +Weak Lensing Massc, MWL +1.2 × 1015M⊙ +1.7 × 1013M⊙ +1.3 × 1013M⊙ +5 +a1: NED; 2: Colless & Dunn (1996); 3: Healy et al. (2021); 4: This study; 5: Okabe et al. (2014). +bCalculated for the velocity dispersion (Healy et al. 2021) using the group virial mass equation: +Mvir/M⊙ = 1.5 × 106h−1σ3 +v in Tully (2015). Note that Colless & Dunn (1996) presented Mvir = +1.3×1015M⊙ for the main cluster, and Mvir = 8.6×1012M⊙ for the NGC 4839 group from galaxy +dynamics. +c Projected masses (M2D) within the truncation radius for the subhalo 2 for the NGC 4816 group, +and the subhalo 9 for the NGC 4839 group derived from the weak lensing analysis in Okabe et al. +(2014), given for h = 0.7. + +Globular Clusters in NGC 4839 +13 +Coma +N +E +1 Mpc +NGC 4839 +NGC 4816 +Figure 1. A gray scale map (4◦ × 4◦) of the r-band SDSS image of NGC 4839 and its environment in the +Coma cluster. Zoom-in fields for NGC 4839 and NGC 4816 (red boxes) are 10′ × 10′. + +: +. +. +:.14 +Oh et al. +200 +400 +Number +NGC 4839(r0 < 26.5) +Background +NGC 4816(r0 < 26.5) +Background +Background +0.5 +0.0 +0.5 +1.0 +1.5 +(g +r)0 +21 +22 +23 +24 +25 +26 +27 +28 +r0 PSF mag +NGC 4839 +0.5 +0.0 +0.5 +1.0 +1.5 +(g +r)0 +NGC 4816 +0.5 +0.0 +0.5 +1.0 +1.5 +(g +r)0 +Background +Figure 2. Color-magnitude diagrams (lower panels) and color distributions (upper panels) of the point +sources with 22.5 < r0 < 26.5 mag in the central regions (1′.2 < Rgal < 3′.3) of NGC 4839, NGC 4816, and +the background region with the same area based on the HSC images. The hatched histograms in the upper +panels for the galaxy regions represent the background region. The red boxes in the lower panels represent +the boundary for GC selection. + +Globular Clusters in NGC 4839 +15 +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +R.A. [deg] +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +Dec. [deg] +Rvir +0.5° +1.0° +NGC 4839 +NGC 4816 +NGC 4854 +NGC 4923 +NGC 4874 +NGC 4889 +IC 4051 +NGC 4798 +Coma GCs +ETG (E+S0, Doi+1995) +LTG (Sa+Im, Doi+1995) +E+A galaxies (Caldwell+1993) +50 +60 +70 +80 +90 +100 +110 +120 +130 +140 +150 +160 +170 +180 +Number density [arcmin +2] +Figure 3. +Spatial number density contour map of GCs in the Coma field including NGC 4839 and +NGC 4816 (see Oh et al. (2023) for details). Dotted line circles represent R = 0.5◦, 1.0◦, and Rvir(=2.8 +Mpc) from NGC 4874 at the Coma center. Red circles and green triangles mark early-type galaxy members, +and late-type galaxy members (Doi et al. 1995). Black boxes mark E+A galaxies (Caldwell et al. 1993). +The contour levels denote 2σbg and larger with an interval of one σbg where σbg denotes the background +fluctuation. The contour maps were smoothed using a Gaussian filter with σG = 1′. The color bar represents +the GC number density. + +16 +Oh et al. +102 +103 +Angular distance from galaxy center [arcsec] +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +Log GC number density [arcsec +2] +NGC 4816 +All GC +Blue GC +Red GC +101 +102 +Linear distance [kpc] +102 +103 +Angular distance from galaxy center [arcsec] +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +Log GC number density [arcsec +2] +NGC 4839 +All GC +Blue GC +Red GC +101 +102 +Linear distance [kpc] +Figure 4. +Radial number density profiles of the GCs in NGC 4839 (upper panel) and NGC 4816 (lower +panel): all GCs (black solid line), blue (metal-poor) GCs (blue dashed line), and red (metal-rich) GCs (red +dashed line). + +Globular Clusters in NGC 4839 +17 +100 +101 +102 +103 +Angular distance from galaxy center [arcsec] +14 +16 +18 +20 +22 +24 +26 +r [mag/arcsec2] +NGC 4839 +NGC 4816 +Galaxy +light +GCS +100 +101 +102 +Linear distance [kpc] +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +Log GC number density [arcsec +2] +Figure 5. +Radial profiles for HSC r-band surface brightness (solid lines) and GC number density (dashed +lines) for NGC 4839 (red lines) and NGC 4816 (blue lines). Dot-dashed lines and dotted lines denote the +results of S´ersic law (n = 4) fitting for galaxy light and GC number density profiles, respectively. Thicker +lines denote the fitting ranges. + +18 +Oh et al. +Figure 6. Comparison of the GC number density map (pseudo color map) with XMM-Newton X-ray map +(Neumann et al. 2001, 2003) after β model subtraction (white contours, based on Figure 3 in Adami et al. +(2005)). Dotted black lines mark the direction of neighboring large scale structures. Green, purple, and +yellow lines denote the primary shock, secondary shock, and contact discontinuity, respectively, in Churazov +et al. (2021) (from their Fig. 11). The red dashed line shows the trajectory of the NGC 4839 group suggested +for the second-infall scenario (Lyskova et al. 2019; Churazov et al. 2021). The color bar represents the GC +number density. + +Adami+2005 X-ray map +175 +29.0 +NGC4839 path +Secondary shock +155 +28.5 +135 +A2199 +Dec. [deg] +115 +A779 +28.0 +95 +N4816 +14839 +C.D. +136 +75 +27.5 +55 +IMpc +27.0 +35 +Primary shock +15 +196.0 +195.5 +195.0 +194.5 +194.0 +R.A. [deg] \ No newline at end of file diff --git a/0dE4T4oBgHgl3EQfyw3J/content/tmp_files/load_file.txt b/0dE4T4oBgHgl3EQfyw3J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f3ae58d3f48700e5e111b1d7c10a897e0a7121d --- /dev/null +++ b/0dE4T4oBgHgl3EQfyw3J/content/tmp_files/load_file.txt @@ -0,0 +1,815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf,len=814 +page_content='Draft version January 16, 2023 Typeset using LATEX preprint2 style in AASTeX63 Globular Clusters in NGC 4839 Falling into Coma: Evidence for the Second Infall?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Seong-A Oh,1 Myung Gyoon Lee,1 and In Sung Jang2 1Astronomy Program, Department of Physics and Astronomy, SNUARC, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea 2Department of Astronomy & Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA ABSTRACT NGC 4839 is the brightest galaxy (cD) of the NGC 4839 group at R ≈ 1 Mpc in the south-west of the Coma cluster, which is known to be falling into Coma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' However, it has been controversial whether it is in the first phase of infall or in the second phase of infall after passing the Coma center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We present a wide field study of globular clusters (GCs) in NGC 4839 and its environment based on Hyper Suprime-Cam gr images in the Subaru archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We compare the GC system of NGC 4839 with that of NGC 4816, which is the brightest member (S0) of the nearby group and lies at a similar distance in the west from the Coma center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Interestingly the spatial distribution of the GCs in NGC 4839 is significantly more compact than that of the GCs in NGC 4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In addition, the radial number density profile of the GCs in NGC 4839 shows an abrupt drop at RN4839 ≈ 80 kpc, while that of the GCs in NGC 4816 shows a continuous slow decline even in the outer region at 80 < RN4816 < 500 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The effective radius of the NGC 4839 GC system is about three times smaller than that of the NGC 4816 GC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This striking difference can be explained if NGC 4839 lost a significant fraction of the GCs in its outskirt when it passed through Coma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This supports strongly the second infall scenario where the NGC 4839 passed the Coma center about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 Gyr ago, and began the second infall after reaching the apocenter in the south-west recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The NGC 4839 Group and the Main Cluster in Coma Coma is the most massive galaxy cluster in the local universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' It is connected with fil- aments from neighboring galaxy clusters and hosts various substructures indicating that it is a complex merger system (Colless & Dunn 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Malavasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thus, Coma is one of the best targets to study how large scale substructures are assembled and Corresponding author: Myung Gyoon Lee sao@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='kr,mglee@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='kr evolve, and has been a focus of many cluster studies in various aspects (see Biviano (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021) and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Two most prominent substructures in Coma are the main cluster core in the center and the NGC 4839 group in the south-west, as shown by galaxy number density maps (Colless & Dunn 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021), X-ray images of hot gas (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021), and radio images of synchrotron emission (Bonafede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Lal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The main cluster core hosts two giant galaxies (NGC 4874 (cD) and NGC 4889 (D)), which are merging now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The NGC 4839 group is at R ≈ 1 Mpc in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='05269v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='GA] 12 Jan 2023 2 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' the south-west of Coma, and it is much smaller and less massive than the main cluster core (Colless & Dunn 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The NGC 4839 group is considered to be falling into Coma and that the two systems will merge to form a more massive system in the future (Bi- viano (1998) and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Merger Scenarios for the NGC 4839 Group: A Pre-merger or a Post-merger?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' It is generally accepted that the NGC 4839 group is merging with the main cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' How- ever, whether it is a pre-merger where the NGC 4839 group is in the first phase of infall (Briel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Colless & Dunn 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Akamatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2013) or a post-merger (Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021) has been controversial (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We summarize the observational features re- lated with the merging of the NGC 4839 group in the previous studies in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' These fea- tures include several substructures seen in X- ray and radio images, an excess of E+A galax- ies in the SW region of the cluster, and sub- structures found in the spatial distribution and kinematics of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Each feature can be ex- plained with either the pre-merger scenario or the post-merger scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Recently the post- merger scenario, which can better explain the existence of X-ray/radio substructures (in par- ticular, bridges and streams), appears to be more supported (Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Bonafede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' How- ever, even in the recent discussions of both sce- narios based on various observations, Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021) state that Nevertheless, the question whether the NGC 4839 group is on its first in- fall or has already passed through the cluster, remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Globular Clusters as a Probe The halos of massive galaxies in galaxy clus- ters grow via numerous mergers of less massive galaxies and host a large number of globular clusters (GCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thus, GCs are an excellent probe for investigating the structure of the outer halos in massive galaxies in the local universe, and they provide a critical clue for revealing the assembly history of galaxy halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In this study, we present a wide field sur- vey of GCs covering the NGC 4839 group and its environment, based on the archival Subaru/Hypersuprime-Cam (HSC) gr images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The primary goals of this study are to derive wide field number density maps of GCs and to use them to constrain the merger scenarios of the NGC 4839 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We adopt the distance to Coma as 100 Mpc (de Grijs & Bono 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Previous Studies of NGC 4839 GCs The main host of the NGC 4839 group is NGC 4839 (MV = −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 mag, vh = 7338 km s−1), which is an elongated cD galaxy (Schombert 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' There are only two previous studies of the GCs in NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Mar´ın-Franch & Aparicio (2002) ap- plied the surface brightness fluctuation (SBF) method to estimate indirectly the total number of GCs in several bright Coma galaxies from r-band images obtained at the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5m Issac New- ton Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' They found that NGC 4839 is the second-most GC-rich in Coma, following NGC 4874 in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Later Jord´an et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2004) presented a F450W(B)/F814W(I) photometry of GCs in NGC 4839 based on HST/WFPC2 images, deriving the total num- ber of GCs to be Ntot(GC) = 3060±850, which is three times smaller than the value given by Mar´ın-Franch & Aparicio (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' These previ- ous studies either covered only the small field of NGC 4839 or used only one band, so lit- tle is known about the GCs in the outskirt of NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Other previous HST surveys of GCs in Coma covered mainly the main cluster Globular Clusters in NGC 4839 3 core, and did not cover the NGC 4839 region (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Madrid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' DATA Utilizing the Subaru/Hyper Suprime-Cam (HSC) archival gr images from the Subaru Mi- taka Okayama Kiso Archive system (SMOKA) (Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019), Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2023) provided a wide field survey of GCs in the entire Coma cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' At the distance of Coma (100 Mpc), one arcsec (arcmin) corresponds to a linear scale of 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='8 pc (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thus GCs at the distance of Coma appear as point sources in the HSC im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2023) obtained photometry of the point sources in the seven HSC fields cover- ing the entire Coma cluster, using DAOPHOT (Stetson 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We adopt the AB magnitudes in the SDSS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The limiting magnitude with 50% completeness of detection derived from ar- tificial star experiments is r ≈ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' De- tailed description of the detection and photom- etry of the point sources is given in Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2023), of which we used the data for NGC 4839 and its environment in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We apply the foreground extinction correction using the extinction maps for Coma given in Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' NGC 4839 in Comparison with NGC 4816 In Figure 1 we show a gray scale map of the r-band SDSS image of the Coma cluster region including the NGC 4839 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The zoom-in images (10′ × 10′) of NGC 4839 and NGC 4816 show that the two galaxies are similar in their luminosity and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In the following analysis of NGC 4839 we chose NGC 4816, a nearby bright S0 galaxy, as a comparison galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' NGC 4839 and NGC 4816 are at similar pro- jected distances from the Coma center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The projected separation between the two galaxies in the sky is 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='63 Mpc at the distance of Coma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021) found 15 groups us- ing the catalog of Coma member galaxies, and provided the number of members and velocity dispersion of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Groups S11 and S14 in their study correspond to the NGC 4816 and NGC 4839 groups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We used these group data in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' NGC 4816 is the brightest member of the S14 group (with N(member) = 17 and σv = 521 km s−1) at R = 49′ in the west of Coma (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 12 in Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Similarly, NGC 4839 is the brightest cD/SA0 member of the NGC 4839 group (the S11 group with N(member) = 24 and σv = 462 km s−1) at R = 43′, but in the south-west of Coma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thus both galaxies are very bright, and the V magnitude of NGC 4816 is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='9 mag fainter than that of NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' While the NGC 4839 group shows a strong X-ray emission, the NGC 4816 group shows little detected X- ray emission even in the recent X-ray images (Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Mi- rakhor & Walker 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Table 2 lists the basic parameters of the NGC 4839 and NGC 4816 groups in compari- son with the main cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We calculated the virial mass from the velocity dispersion of the two groups (Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021) using the group virial mass equation: Mvir/M⊙ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5×106h−1σ3 v in Tully (2015) (adopting h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='7), as listed in Table 2: Mvir = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 × 1014M⊙ for the NGC 4839 group, and Mvir = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 × 1014M⊙ for the NGC 4816 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We also list the mass for the subhalo 2 corresponding to the NGC 4816 group (Mvir = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3 × 1013M⊙), and the sub- halo 9 corresponding to the NGC 4839 group (Mvir = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='7 × 1013M⊙) derived from the weak lensing analysis in Okabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Ok- abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014) derived the mass within the truncation radius of each subhalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The trunca- tion radius of the subhalo 9 is 98 kpc, which is much smaller than the virial radius of the typi- cal galaxy groups (the truncation radius of the subhalo 2 is not given in Okabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 4 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thus weak-lensing masses of the two groups are significantly smaller than the dynamical masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' These results show that the NGC 4839 and NGC 4816 groups have comparable high masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This indicates that the NGC 4816 group should show strong X-ray emission like the NGC 4839 group, but it is not yet detected in any previous X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Not all galaxy groups are detected in X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' About a half of the nearby galaxy groups show X-ray emis- sion (Mulchaey 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' It is not clear why the NGC 4816 group does not show any strong X- ray emission, unlike the NGC 4839 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' It may need a study to investigate this issue fur- ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' CMDs of the GCs In Figure 2 we plot the color-magnitude dia- grams (CMDs) of the point sources in the cen- tral regions (Rgal < 3′ (< 87 kpc)) of NGC 4839 and NGC 4816 as well as a nearby background region with the same area as the galaxy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We also plot the color histograms of the bright sources with r0 < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 mag of each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' To show the net color histograms we display the contribution of the background sources in the galaxy regions using the background histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The color histograms of the sources in the two galaxies clearly show an excess (open his- tograms) with respect to the background re- gion (hatched histograms) in the color range of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 < (g − r)0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In the CMDs the verti- cal structure seen inside the red box represents mainly GCs in NGC 4839 and NGC 4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We select GC candidates from the entire field using the color-magnitude criteria marked by the red box (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='35 < (g − r)0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0, and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 < r0 < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 mag) in the CMDs for the following anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Spatial Distribution of the GCs In Figure 3 we display the spatial number density contour map of the selected GC can- didates in NGC 4839 and its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The region covers out to the virial radius of Coma (96′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='8 Mpc), and NGC 4839 and NGC 4816 are located approximately at the half virial ra- dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The strongest peak of the GC number density is seen at the position of NGC 4874, which is adopted as the Coma center in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Two other strong peaks in the main clus- ter core are visible at the position of NGC 4889 and IC 4051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The main cluster core shows also a large extended distribution of intracluster GCs, the details of which are presented in Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Note that in the south- west outskirt two more strong peaks are found at the positions of NGC 4839 and NGC 4816, similar to those of NGC 4889 and IC 4051 in the main cluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' One striking feature seen in this figure is a clear difference in the spatial distribution of GCs between NGC 4839 and NGC 4816: the spatial extent of the GC system in NGC 4839 is very compact and that of the GC system in NGC 4816 is much more extended, despite both galaxies showing a similarly strong peak at their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' NGC 4839 shows a weak excess tail of GCs in the east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' There is no corresponding galaxies around the center of this excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This may be due to stripped GCs from NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2016) noted that the center of the massive subhalo 9 in Okabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014) is 1′ east of NGC 4839 that is located at the X-ray peak in the XMM-Newton and Suzaku images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This offset of the subhalo 9 might have also pro- duced the east tail of the GCs in NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' No bright galaxies are found in the outer region of NGC 4816, which might have con- tributed to the extended distribution of GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The strong central concentration of the GCs around NGC 4816 (R < 1200′′) in the radial number density profiles, as shown in the fol- lowing section, indicates that a majority of the GCs in this region are bound to the NGC 4816 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In the GC number density map of Fig- ure 3 there are several weak GC clumps in the Globular Clusters in NGC 4839 5 outskirts of the NGC 4816 group, some of which can be due to some non-group member galaxies, but their contribution to the group GC system is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Radial Number Density Profiles of the GCs We derive the radial number density profiles of the GCs in NGC 4839 and NGC 4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We esti- mate the background levels from the surround- ing regions (at Rgal = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2′ for NGC 4839 and Rgal = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4′ for NGC 4816), and subtract them from the original counts for the galaxy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' GC colors such as (g−i) are a useful proxy for metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The (g−r) color in this study is less sensitive than the (g − i) color, but is still use- ful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We divide the GC sample into two subsam- ples according to their color: blue (metal-poor) GCs with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='35 < (g − i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='655, and the red (metal-rich) GCs with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='655 < (g − r) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0, as described in Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We derive the ra- dial number density profiles of the blue GCs and red GCs in NGC 4839 and NGC ,4816, display- ing them as well as that of all GCs in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This figure shows that the blue GC system is slightly more extended than the red GC system in both NGC 4839 and NGC 4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In Figure 5, we compare the radial profiles of the GC number density and surface brightness of galaxy light in NGC 4839 and NG 4816 For comparison with galaxy light, we derive the ra- dial surface brightness profiles of the two galax- ies from the HSC r-band images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' First we mask out several bright sources except for the two galaxies in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Then we obtain surface brightness profiles of the galaxies using annu- lar aperture photometry, and plot them in the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Several interesting features are noted in Fig- ure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' First, the radial number density profiles of the GCs in the two galaxies show a striking difference in the outer region, while they are similar in the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The decline in the central region at Rgal < 20′′ (< 10 kpc) is due to incompleteness of our photometry, so we use only the data for the outer region at Rgal > 20′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We note only the difference in the outer regions between the two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The radial number density profile of the NGC 4839 GCs shows a sudden drop at RN4839 ≈ 80 kpc, and few GCs are found at RN4839 > 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' On the other hand, the radial number density profile of the NGC 4816 GCs shows a slow decline even in the outer region at RN4816 > 100 kpc, and some GCs are found even out to RN4816 ≈ 500 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Second, the surface brightness profiles of the two galaxies are similar in the inner region at 1 < Rgal < 20 kpc, and show a slight difference in the outer region at 20 < Rgal < 50 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The shapes of these profiles are also similar to that of the GC number density profile of NGC 4816, but showing a clear difference against that of the GC number density profile of NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Third, we fit the surface brightness profiles of the galaxies (3′′ < Rgal < 30′′) with a S´ersic law for n = 4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=', a de Vaucouleurs law), as shown by the dot-dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The surface brightness profiles of the galaxy light in the in- ner regions of the two galaxies are reasonably fit by the S´ersic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The effective radius of the NGC 4839 galaxy light, Reff,N4839 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′7 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3 kpc, is similar to that of the NGC 4816 galaxy light, Reff,N4816 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The surface brightness profile of NGC 4839 shows a slight excess over the fitting line at R > 1′, which is a cD envelope, con- sistent with the previous results in Schombert (1988);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' On the other hand, this excess is much weaker in the case of NGC 4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Fourth, we fit the radial number density pro- files of GCs at 50′′ < Rgal < 1260′′ in NGC 4816 with a S´ersic law for n = 4, as shown by the dotted line in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The radial number density profile of the NGC 4816 GCs is approxi- mately fit by the S´ersic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The effective radius of the NGC 4816 GC system derived from this fitting, is Reff,GCS = 124±37 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In the case of 6 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' NGC 4839, the radial number density profile of the GCs in the inner region (50′′ < Rgal < 200′′) is roughly fit by the S´ersic law, but the number density is significantly lower than the fitted line in the outer region at Rgal > 200′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We derive the GC system effective radius of the two galax- ies, from the cumulative radial distribution of GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We assume that the number density pro- file is flat in the central region (Rgal < 25′′) where our data is incomplete (see Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2008) for the radial number density profile of M60 GCs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The effective radius of the GC sys- tem derived from this, is Reff,GCS = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′1 = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 kpc for NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We resample the radial density profiles from the data 1000 times, and repeat the same procedure to de- rive an effective radius from each profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' From this we obtain a standard deviation of resam- pled Reff,GCS as a measuring error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Note that the true error must be larger than this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Similarly we obtain Reff,GCS = 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′2 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='′′8 = 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 kpc for NGC 4816, which is larger than, but consistent, within the error, with the value based on the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thus the effective ra- dius of the NGC 4839 GC system is about three times smaller than that of the NGC 4816 GC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION Coma is an ideal target for investigating not only the general assembly process of galaxy clus- ters but also the details of the merging pro- cess including the infall phase of substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Various substructures related with the merging process in Coma were discovered in previous X- ray images (see Briel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Mirakhor & Walker (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021) and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Early stud- ies based on X-ray observations suggested that the NGC 4839 group is in the first phase of in- fall (Briel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Then Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1994) presented a new scenario, based on hydro/N-body simulations, that Coma already had a lunch (the NGC 4839 group) and the NGC 4839 group is in the second infall, which can explain the optical, radio, and X- ray properties of Coma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Later Colless & Dunn (1996) pointed out the shortcomings of the ar- guments in Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1994), and argued that NGC 4839 is in the first phase of infall, based on dynamics of a large number of Coma galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Most of these substructures could be explained either in pre-merger scenarios or in post-merger scenarios, as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Later Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2019) noted two promi- nent features seen in the XMM-Newton and Chandra images of the NGC 4839 group: a long (600 kpc) bent tail of cool gas of NGC 4839, and a sheath of enhanced X-ray surface brightness due to hotter gas in the southwest, and tried SPH simulations to test both pre-merger and post-merger scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' They concluded that the post-merger scenario can explain better the ob- servational results (X-ray brightness and tem- peratures) than the pre-merger scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Ac- cording to this scenario (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 8), the NGC 4839 group began falling to the main clus- ter from the northeast about 2 Gyr ago, passed the center about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 Gyr ago, and began the second infall after reaching the apocenter in the southwest recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Recently from the X-ray images obtained with the SRG/eROSITA, Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021, 2022) found a faint X-ray bridge connecting the NGC 4839 group with the main cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This bridge may be a remnant of stripped gas while NGC 4839 moves outward from the main clus- ter to the current position, showing that it is strong evidence that NGC 4839 already passed the main cluster core (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Chura- zov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021, 2022) also pointed out that the existence of the bow shock at R ≈ 33′ (960 kpc) in the west and the radio relic at R ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 Mpc in the southwest (Bonafede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021) may cor- respond, respectively, to the secondary shock (produced when crossing the apocenter) and Globular Clusters in NGC 4839 7 the primary shock (produced when crossing the main cluster core) caused by the merging event with NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In Figure 6 we compare the GC number den- sity map (pseudocolor map) with the XMM- Newton X-ray contour map of hot gas ob- tained after β model subtraction (showing sub- structures better, Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2001, 2003)) (based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3 in Adami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In this figure, the X-ray contours around the NGC 4839 region show a slight offset from the cen- ter of the NGC 4839 GC clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This offset is not seen in the recent X-ray data (Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This offset is due to the outdated X-ray data (Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2003) used in Adami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The X- ray map shows three prominent substructures: (a) the NGC 4839 group where a strong con- centration of GCs is seen only at the position of NGC 4839, (b) a large arc-like western substruc- ture where few GCs are found, and (c) a smaller substructure associated with NGC 4911/4921 in the southeast where only a small population of GCs are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Note that the X-ray emission substructure is seen in the NGC 4839 group, but not in the NGC 4816 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The center of the NGC 4839 group (G2 in Adami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2005)) was close to NGC 4839 in the old study by Adami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' However, the recent study based on a much larger sam- ple of Coma members by Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021) shows that the center of the NGC 4839 group (S11) is significantly offset to the southwest from NGC 4839 (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' On the other hand, the recent SRG/eROSITA X-ray data with higher spatial resolution (Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021) (as well as XMM-Newton data) shows clearly an X-ray peak at the position of NGC 4839 which is embedded in a much more diffuse X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' This diffuse component is significantly overlapped with the galaxy dis- tribution of the S11 group (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 11 in Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In the figure we also add the trajectory (red dashed line) of the NGC 4839 group suggested for the second-infall scenario(Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021) (from Figure 11 in Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021)), as well as other known substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The very compact spatial extent of the GC system in NGC 4839, much smaller than the GC system in NGC 4816, can be ex- plained if NGC 4839 lost a significant number of GCs in the outskirt of NGC 4839 when it passed the main cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' On the other hand, the more extended GC system in NGC 4816 indicates that it may be in the first phase of infall, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The radial velocity of the NGC 4839 group is 768 km s−1 larger than that of the main cluster (6853 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Colless & Dunn (1996) suggested that the angle between the observer and the velocity vector of the NGC 4839 group is about 74 deg so the merger is happening with ∆v = 1700 km s−1 almost in the projected sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Which of the main cluster and NGC 4839 is closer to us is not yet known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' On the other hand, the relative velocity of the NGC 4816 group with respect to the main cluster is only +35 km s−1 and the NGC 4816 group is located along the large scale filament connecting with Abell 1367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Consider- ing these we infer that the NGC 4816 group is infalling to the cluster center in the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In addition, the GC system of the NGC 4816 shows an extended structure with a continu- ously declining radial number density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' These results indicate that the NGC 4816 is in its first infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' If it is in its second infall, its ra- dial profile of the GC system would have shown a significant drop in the outer region like the one in the NGC 4839 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' If NGC 4839 is in the first phase of infall, it should show a similar distribution to that of NGC 4816, and it would be difficult to ex- plain the observed difference between NGC 4839 and NGC 4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' When NGC 4839 crosses the 8 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' main cluster core again, it would lose more GCs, which will become part of the intracluster GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' In conclusion, the spatial distribution of GCs in NGC 4839 and its environment supports the second infall scenario where the NGC 4839 passed the Coma center about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 Gyr ago, and began the second infall after reaching the apoc- enter in the southwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Previous simulations on GCs in galaxy clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=', Ramos-Almendares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2018, 2020)) are useful to understand the spatial distribution and kinematics of the GCs in large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' However none of them pro- vide any results on how the motion of individual groups in galaxy clusters affects the size of the GC systems in individual galaxies, which could be compared with the results in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We expect that our results motivate future simula- tions to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the National Re- search Foundation grant funded by the Korean Government (NRF-2019R1A2C2084019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' We thank Brian S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Cho for his help in improving the English in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The authors are grateful to the anonymous referee for use- ful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Facilities: Subaru(Hyper Suprime-Cam) Globular Clusters in 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='L8 Globular Clusters in NGC 4839 11 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Merging-related Features for the NGC 4839 Group Band Features Infall Phasea Referenceb X-ray NGC 4839 inner tail (SW), SW main tail (sheath) 1,2 1,2 SW bridge connecting NGC 4839 and the main cluster 2 W sharp edge, E contact discontinuity 1,2 Radio (cont) Coma radio halo, W halo front 2 3,4 SW bridge, SW streams, SW relic (R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 Mpc) 2 Radio (HI) HI deficiency and old galaxies in NGC 4839 group 1,2 5 Optical E+A galaxies in SWc 2 6 galaxy distribution and kinematics 1,2 7 Compact globular cluster system in NGC 4839 2 This study a1 for the first infall phase (a pre-merger scenario), and 2 for the second infall phase (a post-merger scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021, 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='Bonafede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021, 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='Caldwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='Colless & Dunn (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' cColless & Dunn (1996) pointed out that a few of the E+A galaxies that are the members of the NGC 4839 group, and that these E+A galaxies may be falling recently into Coma like NGC 4839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 12 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Basic Parameters for the Main Cluster, NGC 4839 and NGC 4816 Group in Coma Parameter Main cluster NGC 4839 group NGC 4816 group Referencea Heliocentric galaxy velocity, vh 7167 km s−1 7338 km s−1 6915 km s−1 1,2,3 Heliocentric group velocity, vh 6853 km s−1 7621 km s−1 6898 km s−1 1,2,3 Velocity dispersion, σv 1082 km s−1 462 km s−1 521 km s−1 2,3 Virial Mass (dynamics)b, Mvir 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='7 × 1015M⊙ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='1 × 1014M⊙ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 × 1014M⊙ 4 Weak Lensing Massc, MWL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 × 1015M⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='7 × 1013M⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3 × 1013M⊙ 5 a1: NED;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2: Colless & Dunn (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 3: Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 4: This study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 5: Okabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' bCalculated for the velocity dispersion (Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021) using the group virial mass equation: Mvir/M⊙ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 × 106h−1σ3 v in Tully (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Note that Colless & Dunn (1996) presented Mvir = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3×1015M⊙ for the main cluster, and Mvir = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6×1012M⊙ for the NGC 4839 group from galaxy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' c Projected masses (M2D) within the truncation radius for the subhalo 2 for the NGC 4816 group, and the subhalo 9 for the NGC 4839 group derived from the weak lensing analysis in Okabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2014), given for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Globular Clusters in NGC 4839 13 Coma N E 1 Mpc NGC 4839 NGC 4816 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' A gray scale map (4◦ × 4◦) of the r-band SDSS image of NGC 4839 and its environment in the Coma cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Zoom-in fields for NGC 4839 and NGC 4816 (red boxes) are 10′ × 10′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='14 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 200 400 Number NGC 4839(r0 < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5) Background NGC 4816(r0 < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5) Background Background 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 (g r)0 21 22 23 24 25 26 27 28 r0 PSF mag NGC 4839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 (g r)0 NGC 4816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 (g r)0 Background Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Color-magnitude diagrams (lower panels) and color distributions (upper panels) of the point sources with 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 < r0 < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 mag in the central regions (1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 < Rgal < 3′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='3) of NGC 4839, NGC 4816, and the background region with the same area based on the HSC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The hatched histograms in the upper panels for the galaxy regions represent the background region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The red boxes in the lower panels represent the boundary for GC selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Globular Clusters in NGC 4839 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' [deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='4 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' [deg] Rvir 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0° NGC 4839 NGC 4816 NGC 4854 NGC 4923 NGC 4874 NGC 4889 IC 4051 NGC 4798 Coma GCs ETG (E+S0, Doi+1995) LTG (Sa+Im, Doi+1995) E+A galaxies (Caldwell+1993) 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Number density [arcmin 2] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Spatial number density contour map of GCs in the Coma field including NGC 4839 and NGC 4816 (see Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2023) for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Dotted line circles represent R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5◦, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0◦, and Rvir(=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='8 Mpc) from NGC 4874 at the Coma center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Red circles and green triangles mark early-type galaxy members, and late-type galaxy members (Doi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Black boxes mark E+A galaxies (Caldwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The contour levels denote 2σbg and larger with an interval of one σbg where σbg denotes the background fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The contour maps were smoothed using a Gaussian filter with σG = 1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The color bar represents the GC number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 16 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 102 103 Angular distance from galaxy center [arcsec] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 Log GC number density [arcsec 2] NGC 4816 All GC Blue GC Red GC 101 102 Linear distance [kpc] 102 103 Angular distance from galaxy center [arcsec] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 Log GC number density [arcsec 2] NGC 4839 All GC Blue GC Red GC 101 102 Linear distance [kpc] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Radial number density profiles of the GCs in NGC 4839 (upper panel) and NGC 4816 (lower panel): all GCs (black solid line), blue (metal-poor) GCs (blue dashed line), and red (metal-rich) GCs (red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Globular Clusters in NGC 4839 17 100 101 102 103 Angular distance from galaxy center [arcsec] 14 16 18 20 22 24 26 r [mag/arcsec2] NGC 4839 NGC 4816 Galaxy light GCS 100 101 102 Linear distance [kpc] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 Log GC number density [arcsec 2] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Radial profiles for HSC r-band surface brightness (solid lines) and GC number density (dashed lines) for NGC 4839 (red lines) and NGC 4816 (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Dot-dashed lines and dotted lines denote the results of S´ersic law (n = 4) fitting for galaxy light and GC number density profiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Thicker lines denote the fitting ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 18 Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Comparison of the GC number density map (pseudo color map) with XMM-Newton X-ray map (Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2001, 2003) after β model subtraction (white contours, based on Figure 3 in Adami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Dotted black lines mark the direction of neighboring large scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Green, purple, and yellow lines denote the primary shock, secondary shock, and contact discontinuity, respectively, in Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' (2021) (from their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The red dashed line shows the trajectory of the NGC 4839 group suggested for the second-infall scenario (Lyskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' The color bar represents the GC number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' Adami+2005 X-ray map 175 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 NGC4839 path Secondary shock 155 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 135 A2199 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' [deg] 115 A779 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 95 N4816 14839 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content=' 136 75 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='5 55 IMpc 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} +page_content='0 35 Primary shock 15 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfyw3J/content/2301.05269v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..b870fd6866832bd72c9ab2a37c7a11b8ec979603 --- /dev/null +++ b/1NAzT4oBgHgl3EQfevyy/content/tmp_files/2301.01442v1.pdf.txt @@ -0,0 +1,1749 @@ +Efficient Quantum Simulation of Electron-Phonon Systems by Variational Basis State +Encoder +Weitang Li,1 Jiajun Ren,2 Sainan Huai,1 Tianqi Cai,1 Zhigang Shuai,3, 4, ∗ and Shengyu Zhang1, † +1Tencent Quantum Lab, Tencent, Shenzhen, China +2College of Chemistry, Beijing Normal Univerisity, Beijing, China +3Department of Chemistry, Tsinghua University, Beijing, China +4School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China +(Dated: January 5, 2023) +Digital quantum simulation of electron-phonon systems requires truncating infinite phonon levels +into N basis states and then encoding them with qubit computational basis. Unary encoding and +the more compact binary/Gray encoding are the two most representative encoding schemes, which +demand O(N) and O(log N) qubits as well as O(N) and O(N log N) quantum gates respectively. +In this work, we propose a variational basis state encoding algorithm that reduces the scaling of the +number of qubits and quantum gates to both O(1). The cost for the scaling reduction is a constant +amount of additional measurement. The accuracy and efficiency of the approach are verified by both +numerical simulation and realistic quantum hardware experiments. In particular, we find using 1 +or 2 qubits for each phonon mode is sufficient to produce quantitatively correct results across weak +and strong coupling regimes. +Our approach paves the way for practical quantum simulation of +electron-phonon systems on both near-term hardware and error-corrected quantum computers. +Introduction +Electron-phonon couplings are per- +vasive in quantum materials, governing phenomena such +as charge transport in semiconductors [1], vibrational +spectra [2], polaron formation [3], and superconductiv- +ity [4]. +Classically, expensive numerical methods such +as density matrix renormalization group (DMRG) and +quantum Monte-Carlo (QMC) are required to accurately +simulate electron-phonon systems due to the interior +many-body interaction [5–9]. Quantum computers hold +promise for the simulation of quantum systems with ex- +ponential speedup over classical computers [10]. In the +wake of the tremendous progress in the implementation of +quantum computers [11, 12] and the dawning of the noisy +intermediate-scale quantum (NISQ) era [13], how to solve +electron-phonon coupling problems with quantum com- +puters has attracted a lot of research interest [14–17]. +A prominent problem for the digital quantum simu- +lation of electron-phonon systems is how to encode the +infinite phonon states with finite quantum computational +basis states. The first step is usually truncating the in- +finite phonon states into N basis states {|m⟩} and then +the second step is encoding {|m⟩} into quantum compu- +tational basis {|n⟩}. The phonon basis states are usu- +ally the N lowest harmonic oscillator eigenstates or N +uniformly distributed grid basis states. There are two +established strategies to perform the encoding {|m⟩} �→ +{|n⟩} [18, 19]. The first is unary encoding [20, 21], in +which each |m⟩ is encoded to |00 . . . 1m . . . 00⟩, and the +total number of qubits required scales as O(N). The sec- +ond is binary encoding, in which each |m⟩ is encoded to +� +i |⌊ m +2i ⌋ mod 2⟩ represented by O(log N) qubits [14, 15]. +In terms of two-qubit gates required to simulate quantum +operators such as ˆb† ± ˆb and ˆb†ˆb, unary encoding scales +as O(N) and binary encoding scales as O(N log N) [19]. +The features of unary encoding and binary encoding are +summarized in Table 1. Compared to the simulation of +electrons, the simulation of phonons consumes quantum +resources in a much faster manner, which becomes the +bottleneck for efficient quantum simulation of electron- +phonon systems. +TABLE I. Comparison of traditional encoding schemes and +the proposed variational encoding in terms of encoding for- +mula, the number of qubits Nqubit required and the number +of quantum gates Ngate required to simulate common phonon +operators such as ˆb† ± ˆb and ˆb†ˆb . +Scheme +Formula +Nqubit +Ngate +Unary +|m⟩ �→ |00 . . . 1m . . . 00⟩ +O(N) +O(N) +Binary +|m⟩ �→ � +i |⌊ m +2i ⌋ mod 2⟩ O(log N) O(N log N) +Variational +� +m Cmn |m⟩ �→ |n⟩ +O(1) +O(1) +In this work, we propose a new basis encoding scheme +called variational encoding. Variational encoding maps +linear combinations of |m⟩ that are most entangled to +the simulated system into the computational basis, i.e. +� +m Cmn |m⟩ �→ |n⟩, where Cmn is determined by varia- +tional principle. The advantage of our approach is that, +by encoding only the most entangled states and discard- +ing the ones with little entanglement, the size of {|n⟩} +can be made irrelevant to the size of {|m⟩}. +In other +words, the number of qubits required scales as O(1). +Consequently, the scaling for the number of gates is also +O(1). +Variational encoding is best suited to work in +combination with variational quantum algorithms such +as variational quantum eigensolver (VQE) [22, 23] and +variational quantum dynamics (VQD) [24, 25]. Besides, +the variational encoding is also compatible with Trot- +terized time evolution and quantum phase estimation +(QPE) [10, 26, 27]. +Numerical simulation and experi- +ments on realistic quantum hardware based on the Hol- +arXiv:2301.01442v1 [quant-ph] 4 Jan 2023 + +2 +stein model and spin-boson model shows that using 1 +or 2 qubits for each phonon mode is typically sufficient +for highly accurate results even in the strong coupling +regime. +Variational Basis State Encoder +Encoder coeffi- +cients C are determined by variational principle for both +static and dynamic cases. We start the derivation us- +ing parameterized quantum circuit (PQC) and discuss +how to incorporate the variational encoder in Trotterized +time evolution and QPE later on. We use atomic units +throughout the paper. +More details for the derivation +can be found in the Appendix. +For each phonon mode l, encoded by Nl qubits, define +the variational encoder ˆB[l] +ˆB[l] = +� +m +2Nl +� +n=1 +C[l]mn |n⟩l ⟨m| +l +, +(1) +with orthonormal constraint ˆB[l] ˆB[l]† = ˆI. The original +Hamiltonian in |m⟩ basis can then be encoded to |n⟩ basis +ˆ˜H = � +l ˆB[l] ˆH � +l ˆB[l]†. +Suppose the quantum circuit +is parameterized by |φ⟩ = � +k eiθk ˆ +Rk |φ0⟩, and then the +ground state Lagrangian with multipliers λlnn′ is +L = ⟨φ| ˆ˜H|φ⟩ + +� +lnn′ +λlnn′( +� +m +C[l]mnC[l]mn′ − δnn′) . (2) +Taking the derivative with respect to θk leads to tradi- +tional VQE with encoded Hamiltonian ˆ˜H. Taking the +derivative with respect to C[l] and setting it to 0 yields +(1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ = 0 , +(3) +with projector ˆP[l] = +ˆB[l]† ˆB[l] and the half encoded +Hamiltonian ˆ˜H′[l] = � +k̸=l ˆB[k] ˆH � +k ˆB[k]†. In practice, +θk and C[l] are solved iteratively until convergence. In +the following, this iteration is termed macro-iteration to +avoid confusion with VQE iteration. +Next, we discuss the measurement required to solve +C[l] from Eq. 3. Suppose the Hamiltonian can be written +as ˆH = �M +x ˆhx and ˆhx = � +k ˆh[k]x, where M is the total +number of terms in the Hamiltonian and ˆh[k]x acts on the +kth degree of freedom. The measurement of ⟨φ| ˆ˜H′[l]|φ⟩ +boils down to that of ⟨φ|n⟩l ⟨n′| +l +� +k̸=l +ˆ˜h[k]x |φ⟩, where +ˆ˜h[k]x = ˆB[k]ˆh[k]x ˆB[k]† is the encoded local operator. For +electron degree of freedom a dummy encoder ˆB[k] = ˆI is +used for notational simplicity. The number of additional +measurements for the update of C[l] is thus O +� +2NlM +� +, +which is polynomial to the system size and does not in- +crease with N. If the number of phonon modes is as- +sumed to be linear with M and each C[l] is updated +independently, then the total number of measurements +for all C[l] is O +� +2NlM 2� +. +The measurement overhead +increases exponentially with Nl. Due to arguments pre- +sented later, Nl is usually small and does not increase +with system size. From numerical experiments, we find +Nl ≤ 2 is sufficient to produce excellent results. +For time-dependent problems, it is convenient to define +|ψ⟩ = � +l ˆB[l]† |φ⟩ and use ΘK denote both θk and C[l]. +The Lagrangian with multipliers λlnn′ and γlnn′ is then +L = |i +� +K +∂ |ψ⟩ +∂ΘK +˙ΘK − ˆH |ψ⟩ |2 ++ +� +lnn′ +λlnn′ Re +�� +m +C[l]∗ +mn ˙C[l]mn′ +� ++ +� +lnn′ +γlnn′ Im +�� +m +C[l]∗ +mn ˙C[l]mn′ +� +. +(4) +The constraints ensure that C[l]mn remains orthonormal +during time evolution. Similar to the ground state prob- +lem, the equation of motion for θk is the same as vanilla +VQD with encoded Hamiltonian ˆ˜H +� +j +Re +�∂ ⟨φ| +∂θk +∂ |φ⟩ +∂θj +� +˙θj = Im +�∂ ⟨φ| +∂θk +ˆ˜H |φ⟩ +� +. +(5) +The equation of motion for C[l] reads +iρ[l] ˙C[l]∗ = (1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ , +(6) +where ρ[l]nn′ = Tr{⟨φ|n⟩ ⟨n′|φ⟩} is the reduced density +matrix for the Nl qubits of |φ⟩. +Eq. 3 represents a +˙C[l] = 0 stationary point during real and imaginary +time evolution. +The measurement cost is the same as +the ground state algorithm. +The VQD step described by Eq. 5 can be natu- +rally replaced by a Suzuki-Trotter time evolution step +e−i ˆ˜ +Hτ ≈ �M +x e−iˆ˜hxτ on a digital quantum simulator, so +that Hamiltonian simulation is performed via Trotterized +time evolution instead of VQD. To update C[l] based on +Eq. 6, measurements on the circuit should be performed +for every or every several Trotter steps. The variationally +encoded ground state can then be prepared by adiabatic +state preparation, whose energy is accessible by QPE us- +ing ˆ˜H. +It is instructive to observe that if the variational ba- +sis encoder is viewed as a wavefunction ansatz |ψ⟩, then +the algorithm proposed in this work can be viewed as a +generalization for the local basis optimization method +for DMRG [28, 29], or a special case of the recently +proposed quantum-classical hybrid tensor network [30]. +Thus, ˆB[l] captures the 2Nl phonon states that are most +entangled with the rest of the system. For local Hamil- +tonian obeying the area law, the entanglement entropy +between one phonon mode and the rest of the system S +is a constant [31]. Consequently, | ⟨ψ|Ψ⟩ |2, the fidelity +between the approximated encoded state and the target +state has a lower bound of 2Nl +eS , which lays the theoretical + +3 +0 +1 +2 +3 +Coupling strength g +−8 +−6 +−4 +−2 +E/V +(a) +Exact (DMRG) +Gray encoding +Variational encoding +0 +5 +10 +15 +Iteration +−8 +−6 +−4 +−2 +E/V +(b) +g = 0.3 +g = 1.5 +g = 3.0 +8 +16 +24 +32 +Number of levels N +10−3 +10−2 +10−1 +100 +101 +(E − Eexact)/V +(c) +g = 1.5, 1 qubit +g = 3.0, 1 qubit +g = 1.5, 2 qubits +g = 3.0, 2 qubits +0 +5 +10 +15 +Index +10−14 +10−11 +10−8 +10−5 +10−2 +Singular values +(d) +g = 0.3 +g = 1.5 +g = 3.0 +FIG. 1. +Numerical simulation results for the ground state of +the Holstein model. (a) Ground state energy by numerically +exact DMRG, binary encoding, and variational encoding with +different coupling strength g; (b) Convergence of ground state +energy with respect to the macro-iteration for variational en- +coding; (c) Ground state energy error for the variational en- +coding method at different numbers of phonon basis states +N; (d) The singular values for the Schmidt decomposition +between the last phonon mode and the rest of the system. +foundation for the effectiveness of the variational encod- +ing approach to ground state and low-lying excited state +problems. +Simulations +We first show numerical simulation re- +sults on a noiseless simulator and verify the algorithm +on a superconducting quantum computer at the end of +the section. The variational basis state encoder is first +tested for VQE simulation of the one-dimensional Hol- +stein model [32, 33] +ˆH = − +� +⟨i,j⟩ +V ˆa† +iˆaj + +� +i +ωˆb† +iˆbi + +� +i +gωˆa† +iˆai(ˆb† +i +ˆbi) . (7) +where V is the hopping coefficient, ⟨i, j⟩ denotes near- +est neighbour pairs with periodic boundary condition, ω +is the vibration frequency and g is dimensionless cou- +pling constant. In the following, we assume V = ω = 1 +and adjust g for different coupling strengths. We con- +sider a 3-site system corresponding to 3(Nl + 1) qubits. +We use binary encoding to represent traditional encod- +ing approaches. Unary encoding is expected to produce +similar results with binary encoding only with different +quantum resource budgets. The ansatz used and more +details of the simulation are included in the Appendix. +We first compare the accuracy of the variational encod- +ing and the binary encoding with Nl = 1. It is clear from +Fig. 1(a) that variational encoding is significantly more +accurate than binary encoding, especially at the strong +coupling regime. Within the setup, binary encoding uses +only two phonon basis states to describe each phonon +mode, yet the variational encoding is allowed to use up +to 32 phonon basis states before combining them into the +most entangled states. We note that the quantum circuit +used for variational encoding and binary encoding is es- +sentially the same. The number of macro-iterations to +determine C[l] is found to be rather small, as shown in +Fig. 1(b). +Fully converged results are obtained within +5 iterations. +In Fig. 1(c) we show more details of the +error for the variational approach. +The simulation er- +ror typically decreases exponentially with respect to the +number of phonon levels N included in C[l]. It is worth +noting that quantum computational resources, including +the number of qubits, the number of gates in the circuit, +and the number of measurements remain constant when +N is increased from 2 to 32. Furthermore, by using 2 +qubits to encode each mode, it is possible to further re- +duce the error at the N → ∞ limit. When g = 3.0, the +error is not sensitive to Nl, which implies that the error +is dominated by other sources such as limitations of the +ansatz, instead of the small Nl. Fig. 1(d) shows the sin- +gular values for the Schmidt decomposition between the +last phonon mode and the rest of the system by DMRG. +The exponential decay ensures the fast convergence of +Nl. The von Neumann entropy S for the 3 systems is +found to be 0.01, 0.25, and 0.65 respectively. We also +note the g = 1.5 case has the largest 3rd singular value, +which explains why setting Nl = 2 significantly reduces +the g = 1.5 error in Fig. 1(c). +We now turn to the spin-relaxation dynamics of the +spin-boson model [34], described by the Hamiltonian +ˆH = ϵ +2 ˆσz + ∆ˆσx + +� +j +gjωjˆσz(ˆb† +j +ˆbj) + +� +j +ωjˆb† +jˆbj . (8) +where ϵ is the eigenfrequency and ∆ is the tunnelling rate. +The coupling term has a similar form with Eq. 7 and is +more commonly written as � +j cjˆσzˆxj. For systems in the +condensed phase the coupling is usually characterized by +the spectral density function J (ω) = π +2 +� +j +c2 +j +ωj δ(ω − ωk). +In the following we assume ϵ = 0 and ∆ = 1. We first +use VQD for the simulation and discuss Trotterized time +evolution at last. The variational Hamiltonian ansatz [35] +with 3 layers is used if not otherwise specified. +The performance of variational encoding and binary +encoding is first compared based on a 1-mode spin-boson +model at the strong coupling (ω = 1 and g = 3) regime, +shown in Fig. 2(a). Variational encoding with Nl = 1 +generates much more accurate dynamics than binary en- +coding with fewer qubits and quantum gates. The sim- +ulation of binary encoding with Nl > 4 is prohibited by +the deep circuit depth in the ansatz. The variational en- +coding scheme is exceptionally efficient for this 1-mode +model because Schmidt decomposition guarantees that 2 +variational bases for the phonon mode are sufficient to ex- +actly represent the system. In Fig. 2(b) a 2-mode model +with ωj = 1 +2, 1 and gj = 1 +2, 1 is used. Variational encod- +ing with Nl = 1 is accurate at t < 2 but as the entan- + +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +⟨σz⟩ +(a) +Exact +Binary, N = 2 +Binary, N = 4 +Binary, N = 8 +Binary, N = 16 +Variational, N = 64 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +⟨σz⟩ +(b) +Exact +Variational, 1 qubit +Variational, 2 qubits +0 +1 +2 +3 +4 +5 +Time t +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +⟨σz⟩ +(c) +Exact (DMRG) +Variational +Binary +0 +1 +2 +3 +4 +5 +Time t +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +⟨σz⟩ +(d) +Exact +Trotter+Variational +FIG. 2. +Numerical simulation results for the spin-relaxation +dynamics of the spin-boson model. (a) Comparison between +binary encoding with different numbers of phonon basis states +and variational encoding for a 1-mode spin-boson model; +(b) Variational encoding with different numbers of encoding +qubits Nl for a 2-mode spin-boson model; (c) Comparison +between binary encoding and variational encoding for an 8- +mode spin-boson model with sub-Ohmic spectral density; (d) +Trotterized time evolution with variational encoding based on +a 1-mode spin-boson model. +glement builds up the dynamics deviate from the exact +solution. Increasing Nl to 2 effectively eliminates the er- +ror. Next, we move on to a more challenging model with +8 modes, in which ω and g are determined by discretizing +a sub-Ohmic spectral density J (ω) = π +2 αωsω1−s +c +e−ω/ωc +following the prescription in the literature [36]. The pa- +rameters are s = 1 +4, ωc = 4 and α = 10. As illustrated in +Fig. 2(c) variational encoding with Nl = 1 captures the +localization behavior yet binary encoding with Nl = 1 +completely fails. The number of layers in the variational +Hamiltonian ansatz is 8 and 32 for variational and binary +encoding respectively. Fig. 2(d) demonstrates the possi- +bility to incorporate variational basis state encoder into +Trotterized time evolution with ω = g = 1 and Nl = 1. +The measurement and the evolution of C[l] are performed +at each Trotter step. +Lastly, we verify the accuracy and efficiency of the vari- +ational encoder approach on a superconducting quantum +computer. We consider the ground state problem of a +2-site Holstein model described by Eq. 7 with g = 3 and +Nl = 1. The two electronic sites are represented by 1 +qubit and the total number of qubits for the system is +thus 3. +The quantum circuit for the simulation is de- +picted in Fig. 3(a). The electronic degree of freedom is +mapped to the second qubit, and the two phonon modes +are mapped to the first and the third qubits respectively. +There is one parameter to be determined by VQE in the +circuit and the same ansatz is used for both binary en- +coding and variational encoding. More simulation details +(a) +1 +2 +3 +Coupling strength g +−8 +−6 +−4 +−2 +E/V +(b) +Binary +Variational +Simulation +Exact +1 +2 +3 +4 +5 +Macro-iteration +(c) +Binary +Variational +Simulation +Exact +FIG. 3. +Quantum hardware experiments for the ground state +energy of the Holstein model with variational basis state en- +coder. (a) 3 qubits out of 9 qubits of a superconducting quan- +tum computer and a one-parameter circuit are used for the +simulation; (b) Ground state energy by binary encoding and +variational encoding; (c) Convergence of ground state energy +with respect to the macro-iteration for variational encoding. +can be found in the Appendix. In Fig. 3(b) we show the +ground state energy by variational encoding from weak +to strong coupling, in analog to Fig. 1(a). The simulator +result is based on the parameterized quantum circuit de- +scribed in Fig. 3(a) without considering gate noise and +measurement uncertainty. The results in Fig. 3(b) are +consistent with that in Fig. 1(a). The residual error is +dominated by the intrinsic gate noise in the quantum +computer. +In Fig. 3(c) we show the convergence with +respect to the macro-iteration for variational encoding. +The algorithm is resilient to the presence of quantum +noise and measurement uncertainty. The convergent en- +ergy is reached within 5 iterations. +Conclusion +We proposed variational basis state +encoder to encode phonon basis states into quantum +computational states for efficient quantum simulation +of electron-phonon systems. +The proposed variational +encoding approach requires only O(1) qubits and O(1) +quantum gates, which is significantly better than tradi- +tional encoding schemes. The algorithm enables quan- +tum simulation of electron-phonon systems with smaller +quantum processors and shallower circuits. +The addi- +tional measurement required to implement the approach +is found to be also O(1) with respect to the number of +phonon basis states and it scales quadratically with the +number of Pauli strings in the Hamiltonian. The accu- +racy of the approach is ensured by the finite entangle- +ment entropy between one phonon mode and the rest of +the system in common electron-phonon systems. Vari- +ational basis state encoder most naturally works with +variational quantum algorithms and is compatible with +Trotterized time evolution, adiabatic state preparation, + +Ry(-0) +Ry(-0) +Measurement +Module +Ry(0) +Ry(-0) +D5 +and QPE. Numerical simulation and quantum hardware +experiments based on VQE of the Holstein model and +dynamics of the spin-boson model indicates that varia- +tional encoding is more accurate and resource-efficient +than traditional encoding methods. In particular, using +1 or 2 qubits to represent each phonon mode is suffi- +cient for accurate simulation even at the strong coupling +regime where N = 64 phonon basis states are involved. +The approach could also be extended to other quantum +simulation problems involving an infinite or large local +Hilbert space. +We thank Jinzhao Sun and Shixin Zhang for helpful +discussions. This work is supported by the National Nat- +ural Science Foundation of China through grand numbers +22273005 and 21788102. 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Pe- +dregosa, P. van Mulbregt, and SciPy 1.0 Contributors, +SciPy 1.0: Fundamental Algorithms for Scientific Com- +puting in Python, Nat. Methods 17, 261 (2020). +I. Derivation of variational basis state encoder +Time-independent equation +We start with the Lagrangian defined in the main text, i.e. Eq. 2. Taking the derivative with respect to C[l]mn +and setting it to 0 yields +⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ + +� +n′ +λlnn′C[l]mn′ = 0 . +(9) +Multiply with C[l]mn′′ +� +m +C[l]mn′′ ⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ + +� +n′ +λlnn′ +� +m +C[l]mn′C[l]mn′′ = 0 , +(10) +and use the C[l] orthonormal condition � +m C[l]mnC[l]mn′ = δnn′ to get λ +λlnn′ = − +� +m +C[l]mn′ ⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ . +(11) +Substitute λ into Eq. 9 yields +⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ − +� +n′m +C[l]mn′ ⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ C[l]m′n′ = 0 . +(12) +Using +ˆP = ˆB[l]†B[l] = +� +mm′ +� +n +|m⟩l C[l]mnC[l]m′n ⟨m′| +l +, +(13) +to simplify the notation of the second term +⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ − ⟨φ|n⟩l ⟨m| +l +ˆP ˆ˜H′[l] |φ⟩ = 0 . +(14) +Rearranging and rewriting in matrix form, we get the equation for C[l] +(1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ = 0 . +(15) + +7 +Quantum circuit measurement +In this section, we discuss the quantum circuit measurement required to solve C[l] from Eq. 3. The key quantity +to be computed is matrix G[l]mn, defined as +G[l]mn = ⟨φ|n⟩l ⟨m| +l +ˆ˜H′[l] |φ⟩ . +(16) +Express ˆH in sum-of-product form ˆH = �M +x +� +k ˆh[k]x, using notations in the main text, and we get +G[l]mn = +M +� +x +⟨φ|n⟩l ⟨m| +l +� +k̸=l +ˆB[k] +� +k +ˆh[k]x +� +k +ˆB[k]† |φ⟩ += +M +� +x +⟨φ|n⟩l ⟨m| +l +ˆh[l]x ˆB[l]† � +k̸=l +ˆ˜h[k]x |φ⟩ . +(17) +Here we assume � +k̸=l +ˆ˜h[k]x can be expressed by a constant amount of Pauli strings. Represent ˆh[l]x in operator form +ˆh[l]x = +� +mm′ +h[l]xm′m |m′⟩l ⟨m| +l +. +(18) +G[l]mn then becomes +G[l]mn = +M +� +x +� +m′n′ +h[l]xmm′C[l]m′n′ ⟨φ|n⟩l ⟨n′| +l +� +k̸=l +ˆ˜h[k]x |φ⟩ . +(19) +Thus to evaluate G[l]mn it is sufficient to measure ⟨φ|n⟩l ⟨n′| +l +� +k̸=l +ˆ˜h[k]x |φ⟩. |n⟩l ⟨n′| +l +in general is not Hermitian, +and the real and imaginary parts can be measured by +Re +� +� +�⟨φ|n⟩l ⟨n′| +l +� +k̸=l +ˆ˜h[k]x |φ⟩ +� +� +� = 1 +2 ⟨φ| (|n⟩l ⟨n′| +l ++ |n′⟩l ⟨n| +l +) +� +k̸=l +ˆ˜h[k]x |φ⟩ , +Im +� +� +�⟨φ|n⟩l ⟨n′| +l +� +k̸=l +ˆ˜h[k]x |φ⟩ +� +� +� = 1 +2 ⟨φ| i(|n′⟩l ⟨n| +l +− |n⟩l ⟨n′| +l +) +� +k̸=l +ˆ˜h[k]x |φ⟩ . +(20) +To evaluate all matrix elements in G[l], the total number of measurements required scales as O +� +2NlM +� +. +Time-dependent equation +In this section, we derive the time-dependent equation for C[l]. For time-dependent problems, C[l] in general is +complex +C[l] = D[l] − iE[l] , +(21) +where both D[l] and E[l] are real. The minus sign is for convenience expressing ˆB† |φ⟩. From the definition we have +∂ |ψ⟩ +∂E[l]mn += i +∂ |ψ⟩ +∂D[l]mn +. +(22) +The starting point is the Lagrangian Eq. 4 defined in the main text. Taking the derivative with respect to ˙ΘK + +8 +yields +∂L +∂ ˙ΘK += +� +J +∂ ⟨ψ| +∂ΘJ +∂ |ψ⟩ +∂ΘK +˙ΘJ + +� +J +∂ ⟨ψ| +∂ΘK +∂ |ψ⟩ +∂ΘJ +˙ΘJ ++ i∂ ⟨ψ| +∂ΘK +ˆH |ψ⟩ − i ⟨ψ| ˆH ∂ |ψ⟩ +∂ΘK ++ +� +lnn′ +λlnn′ Re +�� +m +C[l]∗ +mn +∂ ˙C[l]mn′ +∂ ˙ΘK +� ++ +� +lnn′ +γlnn′ Im +�� +m +C[l]∗ +mn +∂ ˙C[l]mn′ +∂ ˙ΘK +� +. +(23) +We first consider the case of ΘK = θk, and then +∂L +∂ ˙θk += +� +J +∂ ⟨ψ| +∂ΘJ +∂ |ψ⟩ +∂θk +˙ΘJ + +� +J +∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂ΘJ +˙ΘJ ++ i∂ ⟨ψ| +∂θk +ˆH |ψ⟩ − i ⟨ψ| ˆH ∂ |ψ⟩ +∂θk += 2 +� +J +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂ΘJ +� +˙ΘJ − 2 Im +�∂ ⟨ψ| +∂θk +ˆH |ψ⟩ +� +, +(24) +which means at the +∂L +∂ ˙θk = 0 minimum, we have +� +J +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂ΘJ +� +˙ΘJ = Im +�∂ ⟨ψ| +∂θk +ˆH |ψ⟩ +� +. +(25) +Substitute ΘJ with θk, D[l]mn and E[l]mn +� +J +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂ΘJ +� +˙ΘJ = +� +j +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂θj +� +˙θj ++ +� +lmn +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂D[l]mn +� +˙D[l]mn ++ +� +lmn +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂E[l]mn +� +˙E[l]mn . +(26) +Using Eq. 22 the last two terms become +� +lmn +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂D[l]mn +� +˙D[l]mn + +� +lmn +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂E[l]mn +� +˙E[l]mn = +� +lmn +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂D[l]mn +˙C[l]∗ +mn +� +, +(27) +which is zero because +� +mn +∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂D[l]mn +˙C[l]∗ +mn = +� +mn +∂ ⟨φ| +∂θk +ˆB[l] |m⟩l ⟨n| +l +˙C[l]∗ +mn |φ⟩ = 0 , +(28) +where the constraint � +m C[l]mn ˙C[l]∗ +mn′ = 0 is used. Thus the simplified equation of motion reads +� +j +Re +�∂ ⟨ψ| +∂θk +∂ |ψ⟩ +∂θj +� +˙θj = Im +�∂ ⟨ψ| +∂θk +ˆH |ψ⟩ +� +, +(29) +or equivalently +� +j +Re +�∂ ⟨φ| +∂θk +∂ |φ⟩ +∂θj +� +˙θj = Im +�∂ ⟨φ| +∂θk +ˆ˜H |φ⟩ +� +. +(30) + +9 +In short, the equation of motion for θk is the same as vanilla VQD with encoded Hamiltonian ˆ˜H . +Next we consider the case of ΘK = D[l] and ΘK = E[l]. After some complex algebra, we have +i +� +J +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂ΘJ +˙ΘJ + i1 +2 +� +n′ +λln′nC[l]∗ +mn′ − 1 +2 +� +n′ +γln′nC[l]∗ +mn′ = +∂ ⟨ψ| +∂D[l]mn +ˆH |ψ⟩ . +(31) +Similar to the case of ΘK = θk, substitute ΘJ with θk, D[l]mn and E[l]mn +� +J +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂ΘJ +˙ΘJ = +� +k +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂θk +˙θk + +� +km′n′ +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂D[k]m′n′ +˙C[k]∗ +m′n′ += +� +k +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂θk +˙θk + +� +n′ +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂D[l]mn′ +˙C[l]∗ +mn′ . +(32) +Here the orthonormal condition is again used. Substitute the equation back into Eq. 31. +i +� +k +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂θk +˙θk + i +� +n′ +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂D[l]mn′ +˙C[l]∗ +mn′ + 1 +2 +� +n′ +(iλln′n − γln′n)C[l]∗ +mn′ = +∂ ⟨ψ| +∂D[l]mn +ˆH |ψ⟩ , +(33) +Following the same strategy with the derivation of the time-independent equation, multiply Eq. 33 with C[l]mn +i +� +k +⟨φ|n⟩l ⟨n′| +l +∂ |φ⟩ +∂θk +˙θk + 1 +2(iλln′n − γln′n) = +� +m +C[l]mn′ +∂ ⟨ψ| +∂D[l]mn +ˆH |ψ⟩ , +(34) +where � +m C[l]∗ +mn′C[l]mn = δn′n and � +m ˙C[l]∗ +mn′C[l]mn = 0 are used. Then, multiply again with C[l]∗ +mn +i +� +k +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂θk +˙θk + 1 +2 +� +n′ +(iλln′n − γln′n)C[l]∗ +mn′ = ˆP[l] +∂ ⟨ψ| +∂D[l]mn +ˆH |ψ⟩ . +(35) +Use this equation to eliminate λ and γ in Eq. 33, we get the equation of motion for C[l] +i +� +n′ +∂ ⟨ψ| +∂D[l]mn +∂ |ψ⟩ +∂D[l]mn′ +˙C[l]∗ +mn′ = (1 − ˆP[l]) +∂ ⟨ψ| +∂D[l]mn +ˆH |ψ⟩ , +(36) +which can be simplified to Eq. 6. The measurement required for time evolution is in the same order as the static VQE +algorithm. +In the end, we note that imaginary time evolution might be another approach to finding the ground state, in addition +to the iterative method described in the main text. Imaginary time evolution might also be a feasible approach to +determine C[l] as an alternative to solving Eq. 3. +II. Numerical noiseless simulations +All numerical quantum circuit simulation is performed using the TensorCircuit [37] package without considering +noise. Classical DMRG simulation is performed using the Renormalizer package [38]. We use harmonic oscillator +eigenstates for phonon basis states. Using positional states might affect the performance of traditional encodings +because of the truncation, however, we expect variational encoding to be insensitive to the choice of phonon basis +states at the N → ∞ limit. We use Gray code for binary encoding as an improvement to the standard approach [18]. +For both ground state simulation and dynamics simulation, C[l] is initialized as C[l]mn = δmn. +For the VQE simulation of the Holstein model, the following ansatz is used +|φ⟩ = +L +� +l +� +� +� +� +⟨j,k⟩ +eθljk(ˆa† +j ˆak−ˆa† +kˆaj) � +j +eθljˆa† +j ˆaj(ˆb† +j−ˆbj) +� +� +� |φ0⟩ . +(37) +where L is the number of layers and L = 3 is adopted. The advantage of Eq. 37 is enforcing real-valued wavefunction. +The circuit parameters ⃗θ are optimized by the L-BFGS-G method implemented in SciPy package [39]. The parameter +gradient is calculated by auto-differentiation. The initial values for the parameters are set to zero at the first round + +10 +of the macro-iteration. In subsequent macro-iterations, the previously optimized parameters are used as the initial +value for faster convergence. Eq. 3 is solved by the DF-SANE method implemented in SciPy [39]. Since this is a +non-linear equation, we provide 3 initial guesses and adopt the one with the lowest energy. The solved C[l] sometimes +does not satisfy the orthonormal condition due to numerical imprecision and the orthonormal condition is enforced +by QR decomposition in each macro-iteration. +For the VQD simulation of the spin-boson model, the variational Hamiltonian ansatz used is more complex than +the VQE simulation. Because C[l] is complex, ˆB[l]ˆh[l]x ˆB[l]† spans the whole Hermitian matrix space. Thus for ˆh[l]x +the whole Pauli matrix set {X, Y, Z, I}⊗Nl is added to the ansatz. To obtain the quantities required to calculate θk, +the Jacobian of the wavefunction φ(⃗θ) is firstly calculated by auto-differentiation, and then the r.h.s and l.h.s of Eq. 5 +is calculated by matrix multiplication. How to measure the quantities in realistic quantum circuits is well described +in the literature [16]. To calculate ˙C[l] it is necessary to take the inverse of ρ[l] which is sometimes ill-conditioned. +We add 1 × 10−5 to the diagonal elements of ρ[l] for regularization. The time evolution of θk and C[l] is carried out +using the RK45 method implemented in SciPy [39]. We observe that the gradient of θk is usually much larger than +C[l]. Thus it is possible to evolve the two sets of parameters separately, which deserves further investigation. For +Trotterized time evolution, N = 16 and a time step of 0.01 are used. +III. Experiments on a superconducting quantum processor +Device parameters +The superconducting quantum processor, as shown in Fig. 3(a), is composed of nine computational transmon +qubits with each pair of neighboring qubits mediated via a tunable coupler, forming a cross-shaped architecture. +Each computational qubit has an independent readout cavity for state measurement and XY /Z control lines for state +operation. High-fidelity simultaneous single-shot readout for all qubits are achieved with the help of the multistage +amplification with the Josephson parametric amplifier (JPA) functioning as the first stage of the amplification. The +fundamental device parameters including qubit parameters and gate parameters are outlined in Table. II and Table. III, +where the parasitic ZZ interaction between qubits is suppressed by the coupler. +TABLE II. Single qubit gate parameters. ωr is the resonant frequency of the readout cavity for each qubit. ωj,max (j = 1 ∼ 9) +are the maximum resonant frequencies when qubits are biased at the sweet spot. ωj,idle (j = 1 ∼ 9) are the idle frequencies +for implementing the single-qubit operations. αj (j = 1 ∼ 9) are the qubits’ anharmonicities. T1, T2,idle and T2E,idle are the +corresponding energy relaxation time, Ramsey dephasing time and echoed dephasing time for the qubits measured at the idle +frequency. The readout fidelities are typically characterized by detecting each qubit in |g⟩ (|e⟩) when it is prepared in |g⟩ (|e⟩), +labeled by F0,j and F1,j. To mitigate the error coming from the readout infidelity, the outcomes are reconstructed with the +calibration matrix through the Bayes’ rule. Single-qubit errors esq are measured with randomized benchmarking (RB). +Q0 +Q1 +Q2 +Q3 +Q4 +Q5 +Q6 +Q7 +Q8 +ωr (GHz) +6.874 +6.825 +6.931 +6.901 +6.845 +6.786 +6.991 +6.961 +6.806 +ωj,max (GHz) +4.003 +4.215 +4.479 +4.689 +4.470 +4.479 +4.657 +4.512 +4.362 +ωj,idle (GHz) +3.988 +4.187 +4.464 +4.668 +4.404 +4.359 +4.641 +4.498 +4.223 +αj/2π (MHz) +−260 +−258 +−255 +−250 +−254 +−258 +−253 +−257 +−264 +T1 (µs) +35.3 +31.6 +29.5 +27.7 +33.9 +34.3 +33.3 +22.1 +31.8 +T2,idle (µs) +11.0 +10.2 +32.6 +38.2 +9.1 +5.6 +43.1 +24.1 +4.3 +T2E,idle (µs) +48.2 +38.4 +47.8 +44.2 +31.6 +21.8 +56.8 +32.9 +18.6 +F0,j (%) +96.9 +97.4 +98.6 +98.9 +98.7 +98.4 +96.3 +97.2 +94.1 +F1,j (%) +93.7 +94.3 +92.5 +94.3 +94.5 +94.6 +92.7 +92.4 +90.9 +esq (%) +0.07 +0.32 +0.06 +0.07 +0.08 +0.05 +0.06 +0.15 +0.08 +Experimental details +We use 3 qubits out of the 9-qubit computer for the 2-site Holstein model +ˆH = −V (a† +1a2 + a† +2a1) + ωb† +1b1 + ωb† +2b2 + gωa† +1a1(b† +1 + b1) + gωa† +2a2(b† +2 + b2) . +(38) +The electronic degree of freedom is mapped to the second qubit. Thus, a† +1a1 is mapped to 1 +2(1 + Z1) and a† +2a2 is +mapped to 1 +2(1 − Z1). The phonon modes are mapped to the first and the third qubit. With binary encoding and + +11 +TABLE III. Two qubits gate parameters. ωc,idle are the idle frequencies for each coupler where the ZZ interaction between +neighboring computational qubits are maximally suppressed. ξZZ is the residual ZZ interaction between each qubit pairs. +Two-qubit gates are implemented with the controlled-Z (CZ) and the corresponding gate errors etq,CZ are characterized with +RB. +Q0 − Q1 +Q0 − Q2 +Q0 − Q3 +Q0 − Q4 +Q1 − Q5 +Q2 − Q6 +Q3 − Q7 +Q4 − Q8 +ωc,idle (GHz) +5.020 +5.445 +5.570 +5.335 +5.325 +5.595 +5.695 +5.355 +|ξZZ| (kHz) +18.0 +10.0 +5.0 +8.0 +2.0 +3.0 +5.0 +2.0 +etq,CZ (%) +1.57 +2.22 +1.99 +2.47 +0.91 +1.04 +1.2 +0.96 +Nl = 1, the Hamiltonian in the Pauli string form reads +ˆH = −V X1 + 1 +2ω(1 − Z0) + 1 +2ω(1 − Z2) + 1 +2gω(1 + Z1)X0 + 1 +2gω(1 − Z1)X2 . +(39) +For variational encoding, we assume C[l] = C. That is, the two modes share the same variational encoder. This is a +reasonable assumption for translational invariant systems. Supposing ˆb†ˆb and ˆb† +ˆb are mapped to the following form +ˆB(ˆb†ˆb) ˆB† = c1iI + c1xX + c1zZ +ˆB(ˆb† + ˆb) ˆB† = c2iI + c2xX + c2zZ , +(40) +the encoded Hamiltonian is then +ˆH = −V X1 + ω(c1iI0 + c1xX0 + c1zZ0) + ω(c1iI2 + c1xX2 + c1zZ2) ++ 1 +2gω(1 + Z1)(c2iI0 + c2xX0 + c2zZ0) + 1 +2gω(1 − Z1)(c2iI2 + c2xX2 + c2zZ2) . +(41) +We use the following ansatz for the parameterized quantum circuit +|φ⟩ = +2 +� +j=1 +eθja† +jaj(b† +j−bj) 1 +√ +2 (|000⟩ + |100⟩) , +(42) +Because C[1] = C[2], the parameter space can be further simplified by setting θ1 = θ2. With binary encoding, the +ansatz transforms to +|φ⟩ = eiθY2e−iθZ1Y2eiθY0eiθZ1Y0H1 |0⟩ . +(43) +The ansatz is compiled into the following quantum circuit with 4 CNOT gates. +q0 : +RZ( −π +2 ) +H +RZ(−θ) +RZ(−θ) +H +RZ( π +2 ) +q1 : +H +• +• +• +• +q2 : +RZ( −π +2 ) +H +RZ(θ) +RZ(−θ) +H +RZ( π +2 ) +Each energy term is measured by 8192 shots, and the uncertainty is obtained by repeating the measurement 5 times +and taking the standard deviation. For the update of C[l], 4096 shots are performed for each term. Local readout +error mitigation is applied for all results presented unless otherwise stated. +In Fig. 4 we plot the energy landscape E(θ)/V in VQE with binary encoding. Both raw data and data with local +readout error mitigation (EM) are presented for the energy expectation from quantum hardware. The mitigated +landscape is in decent agreement with the perfect simulator. A minimum at around θ = 0.6 is clearly visible. We note +that the perfect simulator is also based on the Nl = 1 ansatz and N is far smaller than what is physically demanded. +Thus the minimum presented by the perfect simulator can not be recognized as the ground truth. + +12 +0.00 +0.25 +0.50 +0.75 +1.00 +θ +−3 +−2 +−1 +0 +E/V +Simulator +Hardware (raw) +Hardware (EM) +FIG. 4. +VQE energy landscape for the 2-site Holstein model with binary encoding. For the data from quantum hardware, +both raw data and data with readout error mitigation are presented. The error bar indicates the measurement uncertainty. + diff --git a/1NAzT4oBgHgl3EQfevyy/content/tmp_files/load_file.txt b/1NAzT4oBgHgl3EQfevyy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d297bb52c94b9df74003cc98b7432b096fecaac4 --- /dev/null +++ b/1NAzT4oBgHgl3EQfevyy/content/tmp_files/load_file.txt @@ -0,0 +1,869 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf,len=868 +page_content='Efficient Quantum Simulation of Electron-Phonon Systems by Variational Basis State Encoder Weitang Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 Jiajun Ren,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 Sainan Huai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 Tianqi Cai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 Zhigang Shuai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ∗ and Shengyu Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' † 1Tencent Quantum Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Tencent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' China 2College of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Beijing Normal Univerisity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' China 3Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' China 4School of Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The Chinese University of Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' China (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2023) Digital quantum simulation of electron-phonon systems requires truncating infinite phonon levels into N basis states and then encoding them with qubit computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Unary encoding and the more compact binary/Gray encoding are the two most representative encoding schemes, which demand O(N) and O(log N) qubits as well as O(N) and O(N log N) quantum gates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In this work, we propose a variational basis state encoding algorithm that reduces the scaling of the number of qubits and quantum gates to both O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The cost for the scaling reduction is a constant amount of additional measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The accuracy and efficiency of the approach are verified by both numerical simulation and realistic quantum hardware experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In particular, we find using 1 or 2 qubits for each phonon mode is sufficient to produce quantitatively correct results across weak and strong coupling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Our approach paves the way for practical quantum simulation of electron-phonon systems on both near-term hardware and error-corrected quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Introduction Electron-phonon couplings are per- vasive in quantum materials, governing phenomena such as charge transport in semiconductors [1], vibrational spectra [2], polaron formation [3], and superconductiv- ity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Classically, expensive numerical methods such as density matrix renormalization group (DMRG) and quantum Monte-Carlo (QMC) are required to accurately simulate electron-phonon systems due to the interior many-body interaction [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Quantum computers hold promise for the simulation of quantum systems with ex- ponential speedup over classical computers [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In the wake of the tremendous progress in the implementation of quantum computers [11, 12] and the dawning of the noisy intermediate-scale quantum (NISQ) era [13], how to solve electron-phonon coupling problems with quantum com- puters has attracted a lot of research interest [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' A prominent problem for the digital quantum simu- lation of electron-phonon systems is how to encode the infinite phonon states with finite quantum computational basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The first step is usually truncating the in- finite phonon states into N basis states {|m⟩} and then the second step is encoding {|m⟩} into quantum compu- tational basis {|n⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The phonon basis states are usu- ally the N lowest harmonic oscillator eigenstates or N uniformly distributed grid basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' There are two established strategies to perform the encoding {|m⟩} �→ {|n⟩} [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The first is unary encoding [20, 21], in which each |m⟩ is encoded to |00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 00⟩, and the total number of qubits required scales as O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The sec- ond is binary encoding, in which each |m⟩ is encoded to � i |⌊ m 2i ⌋ mod 2⟩ represented by O(log N) qubits [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In terms of two-qubit gates required to simulate quantum operators such as ˆb† ± ˆb and ˆb†ˆb, unary encoding scales as O(N) and binary encoding scales as O(N log N) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The features of unary encoding and binary encoding are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Compared to the simulation of electrons, the simulation of phonons consumes quantum resources in a much faster manner, which becomes the bottleneck for efficient quantum simulation of electron- phonon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Comparison of traditional encoding schemes and the proposed variational encoding in terms of encoding for- mula, the number of qubits Nqubit required and the number of quantum gates Ngate required to simulate common phonon operators such as ˆb† ± ˆb and ˆb†ˆb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Scheme Formula Nqubit Ngate Unary |m⟩ �→ |00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 00⟩ O(N) O(N) Binary |m⟩ �→ � i |⌊ m 2i ⌋ mod 2⟩ O(log N) O(N log N) Variational � m Cmn |m⟩ �→ |n⟩ O(1) O(1) In this work, we propose a new basis encoding scheme called variational encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Variational encoding maps linear combinations of |m⟩ that are most entangled to the simulated system into the computational basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' � m Cmn |m⟩ �→ |n⟩, where Cmn is determined by varia- tional principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The advantage of our approach is that, by encoding only the most entangled states and discard- ing the ones with little entanglement, the size of {|n⟩} can be made irrelevant to the size of {|m⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In other words, the number of qubits required scales as O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Consequently, the scaling for the number of gates is also O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Variational encoding is best suited to work in combination with variational quantum algorithms such as variational quantum eigensolver (VQE) [22, 23] and variational quantum dynamics (VQD) [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Besides, the variational encoding is also compatible with Trot- terized time evolution and quantum phase estimation (QPE) [10, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Numerical simulation and experi- ments on realistic quantum hardware based on the Hol- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='01442v1 [quant-ph] 4 Jan 2023 2 stein model and spin-boson model shows that using 1 or 2 qubits for each phonon mode is typically sufficient for highly accurate results even in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Variational Basis State Encoder Encoder coeffi- cients C are determined by variational principle for both static and dynamic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We start the derivation us- ing parameterized quantum circuit (PQC) and discuss how to incorporate the variational encoder in Trotterized time evolution and QPE later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We use atomic units throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' More details for the derivation can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For each phonon mode l, encoded by Nl qubits, define the variational encoder ˆB[l] ˆB[l] = � m 2Nl � n=1 C[l]mn |n⟩l ⟨m| l , (1) with orthonormal constraint ˆB[l] ˆB[l]† = ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The original Hamiltonian in |m⟩ basis can then be encoded to |n⟩ basis ˆ˜H = � l ˆB[l] ˆH � l ˆB[l]†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Suppose the quantum circuit is parameterized by |φ⟩ = � k eiθk ˆ Rk |φ0⟩, and then the ground state Lagrangian with multipliers λlnn′ is L = ⟨φ| ˆ˜H|φ⟩ + � lnn′ λlnn′( � m C[l]mnC[l]mn′ − δnn′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (2) Taking the derivative with respect to θk leads to tradi- tional VQE with encoded Hamiltonian ˆ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Taking the derivative with respect to C[l] and setting it to 0 yields (1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ = 0 , (3) with projector ˆP[l] = ˆB[l]† ˆB[l] and the half encoded Hamiltonian ˆ˜H′[l] = � k̸=l ˆB[k] ˆH � k ˆB[k]†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In practice, θk and C[l] are solved iteratively until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In the following, this iteration is termed macro-iteration to avoid confusion with VQE iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Next, we discuss the measurement required to solve C[l] from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Suppose the Hamiltonian can be written as ˆH = �M x ˆhx and ˆhx = � k ˆh[k]x, where M is the total number of terms in the Hamiltonian and ˆh[k]x acts on the kth degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The measurement of ⟨φ| ˆ˜H′[l]|φ⟩ boils down to that of ⟨φ|n⟩l ⟨n′| l � k̸=l ˆ˜h[k]x |φ⟩, where ˆ˜h[k]x = ˆB[k]ˆh[k]x ˆB[k]† is the encoded local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For electron degree of freedom a dummy encoder ˆB[k] = ˆI is used for notational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The number of additional measurements for the update of C[l] is thus O � 2NlM � , which is polynomial to the system size and does not in- crease with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' If the number of phonon modes is as- sumed to be linear with M and each C[l] is updated independently, then the total number of measurements for all C[l] is O � 2NlM 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The measurement overhead increases exponentially with Nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Due to arguments pre- sented later, Nl is usually small and does not increase with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' From numerical experiments, we find Nl ≤ 2 is sufficient to produce excellent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For time-dependent problems, it is convenient to define |ψ⟩ = � l ˆB[l]† |φ⟩ and use ΘK denote both θk and C[l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The Lagrangian with multipliers λlnn′ and γlnn′ is then L = |i � K ∂ |ψ⟩ ∂ΘK ˙ΘK − ˆH |ψ⟩ |2 + � lnn′ λlnn′ Re �� m C[l]∗ mn ˙C[l]mn′ � + � lnn′ γlnn′ Im �� m C[l]∗ mn ˙C[l]mn′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (4) The constraints ensure that C[l]mn remains orthonormal during time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Similar to the ground state prob- lem, the equation of motion for θk is the same as vanilla VQD with encoded Hamiltonian ˆ˜H � j Re �∂ ⟨φ| ∂θk ∂ |φ⟩ ∂θj � ˙θj = Im �∂ ⟨φ| ∂θk ˆ˜H |φ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (5) The equation of motion for C[l] reads iρ[l] ˙C[l]∗ = (1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ , (6) where ρ[l]nn′ = Tr{⟨φ|n⟩ ⟨n′|φ⟩} is the reduced density matrix for the Nl qubits of |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3 represents a ˙C[l] = 0 stationary point during real and imaginary time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The measurement cost is the same as the ground state algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The VQD step described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 5 can be natu- rally replaced by a Suzuki-Trotter time evolution step e−i ˆ˜ Hτ ≈ �M x e−iˆ˜hxτ on a digital quantum simulator, so that Hamiltonian simulation is performed via Trotterized time evolution instead of VQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' To update C[l] based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 6, measurements on the circuit should be performed for every or every several Trotter steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The variationally encoded ground state can then be prepared by adiabatic state preparation, whose energy is accessible by QPE us- ing ˆ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' It is instructive to observe that if the variational ba- sis encoder is viewed as a wavefunction ansatz |ψ⟩, then the algorithm proposed in this work can be viewed as a generalization for the local basis optimization method for DMRG [28, 29], or a special case of the recently proposed quantum-classical hybrid tensor network [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Thus, ˆB[l] captures the 2Nl phonon states that are most entangled with the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For local Hamil- tonian obeying the area law, the entanglement entropy between one phonon mode and the rest of the system S is a constant [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Consequently, | ⟨ψ|Ψ⟩ |2, the fidelity between the approximated encoded state and the target state has a lower bound of 2Nl eS , which lays the theoretical 3 0 1 2 3 Coupling strength g −8 −6 −4 −2 E/V (a) Exact (DMRG) Gray encoding Variational encoding 0 5 10 15 Iteration −8 −6 −4 −2 E/V (b) g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 8 16 24 32 Number of levels N 10−3 10−2 10−1 100 101 (E − Eexact)/V (c) g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5, 1 qubit g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0, 1 qubit g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5, 2 qubits g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0, 2 qubits 0 5 10 15 Index 10−14 10−11 10−8 10−5 10−2 Singular values (d) g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Numerical simulation results for the ground state of the Holstein model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (a) Ground state energy by numerically exact DMRG, binary encoding, and variational encoding with different coupling strength g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (b) Convergence of ground state energy with respect to the macro-iteration for variational en- coding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (c) Ground state energy error for the variational en- coding method at different numbers of phonon basis states N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (d) The singular values for the Schmidt decomposition between the last phonon mode and the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' foundation for the effectiveness of the variational encod- ing approach to ground state and low-lying excited state problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Simulations We first show numerical simulation re- sults on a noiseless simulator and verify the algorithm on a superconducting quantum computer at the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The variational basis state encoder is first tested for VQE simulation of the one-dimensional Hol- stein model [32, 33] ˆH = − � ⟨i,j⟩ V ˆa† iˆaj + � i ωˆb† iˆbi + � i gωˆa† iˆai(ˆb† i +ˆbi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (7) where V is the hopping coefficient, ⟨i, j⟩ denotes near- est neighbour pairs with periodic boundary condition, ω is the vibration frequency and g is dimensionless cou- pling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In the following, we assume V = ω = 1 and adjust g for different coupling strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We con- sider a 3-site system corresponding to 3(Nl + 1) qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We use binary encoding to represent traditional encod- ing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Unary encoding is expected to produce similar results with binary encoding only with different quantum resource budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The ansatz used and more details of the simulation are included in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We first compare the accuracy of the variational encod- ing and the binary encoding with Nl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(a) that variational encoding is significantly more accurate than binary encoding, especially at the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Within the setup, binary encoding uses only two phonon basis states to describe each phonon mode, yet the variational encoding is allowed to use up to 32 phonon basis states before combining them into the most entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We note that the quantum circuit used for variational encoding and binary encoding is es- sentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The number of macro-iterations to determine C[l] is found to be rather small, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Fully converged results are obtained within 5 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(c) we show more details of the error for the variational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The simulation er- ror typically decreases exponentially with respect to the number of phonon levels N included in C[l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' It is worth noting that quantum computational resources, including the number of qubits, the number of gates in the circuit, and the number of measurements remain constant when N is increased from 2 to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Furthermore, by using 2 qubits to encode each mode, it is possible to further re- duce the error at the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' When g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0, the error is not sensitive to Nl, which implies that the error is dominated by other sources such as limitations of the ansatz, instead of the small Nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(d) shows the sin- gular values for the Schmidt decomposition between the last phonon mode and the rest of the system by DMRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The exponential decay ensures the fast convergence of Nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The von Neumann entropy S for the 3 systems is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='25, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='65 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We also note the g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 case has the largest 3rd singular value, which explains why setting Nl = 2 significantly reduces the g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 error in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We now turn to the spin-relaxation dynamics of the spin-boson model [34], described by the Hamiltonian ˆH = ϵ 2 ˆσz + ∆ˆσx + � j gjωjˆσz(ˆb† j +ˆbj) + � j ωjˆb† jˆbj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (8) where ϵ is the eigenfrequency and ∆ is the tunnelling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The coupling term has a similar form with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 7 and is more commonly written as � j cjˆσzˆxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For systems in the condensed phase the coupling is usually characterized by the spectral density function J (ω) = π 2 � j c2 j ωj δ(ω − ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In the following we assume ϵ = 0 and ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We first use VQD for the simulation and discuss Trotterized time evolution at last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The variational Hamiltonian ansatz [35] with 3 layers is used if not otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The performance of variational encoding and binary encoding is first compared based on a 1-mode spin-boson model at the strong coupling (ω = 1 and g = 3) regime, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Variational encoding with Nl = 1 generates much more accurate dynamics than binary en- coding with fewer qubits and quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The sim- ulation of binary encoding with Nl > 4 is prohibited by the deep circuit depth in the ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The variational en- coding scheme is exceptionally efficient for this 1-mode model because Schmidt decomposition guarantees that 2 variational bases for the phonon mode are sufficient to ex- actly represent the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2(b) a 2-mode model with ωj = 1 2, 1 and gj = 1 2, 1 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Variational encod- ing with Nl = 1 is accurate at t < 2 but as the entan- 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 ⟨σz⟩ (a) Exact Binary, N = 2 Binary, N = 4 Binary, N = 8 Binary, N = 16 Variational, N = 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 ⟨σz⟩ (b) Exact Variational, 1 qubit Variational, 2 qubits 0 1 2 3 4 5 Time t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 ⟨σz⟩ (c) Exact (DMRG) Variational Binary 0 1 2 3 4 5 Time t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 ⟨σz⟩ (d) Exact Trotter+Variational FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Numerical simulation results for the spin-relaxation dynamics of the spin-boson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (a) Comparison between binary encoding with different numbers of phonon basis states and variational encoding for a 1-mode spin-boson model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (b) Variational encoding with different numbers of encoding qubits Nl for a 2-mode spin-boson model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (c) Comparison between binary encoding and variational encoding for an 8- mode spin-boson model with sub-Ohmic spectral density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (d) Trotterized time evolution with variational encoding based on a 1-mode spin-boson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' glement builds up the dynamics deviate from the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Increasing Nl to 2 effectively eliminates the er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Next, we move on to a more challenging model with 8 modes, in which ω and g are determined by discretizing a sub-Ohmic spectral density J (ω) = π 2 αωsω1−s c e−ω/ωc following the prescription in the literature [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The pa- rameters are s = 1 4, ωc = 4 and α = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2(c) variational encoding with Nl = 1 captures the localization behavior yet binary encoding with Nl = 1 completely fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The number of layers in the variational Hamiltonian ansatz is 8 and 32 for variational and binary encoding respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2(d) demonstrates the possi- bility to incorporate variational basis state encoder into Trotterized time evolution with ω = g = 1 and Nl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The measurement and the evolution of C[l] are performed at each Trotter step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Lastly, we verify the accuracy and efficiency of the vari- ational encoder approach on a superconducting quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We consider the ground state problem of a 2-site Holstein model described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 7 with g = 3 and Nl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The two electronic sites are represented by 1 qubit and the total number of qubits for the system is thus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The quantum circuit for the simulation is de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The electronic degree of freedom is mapped to the second qubit, and the two phonon modes are mapped to the first and the third qubits respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' There is one parameter to be determined by VQE in the circuit and the same ansatz is used for both binary en- coding and variational encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' More simulation details (a) 1 2 3 Coupling strength g −8 −6 −4 −2 E/V (b) Binary Variational Simulation Exact 1 2 3 4 5 Macro-iteration (c) Binary Variational Simulation Exact FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Quantum hardware experiments for the ground state energy of the Holstein model with variational basis state en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (a) 3 qubits out of 9 qubits of a superconducting quan- tum computer and a one-parameter circuit are used for the simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (b) Ground state energy by binary encoding and variational encoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (c) Convergence of ground state energy with respect to the macro-iteration for variational encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3(b) we show the ground state energy by variational encoding from weak to strong coupling, in analog to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The simulator result is based on the parameterized quantum circuit de- scribed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3(a) without considering gate noise and measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3(b) are consistent with that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The residual error is dominated by the intrinsic gate noise in the quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3(c) we show the convergence with respect to the macro-iteration for variational encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The algorithm is resilient to the presence of quantum noise and measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The convergent en- ergy is reached within 5 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Conclusion We proposed variational basis state encoder to encode phonon basis states into quantum computational states for efficient quantum simulation of electron-phonon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The proposed variational encoding approach requires only O(1) qubits and O(1) quantum gates, which is significantly better than tradi- tional encoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The algorithm enables quan- tum simulation of electron-phonon systems with smaller quantum processors and shallower circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The addi- tional measurement required to implement the approach is found to be also O(1) with respect to the number of phonon basis states and it scales quadratically with the number of Pauli strings in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The accu- racy of the approach is ensured by the finite entangle- ment entropy between one phonon mode and the rest of the system in common electron-phonon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Vari- ational basis state encoder most naturally works with variational quantum algorithms and is compatible with Trotterized time evolution, adiabatic state preparation, Ry(-0) Ry(-0) Measurement Module Ry(0) Ry(-0) D5 and QPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Numerical simulation and quantum hardware experiments based on VQE of the Holstein model and dynamics of the spin-boson model indicates that varia- tional encoding is more accurate and resource-efficient than traditional encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In particular, using 1 or 2 qubits to represent each phonon mode is suffi- cient for accurate simulation even at the strong coupling regime where N = 64 phonon basis states are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The approach could also be extended to other quantum simulation problems involving an infinite or large local Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We thank Jinzhao Sun and Shixin Zhang for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' This work is supported by the National Nat- ural Science Foundation of China through grand numbers 22273005 and 21788102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' This work is also supported by Shenzhen Science and Technology Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ∗ zgshuai@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='cn † shengyzhang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='com [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Bardeen and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Shockley, Deformation potentials and mobilities in non-polar crystals, Phys.' metadata={'source': 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+page_content=' VanderPlas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Laxalde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Perktold, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Cimrman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Henriksen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Quintero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Archibald, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Ribeiro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Pe- dregosa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' van Mulbregt, and SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 Contributors, SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0: Fundamental Algorithms for Scientific Com- puting in Python, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Methods 17, 261 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Derivation of variational basis state encoder Time-independent equation We start with the Lagrangian defined in the main text, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Taking the derivative with respect to C[l]mn and setting it to 0 yields ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ + � n′ λlnn′C[l]mn′ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (9) Multiply with C[l]mn′′ � m C[l]mn′′ ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ + � n′ λlnn′ � m C[l]mn′C[l]mn′′ = 0 , (10) and use the C[l] orthonormal condition � m C[l]mnC[l]mn′ = δnn′ to get λ λlnn′ = − � m C[l]mn′ ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (11) Substitute λ into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 9 yields ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ − � n′m C[l]mn′ ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ C[l]m′n′ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (12) Using ˆP = ˆB[l]†B[l] = � mm′ � n |m⟩l C[l]mnC[l]m′n ⟨m′| l , (13) to simplify the notation of the second term ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ − ⟨φ|n⟩l ⟨m| l ˆP ˆ˜H′[l] |φ⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (14) Rearranging and rewriting in matrix form, we get the equation for C[l] (1 − ˆP[l]) ⟨φ| ˆ˜H′[l]|φ⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (15) 7 Quantum circuit measurement In this section, we discuss the quantum circuit measurement required to solve C[l] from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The key quantity to be computed is matrix G[l]mn, defined as G[l]mn = ⟨φ|n⟩l ⟨m| l ˆ˜H′[l] |φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (16) Express ˆH in sum-of-product form ˆH = �M x � k ˆh[k]x, using notations in the main text, and we get G[l]mn = M � x ⟨φ|n⟩l ⟨m| l � k̸=l ˆB[k] � k ˆh[k]x � k ˆB[k]† |φ⟩ = M � x ⟨φ|n⟩l ⟨m| l ˆh[l]x ˆB[l]† � k̸=l ˆ˜h[k]x |φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (17) Here we assume � k̸=l ˆ˜h[k]x can be expressed by a constant amount of Pauli strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Represent ˆh[l]x in operator form ˆh[l]x = � mm′ h[l]xm′m |m′⟩l ⟨m| l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (18) G[l]mn then becomes G[l]mn = M � x � m′n′ h[l]xmm′C[l]m′n′ ⟨φ|n⟩l ⟨n′| l � k̸=l ˆ˜h[k]x |φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (19) Thus to evaluate G[l]mn it is sufficient to measure ⟨φ|n⟩l ⟨n′| l � k̸=l ˆ˜h[k]x |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' |n⟩l ⟨n′| l in general is not Hermitian, and the real and imaginary parts can be measured by Re � � �⟨φ|n⟩l ⟨n′| l � k̸=l ˆ˜h[k]x |φ⟩ � � � = 1 2 ⟨φ| (|n⟩l ⟨n′| l + |n′⟩l ⟨n| l ) � k̸=l ˆ˜h[k]x |φ⟩ , Im � � �⟨φ|n⟩l ⟨n′| l � k̸=l ˆ˜h[k]x |φ⟩ � � � = 1 2 ⟨φ| i(|n′⟩l ⟨n| l − |n⟩l ⟨n′| l ) � k̸=l ˆ˜h[k]x |φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (20) To evaluate all matrix elements in G[l], the total number of measurements required scales as O � 2NlM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Time-dependent equation In this section, we derive the time-dependent equation for C[l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For time-dependent problems, C[l] in general is complex C[l] = D[l] − iE[l] , (21) where both D[l] and E[l] are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The minus sign is for convenience expressing ˆB† |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' From the definition we have ∂ |ψ⟩ ∂E[l]mn = i ∂ |ψ⟩ ∂D[l]mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (22) The starting point is the Lagrangian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 4 defined in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Taking the derivative with respect to ˙ΘK 8 yields ∂L ∂ ˙ΘK = � J ∂ ⟨ψ| ∂ΘJ ∂ |ψ⟩ ∂ΘK ˙ΘJ + � J ∂ ⟨ψ| ∂ΘK ∂ |ψ⟩ ∂ΘJ ˙ΘJ + i∂ ⟨ψ| ∂ΘK ˆH |ψ⟩ − i ⟨ψ| ˆH ∂ |ψ⟩ ∂ΘK + � lnn′ λlnn′ Re �� m C[l]∗ mn ∂ ˙C[l]mn′ ∂ ˙ΘK � + � lnn′ γlnn′ Im �� m C[l]∗ mn ∂ ˙C[l]mn′ ∂ ˙ΘK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (23) We first consider the case of ΘK = θk, and then ∂L ∂ ˙θk = � J ∂ ⟨ψ| ∂ΘJ ∂ |ψ⟩ ∂θk ˙ΘJ + � J ∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂ΘJ ˙ΘJ + i∂ ⟨ψ| ∂θk ˆH |ψ⟩ − i ⟨ψ| ˆH ∂ |ψ⟩ ∂θk = 2 � J Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂ΘJ � ˙ΘJ − 2 Im �∂ ⟨ψ| ∂θk ˆH |ψ⟩ � , (24) which means at the ∂L ∂ ˙θk = 0 minimum, we have � J Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂ΘJ � ˙ΘJ = Im �∂ ⟨ψ| ∂θk ˆH |ψ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (25) Substitute ΘJ with θk, D[l]mn and E[l]mn � J Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂ΘJ � ˙ΘJ = � j Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂θj � ˙θj + � lmn Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂D[l]mn � ˙D[l]mn + � lmn Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂E[l]mn � ˙E[l]mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (26) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 22 the last two terms become � lmn Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂D[l]mn � ˙D[l]mn + � lmn Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂E[l]mn � ˙E[l]mn = � lmn Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂D[l]mn ˙C[l]∗ mn � , (27) which is zero because � mn ∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂D[l]mn ˙C[l]∗ mn = � mn ∂ ⟨φ| ∂θk ˆB[l] |m⟩l ⟨n| l ˙C[l]∗ mn |φ⟩ = 0 , (28) where the constraint � m C[l]mn ˙C[l]∗ mn′ = 0 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Thus the simplified equation of motion reads � j Re �∂ ⟨ψ| ∂θk ∂ |ψ⟩ ∂θj � ˙θj = Im �∂ ⟨ψ| ∂θk ˆH |ψ⟩ � , (29) or equivalently � j Re �∂ ⟨φ| ∂θk ∂ |φ⟩ ∂θj � ˙θj = Im �∂ ⟨φ| ∂θk ˆ˜H |φ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (30) 9 In short, the equation of motion for θk is the same as vanilla VQD with encoded Hamiltonian ˆ˜H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Next we consider the case of ΘK = D[l] and ΘK = E[l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' After some complex algebra, we have i � J ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂ΘJ ˙ΘJ + i1 2 � n′ λln′nC[l]∗ mn′ − 1 2 � n′ γln′nC[l]∗ mn′ = ∂ ⟨ψ| ∂D[l]mn ˆH |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (31) Similar to the case of ΘK = θk, substitute ΘJ with θk, D[l]mn and E[l]mn � J ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂ΘJ ˙ΘJ = � k ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂θk ˙θk + � km′n′ ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂D[k]m′n′ ˙C[k]∗ m′n′ = � k ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂θk ˙θk + � n′ ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂D[l]mn′ ˙C[l]∗ mn′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (32) Here the orthonormal condition is again used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Substitute the equation back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' i � k ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂θk ˙θk + i � n′ ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂D[l]mn′ ˙C[l]∗ mn′ + 1 2 � n′ (iλln′n − γln′n)C[l]∗ mn′ = ∂ ⟨ψ| ∂D[l]mn ˆH |ψ⟩ , (33) Following the same strategy with the derivation of the time-independent equation, multiply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 33 with C[l]mn i � k ⟨φ|n⟩l ⟨n′| l ∂ |φ⟩ ∂θk ˙θk + 1 2(iλln′n − γln′n) = � m C[l]mn′ ∂ ⟨ψ| ∂D[l]mn ˆH |ψ⟩ , (34) where � m C[l]∗ mn′C[l]mn = δn′n and � m ˙C[l]∗ mn′C[l]mn = 0 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Then, multiply again with C[l]∗ mn i � k ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂θk ˙θk + 1 2 � n′ (iλln′n − γln′n)C[l]∗ mn′ = ˆP[l] ∂ ⟨ψ| ∂D[l]mn ˆH |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (35) Use this equation to eliminate λ and γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 33, we get the equation of motion for C[l] i � n′ ∂ ⟨ψ| ∂D[l]mn ∂ |ψ⟩ ∂D[l]mn′ ˙C[l]∗ mn′ = (1 − ˆP[l]) ∂ ⟨ψ| ∂D[l]mn ˆH |ψ⟩ , (36) which can be simplified to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The measurement required for time evolution is in the same order as the static VQE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In the end, we note that imaginary time evolution might be another approach to finding the ground state, in addition to the iterative method described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Imaginary time evolution might also be a feasible approach to determine C[l] as an alternative to solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Numerical noiseless simulations All numerical quantum circuit simulation is performed using the TensorCircuit [37] package without considering noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Classical DMRG simulation is performed using the Renormalizer package [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We use harmonic oscillator eigenstates for phonon basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Using positional states might affect the performance of traditional encodings because of the truncation, however, we expect variational encoding to be insensitive to the choice of phonon basis states at the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We use Gray code for binary encoding as an improvement to the standard approach [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For both ground state simulation and dynamics simulation, C[l] is initialized as C[l]mn = δmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For the VQE simulation of the Holstein model, the following ansatz is used |φ⟩ = L � l � � � � ⟨j,k⟩ eθljk(ˆa† j ˆak−ˆa† kˆaj) � j eθljˆa† j ˆaj(ˆb† j−ˆbj) � � � |φ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (37) where L is the number of layers and L = 3 is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The advantage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 37 is enforcing real-valued wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The circuit parameters ⃗θ are optimized by the L-BFGS-G method implemented in SciPy package [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The parameter gradient is calculated by auto-differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The initial values for the parameters are set to zero at the first round 10 of the macro-iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In subsequent macro-iterations, the previously optimized parameters are used as the initial value for faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3 is solved by the DF-SANE method implemented in SciPy [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Since this is a non-linear equation, we provide 3 initial guesses and adopt the one with the lowest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The solved C[l] sometimes does not satisfy the orthonormal condition due to numerical imprecision and the orthonormal condition is enforced by QR decomposition in each macro-iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For the VQD simulation of the spin-boson model, the variational Hamiltonian ansatz used is more complex than the VQE simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Because C[l] is complex, ˆB[l]ˆh[l]x ˆB[l]† spans the whole Hermitian matrix space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Thus for ˆh[l]x the whole Pauli matrix set {X, Y, Z, I}⊗Nl is added to the ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' To obtain the quantities required to calculate θk, the Jacobian of the wavefunction φ(⃗θ) is firstly calculated by auto-differentiation, and then the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='s and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='s of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 5 is calculated by matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' How to measure the quantities in realistic quantum circuits is well described in the literature [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' To calculate ˙C[l] it is necessary to take the inverse of ρ[l] which is sometimes ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We add 1 × 10−5 to the diagonal elements of ρ[l] for regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The time evolution of θk and C[l] is carried out using the RK45 method implemented in SciPy [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We observe that the gradient of θk is usually much larger than C[l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Thus it is possible to evolve the two sets of parameters separately, which deserves further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For Trotterized time evolution, N = 16 and a time step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='01 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Experiments on a superconducting quantum processor Device parameters The superconducting quantum processor, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 3(a), is composed of nine computational transmon qubits with each pair of neighboring qubits mediated via a tunable coupler, forming a cross-shaped architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Each computational qubit has an independent readout cavity for state measurement and XY /Z control lines for state operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' High-fidelity simultaneous single-shot readout for all qubits are achieved with the help of the multistage amplification with the Josephson parametric amplifier (JPA) functioning as the first stage of the amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The fundamental device parameters including qubit parameters and gate parameters are outlined in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' II and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' III, where the parasitic ZZ interaction between qubits is suppressed by the coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Single qubit gate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ωr is the resonant frequency of the readout cavity for each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ωj,max (j = 1 ∼ 9) are the maximum resonant frequencies when qubits are biased at the sweet spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ωj,idle (j = 1 ∼ 9) are the idle frequencies for implementing the single-qubit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' αj (j = 1 ∼ 9) are the qubits’ anharmonicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' T1, T2,idle and T2E,idle are the corresponding energy relaxation time, Ramsey dephasing time and echoed dephasing time for the qubits measured at the idle frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The readout fidelities are typically characterized by detecting each qubit in |g⟩ (|e⟩) when it is prepared in |g⟩ (|e⟩), labeled by F0,j and F1,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' To mitigate the error coming from the readout infidelity, the outcomes are reconstructed with the calibration matrix through the Bayes’ rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Single-qubit errors esq are measured with randomized benchmarking (RB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Q0 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 ωr (GHz) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='874 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='825 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='931 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='901 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='845 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='786 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='991 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='961 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='806 ωj,max (GHz) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='003 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='215 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='479 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='689 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='470 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='479 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='657 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='512 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='362 ωj,idle (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='988 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='187 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='464 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='668 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='404 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='359 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='641 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='498 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='223 αj/2π (MHz) −260 −258 −255 −250 −254 −258 −253 −257 −264 T1 (µs) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 T2,idle (µs) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 T2E,idle (µs) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 F0,j (%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='9 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='1 F1,j (%) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='9 esq (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='08 Experimental details We use 3 qubits out of the 9-qubit computer for the 2-site Holstein model ˆH = −V (a† 1a2 + a† 2a1) + ωb† 1b1 + ωb† 2b2 + gωa† 1a1(b† 1 + b1) + gωa† 2a2(b† 2 + b2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (38) The electronic degree of freedom is mapped to the second qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Thus, a† 1a1 is mapped to 1 2(1 + Z1) and a† 2a2 is mapped to 1 2(1 − Z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The phonon modes are mapped to the first and the third qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' With binary encoding and 11 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Two qubits gate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ωc,idle are the idle frequencies for each coupler where the ZZ interaction between neighboring computational qubits are maximally suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' ξZZ is the residual ZZ interaction between each qubit pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Two-qubit gates are implemented with the controlled-Z (CZ) and the corresponding gate errors etq,CZ are characterized with RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Q0 − Q1 Q0 − Q2 Q0 − Q3 Q0 − Q4 Q1 − Q5 Q2 − Q6 Q3 − Q7 Q4 − Q8 ωc,idle (GHz) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='020 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='445 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='570 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='335 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='325 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='595 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='695 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='355 |ξZZ| (kHz) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='0 etq,CZ (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='96 Nl = 1, the Hamiltonian in the Pauli string form reads ˆH = −V X1 + 1 2ω(1 − Z0) + 1 2ω(1 − Z2) + 1 2gω(1 + Z1)X0 + 1 2gω(1 − Z1)X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (39) For variational encoding, we assume C[l] = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' That is, the two modes share the same variational encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' This is a reasonable assumption for translational invariant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Supposing ˆb†ˆb and ˆb† +ˆb are mapped to the following form ˆB(ˆb†ˆb) ˆB† = c1iI + c1xX + c1zZ ˆB(ˆb† + ˆb) ˆB† = c2iI + c2xX + c2zZ , (40) the encoded Hamiltonian is then ˆH = −V X1 + ω(c1iI0 + c1xX0 + c1zZ0) + ω(c1iI2 + c1xX2 + c1zZ2) + 1 2gω(1 + Z1)(c2iI0 + c2xX0 + c2zZ0) + 1 2gω(1 − Z1)(c2iI2 + c2xX2 + c2zZ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (41) We use the following ansatz for the parameterized quantum circuit |φ⟩ = 2 � j=1 eθja† jaj(b† j−bj) 1 √ 2 (|000⟩ + |100⟩) , (42) Because C[1] = C[2], the parameter space can be further simplified by setting θ1 = θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' With binary encoding, the ansatz transforms to |φ⟩ = eiθY2e−iθZ1Y2eiθY0eiθZ1Y0H1 |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' (43) The ansatz is compiled into the following quantum circuit with 4 CNOT gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' q0 : RZ( −π 2 ) H RZ(−θ) RZ(−θ) H RZ( π 2 ) q1 : H q2 : RZ( −π 2 ) H RZ(θ) RZ(−θ) H RZ( π 2 ) Each energy term is measured by 8192 shots, and the uncertainty is obtained by repeating the measurement 5 times and taking the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For the update of C[l], 4096 shots are performed for each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Local readout error mitigation is applied for all results presented unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 4 we plot the energy landscape E(θ)/V in VQE with binary encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Both raw data and data with local readout error mitigation (EM) are presented for the energy expectation from quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The mitigated landscape is in decent agreement with the perfect simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' A minimum at around θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='6 is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' We note that the perfect simulator is also based on the Nl = 1 ansatz and N is far smaller than what is physically demanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' Thus the minimum presented by the perfect simulator can not be recognized as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content='00 θ −3 −2 −1 0 E/V Simulator Hardware (raw) Hardware (EM) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' VQE energy landscape for the 2-site Holstein model with binary encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' For the data from quantum hardware, both raw data and data with readout error mitigation are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} +page_content=' The error bar indicates the measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfevyy/content/2301.01442v1.pdf'} diff --git a/1tFQT4oBgHgl3EQf1jZM/content/tmp_files/2301.13420v1.pdf.txt b/1tFQT4oBgHgl3EQf1jZM/content/tmp_files/2301.13420v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb3905ce76db3dd7a08a68da0747a504cd73e43f --- /dev/null +++ b/1tFQT4oBgHgl3EQf1jZM/content/tmp_files/2301.13420v1.pdf.txt @@ -0,0 +1,2265 @@ +Superhuman Fairness +Omid Memarrast 1 Linh Vu 1 Brian Ziebart 1 +Abstract +The fairness of machine learning-based decisions +has become an increasingly important focus in +the design of supervised machine learning meth- +ods. Most fairness approaches optimize a spec- +ified trade-off between performance measure(s) +(e.g., accuracy, log loss, or AUC) and fairness met- +ric(s) (e.g., demographic parity, equalized odds). +This begs the question: are the right performance- +fairness trade-offs being specified? We instead re- +cast fair machine learning as an imitation learning +task by introducing superhuman fairness, which +seeks to simultaneously outperform human de- +cisions on multiple predictive performance and +fairness measures. We demonstrate the benefits +of this approach given suboptimal decisions. +1. Introduction +The social impacts of algorithmic decisions based on ma- +chine learning have motivated various group and individ- +ual fairness properties that decisions should ideally satisfy +(Calders et al., 2009; Hardt et al., 2016). Unfortunately, im- +possibility results prevent multiple common group fairness +properties from being simultaneously satisfied (Kleinberg +et al., 2016). Thus, no set of decisions can be universally fair +to all groups and individuals for all notions of fairness. In- +stead, specified weightings, or trade-offs, of different criteria +are often optimized (Liu & Vicente, 2022). Identifying an +appropriate trade-off to prescribe to these fairness methods +is a daunting task open to application-specific philosophical +and ideological debate that could delay or completely derail +the adoption of algorithmic methods. +We consider the motivating scenario of a fairness-aware deci- +sion task currently being performed by well-intentioned, but +inherently error-prone human decision makers. Rather than +seeking optimal decisions for specific performance-fairness +trade-offs, which may be difficult to accurately elicit, we +propose a more modest, yet more practical objective: out- +perform human decisions across all performance and +1Computer Science Department, University of Illinois Chicago. +Correspondence to: O. Memarrast . +Figure 1. Three sets of +decisions (black dots) +with different predictive +performance and group +disparity values defining +the sets of 100%-, 67%-, +and +33%-superhuman +fairness-performance +values (red shades) based +on Pareto dominance. +fairness measures with maximal frequency. We implic- +itly assume that available human decisions reflect desired +performance-fairness trade-offs, but are often noisy and sub- +optimal. This provides an opportunity for superhuman +decisions that Pareto dominate human decisions across pre- +dictive performance and fairness metrics (Figure 1) without +identifying an explicit desired trade-off. +To the best of our knowledge, this paper is the first to define +fairness objectives for supervised machine learning with +respect to noisy human decisions rather than using prescrip- +tive trade-offs or hard constraints. We leverage and extend a +recently-developed imitation learning method for subdomi- +nance minimization (Ziebart et al., 2022). Instead of using +the subdominance to identify a target trade-off, as previous +work does in the inverse optimal control setting to estimate +a cost function, we use it to directly optimize our fairness- +aware classifier. We develop policy gradient optimization +methods (Sutton & Barto, 2018) that allow flexible classes +of probabilistic decision policies to be optimized for given +sets of performance/fairness measures and demonstrations. +We conduct extensive experiments on standard fairness +datasets (Adult and COMPAS) using accuracy as a per- +formance measure and three conflicting fairness definitions: +Demographic Parity (Calders et al., 2009), Equalized Odds +(Hardt et al., 2016), and Predictive Rate Parity (Choulde- +chova, 2017)). Though our motivation is to outperform hu- +man decisions, we employ a synthetic decision-maker with +differing amounts of label and group membership noise to +identify sufficient conditions for superhuman fairness of +varying degrees. We find that our approach achieves high +levels of superhuman performance that increase rapidly with +reference decision noise and significantly outperform the +superhumanness of other methods that are based on more +arXiv:2301.13420v1 [cs.LG] 31 Jan 2023 + +Superhuman Fairness +narrow fairness-performance objectives. +2. Fairness, Elicitation, and Imitation +2.1. Group Fairness Measures +Group fairness measures are primarily defined by confu- +sion matrix statistics (based on labels yi ∈ {0, 1} and +decisions/predictions ˆyi ∈ {0, 1} produced from inputs +xi ∈ RM) for examples belonging to different protected +groups (e.g., ai ∈ {0, 1}). +We focus on three prevalent fairness properties in this paper: +• Demographic Parity (DP) (Calders et al., 2009) requires +equal positive rates across protected groups: +P( ˆY = 1|A = 1) = P( ˆY = 1|A = 0); +• Equalized Odds (EqOdds) (Hardt et al., 2016) requires +equal true positive rates and false positive rates across +groups, i.e., +P( ˆY =1|Y =y, A=1) = P( ˆY =1|Y =y, A=0), y ∈ {0, 1}; +• Predictive Rate Parity (PRP) (Chouldechova, 2017) re- +quires equal positive predictive value (ˆy = 1) and negative +predictive value (ˆy = 0) across groups: +P(Y =1|A=1, ˆY = ˆy) = P(Y =1|A=0, ˆY = ˆy), +ˆy ∈ {0, 1}. +Violations of these fairness properties can be measured as +differences: +D.DP(ˆy, a) = +����� +�N +i=1 I [ˆyi =1, ai =1] +�N +i=1 I [ai =1] +(1) +− +�N +i=1 I [ˆyi =1, ai =0] +�N +i=1 I [ai =0] +�����; +D.EqOdds(ˆy, y, a) = max +y∈{0,1} +����� +�N +i=1 I [ˆyi =1, yi =y, ai =1] +�N +i=1 I [ai =1, yi =y] +− +�N +i=1 I [ˆyi =1, yi =y, ai =0] +�N +i=1 I [ai =0, yi =y] +�����; +(2) +D.PRP(ˆy, y, a) = max +y∈{0,1} +����� +�N +i=1 I [yi =1, ˆyi =y, ai =1] +�N +i=1 I [ai =1, ˆyi =y] +− +�N +i=1 I [yi =1, ˆyi =y, ai =0] +�N +i=1 I [ai =0, ˆyi =y] +�����. +(3) +2.2. Performance-Fairness Trade-offs +Numerous fair classification algorithms have been devel- +oped over the past few years, with most targeting one fair- +ness metric (Hardt et al., 2016). With some exceptions +(Blum & Stangl, 2019), predictive performance and fairness +are typically competing objectives in supervised machine +learning approaches. Thus, though satisfying many fairness +properties simultaneously may be na¨ıvely appealing, doing +so often significantly degrades predictive performance or +even creates infeasibility (Kleinberg et al., 2016). +Given this, many approaches seek to choose parameters θ +for (probabilistic) classifier Pθ that balance the competing +predictive performance and fairness objectives (Kamishima +et al., 2012; Hardt et al., 2016; Menon & Williamson, 2018; +Celis et al., 2019; Martinez et al., 2020; Rezaei et al., 2020). +Recently, Hsu et al. (2022) proposed a novel optimization +framework to satisfy three conflicting fairness metrics (de- +mographic parity, equalized odds, and predictive rate parity) +to the best extent possible: +min +θ +Eˆy∼Pθ +� +loss(ˆy, y) + αDPD.DP(ˆy, a) +(4) ++ αOddsD.EqOdds(ˆy, y, a) + αPRPD.PRP(ˆy, y, a) +� +. +2.3. Preference Elictation & Imitation Learning +Preference elicitation (Chen & Pu, 2004) is one natural ap- +proach to identifying desirable performance-fairness trade- +offs. Preference elicitation methods typically query users +for their pairwise preference on a sequence of pairs of op- +tions to identify their utilities for different characteristics of +the options. This approach has been extended and applied to +fairness metric elicitation (Hiranandani et al., 2020), allow- +ing efficient learning of linear (e.g., Eq. (4)) and non-linear +metrics from finite and noisy preference feedback. +Imitation learning (Osa et al., 2018) is a type of supervised +machine learning that seeks to produce a general-use policy +ˆπ based on demonstrated trajectories of states and actions, +˜ξ = (˜s1, ˜a1, ˜s2, . . . , ˜sT ). Inverse reinforcement learning +methods (Abbeel & Ng, 2004; Ziebart et al., 2008) seek +to rationalize the demonstrated trajectories as the result +of (near-) optimal policies on an estimated cost or reward +function. Feature matching (Abbeel & Ng, 2004) plays a key +role in these methods, guaranteeing if the expected feature +counts match, the estimated policy ˆπ will have an expected +cost under the demonstrator’s unknown fixed cost function +weights ˜w ∈ RK equal to the average of the demonstrated +trajectories: +Eξ∼ˆπ [fk(ξ)] = 1 +N +N +� +i=1 +fk +� +˜ξi +� +, ∀k +(5) +=⇒ Eξ∼ˆπ [cost ˜ +w(ξ)] = 1 +N +N +� +i=1 +cost ˜ +w +� +˜ξi +� +, +where fk(ξ) = � +st∈ξ fk (st). +Syed & Schapire (2007) seeks to outperform the set of +demonstrations when the signs of the unknown cost function + +Superhuman Fairness +are known, ˜wk ≥ 0, by making the inequality, +Eξ∼π [fk(ξ)] ≤ 1 +N +N +� +i=1 +fk +� +˜ξi +� +, ∀k, +(6) +strict for at least one feature. Subdominance minimization +(Ziebart et al., 2022) seeks to produce trajectories that out- +perform each demonstration by a margin: +fk(ξ) + mk ≤ fk(˜ξi), ∀i, k, +(7) +under the same assumption of known cost weight signs. +However, since this is often infeasible, the approach in- +stead minimizes the subdominance, which measures the +α-weighted violation of this inequality: +subdomα(ξ, ˜ξ) ≜ +� +k +� +αk +� +fk(ξ) − fk(˜ξ) +� ++ 1 +� ++ , (8) +where [f(x)]+ ≜ max(f(x), 0) is the hinge function and +the per-feature margin has been reparameterized as α−1 +k . +Previous work (Ziebart et al., 2022) has employed subdom- +inance minimization in conjunction with inverse optimal +control: +min +w min +α +N +� +i=1 +K +� +k=1 +subdomα(ξ∗(w), ˜ξi), where: +ξ∗(w) = argmin +ξ +� +k +wkfk(ξ), +learning the cost function parameters w for the optimal tra- +jectory ξ∗(w) that minimizes subdominance. One contribu- +tion of this paper is extending subdominance minimization +to the more flexible prediction models needed for fairness- +aware classification that are not directly conditioned on cost +features or performance/fairness metrics. +3. Subdominance Minimization for Improved +Fairness-Aware Classification +We approach fair classification from an imitation learning +perspective. We assume vectors of (human-provided) ref- +erence decisions are available that roughly reflect desired +fairness-performance trade-offs, but are also noisy. Our +goal is to construct a fairness-aware classifier that outper- +forms reference decisions on all performance and fairness +measures on withheld data as frequently as possible. +3.1. Superhumanness and Subdominance +We consider reference decisions ˜y = {˜yj}M +j=1 that are +drawn from a human decision-maker or baseline method ˜P, +on a set of M items, XM×L = {xj}M +j=1, where L is the num- +ber of attributes in each of M items xj. Group membership +Figure 2. A Pareto fron- +tier for possible ˆPθ (blue) +optimally trading off pre- +dictive performance (e.g., +inaccuracy) and group +unfairness. The model- +produced decision (red +point) defines dominance +boundaries (solid red) +and margin boundaries +(dashed red), which in- +cur subdominance (green +lines) on three examples. +attributes am from vector a indicate to which group item m +belongs. +The predictive performance and fairness of decisions ˆy for +each item are assessed based on ground truth y and group +membership a using a set of predictive loss and unfairness +measures {fk(ˆy, y, a)}. +Definition 3.1. A fairness-aware classifier is considered γ- +superhuman for a given set of predictive loss and unfairness +measures {fk} if its decisions ˆy satisfy: +P (f (ˆy, y, a) ⪯ f (˜y, y, a)) ≥ γ. +If strict Pareto dominance is required to be γ-superhuman, +which is often effectively true for continuous domains, then +by definition, at most (1 − γ)% of human decision makers +could be γ-superhuman. However, far fewer than (1 − γ) +may be γ−superhuman if pairs of human decisions do not +Pareto dominate one another in either direction (i.e., neither +f (˜y, y, a) ⪯ f (˜y′, y, a) nor f (˜y′, y, a) ⪯ f (˜y, y, a) +for pairs of human decisions ˜y and ˜y′). From this perspec- +tive, any decisions with γ−superhuman performance more +than (1 − γ)% of the time exceed the performance limit for +the distribution of human demonstrators. +Unfortunately, directly maximizing γ is difficult in part +due to the discontinuity of Pareto dominance (⪯). The +subdominance (Ziebart et al., 2022) serves as a convex upper +bound for non-dominance in each metric {fk} and on 1 − γ +in aggregate: +subdomk +αk(ˆy, ˜y, y, a) ≜ [αk (fk(ˆy, y, a) − fk(˜y, y, a)) + 1]+ . +subdomα(ˆy, ˜y, y, a) ≜ +� +k +subdomk +αk(ˆy, ˜y, y, a). +(9) +Given N vectors of reference decisions as demonstrations, +˜Y = {˜yi}N +i=1, the subdominance for decision vector ˆy with +respect to the set of demonstrations is1 +subdomα(ˆy, ˜Y, y, a) = 1 +N +� +˜y∈ ˜ +Y +subdomα(ˆy, ˜y, y, a), +1For notational simplicity, we assume all demonstrated deci- +sions ˜y ∈ ˜Y correspond to the same M items represented in X. +Generalization to unique X for each demonstration is straightfor- +ward. + +Superhuman Fairness +where ˆyi is the predictions produced by our model for the +set of items Xi, and ˆY is the set of these prediction sets, +ˆY = {ˆyi}N +i=1. The subdominance is illustrated by Figure 2. +Following concepts from support vector machines (Cortes & +Vapnik, 1995), reference decisions ˜y that actively constrain +the predictions ˆy in a particular feature dimension, k, are +referred to as support vectors and denoted as: +˜YSVk(ˆy, αk) = +� +˜y|αk(fk(ˆy) − fk(˜y)) + 1 ≥ 0 +� +. +3.2. Performance-Fairness Subdominance +Minimization +We consider probabilistic predictors Pθ : X M → ∆YM +that make structured predictions over the set of items in +the most general case, but can also be simplified to make +conditionally independent decisions for each item. +Definition 3.2. The minimally subdominant fairness-aware +classifier ˆPθ has model parameters θ chosen by: +argmin +θ +min +α⪰0 Eˆy|X∼Pθ +� +subdomα,1 +� +ˆy, ˜Y, y, a +�� ++ λ∥α∥1. +Hinge loss slopes α ≜ {αk}K +k=1 are also learned from the +data during training. For subdominance of kth feature, αk +indicates the degree of sensitivity to how much the algo- +rithm fails to sufficiently outperform demonstrations in that +feature. When αk value is higher, the algorithm chooses that +feature to minimize subdominance. In our setting, features +are loss/violation metrics defined to measure the perfor- +mance/fairness of a set of reference decisions. +We use the subgradient of subdominance with respect to θ +and α to update these variables iteratively, and after con- +vergence, the best learned weights θ∗ are used in the final +model ˆPθ∗. A commonly used model like logistic regression +can be used for ˆPθ. +Theorem 3.3. The gradient of expected subdominance un- +der Pθ with respect to the set of reference decisions {˜yi}N +i=1 +is: +∇θEˆy|X∼ ˆ +Pθ +� +���� +� +k +Γk(ˆy, ˜ +Y,y,a) +� +�� +� +min +αk +� +subdomk +αk +� +ˆy, ˜Y, y, a +� ++ λkαk +� +� +���� += Eˆy|X∼ ˆ +Pθ +� �� +k +Γk(ˆy, ˜Y, y, a) +� +∇θ log ˆPθ(ˆy|X) +� +, +where the optimal αk for each γk is obtained from: +αk = argmin +α(m) +k +m such that fk (ˆy) + λ ≤ 1 +m +m +� +j=1 +fk +� +˜y(j)� +, +using α(j) +k += +1 +fk(ˆy(j))−fk(˜y(j)) to represent the αk value +that would make the demonstration with the jth smallest fk +feature, ˜y(j), a support vector with zero subdominance. +Using gradient descent, we update the model weights θ +using an approximation of the gradient based on a set of +sampled predictions ˆy ∈ ˆY from the model ˆPθ: +θ ← θ + η +� +�� +ˆy∈ ˆ +Y +�� +k +Γk(ˆy, ˜Y, y, a) +� +∇θ log ˆPθ(ˆy|X) +� +� , +We show the steps required for the training of our model in +Algorithm 1. Reference decisions {˜yi}N +i=1 from a human +decision-maker or baseline method ˜P are provided as input +to the algorithm. +θ is set to an initial value. In each +iteration of the algorithm, we first sample a set of model +predictions {ˆyi}N +i=1 from ˆPθ(.|Xi) for the matching items +used for reference decisions {˜yi}N +i=1. We then find the new +θ (and α) based on the algorithms discussed in Theorem +3.3. +Algorithm 1: Subdominance policy gradient opti- +mization +Draw N set of reference decisions {˜yi}N +i=1 from a +human decision-maker or baseline method ˜P. +Initialize: θ ← θ0; +while θ not converged do +Sample model predictions {ˆyi}N +i=1 from +ˆPθ(.|Xi) for the matching items used in +reference decisions {˜yi}N +i=1; +for k ∈ {1, ..., K} do +Sort reference decisions {˜yi}N +i=1 in +ascending order based on their kth feature +value fk(˜yi): {˜y(j)}N +j=1; +Compute α(j) +k += +1 +fk(˜y(j))−fk(ˆy(j)); +αk = argmin +α(m) +k +m +such that fk +�ˆy(j)� +≤ 1 +m +�m +j=1 fk +�˜y(j)� +; +Compute Γk(ˆyi, ˜Y, y, a); +θ ← θ + +η +N +� +i +�� +k Γk(ˆyi, ˜Y, y, a) +� +∇θ log ˆPθ(ˆyi|Xi); +3.3. Generalization Bounds +With a small effort, we extend the generalization bounds +based on support vectors developed for inverse optimal con- +trol subdominance minimization (Ziebart et al., 2022). +Theorem 3.4. +A classifier +ˆPθ +trained to minimize +subdomα (ˆy, ˜yi) on a set of N iid reference decisions +has the support vector set +�� +ˆy:Pθ(ˆy|X)>0 ˜YSVk (ˆy, αk) +� +defined by the union of support vectors for any decision +with support under ˆPθ. Such a classifier is on average γ- +superhuman on the population distribution with: γ = 1− +1 +N +����K +k=1 +� +ˆy:Pθ(ˆy|X)>0 ˜YS Vk (ˆy, αk) +���. +This generalization bound requires overfitting to the training + +Superhuman Fairness +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +DP +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +EqOdds +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +D.PRP +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +PRP +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +D.DP +0.35 +0.40 +0.45 +0.50 +0.55 +Prediction error +1/ +DP +1/ +error +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +D.EqOdds +0.35 +0.40 +0.45 +0.50 +0.55 +Prediction error +1/ +EqOdds +1/ +error +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +D.PRP +0.35 +0.40 +0.45 +0.50 +0.55 +Prediction error +1/ +PRP +1/ +error +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +Figure 3. Prediction error versus difference of: Demographic Parity (D.DP), Equalized Odds (D.EqOdds) and Predictive Rate Parity +(D.PR) on test data using noiseless training data (ϵ = 0) for Adult (top row) and COMPAS (bottom row) datasets. +data so that the ˆPθ has restricted support (i.e., ˆPθ(ˆy|X) = 0 +for many ˆy) or becomes deterministic. +4. Experiments +The goal of our approach is to produce a fairness-aware +prediction method that outperforms reference (human) de- +cisions on multiple fairness/performance measures. In this +section, we discuss our experimental design to synthesize +reference decisions with varying levels of noise, evaluate +our method, and provide comparison baselines. +4.1. Training and Testing Dataset Construction +To emulate human decision-making with various levels of +noise, we add noise to the ground truth data of benchmark +fairness datasets and apply fair learning methods over re- +peated randomized dataset splits. We describe this process +in detail in the following section. +Datasets +We perform experiments on two benchmark fair- +ness datasets: +• UCI Adult dataset (Dheeru & Karra Taniskidou, 2017) +considers predicting whether a household’s income is +higher than $50K/yr based on census data. Group mem- +bership is based on gender. The dataset consists of 45,222 +items. +• COMPAS dataset (Larson et al., 2016) considers predict- +ing recidivism with group membership based on race. It +consists of 6,172 examples. +Partitioning the data +We first split entire dataset +randomly into a disjoint train (train-sh) and test +(test-sh) set of equal size. The test set (test-sh) is +entirely withheld from the training procedure and ultimately +used solely for evaluation. To produce each demonstration +(a vector of reference decisions), we split the (train-sh) +set, randomly into a disjoint train (train-pp) and test +(test-pp) set of equal size. +Noise insertion +We randomly flip ϵ% of the ground truth +labels y and group membership attributes a to add noise to +our demonstration-producing process. +Fair classifier ˜P: +Using the noisy data, we provide ex- +isting fairness-aware methods with labeled train-pp +data and unlabeled test-pp to produce decisions on the +test-pp data as demonstrations ˜y. Specifically, we em- +ploy: +• The Post-processing method of Hardt et al. (2016), which +aims to reduce both prediction error and {demographic +parity or equalized odds} at the same time. We use de- +mographic parity as the fairness constraint. We produce +demonstrations using this method for Adult dataset. +• Robust fairness for logloss-based classification (Rezaei +et al., 2020) employs distributional robustness to match +target fairness constraint(s) while robustly minimizing the +log loss. We use equalized odds as the fairness constraint. + +Superhuman Fairness +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.DP +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Prediction error +1/ +DP +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Prediction error +1/ +EqOdds +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +D.PRP +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Prediction error +1/ +PRP +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +D.DP +0.35 +0.40 +0.45 +0.50 +0.55 +Prediction error +1/ +DP +1/ +error +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +D.EqOdds +0.35 +0.40 +0.45 +0.50 +0.55 +Prediction error +1/ +EqOdds +1/ +error +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +D.PRP +0.35 +0.40 +0.45 +0.50 +0.55 +Prediction error +1/ +PRP +1/ +error +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +Figure 4. Experimental results on the Adult and COMPAS datasets with noisy demonstrations (ϵ = 0.2). Margin boundaries are shown +with dotted red lines. Each plot shows the relationships between two features. +We employ this method to produce demonstrations for +COMPAS dataset. +We repeat the process of partitioning train-sh N = +50 times to create randomized partitions of train-pp +and test-pp and then produce a set of demonstrations +{˜y}50 +i=1. +4.2. Evaluation Metrics and Baselines +Predictive Performance and Fairness Measures +Our +focus for evaluation is on outperforming demonstrations in +multiple fairness and performance measures. We use K = 4 +measures: inaccuracy (Prediction error), difference +from demographic parity (D.DP), difference from equalized +odds (D.EqOdds), difference from predictive rate parity +(D.PRP). +Baseline methods +As baseline comparisons, we train five +different models on the entire train set (train-sh) and +then evaluate them on the withheld test data (test-sh): +• The Post-processing model of (Hardt et al., 2016) +with demographic parity as the fairness constraint +(post proc dp). +• The Post-processing model of (Hardt et al., 2016) +with +equalized +odds +as +the +fairness +constraint +(post proc eqodds). +• The Robust Fair-logloss model of (Rezaei et al., 2020) +with demographic parity as the fairness constraint +(fair logloss dp). +• The Robust Fair-logloss model of (Rezaei et al., +2020) +equalized +odds +as +the +fairness +constraint +(fair logloss eqodds). +• The Multiple Fairness Optimization framework of Hsu +et al. (2022) which is designed to satisfy three conflict- +ing fairness metrics (demographic parity, equalized odds +and predictive rate parity) to the best extent possible +(MFOpt). +Hinge Loss Slopes +As discussed previously, αk value cor- +responds to the hinge loss slope, which defines by how far +a produced decision does not sufficiently outperform the +demonstrations on the kth feature. When the αk is large, the +model chooses heavily weights support vector reference de- +cisions for that particular k when minimizing subdominance. +We report these values in our experiments. +4.3. Superhuman Model Specification and Updates +We use a logistic regression model Pθ0 with first-order mo- +ments feature function, φ(y, x) = [x1y, x2y, . . . xmy]⊤, +and weights θ applied independently on each item as our +decision model. During the training process, we update the +model parameter θ to reduce subdominance. +Sample from Model ˆPθ +In each iteration of the algorithm, +we first sample prediction vectors {ˆyi}N +i=1 from ˆPθ(.|Xi) + +Superhuman Fairness +Table 1. Experimental results on noise-free datasets, along with the αk values learned for each feature in subdominance minimization. +Method +Dataset +Adult +COMPAS +Prediction error +DP diff +EqOdds diff +PRP diff +Prediction error +DP diff +EqOdds diff +PRP diff +αk +62.62 +35.93 +6.46 +4.98 +82.5 +4.27 +3.15 +12.72 +γ-superhuman +98% +94% +100% +100% +100% +100% +100% +100% +MinSub-Fair (ours) +0.210884 +0.025934 +0.006690 +0.183138 +0.366806 +0.040560 +0.124683 +0.171177 +MFOpt +0.195696 +0.063152 +0.077549 +0.209199 +0.434743 +0.005830 +0.069519 +0.161629 +post proc dp +0.212481 +0.030853 +0.220357 +0.398278 +0.345964 +0.010383 +0.077020 +0.173689 +post proc eqodds +0.213873 +0.118802 +0.007238 +0.313458 +0.363395 +0.041243 +0.060244 +0.151040 +fair logloss dp +0.281194 +0.004269 +0.047962 +0.124797 +0.467610 +0.000225 +0.071392 +0.172418 +fair logloss eqodds +0.254060 +0.153543 +0.030141 +0.116579 +0.451496 +0.103093 +0.029085 +0.124447 +for the matching items used in demonstrations {˜yi}N +i=1. In +the implementation, to produce the ith sample, we look up +the indices of the items used in ˜yi, which constructs item set +Xi. Now we make predictions using our model on this item +set ˆPθ(.|Xi). The model produces a probability distribution +for each item which can be sampled and used as a prediction +{ˆyi}N +i=1. +Update model parameters θ +We update θ until conver- +gence using Algorithm 1. For our logistic regression model, +the gradient is: +∇θ log ˆPθ(ˆy|X) = φ(ˆy, X) − Eˆy|X∼ ˆ +Pθ [φ(ˆy, X)] , +where φ denotes the feature function and φ(ˆy, X) = +�M +m=1 φ(ˆym, xm) is the corresponding feature function +for the ith set of reference decisions. +4.4. Experimental Results +After training each model, e.g., obtaining the best +model weight θ∗ from the training data (train-sh) +for superhuman, we evaluate each on unseen test data +(test-sh). We employ hard predictions (i.e., the most +probable label) using our approach at time time rather than +randomly sampling. +Noise-free reference decisions +Our first set of experi- +ments considers learning from reference decisions with no +added noise. The results are shown in Figure 3. We ob- +serve that our approach outperforms demonstrations in all +fairness metrics and shows comparable performance in accu- +racy. The (post proc dp) performs almost as an average +of demonstrations in all dimensions, hence our approach +can outperform it in all fairness metrics. In comparison +to (post proc dp), our approach can outperform in all +fairness metrics but is slightly worse in prediction error. +We show the experiment results along with αk values in +Table 1. Note that the margin boundaries (dotted red lines) +in Figure 3 are equal to +1 +αk for feature k, hence there is re- +verse relation between αk and margin boundary for feature +k. We observe larger values of αk for prediction error and +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Noise Ratio +0.86 +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +-Superhumn +Adult +Predictive value difference +Equalized odds difference +Demographic parity difference +ZeroOne +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Noise Ratio +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +-Superhumn +Adult +Predictive value difference +Equalized odds difference +Demographic parity difference +ZeroOne +Figure 5. The relationship between the ratio of augmented noise +in the label and the protected attribute of reference decisions +produced by post-processing (upper) and fair-logloss (lower) +and achieving γ-superhuman performance in our approach. +demographic parity difference. The reason is that these fea- +tures are already optimized in demonstrations and our model +has to increase αk values for those features to sufficiently +outperform them. +Noisy reference decisions +In our second set of experi- +ments, we introduce significant amounts of noise (ϵ = 0.2) +into our reference decisions. The results for these experi- +ments are shown in Figure 4. We observe that in the case of +learning from noisy demonstrations, our approach still out- +performs the reference decisions. The main difference here +is that due to the noisy setting, demonstrations have worse +prediction error but regardless of this issue, our approach + +Superhuman Fairness +Table 2. Experimental results on datasets with noisy demonstrations, along with the αk values learned for each feature. +Method +Dataset +Adult +COMPAS +Prediction error +DP diff +EqOdds diff +PRP diff +Prediction error +DP diff +EqOdds diff +PRP diff +αk +29.63 +10.77 +5.83 +13.42 +29.33 +4.51 +3.34 +57.74 +γ-superhuman +100% +100% +100% +100% +100% +100% +100% +98% +MinSub-Fair (ours) +0.202735 +0.030961 +0.009263 +0.176004 +0.359985 +0.031962 +0.036680 +0.172286 +MFOpt +0.195696 +0.063152 +0.077549 +0.209199 +0.459731 +0.091892 +0.039745 +0.153257 +post proc dp +0.225462 +0.064232 +0.237852 +0.400427 +0.353164 +0.087889 +0.088414 +0.160538 +post proc eqodds +0.224561 +0.103158 +0.010552 +0.310070 +0.351269 +0.144190 +0.158372 +0.148493 +fair logloss dp +0.285549 +0.007576 +0.057659 +0.115751 +0.484620 +0.005309 +0.145502 +0.183193 +fair logloss eqodds +0.254577 +0.147932 +0.012778 +0.118041 +0.487025 +0.127163 +0.011918 +0.153869 +Table 3. Percentage of reference demonstrations that each method outperforms in all prediction/fairness measures. +Method +Adult(ϵ = 0.0) +Adult(ϵ = 0.2) +COMPAS(ϵ = 0.0) +COMPAS(ϵ = 0.2) +MinSub-Fair (ours) +96% +100% +100% +98% +MFOpt +42% +0% +18% +18% +post proc dp +16% +86% +100% +80% +post proc eqodds +0% +66% +100% +88% +fair logloss dp +0% +0% +0% +0% +fair logloss eqodds +0% +0% +0% +0% +still can achieve a competitive prediction error. We show +the experimental results along with αk values in Table 2. +Relationship of noise to superhuman performance +We +also evaluate the relationship between the amount of aug- +mented noise in the label and protected attribute of demon- +strations, with achieving γ-superhuman performance in our +approach. As shown in Figure 5, with slightly increasing the +amount of noise in demonstrations, our approach can outper- +form 100% of demonstrations and reach to 1-superhuman +performance. In Table 3 we show the percentage of demon- +strations that each method can outperform across all predic- +tion/fairness measures (i.e., the γ−superhuman value). +5. Conclusions +In this paper, we introduce superhuman fairness, an ap- +proach to fairness-aware classifier construction based on im- +itation learning. Our approach avoids explicit performance- +fairness trade-off specification or elicitation. Instead, it +seeks to unambiguously outperform human decisions across +multiple performance and fairness measures with maximal +frequency. We develop a general framework for pursuing +this based on subdominance minimization (Ziebart et al., +2022) and policy gradient optimization methods (Sutton +& Barto, 2018) that enable a broad class of probabilistic +fairness-aware classifiers to be learned. Our experimental +results show the effectiveness of our approach in outper- +forming synthetic decisions corrupted by small amounts of +label and group-membership noise when evaluated using +multiple fairness criteria combined with predictive accuracy. +Societal impacts +By design, our approach has the po- +tential to identify fairness-aware decision-making tasks in +which human decisions can frequently be outperformed by +a learned classifier on a set of provided performance and +fairness measures. This has the potential to facilitate a tran- +sition from manual to automated decisions that are preferred +by all interested stakeholders, so long as their interests are +reflected in some of those measures. However, our approach +has limitations. First, when performance-fairness tradeoffs +can either be fully specified (e.g., based on first principles) +or effectively elicited, fairness-aware classifiers optimized +for those trade-offs should produce better results than our +approach, which operates under greater uncertainty cast by +the noisiness of human decisions. Second, if target fair- +ness concepts lie outside the set of metrics we consider, +our resulting fairness-aware classifier will be oblivious to +them. Third, our approach assumes human-demonstrated +decision are well-intentioned, noisy reflections of desired +performance-fairness trade-offs. If this is not the case, then +our methods could succeed in outperforming them across all +fairness measures, but still not provide an adequate degree +of fairness. +Future directions +We have conducted experiments with +a relatively small number of performance/fairness measures +using a simplistic logistic regression model. Scaling our ap- +proach to much larger numbers of measures and classifiers +with more expressive representations are both of great inter- +est. Additionally, we plan to pursue experimental validation +using human-provided fairness-aware decisions in addition +to the synthetically-produced decisions we consider in this +paper. + +Superhuman Fairness +References +Abbeel, P. and Ng, A. Y. Apprenticeship learning via inverse +reinforcement learning. In Proceedings of the Interna- +tional Conference on Machine Learning, pp. 1–8, 2004. +Blum, A. and Stangl, K. Recovering from biased data: Can +fairness constraints improve accuracy? arXiv preprint +arXiv:1912.01094, 2019. +Boyd, S. and Vandenberghe, L. Convex optimization. Cam- +bridge University Press, 2004. +Calders, T., Kamiran, F., and Pechenizkiy, M. Building +classifiers with independency constraints. In 2009 IEEE +International Conference on Data Mining Workshops, pp. +13–18. IEEE, 2009. +Celis, L. 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The gradient of the training objective with respect to model parameters θ is: +∇θEˆy|X∼ ˆ +Pθ +� +���� +� +k +Γk(ˆy, ˜ +Y,y,a) +� +�� +� +min +αk +� +subdomk +αk +� +ˆy, ˜Y, y, a +� ++ λkαk +� +� +���� = Eˆy|X∼ ˆ +Pθ +� �� +k +Γk(ˆy, ˜Y, y, a) +� +∇θ log ˆPθ(ˆy|X) +� +, +which follows directly from a property of gradients of logs of function: +∇θ log ˆP(ˆy|X) = +1 +ˆP(ˆy|X) +∇θˆP(ˆy|X) =⇒ ∇θˆPθ(ˆy|X) = ˆP(ˆy|X)∇θ log ˆP(ˆy|X). +(10) +We note that this is a well-known approach employed by policy-gradient methods in reinforcement learning (Sutton & Barto, +2018). +Next, we consider how to obtain the α−minimized subdominance for a particular tuple (ˆy, ˜Y,y,a), Γk +� +ˆy, ˜Y, y, a +� += +minαk +� +subdomk +αk +� +ˆy, ˜Y, y, a +� ++ λkαk +� +, analytically. +First, we note that subdomk +αk +� +ˆy, ˜Y, y, a +� ++ λkαk is comprised of hinged linear functions of αk, making it a convex +and piece-wise linear function of αk. This has two important implications: (1) any point of the function for which the +subgradient includes 0 is a global minimum of the function (Boyd & Vandenberghe, 2004); (2) an optimum must exist at a +corner of the function: αk = 0 or where one of the hinge functions becomes active: +αk(fk(ˆyi) − fk(˜yi)) + 1 = 0 =⇒ αk = +1 +fk(˜yi) − fk(ˆyi). +(11) +The subgradient for the jth of these points (ordered by fk value from smallest to largest and denoted fk(˜y(j)) for the +demonstration) is: +∂αk subdomk +αk +� +ˆy, ˜Y, y, a +� ��� +αk=(fk(ˆy)−fk(˜y(j)))−1 = ∂αk +� +1 +N +j +� +i=1 +� +αk +� +fk(ˆy) − fk(˜y(i)) +� ++ 1 +� ++ ++ λαk +� += λ + 1 +N +j−1 +� +i=1 +� +fk(ˆy) − fk(˜y(i)) +� ++ +� +0, fk(ˆy) − fk(˜y(j)) +� +, +where the final bracketed expression indicates the range of values added to the constant value preceding it. +The smallest j for which the largest value in this range is positive must contain the 0 in its corresponding range, and is thus +the provides the j value for the optimal αk value. +Proof of Theorem 3.4. We extend the leave-one-out generalization bound of Ziebart et al. (2022) by considering the set of +reference decisions that are support vectors for any learner decisions with non-zero probability. For the remaining reference +decisions that are not part of this set, removing them from the training set would not change the optimal model choice +and thus contribute zero error to the leave-one-out cross validation error, which is an almost unbiased estimate of the +generalization error (Vapnik & Chapelle, 2000). +B. Additional Results +In the main paper, we only included plots that show the relationship of a fairness metric with prediction error. To show the +relation between each pair of fairness metrics, in Figures 6 and 7 we show the remaining plots removed from Figures 3 and +4 respectively. + +Superhuman Fairness +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +1/ +EqOdds +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +D.PRP +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +1/ +PRP +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +D.PRP +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +1/ +PRP +1/ +EqOdds +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +D.EqOdds +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +D.DP +1/ +EqOdds +1/ +DP +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +D.PRP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +D.DP +1/ +PRP +1/ +DP +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +D.PRP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +D.EqOdds +1/ +PRP +1/ +EqOdds +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +Figure 6. The trade-off between each pair of: difference of Demographic Parity (D.DP), Equalized Odds (D.EqOdds) and Predictive +Rate Parity (D.PR) on test data using noiseless training data (ϵ = 0) for Adult (top row) and COMPAS (bottom row) datasets. +B.1. Experiment with more measures +Since our approach is flexible enough to accept wide range of fairness/performance measures, we extend the experiment on +Adult to K = 5 features. In this experiment we use Demographic Parity (D.DP), Equalized Odds (D.EqOdds), False +Negative Rate (D.FNR), False Positive Rate (D.FPR) and Prediction Error as the features to outperform reference decisions +on. The results are shown in Figure 8. + +Superhuman Fairness +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.DP +1/ +EqOdds +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +D.PRP +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.DP +1/ +PRP +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +D.PRP +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +1/ +PRP +1/ +EqOdds +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +D.EqOdds +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +D.DP +1/ +EqOdds +1/ +DP +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +D.PRP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +D.DP +1/ +PRP +1/ +DP +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.15 +0.20 +0.25 +0.30 +0.35 +D.PRP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +D.EqOdds +1/ +PRP +1/ +EqOdds +COMPAS +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +Figure 7. The trade-off between each pair of: difference of Demographic Parity (D.DP), Equalized Odds (D.EqOdds) and Predictive +Rate Parity (D.PR) on test data using noiseless training data (ϵ = 0.2) for Adult (top row) and COMPAS (bottom row) datasets. + +Superhuman Fairness +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +DP +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +EqOdds +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.FNR +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +FNR +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.FPR +0.20 +0.22 +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +Prediction error +1/ +FPR +1/ +error +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +1/ +EqOdds +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.FNR +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +1/ +FNR +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.FPR +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +D.DP +1/ +FPR +1/ +DP +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.FNR +1/ +EqOdds +1/ +FNR +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.FPR +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.FNR +1/ +FPR +1/ +FNR +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +D.EqOdds +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +D.FPR +1/ +EqOdds +1/ +FPR +Adult +fair_logloss_eqodds +fair_logloss_dp +post_proc_eqodds +post_proc_dp +MFOpt +post_proc_demos +superhuman_train +superhuman_test +Figure 8. The trade-off between each pair of: difference of Demographic Parity (D.DP), Equalized Odds (D.EqOdds), False Negative +Rate (D.FNR), False Positive Rate (D.FPR) and Prediction Error on test data using noiseless training data (ϵ = 0) for Adult dataset. + diff --git a/1tFQT4oBgHgl3EQf1jZM/content/tmp_files/load_file.txt b/1tFQT4oBgHgl3EQf1jZM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9a353c7a19c19137824f20ffde4144537569463 --- /dev/null +++ b/1tFQT4oBgHgl3EQf1jZM/content/tmp_files/load_file.txt @@ -0,0 +1,1182 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf,len=1181 +page_content='Superhuman Fairness Omid Memarrast 1 Linh Vu 1 Brian Ziebart 1 Abstract The fairness of machine learning-based decisions has become an increasingly important focus in the design of supervised machine learning meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Most fairness approaches optimize a spec- ified trade-off between performance measure(s) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=', accuracy, log loss, or AUC) and fairness met- ric(s) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=', demographic parity, equalized odds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' This begs the question: are the right performance- fairness trade-offs being specified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' We instead re- cast fair machine learning as an imitation learning task by introducing superhuman fairness, which seeks to simultaneously outperform human de- cisions on multiple predictive performance and fairness measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' We demonstrate the benefits of this approach given suboptimal decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Introduction The social impacts of algorithmic decisions based on ma- chine learning have motivated various group and individ- ual fairness properties that decisions should ideally satisfy (Calders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Unfortunately, im- possibility results prevent multiple common group fairness properties from being simultaneously satisfied (Kleinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Thus, no set of decisions can be universally fair to all groups and individuals for all notions of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' In- stead, specified weightings, or trade-offs, of different criteria are often optimized (Liu & Vicente, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Identifying an appropriate trade-off to prescribe to these fairness methods is a daunting task open to application-specific philosophical and ideological debate that could delay or completely derail the adoption of algorithmic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' We consider the motivating scenario of a fairness-aware deci- sion task currently being performed by well-intentioned, but inherently error-prone human decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Rather than seeking optimal decisions for specific performance-fairness trade-offs, which may be difficult to accurately elicit, we propose a more modest, yet more practical objective: out- perform human decisions across all performance and 1Computer Science Department, University of Illinois Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Correspondence to: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFQT4oBgHgl3EQf1jZM/content/2301.13420v1.pdf'} +page_content=' Memarrast 0, depending on the class. +We can compute with arbitrary precision the numerical values of R and C (see Sec- +tion 4.2). All the numerical values of R and C vary according to each class which confirms +that all these classes are significantly different. +Theorem 1.1 provides a precise estimation of OccH(G(n)) for every cograph H. But for +every graph H which is not a cograph, the only information given by the convergence in +the sense of graphon is that the number of induced H in G(n) is typically o(n|H|). Quite +unexpectedly, thanks to the tools developed to prove Theorem 1.1, we are able to estimate +the expected number of induced subgraphs isomorphic to a specific class of graphs H in +G(n): the graphs that are called ”prime” for the modular decomposition (see Definition 2.8). +Theorem 1.3. Let G(n) be a uniform graph of size n taken uniformly at random in one of +the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible or P4-reducible. Let H be +a prime graph, denote by OccH(G(n)) the number of labeled subgraphs of G(n) isomorphic +to H. + +回4 +TH´EO LENOIR +Then there exists KH ≥ 0 such that: +E[OccH(G(n))] ∼ +� +� +� +KHn +3 +2 +if H verifies condition (A) +KHn +otherwise +where (A) is defined p.38 and constant KH is given in Theorem 6.9. +This results follows from Theorem 6.9 which is stated in a more general setting. The +condition (A) depends on the class of graphs, checking if H verifies condition (A) and if +KH is positive is quite straightforward. +To make things more concrete, let us apply Theorem 1.3 to the example of H = P4. We +can check that for each class P4 does not verify condition (A). Thus a uniform random +graph contains in average a linear number of induced P4, while Theorem 1.1 only implies +that this number is o(n4). The different numerical values of KP4 are explicitly computed +p.41, and happen to take different values for each class. For each class, the graph called +”bull” (see Fig. 7) does not verify condition (A). Thus a uniform random graph contains +in average a number of induced bulls growing as n3/2, while Theorem 1.1 only implies that +this number is o(n5). However, for non prime graphs H, the behavior of the expected +value of induced subgraphs of G(n) isomorphic to H is not well-understood, which leads to +interesting open questions. +1.3. Proof strategy. The proof is essentially combinatorial and is based on modular de- +composition, which allows to encode a graph with a decorated tree. Modular decomposition +is a standard tool in graph theory (it was introduced in the 60’s by Gallai [9]) but to our +knowledge it has been very little used in the context of random graphs. In this paper we +introduce an enriched modular decomposition which enables us to obtain exact enumer- +ations for a large family of graph classes. The five classes mentioned before fit in this +framework. We exploit those enumerative results with tools from analytic combinatorics +to get asymptotic estimates in order to prove Theorem 1.2. +The more technical part of the proof is, for every finite graph H, to estimate the number +of induced subgraphs of G(n) isomorphic to H. The enriched modular decomposition allows +us to count the number of graphs with a specific induced subgraph H. Again asymptotics +are derived with tools from combinatorics to prove Theorem 1.1 and Theorem 1.3. +1.4. Outline of the paper. +• In Section 2 we define the encoding of graphs with trees, the modular decomposition +and the enriched modular decomposition which will be used throughout the different +proofs. +• Section 3 presents the necessary material on the different classes of graphs studied: +results are already widely known, most of them are quoted from the litterature and +reformulated to suit our enriched modular decomposition. +• Sections 4 and 5 are about calculating generating series related to our graph classes: +in Section 4 we prove Theorem 1.2 and Section 5 deals with the generating series +of graphs with a given induced subgraph. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +5 +• Section 6 presents the necessary material on graphons, and the proofs of Theo- +rem 1.1 and Theorem 1.3. +2. Modular decomposition of graphs: old and new +2.1. Labeled graphs. In the following all the graphs considered are simple and finite. +Each time a graph G is defined, we denote by V its set of vertices and E its set of edges. +Whenever there is an ambiguity, we denote by VG (resp. EG) the set of vertices (resp. edges) +of G. +Definition 2.1. We say that G = (V, E) is a weakly-labeled graph if every element of V +has a distinct label in N and that G = (V, E) is a labeled graph if every element of V has +a distinct label in {1, . . . , |V |}. +The size of a graph G, denoted by |G|, is its number of vertices. +The minimum of a graph G, denoted min(G), is the minimal label of its vertices. +In the following, every graph will be labeled, otherwise we will mention explicitly that +the graph is weakly-labeled. +Remark. We do not identify a vertex with its label. A vertex of label i will be denoted vi. +The label of a vertex v will be denoted ℓ(v). +Definition 2.2. For any weakly-labeled object (graph or tree) of size n, we call reduction +the operation that reduces its labels to the set {1, . . . , n} while preserving the relative order +of the labels. +For example if G labels 2, 4, 12, 63 then the reduced version of G is a copy of G in which +2, 4, 12, 63 are respectively replaced by 1, 2, 3, 4. +2.2. Encoding graphs with trees. +Definition 2.3. Let G be a graph of size n and H1, . . . , Hn be weakly-labeled graphs such +that no label is given to two distinct vertices of �n +i=1 Hi. The graph G[H1, . . . , Hn] = (V, E) +is the graph whose set of vertices is V = �n +i=1 VHi and such that: +• for every i ∈ {1, . . . , n} and every pair (v, v′) ∈ V 2 +Hi, {v, v′} ∈ E if and only if +{v, v′} ∈ EHi; +• For every (i, j) ∈ {1, . . . , n} with i ̸= j, and every pair (v, v′) ∈ VHi ×VHj, {v, v′} ∈ +E if and only if {vi, vj} ∈ EG. +Notation. In Definition 2.3 we will use the shortcut ⊕ for the complete graph of size n. +Thus ⊕[H1, . . . , Hn] is the graph obtained from copies of H1, . . . , Hn in which for every +i ̸= j every vertex of Hi is connected to every vertex of Hj. This graph is called the join +of H1, . . . , Hn +We use the shortcut ⊖ for the empty graph of size n. Thus ⊖[H1, . . . , Hn] is the graph +given by the disjoint union of H1, . . . , Hn This graph is called the union of H1, . . . , Hn. +This construction allows us to transform non-plane labeled trees with internal nodes +decorated with graphs, ⊕ and ⊖ into graphs. + +6 +TH´EO LENOIR +Definition 2.4. Let T0 be the set of rooted non-plane trees whose leaves have distinct labels +in N and whose internal nodes carry decorations satisfying the following constraints: +• internal nodes are decorated with ⊕, ⊖ or a graph; +• If a node is decorated with some graph G then |G| ≥ 2 and this node has |G| +children. If a node is decorated with ⊕ or ⊖ then it has at least 2 children. +A tree t ∈ T0 is called a substitution tree if the labels of its leaves are in {1, . . . , |t|}. +We call linear the internal nodes decorated with ⊕ or ⊖ and non-linear the other ones. +Notation. For a non-plane rooted tree t, and an internal node v of t, let tv be the multiset +of trees attached to v and let t[v] be the non-plane tree rooted at v containing only the +descendants of v in t. +Convention. We only consider non-plane trees. However it is sometimes convenient to +order the subtrees of a given node. The convention is that for some v in a tree t the trees +of tv are ordered according to their minimal leaf labels. +Definition 2.5. Let t be an element of T0, the weakly-labeled graph Graph(t) is inductively +defined as follows: +• if t is reduced to a single leaf labeled j, Graph(t) is the graph reduced to a single +vertex labeled j; +• otherwise, the root r of t is decorated with a graph H, and +Graph(t) = H[Graph(t1), . . . , Graph(t|H|)] +where ti is the i-th tree of tr. +1 +2 +7 +5 +8 +3 +4 +6 +9 +Root +1 +9 +6 +2 +3 +8 +7 +5 +4 +t0 +Graph(t0) +Figure 3. A substitution tree t0 and the corresponding graph Graph(t0) +Note that if t is a substitution tree then Graph(t) is a labeled graph. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +7 +The following simple Lemma is essential to the study of the enriched decomposition of +graphs introduced in Section 2.4. +Lemma 2.6. Let t be a substitution tree such that the decoration of the root of t (resp. its +complementary) is connected. Then Graph(t) (resp. its complementary) is connected. +Proof. Since both cases are similar, we only deal with the case of a connected decoration. +Let r be the root of t, H its decoration and k the size of H. Let w1, . . . , wk be vertices of +Graph(t) such that for each i ∈ {1, . . . , k} there is a leaf labeled ℓ(wi) in the i-th tree of tr. +Since the unlabeled graph induced by {wi | 1 ≤ i ≤ k} is isomorphic to H, it is connected. +Let C be the connected component of Graph(t) containing all wi’s. Note that for every +vertex v of Graph(t), there exists p ∈ {1, . . . k} such that the leaf labeled ℓ(v) belongs to +the p-th tree of tr. Since H is connected and of size at least 2, there exists q ̸= p such that +the vertices of label q and p are connected by an edge in H. Thus v and wq are connected +by an edge in Graph(t), which means that v ∈ C. This implies that C = V , thus Graph(t) +is connected. +□ +2.3. Modular decomposition. In this short section we gather the main definitions and +properties of modular decomposition. The historical reference is [9], the interested reader +may also look at [3] or [20]. +The next definitions and theorems allows to get a unique recursive decomposition of any +graph in the sense of Definition 2.5, the modular decomposition, and to encode it by a +tree. +Definition 2.7. Let G be a graph (labeled or not). A module M of G is a subset of V +such that for every (x, y) ∈ M 2, and every z ∈ V \M, {x, z} ∈ E if and only if {y, z} ∈ E. +Remark. Note that ∅, V and {v} for v ∈ V are always modules of G. Those sets are called +the trivial modules of G. +Definition 2.8. A graph G is prime if it has at least 3 vertices and its only modules are +the trivial ones. +Definition 2.9. A graph is called ⊖-indecomposable (resp. ⊕-indecomposable) if it cannot +be written as ⊖[G1, . . . , Gk] (resp. ⊕[G1, . . . , Gk]) for some k ≥ 2 and weakly-labeled graphs +G1, . . . , Gk. +Note that a graph is ⊖-indecomposable if and only if it is connected, and ⊕-indecomposable +if and only if its complementary is connected. +Theorem 2.10 (Modular decomposition, [9]). Let G be a graph with at least 2 vertices, +there exists a unique partition M = {M1, . . . , Mk} for some k ≥ 2, where each Mi is a +module of G and such that either +• G = ⊕[M1, . . . , Mk] and the (Mi)1≤i≤k are ⊕-indecomposable; +• G = ⊖[M1, . . . , Mk] and the (Mi)1≤i≤k are ⊖-indecomposable; +• G = P[M1, . . . , Mk] for some prime graph P. +Moreover, only one of the possibilities occurs. + +8 +TH´EO LENOIR +This decomposition can be used to encode graphs by specific trees to get a one-to-one +correspondence. +Definition 2.11. Let t be a substitution tree. We say that t is a canonical tree if its +internal nodes are either ⊕, ⊖ or prime graphs, and if there is no child of a node decorated +with ⊕ (resp. ⊖) which is decorated with ⊕ (resp. ⊖). +To a graph G we associate a canonical tree by recursively applying the decomposition +of Theorem 2.10 to the modules (Mi)1≤i≤k, until they are of size 1. First of all, at each +step, we order the different modules increasingly according to their minimal vertex labels. +Doing so, a labeled graph G can be encoded by a canonical tree. The internal nodes are +decorated with the different graphs that are encountered along the recursive decomposition +process (⊕ if G = ⊕[M1, . . . , Mk], ⊖ if G = ⊖[M1, . . . , Mk], P if G = P[M1, . . . , Mk]). +At the end, every module of size 1 is converted into a leaf labeled by the label of the vertex. +This construction provides a one-to-one correspondence between labeled graphs and +canonical trees that maps the size of a graph to the size of the corresponding tree. +Proposition 2.12. Let G be a graph, and t its canonical tree, then t is the only canonical +tree such that Graph(t) = G. +Remark. It is crucial to consider canonical trees as non-plane: otherwise, since prime graphs +can have several labelings, there would be several canonical trees associated with the same +graph. +2.4. Enriched modular decomposition. Unfortunately the modular decomposition alone +does not provide usable decompositions for the graph classes that we consider. The aim of +this section is to solve this issue: we will state and prove Proposition 2.18 which provides +in a very general settings a one-to-one encoding of graphs with substitution trees with +constraints. In Section 3 we will show that P4-reducible graphs, P4-sparse graphs, P4-lite +graphs, P4-extendible graphs, P4-tidy graphs fit in the settings of Proposition 2.18. +Definition 2.13. We say that G is a graph with blossoms if there exists k ∈ {0, . . . , |V |} +such that exactly k vertices of G are labeled ∗, and the others ones have a distinct label in +{1, . . . , |V | − k}. +The vertices labeled ∗ are called the blossoms of G. Let BG the set of vertices that are +blossoms of G and N(G) := |V | − |BG| the number of vertices that are not a blossom of G. +Remark. In the above definition, we allow k = 0, then the definition reduces to the one of +a labeled graph. +Definition 2.14. Let G be a graph with blossoms and π be a permutation of {1, . . . , N(G)}. +The π-relabeling of G is the graph G′ such that: +• VG′ = VG and BG′ = BG; +• for every vertex v in VG′\BG′, we replace the label of the leaf v by π(ℓ(v)). +We write G ∼ G′ if there exists a permutation π of {1, . . . , N(G)} such that G is iso- +morphic to the π-relabeling of G′. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +9 +Note that ∼ is an equivalence relation. +Definition 2.15. Let G be a graph with blossoms, a permutation π of {1, . . . , N(G)} is an +automorphism of G if the π-relabeling of G is G. +Definition 2.16. A module of a graph with blossoms is called flowerless if it does not +contain any blossom. +Let G be a graph with blossoms and M a non-empty flowerless module of G. We define +bloM(G) to be the labeled graph obtained after the following transformations: +• M is replaced by a new vertex v, that is now labeled ∗; +• for every vertex w ∈ G\M, {w, v} is an edge if and only if {w, m} is an edge of G +for every m ∈ M; +• the graph obtained is replaced by its reduction as defined in Definition 2.2. +If G is a graph with one blossom and M is a non-empty flowerless module of G, we +define bloM,0(G) (resp. bloM,1(G)) to be the graph bloM(G) where the label of the initial +blossom of G is replaced by ∗0 (resp. ∗1) and the label of the new blossom is replaced by ∗1 +(resp. ∗0). +1 +2 +3 +4 +5 +∗ +1 +2 +4 +5 +6 +3 +7 +8 +M = {v3, v7, v8} +bloM(G) +Figure 4. Illustration of Definition 2.16 Left: A graph G in which we have +highlighted the module M += {v3, v7, v8}. +Right: +The corresponding +bloM(G). +In this paper, we only consider the construction bloM(G) for graphs with 0 or 1 blossom. +We are now ready to precise the general framework of our study. One of the key ingredient +is the following recursive definition of families of graphs. +Definition 2.17. Let P be a set of graphs with no blossom and P• be a set of graphs with +one blossom. A tree t ∈ T0 is called (P, P•)-consistent if one of the following conditions +holds: +(D1) The tree t is a single leaf. +(D2) The root r of t is decorated with a graph H ∈ P and tr (the multiset of trees attached +to r) is a union of leaves. +(D3) The root r of t is decorated with ⊕ (resp. ⊖) and all the elements of tr are (P, P•)- +consistent and their roots are not decorated with ⊕ (resp. ⊖). + +10 +TH´EO LENOIR +(D4) The root r of t is decorated with a graph H /∈ {⊕, ⊖} and there exists at least +one index i ∈ {1, . . . , |H|} such that the i-th tree of tr is (P, P•)-consistent, the +remaining subtrees of tr are reduced to a single leaf and blo{vi}(H) ∈ P•. +We define TP,P• to be the set of trees t that are (P, P•)-consistent and such that each leaf +has a distinct label in {1, . . . , |t|}. +1 +2 +5 +7 +4 +9 8 +3 +6 +10 +12 +11 +(D2) +(D3) +(D4) +(D3) +(D3) +Figure 5. An example of tree in some TP,P•. The different colours illustrate +the different cases of Definition 2.17. The subtree with leaves {5, 6} on the +top-right is attached to the vertex which is circled in red inside the vertex of +case (D4). This corresponds to the i-th subtree of case (D4) +A graph G is called (P, P•)-consistent if there exists a (P, P•)-consistent tree t such +that G = Graph(t). We let GP,P• be the set of Graph(t) for t ∈ TP,P•. +The map t �→ Graph(t) from TP,P• to GP,P• is surjective, but without conditions on +(P, P•) this map is not one-to-one. To solve this issue, we introduce the following additional +constraints on the set P, P•: +Condition (C). +(C1) P and P• do not contain a graph of size 1. +(C2) For every F ∈ P and every module M of F, either bloM(F) ̸∈ P• or the subgraph +of F induced by M is not (P, P•)-consistent. +(C3) For every F and F ′ in P•, and every flowerless modules M and M ′ of respectively +F and F ′ one of the following conditions is verified: +• bloM,0(F) ̸= bloM′,1(F ′) +• The subgraph of F induced by M is not (P, P•)-consistent. +• The subgraph of F ′ induced by M ′ is not (P, P•)-consistent. +(C4) Every element of P and P• is ⊕-indecomposable and ⊖-indecomposable. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +11 +(C5) For every G ∈ P•, the only modules of G containing the blossom are {∗} and G. +We say that (P, P•) verifies condition (C) if (C1) − (C5) hold. +Remark. The last two constraints are not necessary to ensure that the map is bijective. +However, giving necessary and sufficient conditions to have unicity that can be checked +easily is quite complicated. +Note that if condition (C) is satisfied for a pair of sets (P, P•) and Q ⊂ P and Q• ⊂ P•, +it is also verified by (Q, Q•). +Proposition 2.18. Let P be a set of graphs with no blossom and P• a set of graphs with +one blossom. Assume that (P, P•) verifies condition (C). For any G ∈ GP,P•, there exists +a unique t ∈ TP,P• such that G = Graph(t). Moreover, for any element of TP,P• satifying +case (D4) in Definition 2.17, the index i such that case (D4) holds is unique. +Proof. Existence is guaranted by definition of GP,P•. +We proceed by contradiction to prove the uniqueness of t. +Let t be a smallest tree +in TP,P• such that there exists another t′ in TP,P• verifying Graph(t) = Graph(t′). Let +G = Graph(t). +The graph G cannot be reduced to a single vertex due to (C1), otherwise t and t′ would +be a single leaf with label 1. Thus we can assume that t and t′ are not in case (D1). +By Lemma 2.6 and (C4), G is ⊕-indecomposable (resp. ⊖-indecomposable) if and only +if t is not in case (D3) with a root decorated with ⊕ (resp. ⊖). Thus either t and t′ are +both in case (D3) and their roots are both decorated ⊕ or ⊖, or they are both in case +(D2) or (D4). +Case (i): t, t′ are both in case (D3) and their are both decorated ⊕ or ⊖. +Let r and r′ be the roots of respectively t and t′. Assume that both decorations are +⊖, the other case is similar. The elements of tr induce connected graphs by Lemma 2.6 +as their roots are either decorated with ⊕, or ⊖-indecomposable by (C4). Since the roots +of t and t′ are decorated with ⊖, we have a one-to-one correspondence between trees +of tr and connected components of G. The same is true for t′ +r′. Assume that two trees +corresponding to the same connected component of G are different. Since their set of labels +are the same (they correspond to the labels of the vertices in the connected component) +after reduction, one would obtain two trees t1, t2 that are different, (P, P•)-consistent and +such that Graph(t1) = Graph(t2) since both are equal to the reduction of the corresponding +connected component of G. This contradicts the minimality of t. Therefore tr = t′ +r′ and +t = t′. +Case (ii): t, t′ are both in case (D2). +The graph G is simply the decoration of the root of t so t = t′. +Case (iii): t is in case (D4), t′ is in case (D2). +Let r be the root of t and H its decoration. Let i be one of the elements of {1, . . . |VH|} +such that (D4) holds for t, H and i. Let M be the set of vertices of G whose labels are +labels of leaves that belong to the i-th tree of tr: M is a module of G. Then bloM(G) is +equal to blo{vi}(H) and thus belongs to P•. Moreover the subgraph of G induced by M is +(P, P•)-consistent as the i-th subtree of t is also (P, P•)-consistent. This contradicts (C2). + +12 +TH´EO LENOIR +Case (iv): t, t′ are both in case (D4). +Let r and r′ be the roots of respectively t and t′ and H and H′ be their decorations. Let +i be an element of {1, . . . , |VH|} such that (D4) is true for t, H and i, and i′ be an element +of {1, . . . , |VH′|} such that (D4) is true for t′, H′ and i′. Consider M (resp. M ′) the set of +vertices of G whose labels are labels of leaves that belong to the i-th tree of tr (resp. i′-th +tree of t′ +r′): M (resp. M ′) is a module of G. Since the i-th tree of tr (resp. the i′-th tree of +t′ +r′) is (P, P•)-consistent the subgraph of G induced by M (resp. M ′) is (P, P•)-consistent. +We now prove by contradiction that M = M ′. +By symmetry we can assume that +M ′ ̸⊂ M. +First assume that M ∩ M ′ = ∅. Note that bloM,1(bloM′(G)) = bloM′,0(bloM(G)). Since +bloM(G) = blo{vi}(H) and bloM′(G) = blo{vi′}(H′), we get that bloM′,0(blo{vi}(H)) = +bloM,1(blo{vi′}(H′)) which contradicts (C3) as both subgraphs of G induced by M and M ′ +are (P, P•)-consistent. +Now assume that M ∩ M ′ ̸= ∅. Let L be the subset of VH such that v ∈ L if and only if +the ℓ(v)-th tree of tr contains a leaf labeled with the label of an element of M ′. Since M ′ +is a module of G and M ∩ M ′ ̸= ∅, L is a module of blo{vi}(H) containing the blossom. +Since M ′ is not included in M, by (C5), L = H. Since M ′ ̸= G, there exists a vertex w +in G such that w ̸∈ M ′. Let w′ be the vertex of H such that w is in the ℓ(w′)-th tree +of tr. Since M ′ is a module, every vertex of M ′ is either connected or not to w, thus w′ +is connected to every vertex of H (except w′) or to none of them. This means that H is +either ⊕-decomposable or ⊖-decomposable, which is a contradiction. +Thus M = M ′ and blo{vi}(H) = bloM(G) = bloM′(G) = blo{vi′}(H′), and we get that +H = H′, and that i = i′: thus i is unique. +We know that the i-th tree of tr and the i-th tree of t′ +r′ are (P, P•)-consistent and the +associated graph is the one induced by M. By taking the reduction of the trees and the +graph, we get by minimality of t that the reductions of both trees are equal. Since M = M ′, +it implies that both subtrees are the same: thus t = t′. +□ +3. Zoology of graph classes with few P4’s +Several classes have been defined as generalizations of the class of P4-free graphs, the +cographs. Here the classes we will focus on are the following: P4-reducible graphs [15,18], +P4-sparse graphs [13,17] P4-lite graphs [14], P4-extendible graphs [16], P4-tidy graphs [10]. +The aim of this section is to give explicit sets P and P• such that GP,P• is one of the +previously mentioned classes. +3.1. Basic definitions. The following results and definitions are from [3, Section 11.3]. +Definition 3.1. A graph G is a Pk if it is a path of k vertices, and a Ck if it is a cycle of +k vertices. +The two vertices of degree one of a P4 are called the endpoints, the two vertices of degree +two are called the midpoints. +Notation. For a graph G, we denote by G its complementary. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +13 +The modular decompositions of classes of graphs we consider are already well-known [10]. +To explain the different properties, we need the notion of spider and bull. +Definition 3.2. A spider is a graph G, such that there exists a partition of VG in three +parts, K, S, R, verifying: +• |K| ≥ 2; +• K induces a clique; +• S induces a graph without edges; +• every element of R is connected to every element of K but to none of S; +• there exists a bijection f from K to S such that for every k ∈ K, k is only connected +to f(k) in S, or such that for every k ∈ K, k is connected to every element of S +except f(k). In the first case the spider is called thin, in the second one it is called +fat. +K +S +K +S +R +R +1 +1 +2 +2 +3 +3 +4 +4 +5 +5 +6 +6 +7 +7 +Figure 6. Left: a thin spider. Right: a fat spider. Both with |K| = 3. +Remark. For every spider G, the partition (K, S, R) is uniquely determined by G. Moreover, +the bijection f given by the definition is unique, except in the case |K| = 2. In this case, +since there is no difference between a thin and a fat spider, a spider with |K| = 2 is called +thin. A spider with |K| = 2 and |R| = 1 is called a bull, and a spider with |K| = 2 and +|R| = 0 is simply a P4. +1 +2 +3 +4 +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +Figure 7. From left to right: a P4, a bull, a C5 +Proposition 3.3. A spider is prime if and only if |R| ≤ 1. + +14 +TH´EO LENOIR +In the following, if |R| = 1, the vertex belonging to R will be a blossom of the spider, +and it will be its only blossom: such spiders will be called blossomed spiders. If |R| = 0, +the spider will have no blossom. This also applies for bulls and P4. +Definition 3.4. We call a graph H a pseudo-spider if there exists a prime spider G such +that, if we duplicate a vertex that is not a blossom of G (his label is the new number of +vertices), and if either by adding or not an edge between the vertex and its duplicate, the +graph obtained is a relabeling of H. If |K| = 2, we also call H a pseudo-P4. +Moreover, we say that H is a blossomed pseudo-spider if G is a blossomed spider. If +|K| = 2, we also call H a pseudo-bull. +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +∗ +K +S +R +1 +2 +∗ +3 +4 +5 +6 +Duplicate +7 +Figure 8. A blossomed pseudo-spider, a pseudo-bull, a pseudo P4 +Lemma 3.5. A prime spider with 0 or 1 blossom has |K|! automorphisms (as there is a +natural bijection between the automorphisms of the spider and the automorphisms of K). +A pseudo-spider with 0 or 1 blossom has 2 × (|K| − 1)! automorphisms. +3.2. P4-tidy graphs. +Definition 3.6. A graph G is said to be a P4-tidy graph if, for every subgraph H of G +inducing a P4, there exists at most one vertex y ∈ VG\VH such that y is connected to at +least one element of H but not all, and y is not connected to exactly both midpoints of H. +Theorem 3.7. Let Ptidy be the set containing all C5, P5, P5, all prime spiders without +blossom and all pseudo-spiders without blossom. +Let P• +tidy be the set of all blossomed +prime spiders and all blossomed pseudo-spiders. Then the set of graphs that are P4-tidy is +GPtidy,P• +tidy. +Proof. It is simply a reformulation in our setting of [10, Theorem 3.3] that states that a +graph G is P4-tidy if and only if its canonical tree t verifies the following conditions: +• Every node in t is labeled with ⊕, ⊖, C5, P5, P5 or a prime spider. +• If a node w in t is decorated with C5, P5 or P5, every element of tw is reduced to a +single leaf. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +15 +• If a node w in t is decorated with a prime spider with |R| = 0, every element of tw +is a tree of size at most two, and at most one is of size two. +• If a node w in t is decorated with a prime spider H with |R| = 1, let v be the vertex +of H in R, and t′ the ℓ(v)-th tree of tw. Every element of tw\{t′} is a tree of size +at most two, and at most one is of size two. +□ +Proposition 3.8. The pair (Ptidy, P• +tidy) verifies (C) +Proof. Note that all the graph in Ptidy or P• +tidy are prime except the pseudo-spiders. The +only modules of the pseudo-spiders are the trivial ones, and the module formed by the +vertex that was duplicated and its duplicate, which implies (C5). +(C2) is also verified with the previous observation, as the modules of every graph in Ptidy +are trivial. +(C1) is clearly verified and (C4) can be checked easily as all the graphs in P ∪ P• are +connected, and their complementary is also connected. +For (C3), assume that for (F, F ′)2 ∈ P• +tidy and M, M ′ are respectively flowerless modules +of F and F ′, bloM,0(G) = bloM′,1(G). +By cardinality argument, F and F ′ are either +both spiders, or both pseudo-spiders of same size. If both are spiders, as R is uniquely +determined by the spiders, and the only element of R does not have the same label in +bloM,0(G) and in bloM,1(G), we get a contradiction. If both are pseudo-spiders, note that +the original node and its duplicate form the only module of size 2 of bloM,0(G). Thus the +only element of R (in the original spiders) is uniquely determined by the pseudo-spiders, +and the only element of R does not have the same label in bloM,0(G) and in bloM,1(G), we +get a contradiction. +□ +3.3. P4-lite graphs. +Definition 3.9. A graph G is said to be a P4-lite graph if every subgraph of G of size at +most 6 does not contain three induced P4. +Theorem 3.10. Let Plite be the set containing all P5, P5, all prime spiders without blossom +and all pseudo-spiders without blossom. Let P• +lite to be the set containing all blossomed +prime spiders and all blossomed pseudo-spiders. Then the set of graphs that are P4-lite is +GPlite,P• +lite. +Proof. It is simply a reformulation in our setting of [10, Theorem 3.8] that states that a +graph G is P4-lite if and only if its canonical tree t verifies the following conditions: +• Every node in t is labeled with ⊕, ⊖, P5, P5 or a prime spider. +• If a node w in t is decorated with P5 or P5, every element of tw is reduced to a +single leaf. +• If a node w in t is decorated with a prime spider with |R| = 0, every element of tw +is a tree of size at most two, and at most one is of size two. +• If a node w in t is decorated with a prime spider H with |R| = 1, let v be the vertex +of H in R, and t′ the ℓ(v)-th tree of tw. Every element of tw\{t′} is a tree of size +at most two, and at most one is of size two. +□ + +16 +TH´EO LENOIR +By Proposition 3.8 since Plite ⊂ Ptidy, P• +lite ⊂ P• +tidy we get that the pair (Plite, P• +lite) +verifies (C). +3.4. P4-extendible graphs. +Definition 3.11. A graph G is said to be a P4-extendible graph if, for every subgraph H +of G inducing a P4, there exists at most one vertex y ∈ VG\VH such that y belongs to an +induced P4 sharing at least one vertex with H. +Theorem 3.12. Let Pext be the set containing all C5, P5, P5, P4 and all pseudo-P4. Let +P• +ext be the set containing all bulls and all pseudo-bulls. Then the set of graphs that are +P4-extendible is GPext,P• +ext. +Proof. It is simply a reformulation in our setting of [10, Theorem 3.7] that states that a +graph G is P4-extendible if and only if its canonical tree t verifies the following conditions: +• Every node in t is labeled with ⊕, ⊖, C5, P5, P5, P4 or a bull. +• If a node w in t is decorated with C5, P5 or P5, every element of tw is reduced to a +single leaf. +• If a node w in t is decorated with P4, every element of tw is a tree of size at most +two, and at most one is of size two. +• If a node w in t is decorated with a bull G, let v be the vertex of G in R, and t′ +the ℓ(v)-th tree of tn. Every element of tw\{t′} is a tree of size at most two, and at +most one is of size two. +□ +By Proposition 3.8 since Pext ⊂ Ptidy, P• +ext ⊂ P• +tidy we get that the pair (Pext, P• +ext) +verifies (C). +3.5. P4-sparse graphs. +Definition 3.13. A graph G is said to be a P4-sparse graph if every subgraph of G of size +5 does not contain two induced P4. +Theorem 3.14. Let P be the set containing all prime spiders without blossom. Let P• be +the set containing all blossomed prime spiders. Then the set of graphs that are P4-sparse +is GP,P•. +Proof. It is simply a reformulation in our setting of [11, Theorem 3.4] that states that a +graph G is P4-sparse if and only if its canonical tree t verifies the following conditions: +• Every node in t is labeled with ⊕, ⊖ or a prime spider. +• If a node w in t is decorated with a prime spider with |R| = 0, every element of tw +is reduced to a single leaf. +• If a node w in t is decorated with a prime spider h with |R| = 1, let v be the vertex +of H in R, and t′ the ℓ(v)-th tree of tw. Every element of tw\{t′} is reduced to a +single leaf. +□ +By Proposition 3.8 since Pspa ⊂ Ptidy, P• +spa ⊂ P• +tidy we get that the pair (Pspa, P• +spa) +verifies (C). + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +17 +3.6. P4-reducible graphs. +Definition 3.15. A graph G is said to be a P4-reducible graph if every vertex of G belongs +to at most one induced P4. +Theorem 3.16. Let Pred be the set containing all P4. Let P• +red be the set containing all +bulls. Then the set of graphs that are P4-reducible is GPred,P• +red. +Proof. It is simply a reformulation in our setting of [11, Theorem 4.2] that states that a +graph G is P4-reducible if and only if its canonical tree t verifies the following conditions: +• Every node in t is labeled with ⊕, ⊖, P4 or a bull. +• If a node w in t is decorated with a P4, every element of tw is reduced to a single +leaf. +• If a node w in t is decorated with a bull H, let v be the vertex of H in R, and t′ +the ℓ(v)-th tree of tn. Every element of tw\{t′} is reduced to a single leaf. +□ +By Proposition 3.8 since Pred ⊂ Ptidy, P• +red ⊂ P• +tidy we get that the pair (Pred, P• +red) +verifies (C). +3.7. P4-free graphs (cographs). +Definition 3.17. A graph G is said to be a cograph if no subgraph of G induces a P4. +Theorem 3.18. Set Pcog = ∅ and P• +cog = ∅. Then the set of graphs that are cographs is +GPcog,P•cog. +Proof. It is simply a reformulation in our setting of [5, Theorem 7] that states that a graph +G is a cograph if and only if its canonical tree t has no internal node decorated with a +prime graph. +□ +Clearly the pair (Pcog, P• +cog) verifies (C). +4. Enriched modular decomposition: enumerative results +4.1. Exact enumeration. In the following, we establish combinatorial identities between +formal power series involving subsets of P and P•. +Throughout this section, we consider generic pairs (P, P•) where P (resp. P•) is a set +of graphs with no blossom (resp. with one blossom) verifying condition (C) defined p.10. +Recall that for a graph G with blossoms, N(G) is the number of vertices that are not +a blossom: this will be the crucial parameter in the subsequent analysis. Let P •(z) := +� +s∈P• +zN(s) +N(s)! and P(z) := � +s∈P +zN(s) +N(s)!. +For n ∈ N, let Pn (resp. P• +n) be the set of graphs G in P (resp. P•) such that N(G) = n. +Note that, if both P and P• are stable under relabeling (which is the case for the classes +of graphs mentioned in Section 3), for each n ∈ N, there is a natural action Φn of the + +18 +TH´EO LENOIR +permutations of {1, . . . , n} over Pn and P• +n. Let RPn and RP•n be a system of representants +of every orbit under this action, then +P •(z) = +� +n∈N +|P• +n|zn +n! = +� +n∈N +� +s∈RPn +|RP•n| +n! +|Aut(s)| +zn +n! = +� +n∈N +� +s∈RPn +|RP•n| +zn +|Aut(s)| +Similarly, we have: +P(z) = +� +n∈N +� +s∈RPn +|RPn| +zn +|Aut(s)| +Theorem 4.1. For each graph class introduced in Section 3, we have the following expres- +sions for P and P •: +P4-tidy +P • +tidy(z) = (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4 +2 − 2z5 +Ptidy(z) = P • +tidy(z) + z5 + z5 +10 +P4-lite +P • +lite(z) = (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4 +2 − 2z5 +Plite(z) = P • +lite(z) + z5 +P4-extendible +P • +ext(z) = z4 +2 + 2z5 +Pext(z) = P • +ext(z) + z5 + z5 +10 +P4-sparse +P • +spa(z) = Pspa(z) = 2(exp(z2) − 1 − z2 − z4 +4 ) +P4-reducible +P • +red(z) = Pred(z) = z4 +2 +P4-free +P • +cog(z) = Pcog(z) = 0 +Proof. We only detail the computation of Ptidy and P • +tidy for P4-tidy graphs as this is the +most involved case. According to Theorem 3.7, Ptidy is composed of one C5 that has 10 +automorphisms and all its relabelings, one P5, and one P5 that both have 2 automorphisms +and all their relabelings. +For k ≥ 3 (resp. k = 2), there are thin and fat spiders corresponding to the 2 (resp. 1) +different orbits of the action Φ2k over prime spiders of size 2k, each having k! automor- +phisms. +For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can +come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4) +cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over pseudo-spiders +of size 2k + 1, each having 2(k − 1)! automorphisms. +Thus we have +Ptidy(z) = z5 +10 + 2z5 +2 + z4 +2 + 2 +� +k≥3 +z2k +k! + 4z5 +2 + 8 +� +k≥3 +z2k+1 +2(k − 1)! +Hence +Ptidy(z) = z5 + z5 +10 + (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4 +2 − 2z5. +Now let’s compute P • +tidy. For k ≥ 3 (resp. k = 2), there are thin and fat spiders with +blossom corresponding to the 2 (resp. 1) different orbits of the action Φ2k over blossomed +prime spiders G with 2k non blossomed vertices, each having k! automorphisms. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +19 +For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can +come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4) +cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over blossomed +pseudo-spiders with 2k + 1 non blossomed vertices, each having 2(k − 1)! automorphisms. +Hence +P • +tidy(z) = z4 +2 +2 +� +k≥3 +z2k +k! +4z5 +2 +8 +� +k≥3 +z2k+1 +2(k − 1)! = (2+4z3) exp(z2)−2−2z2−4z3− z4 +2 −2z5, +which gives the announced result. +□ +Let T be the exponential generating function of TP,P•, the set of trees defined in Def- +inition 2.17 counted by their number of leaves. Denote by Tnot⊕ (resp. Tnot⊖) the set of +all t ∈ TP,P• whose root is not decorated with ⊕ (resp. ⊖) and by Tnot⊕ (resp. Tnot⊖) the +corresponding exponential generating function. +Theorem 4.2. Together with Tnot⊕ = 0, the exponential generating function Tnot⊕ is de- +termined (as a formal series) by the following equation: +Tnot⊕ = z + P + (exp(Tnot⊕) − 1)P • + exp(Tnot⊕) − 1 − Tnot⊕, +(1) +and the series T and Tnot⊖ are simply given by the following equations: +T = exp(Tnot⊕) − 1 +(2) +Tnot⊖ = Tnot⊕ +(3) +Moreover, Eq. (1) with Tnot⊕(0) = 0 determines uniquely the generating function Tnot⊕. +Proof. Note that there is a natural involution on TP,P•: the decoration of every linear node +can be changed to its opposite: ⊕ to ⊖, and ⊖ to ⊕. Therefore Tnot⊕ = Tnot⊖. +First, we prove that +T = z + T × P • + P + 2 × (exp(Tnot⊕) − 1 − Tnot⊕) +(4) +We split the enumeration of the trees t ∈ TP,P• according to the different cases of +Definition 2.17. +(D1) The tree t is a single leaf (which gives the z in Eq. (4)). +(D2) The tree t has a root decorated with a graph H belonging to P. The exponential +generating function for a fixed H is zN(H) +N(H)!. Summing over all H and all n gives the +term P in Eq. (4). +(D3) The tree t has a root r decorated with ⊕ and having k children with k ≥ 2. In this +case, the generating function of the set of the k subtrees of tr is +T k +not⊕ +k! . Summing +over all k implies that the exponential generating function of all trees in case (D3) +with a root labeled ⊕ is exp(Tnot⊕) − 1 − Tnot⊕. +The tree t can also have a root r decorated with ⊖. Since Tnot⊕ = Tnot⊖, the +exponential generating function of all trees in case (D3) with a root labeled ⊖ is +exp(Tnot⊕) − 1 − Tnot⊕. + +20 +TH´EO LENOIR +(D4) The tree t has a root r decorated with a graph H and there exists v ∈ VH such that +blov(H) = W where W ∈ P•. Denote t′ the ℓ(v)-th tree of tr. +The exponential generating function corresponding to the set of leaves in t\t′ +is +zN(W ) +N(W)!, and the exponential generating function corresponding to t′ is T. Note +that the tree t is uniquely determined by W, the labeled product of t′ and the +set of leaves of t\t′. Thus the corresponding generating function for a fixed W is +T × zN(W ) +N(W)!. Summing over all W and all n gives the term T × P • in Eq. (4). +Summing all terms gives Eq. (4). +Similarly, we get +Tnot⊕ = z + T × P • + P + exp(Tnot⊕) − 1 − Tnot⊕. +(5) +Substracting Eq. (5) to Eq. (4) gives Eq. (2). Then Eq. (1) is an easy consequence from +Eqs. (2) and (5). +Note that Eq. (1) can be rewritten as: +Tnot⊕ = z + P + (exp(Tnot⊕) − 1)P • + +� +k≥2 +T k +not⊕ +k! . +(6) +For every n ≥ 1, the coefficient of degree n of Tnot⊕ only depends on coefficients of lower +degree as P •(z) has no term of degree 0 or 1 and Tnot⊕(0) = 0. Thus Eq. (1) combined +with Tnot⊕(0) = 0 determines uniquely Tnot⊕. +□ +We are going to define the notions of trees with marked leaves, and of blossomed trees, +which will be crucial in the next section. We insist on the fact that the size parameter will +count the number of leaves including the marked ones but not the blossoms. +Definition 4.3. A marked tree is a pair (t, I) where t is a tree and I a partial injection +from the set of labels of leaves of t to N. The number of marked leaves is the size of the +domain of I denoted by |(t, I)|, and a leaf is marked if its label j is in the domain, its mark +being I(j). +Remark. In the following, we will consider marked trees (t, I), and subtrees t′ of t. The +marked tree (t′, I) will refer to the marked tree (t′, I′) where I′ is the restriction of I to +the set of labels of leaves of t′. +Remark. Let F ∈ {TP,P•, Tnot⊖, Tnot⊕}, and F be its generating exponential function. The +exponential generating function of trees in F with a marked leaf is zF ′(z): if there are fn +trees of size n in F, there are nfn trees with a marked leaf. Thus the generating exponential +function is � +n≥1 +nfn +n! zn = zF ′(z). +Blossoming transformation. Let t be a tree not reduced to a leaf in TP,P•, ℓ a leaf of t +and n the parent of ℓ. If n is a linear node, we replace the label of ℓ by ∗, and do the +reduction on t. If v is a non-linear node, and ℓ is in the i-th tree of tv (where i is the +element such that (D4) holds in Definition 2.17), we replace the label of ℓ by ∗ and i by +∗ in the decoration of v, and do the reduction on both t and the decoration of v. If t is + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +21 +reduced to a leaf, we replace the leaf by a blossom. We call such this transformation the +blossoming of (t, ℓ). +We extend this operation to internal node: if n is a internal node, we replace t[n] by its +leaf of smallest label, and do the blossoming operation on the tree obtained. The resulting +tree is still called the blossoming of (t, n). +Definition 4.4 (Blossomed tree). A blossomed tree is a tree that can be obtained by the +blossoming of a tree in TP,P•. Its size is its number of leaves without blossom. +A blossom is ⊕-replaceable (resp. ⊖-replaceable) if its parent is not decorated with ⊕ +(resp. ⊖). +Remark. Similarly to a tree, a blossomed tree can be marked by a partial injection I. +We will denote T b and T b +a with a = not⊕, not⊖, and b = ⊕, ⊖ or blo the set of trees +whose root is not ⊕ (resp. ⊖) if a = not⊕ (resp. a = not⊖), and with one blossom that is +b-replaceable if b = ⊕ or ⊖, or just with one blossom if b = blo. +We define T b +a to be the corresponding exponential generating function of trees, counted +by the number of non blossomed leaves. +However, we take the convention that T ⊕ +not⊕(0) = 0 = T ⊖ +not⊖. In other words, a single leaf +is neither in T ⊕ +not⊕ nor in T ⊖ +not⊖. The other series have constant coefficient 1. +Remark. From the previously defined involution, it follows that T ⊖ +not⊕ = T ⊕ +not⊖, T ⊕ +not⊕ = +T ⊖ +not⊖ et T ⊕ = T ⊖ and T blo +not⊕ = T blo +not⊖. +Theorem 4.5. The functions T ⊕, T ⊕ +not⊕, T ⊕ +not⊖ are given by the following equations: +T ⊕ = +1 +2 − exp(Tnot⊕) − P • exp(Tnot⊕) +(7) +T ⊖ +not⊕ = +T ⊕ +exp(Tnot⊕) +(8) +T ⊕ +not⊕ = +T ⊕ − 1 +exp(Tnot⊕) +(9) +Proof. Let t be a tree in T ⊕ +not⊕. Note that it cannot be reduced to a single leaf, have a root +decorated with ⊕ or be in case (D2) of Definition 2.17. +(D3) The tree t can have a root r decorated with ⊖ and having k children with k ≥ 2. +There are k−1 subtrees without blossom, and 1 with a blossom. Thus the generating +function of the set of the k subtrees of tr is +T k−1 +not⊖ +(k−1)!T ⊖ +not⊕. Summing over all k gives +that the exponential generating function of all trees in case (D3) with a root labeled +⊖ is +� +k≥2 +T k−1 +not⊖ +(k − 1)!T ⊕ +not⊖ = (exp(Tnot⊖) − 1)T ⊕ +not⊖ + +22 +TH´EO LENOIR +⊕ or (D4) +⊖ or (D4) +⊖ or (D4) +not (D2) +(D4) +2 +9 +6 +or +At least one tree, each does not +have a root decorated with ⊕ +If the previous node is in (D4) the marked leaf must be in the i-th tree +Figure 9. Illustration of both cases in the proof of Theorem 4.5 +(D4) The tree t can have a root r decorated with H and v ∈ VH such that blov(H) = W +with W ∈ P•. Then the blossom must be in the ℓ(v)-th tree of tr that will be +denoted t′. +The exponential generating function corresponding to the set of leaves in t\t′ +is zN(W ) +N(W)!, and the exponential generating function corresponding to t′ is T ⊕. Note +that the tree t is uniquely determined by W, the labeled product of t′ and the +set of leaves of t\t′. Thus the corresponding generating function for a fixed W +is T ⊕ × zN(W ) +N(W)!. Summing over all W and all n gives the exponential generating +function T ⊕ × P •. +This implies the following equation: +T ⊕ +not⊕ = (exp(Tnot⊖) − 1)T ⊕ +not⊖ + P •T ⊕ = (exp(Tnot⊕) − 1)T ⊖ +not⊕ + P •T ⊕ +(10) +We have similarly: +T ⊖ +not⊕ = 1 + (exp(Tnot⊖) − 1)T ⊖ +not⊖ + P •T ⊖ = 1 + (exp(Tnot⊕) − 1)T ⊕ +not⊕ + P •T ⊕ +(11) +T ⊕ = 1 + (exp(Tnot⊖) − 1)T ⊕ +not⊖ + (exp(Tnot⊕) − 1)T ⊕ +not⊕ + P •T ⊕ +(12) +Thus: +T ⊕ = 1 + (exp(Tnot⊕) − 1)(T ⊕ +not⊕ + T ⊖ +not⊕) + P •T ⊕ +(13) +By substracting Eq. (11) to Eq. (13), we get T ⊕ − T ⊖ +not⊕ = (exp(Tnot⊕) − 1)T ⊖ +not⊕ which +implies Eq. (8). +Using Eqs. (10) and (13), we get +T ⊕ = 1 + (exp(Tnot⊕) − 1)T ⊕ +not⊕ + T ⊕ +not⊕ = 1 + exp(Tnot⊕)T ⊕ +not⊕ +which implies Eq. (9). + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +23 +Substituting T ⊕ +not⊕ and T ⊖ +not⊕ with Eqs. (9) and (10) ins Eq. (8), it follows that: +T ⊕ − 1 = (exp(Tnot⊕) − 1)T ⊕ + exp(Tnot⊕)P •T ⊕ +and T ⊕(2 − exp(Tnot⊕) − P • exp(Tnot⊕)) = 1 which implies Eq. (7). +□ +Theorem 4.6. We also have the following equations: +T blo = +exp(Tnot⊕) +2 − exp(Tnot⊕) − P • exp(Tnot⊕) +(14) +T blo +not⊕ = +1 +exp(Tnot⊕)T blo +(15) +Proof. By the same techniques used as those of the previous proof, we establish that: +T blo = 1 + 2(exp(Tnot⊕) − 1)T blo +not⊕ + P •T blo; +(16) +T blo +not⊕ = 1 + (exp(Tnot⊕) − 1)T blo +not⊕ + P •T blo. +(17) +By substracting Eq. (17) to Eq. (16), we get that: +T blo − T blo +not⊕ = (exp(Tnot⊕) − 1)T blo +not⊕ +which implies Eq. (15). +By multiplying Eq. (17) by exp(Tnot⊕) and using Eq. (15) we get that: +T blo (2 − P • exp(Tnot⊕) − exp(Tnot⊕)) = exp(Tnot⊕) +which implies Eq. (14). +□ +Combining Theorem 4.5 and Theorem 4.6 we obtain: +Corollary 4.7. We have the following equations: +T blo = exp(Tnot⊕)T ⊕; +(18) +T blo +not⊕ = T ⊕. +(19) +4.2. Asymptotic enumeration. In the following, we derive from the previously obtained +equations the radii of the different series introduced, the asymptotic behavior of the dif- +ferent series in R and an equivalent of the number of graphs in GP,P• +From now on, we assume that P and P • have a positive radius of convergence. +Let R0 +be the minimum of their radii of convergence. Denote by P(R0) and P •(R0) the limit in +[0, +∞] of P and P • at R− +0 . +In the following, we assume that one of the conditions below is verified: +• P •(R0) ≥ 1 +• R0 + P(R0) + 2 ln(1 + P •(R0)) − P •(R0) > 2 ln(2) − 1 + +24 +TH´EO LENOIR +Note that one of these conditions is verified in the different classes of graphs we study, +as R0 = +∞. +Denote by R the only solution in [0, R0) of the equation: +R + P(R) + 2 ln(1 + P •(R)) − P •(R) = 2 ln(2) − 1 +(20) +such that P •(R) < 1 (unicity comes from the fact that z �→ 2 ln(1 + z) − z is increasing in +[0, 1]). Note that by definition, 0 < R < R0. +Recall that a formal series A is aperiodic if there does not exist two integers r ≥ 0 and +d ≥ 2 and B a formal series such that A(z) = zrB(zd). +Lemma 4.8. The functions T, Tnot⊕, T ⊕, T ⊖ +not⊕, T ⊕ +not⊕, T blo, T blo +not⊕ are aperiodic. +Proof. One can easily check that for each of the previous series, the coefficients of degree +3 and 4 are positive, and thus all the series are aperiodic. +□ +Definition 4.9. A set ∆ is a ∆-domain at 1 if there exist two positive numbers R and +π +2 < φ < π such that +∆ = {z ∈ C||z| ≤ R, z ̸= 1, |arg(1 − z)| < φ} +For every w ∈ C∗, a set is a ∆-domain at w if it is the image of a ∆-domain by the +mapping z �→ zw. +Definition 4.10. A power series U is said to be ∆-analytic if it has a positive radius of +convergence ρ and there exists a ∆-domain D at ρ such that U has an analytic continuation +on D. +Theorem 4.11. Both T and Tnot⊕ have R as radius of convergence and a unique dominant +singularity at R. They are ∆-analytic. Their asymptotic expansions near R are: +Tnot⊕(z) = ln +� +2 +1 + P •(R) +� +− κ +� +1 − z +R + o +�� +1 − z +R +� +(21) +T(z) = +2 +1 + P •(R) − 1 − +2 +1 + P •(R)κ +� +1 − z +R + o +�� +1 − z +R +� +(22) +where κ is the constant given by: +κ = +� +� +� +�R +� +1 + P ′(R) + (1 − P •(R))(P •)′(R) +1 + P •(R) +� +Proof. We begin with the expansion of Tnot⊕ for which we apply the smooth implicit the- +orem [8, Theorem VII.3, p.467]. Following [8, Sec VII.4.1] we claim that Tnot⊕ satifies the +settings of the so-called smooth implicit-function schema: Tnot⊕ is solution of +T = G(z, T), +where G(z, w) = z + P(z) + (exp(w) − 1)P •(z) + (exp(w) − 1 − w). + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +25 +The singularity analysis of Tnot⊕ will go through the study of the characteristic system: +� +� +� +G(r, s) = s, +Gw(r, s) = 1 +with 0 < r < R, s > 0 +where Fx = ∂F +∂x . +Note that (r, s) = +� +R, ln +� +2 +1+P •(R) +�� +is a solution of the characteristic system of G since +• Gw(r, s) = exp(s)(1 + P •(R)) − 1 = 2 − 1 = 1 +• G(r, s) = R+P(R)−P •(R)+∂wG(r, s)−s = 2 ln(2)−1−2 ln(1+P •(R))+1−s = +2s − s = s +Moreover +• Gz(r, s) = 1 + P ′(R) + (exp(s) − 1)(P •)′(R) = 1 + P ′(R) + (1−P •(R))(P •)′(R) +(1+P •(R)) +• Gw,w(r, s) = exp(s)(1 + P •(r)) = 2 +The expansion of T is then a consequence of Eq. (2) p.19 and of the expansion of +Tnot⊕. +□ +Corollary 4.12. The radius of convergence of T ⊕, T ⊖ +not⊕, T ⊕ +not⊕, T blo, and T blo +not⊕ is R +and R is the unique dominant singularity of these series. They are ∆-analytic and their +asymptotic expansions near R are: +T ⊕ = 1 +2κ +� +1 − z +R +�− 1 +2 + o +�� +1 − z +R +�− 1 +2 +� +(23) +T ⊖ +not⊕ = (1 + P•(R)) +4κ +� +1 − z +R +�− 1 +2 + o +�� +1 − z +R +�− 1 +2 +� +(24) +T ⊕ +not⊕ = (1 + P•(R)) +4κ +� +1 − z +R +�− 1 +2 + o +�� +1 − z +R +�− 1 +2 +� +(25) +T blo = +1 +(1 + P•(R))κ +� +1 − z +R +�− 1 +2 + o +�� +1 − z +R +�− 1 +2 +� +(26) +T blo +not⊕ = 1 +2κ +� +1 − z +R +�− 1 +2 + o +�� +1 − z +R +�− 1 +2 +� +(27) +Proof. note that, if |z| ≤ R, +|(1 + P •(z)) exp(Tnot⊕(z))| ≤ (1 + P •(|z|)) exp(|Tnot⊕(z)|) ≤ (1 + P •(R)) exp(Tnot⊕(R)) = 2 +with equality if and only if z = R by aperiodicity from Daffodil lemma [8, Lemma IV.1] +and since Tnot⊕(R) ∈ R+. +Hence, by Theorem 4.11 and by compacity, 2−(1+P •(z)) exp(Tnot⊕(z)) can be extended +to a ∆-domain D at R with 2 − (1 + P •(z)) exp(Tnot⊕)(z) ̸= 0 for every z ∈ D. +Eq. (7) shows that T ⊕ can be extended to D and yields the announced expansions when +z tends to R. These expansions show that all these series have a radius of convergence +exactly equal to R. +□ + +26 +TH´EO LENOIR +Applying the Transfer Theorem [8, Corollary VI.1 p.392] to the results of Theorem 4.11, +we obtain an equivalent of the number of trees of size n in TP,P•. Since there is a one-to-one +correspondence between graphs in GP,P• and trees in TP,P•, we get the following result: +Corollary 4.13. The number of graphs in GP,P• of size n is asymptotically equivalent to +C +n! +Rnn +3 +2 +where +C = +κ +√π(1 + P •(R)). +Here are the numerical approximations of R and C in the different cases: +class of graph +R−1 +R +C +P4-tidy +2.90405818 +0.34434572 +0.40883495 +P4-lite +2.90146936 +0.34465296 +0.40833239 +P4-extendible +2.88492066 +0.34662998 +0.40351731 +P4-sparse +2.72743550 +0.36664478 +0.37405701 +P4-reducible +2.71715531 +0.36803196 +0.37115484 +P4-free +1 +2 ln(2)−1 ≈ 2.58869945 +2 ln(2) − 1 ≈ 0.38629436 +0.35065840 +5. Enumeration of graphs with a given induced subgraph +5.1. Induced subtrees and subgraphs. We recall that the size of a graph is its number +of vertices, and the size of a tree is its number of leaves. +Definition 5.1 (Induced subgraph). Let G be a graph, k a positive integer and I a partial +injection from the set of labels of G to N. The labeled subgraph GI of G induced by I is +defined as: +• The vertices of GI are the vertices of G whose label ℓ is in the domain of I. For +every such vertex, we replace the label ℓ of the vertex by I(ℓ); +• For two vertices v and v′ of GI, (v, v′) is an edge of GI if and only if it is an edge +of G. +Definition 5.2 (First common ancestor). Let t be a rooted tree and let ℓ1, ℓ2 be two distinct +leaves of t. The first common ancestor of ℓ1 and ℓ2 is the internal node of t that is the +furthest from the root and that belongs to the shortest path from the root to ℓ1, and the +shortest path from the root to ℓ2. +Definition 5.3 (Induced subtree). Let (t, I) be a marked tree in T0 (T0 is defined in +Definition 2.4, and the notion of marked tree in Definition 4.3). The induced subtree tI +of t induced by I is defined as: +• The leaves of tI are the leaves of t that are marked. For every such leaf labeled with +an integer ℓ, the new label of ℓ is I(ℓ); +• The internal nodes of tI are the internal nodes of t that are first common ancestors +of two or more leaves of tI; +• The ancestor-descendent relation in tI is inherited from the one in t; + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +27 +• For every internal node v of t that appears in tI, let H be its decoration in t. Denote +by J the set of positive integers k such that the k-th tree of tv contains a leaf of tI. +For every k in J, we define L(k) as the smallest image by I of a marked leaf label +in the k-th tree of tv. The decoration of v in tI is the reduction of HL. +For every internal node v (resp. leaf ℓ) of tI, we also define φ(v) to be the only internal +node (resp. leaf) of t corresponding to v. +Remark. When (t, I) is a marked tree and t′ is a subtree of t, we will denote t′ +I the tree +induced by the restriction of I to the set of labels of leaves of t′. +As a consequence of Definitions 5.1 and 5.3, we obtain: +Lemma 5.4. Let (t, I) be a marked tree in T0. Then +Graph(t)I = Graph(tI). +8 +3 +7 +2 +6 +4 +5 +1 +4 +1 +3 +2 +Graph(t) +3 +7 +5 +6 +1 +2 +4 +1 +3 +4 +2 +Figure 10. Relations between induced subgraph and induced subtree. +Definition 5.5. For every pair of graphs (G, H) such that G has no blossom and H has at +most one blossom, let OccG(H) be the number of partial injection I from the vertex labels +of G to N such that no blossom is marked and HI is isomorphic to G. +Definition 5.6. For every pair of graphs (G, H) and a ∈ N such that G has no blossom, +H has exactly one blossom and a is the label of a vertex of G, let OccG,a(H) be the number + +(6) = 1, J(7) = 2, J(3) = 1, J(4) = 3tGraph(ts)Graph(t)>J(6) = 1, (7) = 2, J(1) = 3, J(3) = 428 +TH´EO LENOIR +of partial injection I from the vertex labels of G to N such that the image of the blossom +by I is a and HI is isomorphic to G. +2 +3 +1 +∗ +8 +5 +7 +9 +6 +6 +Figure 11. Two occurences of a P4 in a blossomed graph H. If G is a P4, +the blue one is counted twice in OccG(H), the red one in counted once in +OccG,a(H) iff a is the label of an extremity of G. +Definition 5.7. For every graph G without blossom, and every a ∈ {1, . . . , N(G) = |G|}, +set: +OccG,P(z) := +� +H∈P +OccG(H)zN(H)−N(G) +N(H)! +; +OccG,P•(z) := +� +H∈P• +OccG(H)zN(H)−N(G) +N(H)! +OccG,a,P•(z) := +� +H∈P• +OccG,a(H)zN(H)−N(G)+1 +N(H)! +Notation. OccG,... will only be used for graphs G with no blossom. +Proposition 5.8. For every k ≥ 1 and every a ∈ {1, . . . , k}: +� +G: N(G)=k +OccG,P(z) = P (k)(z) +(28) +� +G: N(G)=k +OccG,P•(z) = (P •)(k)(z) +(29) +� +G: N(G)=k +OccG,a,P•(z) = (P •)(k−1)(z) +(30) +Thus for every graph G with no blossom and every a ∈ {1, . . . , N(G)}, OccG,P, OccG,P• +and OccG,a,P• have a radius of convergence strictly greater than R, the radius of convergence +of T. +Proof. Let H be an element of P. Since there are +N(H)! +(N(H)−k)! choices of partial injection +whose image is {1, . . . , k}, we have: +� +G: N(G)=k +OccG,P(z) = +� +H∈P +� +G: N(G)=k +OccG(H)zN(H)−k +N(H)! += +� +H∈P +zN(H)−k +(N(H) − k)! = P (k)(z) + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +29 +The proofs of Eqs. (29) and (30) are similar. In Eq. (30), since I−1(a) must be ∗, there +are exactly +N(H)! +(N(H)−(k−1))! choices for the partial injection. +For every graph G, OccG,P has nonnegative coefficients and for every k ≥ 0, as mentioned +in Section 4.2, P (k) has a radius of convergence at least R0, the minimum of the radii of +convergence P and P •, which is greater than R. This implies that OccG,P has a radius of +convergence greater than R. The proof for the other series is similar. +□ +5.2. Enumerations of trees with a given induced subtree. The key step in the proof +of our main theorem is to compute the limiting probability (when n → ∞) that a uniform +induced subtree of a uniform tree in TP,P• with n leaves is a given substitution tree. +In the following, let τ ∈ T0 be a fixed substitution tree of size at least 2. +Definition 5.9. We define Tτ to be the set of marked trees (t, I) where t ∈ TP,P• and I +is such that tI is isomorphic to τ. We also define Tτ to be the corresponding exponential +generating function (where the size parameter is the total number of leaves, including the +marked ones). +The aim now is to decompose a tree admitting τ as a subtree in smaller trees. Let +(t, I) be in Tτ. A prime node v of τ is such that t[φ(v)] is either in case (D2) or (D4) of +Definition 2.17: in other word, φ(v) must be a prime node. In constrast, knowing that an +internal node v′ of τ is decorated with ⊕ or ⊖ does not give any information about the +decoration of φ(v′). +In order to state Theorem 5.11 below, we need to partition the internal nodes of τ: +Definition 5.10. Let (t, I) be in Tτ. We denote by V(t, I) the set of internal nodes v of +τ such that φ(v) is non-linear. The set V(t, I) can be partitioned in 4 subsets: +• V0(t, I) the set of internal nodes v such that t[φ(v)] is in case (D2); +• V1(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and no marked +leaf is in the i-th tree of tφ(v) (where i is the element such that (D4) holds in +Definition 2.17); +• V2(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and exactly +one marked leaf is in the i-th tree of tφ(v) (where i is the element such that (D4) +holds in Definition 2.17); +• V3(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and at least +two marked leaves are in the i-th tree of tφ(v) (where i is the element such that (D4) +holds in Definition 2.17). +Note that the set of non-linear nodes of τ must be included in V(t, I). Since for every +element v of V(t, I) at most one element of tφ(v) is non trivial, at most one element of τv +is non trivial. Thus if τ has some non-linear nodes v such that two or more elements of τv +are not reduced to a single leaf, Tτ = ∅. In the following, we assume that it is not the case +for τ. If τv has exactly one non trivial subtree, then v ∈ V3(t, I). Otherwise, τv is a union +of leaves. +Notation. We denote by U0 (resp. U1) the set of internal nodes v of τ such that no tree +(resp. exactly one tree) of τv has size greater or equal to 2. + +30 +TH´EO LENOIR +Note that by definition V0(t, I) ∪ V1(t, I) ∪ V2(t, I) ⊂ U0 and V3(t, I) ⊂ U1. +We also define rkt,I : V2(t, I) �→ N as follows. Let v ∈ V2(t, I), we define rkt,I(v) to +be the only integer k such that, if ℓ is the label of the k-th leaf of τv then the leaf of label +I−1(ℓ) in t belongs to the i-th tree of tφ(v) (where i is the element such that (D4) holds in +Definition 2.17). For every v ∈ V2(t, I), we have 1 ≤ rkt,I(v) ≤ |τv|. +Theorem 5.11. Let τ be a substitution tree of size at least 2 such that every non-linear +node of τ is in U0 ∪ U1. Let V0, V1 and V2 be three disjoint subsets of U0 and let V3 be a +subset of U1 such that every non-linear node of τ is in V := V0 ∪ V1 ∪ V2 ∪ V3. Let rk: +V2 → N be such that 1 ≤ rk(w) ≤ |τw| for every w ∈ V2. +Let Tτ,V0,V1,V2,V3,rk be the set of marked trees (t, I) in Tτ such that V0(t, I) = V0, V1(t, I) = +V1, V2(t, I) = V2, V3(t, I) = V3, rkt,I = rk, and let Tτ,V0,V1,V2,V3,rk be its exponential gener- +ating function. +Then +Tτ,V0,V1,V2,V3,rk = z|t|T root � +T ⊕ +not⊕ +�d= � +T ⊖ +not⊕ +�d̸= � +T blo +not⊕ +�dV →V � +T +′ +not⊕ +�dV →ℓ exp(nLTnot⊕) +× T |V1|T ′|V2|(T ⊕)n1(T blo)n2F +where +F := +� +v∈V0 +Occdec(v),P +� +v∈V3 +Occdec(v),br(v),P• +� +v∈V1 +Occdec(v),P• +� +v∈V2 +Occdec(v),rk(v),P• +and: +• d= is the number of edges between two internal nodes not in V with the same +decoration (⊕ and ⊕, or ⊖ and ⊖); +• d̸= is the number of edges between two internal nodes not in V decorated with +different decorations (⊕ and ⊖); +• dV →V is the number of edges between an internal node not belonging to V and one +of its children belonging to V ; +• dV →ℓ is the number of edges between an internal node not in V and a leaf; +• nL is the number of internal nodes not in V ; +• dec(v) is the decoration of v; +• for every v ∈ V3, br(v) is the position of the subtree of τv not reduced to a leaf; +• n1 (resp. n2) is the number of internal nodes v in V3 such that the root of the +br(v)-th tree of τv is not in V (resp. is in V ); +• T root = T ⊕ if the root of τ is not in V , T root = T blo otherwise. +Proof. Let t be a tree in Tτ,V0,V1,V2,V3,rk. We decompose t into several disjoints subtrees. +The blossoms are nodes where (the root of) an other tree will be glued (and thus they are +not counted in the generating series, to avoid counting them twice). +We define t→root to be the tree t blossomed at φ(r0), where r0 is the root of τ. +We define the tree tv→ in the following way: +• If v is not in V , tv→ is the subtree of t containing φ(v) and all the subtrees of tφ(v) +that do not contain a marked leaf of t. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +31 +V0 +V0 +V1 +V2 +V3 +V +V +Figure 12. A possible τ and choices of V0, V1, V2, V3 +• If v is in V0 ∪ V1 ∪ V2, tv→ is the tree t[φ(v)]. +• If v is in V3, tv→ is the tree t[φ(v)] obtained after blossoming the root of the non +trivial tree of tφ(v). The blossom is marked with the smallest mark in the non trivial +tree of tφ(v). +For every internal nodes v, v′ in τ such that v is not in V and v′ is a child of v, let tv→v′ +be the unique tree of tφ(v) containing φ(v′), blossomed at φ(v′). +For every internal node v in τ not in V , and every leaf f which is a child of v in τ, we +define tv→f to be the subtree of tφ(v) containing φ(f). +For every internal node v in V3, we define tv→br(v) to be the non trivial tree of tφ(v) blossomed +at φ(v′), where v′ is the root of the br(v)-th tree of τv. +Now we need to analyze the properties of the trees that appear in this decomposition +and compute the corresponding exponential generating function. In the rest of the proof, +we will say abusively that every blossomed tree belongs to TP,P•, and that two nodes both +decorated with ⊕ or ⊖ have the same decoration, even if they do not have the same number +of children. +(i): analysis of t→root where v ̸∈ V +The tree t→root is a tree in TP,P•, it has no marked leaf and a unique blossom. If the root +is not in V and decorated with ⊕ (resp. ⊖), the blossom is ⊕-replaceable (see Definition 4.4) +(resp. ⊖-replaceable). If the root is in V , the blossom is replaceable. + +32 +TH´EO LENOIR +root +ii +iii +iv +iv +iv +iv +v +v +vii +vii +vii +vi +ix +tv,v′ +i +viii +viii +x +xi +tv,f +tv→ +t→root +tv→br(v) +Figure 13. The decomposition of a tree admitting the graph τ of Fig. 12 +as an induced tree. The different notations correspond to the different cases +of the proof of Theorem 5.11. +The corresponding exponential generating function is equal to T ⊕ if the root is not in +V and equal to T blo otherwise. +(ii): analysis of tv→v′ where v ̸∈ V and v′ is a child of v not in V with the same +decoration +The tree tv→v′ is a tree in TP,P• whose root is not decorated with the same decoration as +v and with one blossom ⊕-replaceable if v′ is decorated with ⊕, ⊖-replaceable otherwise +and no marked leaf. +The exponential generating function of such trees is either T ⊕ +not⊕ if both nodes are deco- +rated with ⊕ or T ⊖ +not⊖ if both nodes are decorated with ⊖, which are both equal. +(iii): analysis of tv→v′ where v ̸∈ V and v′ is a child of v not in V with a different +decoration +The tree tv→v′ is a tree in TP,P• whose root is not decorated with the same decoration as +v and with one blossom ⊕-replaceable if v′ is decorated with ⊕, ⊖-replaceable otherwise +and no marked leaf. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +33 +The exponential generating function of such trees is either T ⊖ +not⊕ if v is decorated with +⊕ and v′ with ⊖ or T ⊖ +not⊖ if v is decorated with ⊖ and v′ with ⊕, which are both equal. +(iv): analysis of tv→v′ where v ̸∈ V and v′ is a child of v in V +The tree tv→v′ is a tree in TP,P• whose root is not decorated with the decoration of v +with one blossom and no marked leaf. +The corresponding exponential generating function is T blo +not⊕. +(v): analysis of tv→f where v ̸∈ V and f is a leaf which is a child of v +The tree tv→f is a tree in TP,P• whose root is not decorated with the decoration of v +with one marked leaf and no blossom. +The corresponding exponential generating function is zT ′ +not⊕. +(vi): analysis of tv→br(v) where v ∈ V3 +The tree tv→br(v) is a tree with a blossom that is replaceable if the root of the br(v)-th +subtree of t[v] is in V , ⊕-replaceable (resp. ⊖-replaceable) if the root is not in V and +labeled ⊕ (resp. ⊖), with no marked leaf. +The corresponding exponential generating function is equal to T ⊕ if the root of the +br(v)-th tree of τv is not in V and equal to T blo otherwise. +(vii): analysis of tv→ where v ̸∈ V +The tree tv→ is a tree whose root denoted is decorated with the same decoration as v, who +has no marked leaf and no blossom. It verifies all the conditions of being (P, P•)-consistent, +except that the root can have 0 or 1 child. +The corresponding exponential generating function is � +k≥0 T k +not⊕ = exp(Tnot⊕). +(viii): analysis of tv→ where v ∈ V0 +The tree tv→ is a tree in TP,P• whose root is decorated with an element of P. The subtree +induced by the marked leaves of tv→ is τ[v]. Moreover tv→ has only one internal node. +The corresponding exponential generating function is +� +H∈P +Occdec(v)(H)zN(H) +N(H)! += zN(dec(v))Occdec(v),P. +Indeed, for a given H ∈ P, the term zN(H) +N(H)! correspond to the set of leaves and the term +Occdec(v)(H) to the possible markings. +(ix): analysis of tv→ where v ∈ V3 +The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.17. The subtree +induced by the marked leaves of tv→ is τ[v], where the non-trivial tree of τv is replaced by +a blossom, marked with the smallest mark in the non-trivial tree of τv. Moreover tv→ has +only one internal node. +Similarly to case (viii), the corresponding exponential generating function is: +� +H∈P• +Occdec(v),br(v)(H)zN(H) +N(H)! += zN(dec(v))−1Occdec(v),rk(v),P•. +(x): analysis of tv→ where v ∈ V1 + +34 +TH´EO LENOIR +The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.17. The subtree +induced by the marked leaves of tv→ is τ[v] and no marked leaf belongs to the i-th tree of +tφ(v) (where i is the element such that (D4) holds in Definition 2.17). +The corresponding exponential generating function is: +� +H∈P• +Occdec(v)(H)zN(H) +N(H)! +× T = zN(dec(v))Occdec(v),P• × T. +The sum corresponds to the choice of the root (as in the previous cases), and the factor +T to the potential non trivial tree of tv. +( +¯ +xi): analysis of tv→ where v ∈ V2 +The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.17. The subtree +induced by the marked leaves of tv→ is τ[v] and there is only one marked leaf ℓ in the i-th +tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.17). Moreover, if +we denote by j the label of ℓ, the label of the rk(v)-th leaf of τv is I(j). +Similarly to case (x), the corresponding exponential generating function is: +� +H∈P• +Occdec(v),rk(v)(H)zN(H) +N(H)! +× zT ′ = zN(dec(v))Occdec(v),rk(v),P• × T ′. +All these conditions ensure that we can recover t by gluing all the different trees and that +the subtree of t induced by I is τ. Thus, Tτ,V0,V1,V2,V3,rk is the product of the generating +functions corresponding to labeled such trees and concludes the proof of the theorem. +□ +Corollary 5.12. The series Tτ,V0,V1,V2,V3,rk has radius at least R, is ∆-analytic and its +asymptotic expansion near R is: +Tτ,V0,V1,V2,V3,rk = Cτ,V0,V1,V2,V3,rk +� +1 − z +R +�β +(1 + o(1)) +where +Cτ,V0,V1,V2,V3,rk := ακγ(1 + P •(R))θ(1 − P •(R))|V1|2λRµ × F(R) +with +β = −1 + d= + d̸= + dV →V + dV →ℓ + |V2| + |V3| +2 +γ = dV →ℓ + |V2| − d= − d̸= − dV →V − |V3| − 1 +θ = d= + d̸= − |V1| − |V2| − n2 − nL +λ = −dV →ℓ − n1 − 2d= − 2d̸= + dV →V + nL +µ = −dV →ℓ − |V2| + l +and α = 1 +2 if the root is not in V , +1 +1+P •(κ) otherwise. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +35 +6. Proof of the main theorems +6.1. Background on graphons. We now review the necessary material on graphons. +We refer the reader to [19] for a comprehensive presentation of deterministic graphons, +while [7] studies specifically the convergence of random graphs in the sens of graphons. +Here we will only recall the properties needed to prove the convergence of random graphs +toward the Brownian cographon (see [1]). +Definition 6.1. A graphon is an equivalence class of symmetric functions f : [0, 1]2 �→ +[0, 1], under the equivalence relation ∼, where f ∼ g if there exists a measurable function +φ : [0, 1] �→ [0, 1] that is invertible and measure preserving such that, for almost every +(x, y) ∈ [0, 1]2, f(φ(x), φ(y)) = g(x, y). We denote by ˜ +W the set of graphons. +Intuitively graphons can be seen as continuous analogous of graph adjacency matrices, +where graphs are considered up to relabeling (hence the quotient by ∼). There is a natural +way to embed a finite graph into graphons: +Definition 6.2. Let G be a (random) graph of size n. We define the (random) graphon +WG to be the equivalence class of wG : [0, 1]2 �→ [0, 1] defined by: +∀(x, y) ∈ [0, 1]2 +wG(x, y) := 1⌈nx⌉connected to⌈ny⌉ +There exists a metric δ□ on the set of graphons ˜ +W such that ( ˜ +W, δ□) is compact [19, +Chapter 8], thus we can define for δ□ the convergence in distribution of a random graphon. +If (G(n))n≥1 is a sequence of random graphs, there exists a simple criterion [7, Theorem +3.1] characterizing the convergence in distribution of (WG(n)) with respect to δ□: +Theorem 6.3 (Rephrasing of [7], Theorem 3.1). For any n, let G(n) be a random graph of +size n. Denote by WG(n) the random graphon associated to G(n). The following assertions +are equivalent: +(a) The sequence of random graphons (WG(n))n≥1 converges in distribution to some +random graphon W. +(b) The random infinite vector +� +OccG(n)(H) +n(n−1)...(n−|H|+1) +� +H finite graph converges in distribution +in the product topology to some random infinite vector (ΛH)H finite graph. +For a finite graph H, the random variable ΛH can be seen as the density of the pattern +H in the graphon W: the variables (ΛH)H play the roles of margins of W in the space of +graphons. +For k ≥ 1 and W a random graphon, we denote by Samplek(W) the unlabeled random +graph built as follows: Samplek(W) has vertex set {v1, v2, . . . , vk} and, letting (X1, . . . , Xk) +be i.i.d. uniform random variables in [0, 1], we connect vertices vi and vj with probability +w(Xi, Xj) (these events being independent, conditionally on (X1, · · · , Xk) and W). The +construction does not depend on the representation of the graphon. +With the notations of Theorem 6.3, we have for any finite graph H +ΛH = P(Sample|H|(W) = H | W). + +36 +TH´EO LENOIR +The article [1] introduces a random graphon W1/2 called the Brownian cographon which +can be explicitly constructed as a function of a realization of a Brownian excursion. Besides, +[1, Proposition 5] states that the distribution of the Brownian cographon is characterized2 +by the fact that for every k ≥ 2, Samplek(W1/2) has the same law as the unlabeled +version of Graph(bk) with bk a uniform labeled binary tree with k leaves and i.i.d. uniform +decorations in {⊕, ⊖}. +A consequence of this characterization is a simple criterion for convergence to the Brow- +nian cographon. +Lemma 6.4 (Rephrasing of [1] Lemma 4.4). For every positive integer n, let T(n) be a +uniform random tree in TP,P• with n vertices. +For every positive integer ℓ, Iℓ +(n) be a +uniform partial injection from {1, . . . , n} to N whose image is {1, . . . , ℓ} and independent +of T(n). Denote by T(n) +Iℓ(n) the subtree induced by Iℓ +(n). +Suppose that for every ℓ and for every binary tree τ with ℓ leaves, +(31) +P(T(n) +I(n) = τ) −−−→ +n→∞ +(ℓ − 1)! +(2ℓ − 2)!. +Then WGraph(T(n)) converges as a graphon to the Brownian cographon W1/2 of parameter +1/2. +6.2. Conclusion of the proof of Theorem 1.1. +Proposition 6.5. Let τ be a binary tree with ℓ ≥ 2 leaves. The series Tτ has radius of +convergence R, is ∆-analytic and its asymptotic expansion near R is: +Tτ = +κ +(1 + P •(R))22ℓ−2 +� +1 − z +R +�− 2ℓ−1 +2 +(1 + o(1)) . +(32) +Proof. As +Tτ = +� +τ,V0,V1,V2,V3,rk +Tτ,V0,V1,V2,V3,rk, +the asymptotic expansions of the different series Tτ,V0,V1,V2,V3,rk yield the ∆-analyticity of +Tτ, its asymptotic expansion and its radius of convergence. +Note that β ≤ +1+e +2 +where e is the number of edge of τ, with equality if and only if +V0, V1, V2 and V3 are all empty. +Therefore, only the series Tτ,∅,∅,∅,∅,rk contributes to the leading term of the asymptotic +expansion. In this case, dV →ℓ = ℓ, d= + d̸= = ℓ − 2 and nL = ℓ − 1 which gives the +announced expansion. +□ +Theorem 6.6. Let τ be a binary tree with ℓ ≥ 2 leaves. +For n ≥ ℓ and T(n) be a +uniform random tree in TP,P• with n vertices. Let Iℓ +(n) be a uniform partial injection from +{1, . . . , n} to N whose image is {1, . . . , ℓ} and independent of T(n). Denote by T(n) +Iℓ(n) the +subtree induced by Iℓ +(n). +2This characterization is strongly linked to the remarkable property that k uniform leaves in the CRT +induce a uniform binary tree with k leaves, see again [1, Section 4.2]. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +37 +Then +P(T(n) +Iℓ(n) = τ) −−−→ +n→∞ +(ℓ − 1)! +(2(ℓ − 1))!. +Proof. Since Iℓ +(n) is independent of T(n), +P(T(n) +Iℓ(n) = τ) = +n![zn]Tτ +n(n − 1) . . . (n − ℓ + 1)n![zn]T = +[zn]Tτ +n(n − 1) . . . (n − ℓ + 1)[zn]T +By applying the Transfer Theorem [8, Corollary VI.1 p.392] to Eq. (32), we get +[zn]Tτ ∼ +κ +(1 + P •(R))22ℓ−2 +n +2ℓ−3 +2 +Γ +� +2ℓ−1 +2 +� +Rn +and by Corollary 4.13 we obtain +n × · · · × (n − ℓ + 1)[zn]T ∼ nℓ +κ +√π(1 + P•(R)) +1 +Rnn +3 +2 . +Thus when n goes to infinity +P(T(n) +Iℓ(n) = τ) → +√π +22ℓ−2Γ +� +2ℓ−1 +2 +� = +(ℓ − 1)! +(2(ℓ − 1))! +□ +Combining Lemma 6.4 and Theorem 6.6 prove Theorem 6.7 of which Theorem 1.1 is a +particular case. +Theorem 6.7. Let G(n) be a uniform random graph in GP,P• with n vertices. We have the +following convergence in distribution in the sense of graphons: +WG(n) +n→∞ +−→ W +1 +2 +where W +1 +2 is the Brownian cographon of parameter 1 +2. +6.3. Number of induced prime subgraphs. We now estimate for a prime graph H the +number OccH(G(n)) of induced occurences of H in G(n) and show that in average it is null, +linear or of order n +3 +2. +We first observe that substitution trees encoding prime graphs have a very simple struc- +ture. +Lemma 6.8. Let H be a prime graph. If t is a substitution tree such that H = Graph(t), +t is reduced to a single internal node decorated with a relabeling of H with |H| leaves. +Proof. Let t be such a tree and r its root. To every element t′ of tr we can associate a +module of H by taking the vertices whose labels are the labels of the leaves of t′. Thus tr +is a union of leaves, and the decoration of the root is a relabeling of H. +□ +We say that H verifies (A) if there exists a ∈ {1, . . . , ℓ} such that OccG,a,P•(R) > 0. + +38 +TH´EO LENOIR +Theorem 6.9. Let H be a prime graph and let ℓ be its size. For n ≥ ℓ, let G(n) be a +uniform random graph in GP,P• with n vertices. +Then if H verifies (A), +E[OccH(G(n))] ∼ KHn +3 +2 +with +KH = +Rℓ−1√π +� +a∈{1,...,ℓ} +OccH,a,P•(R) +κ(1 + P •(R)) +otherwise, +E[OccH(G(n))] ∼ KHn +with +KH = +�1 − P •(R) +1 + P •(R)OccH,P•(R) + OccH,P(R) +� Rℓ +κ2 +Proof. Let T(n) be a uniform random tree in TP,P• with n vertices . +Let τ be the canonical tree of H and NT(n),τ the number of induced subtrees of Tn +isomorphic to τ. Since τ is the unique substitution tree of G, E[OccH(G(n))] = E[NT(n),τ]. +By independence +E[OccH(G(n))] = n![zn]Tτ +n![zn]T = [zn]Tτ +[zn]T . +From Theorem 5.11, since in this case the only node of τ is either in V0, V1 or V2, we +have that: +Tτ = zℓT blo +� +�T ′ +� +� +� +a∈{1,...,ℓ} +OccG,a,P• +� +� + TOccG,P• + OccG,P +� +� . +Thus +• in case (A), with Eqs. (22) and (26) +Tτ ∼ +Rℓ +R(1 + P•(R))2 +� +� +� +a∈{1,...,ℓ} +OccH,a,P•(R) +� +� +� +1 − z +R +�−1 +; +• Otherwise, Tτ ∼ +� 1−P •(R) +1+P •(R)OccH,P•(R) + OccH,P(R) +� +Rℓ +κ(1+P •(R)) +� +1 − z +R +�− 1 +2 . +By applying the Transfer Theorem [8, Corollary VI.1 p. 392], +• In case (A), +[zn]Tτ ∼ +Rℓ +Rn+1(1 + P•(R))2 +� +a∈{1,...,ℓ} +OccH,a,P•(R) +• Otherwise, +[zn]Tτ ∼ +�1 − P •(R) +1 + P •(R)OccH,P•(R) + OccH,P(R) +� +Rℓ +√πκ(1 + P•(R)) +1 +Rnn +1 +2 +. + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +39 +By Corollary 4.13, +[zn]T ∼ +κ +√π(1 + P•(R)) +1 +Rnn +3 +2 . +Thus: +• In case (A), +E[OccG(G(n))] ∼ +Rℓ−1√π +� +a∈{1,...,ℓ} +OccG,a,P•(R) +κ(1 + P •(R)) +n +3 +2, +• Otherwise, +E[OccG(G(n))] ∼ +�1 − P •(R) +1 + P •(R)OccG,P•(R) + OccG,P(R) +� Rℓ +κ2 n, +concluding the proof. +□ +An interesting application of this theorem is the computation of the asymptotic number +of ˜P4’s in a random uniform graph of each of the graph classes of Section 3, where ˜P4 is +the only labeling of P4 with endpoints 1 and 4 and 2 connected to 1. +Lemma 6.10. A prime spider has exactly |K|(|K| − 1) induced ˜P4. A pseudo-spider of +size k has exactly (|K| + 2)(|K| − 1) induced ˜P4. +Proof. One can check that for a prime spider, the P ′ +4s are induced by the partial injections +I whose domain is {k, k′, f(k), f(k′)} for every (k, k′) ∈ K2 with k ̸= k′. In the 24 such +partial injections, only 2 are such that the graph induced is ˜P4. Since every induced ˜P4 is +counted twice, we have |K|(|K| − 1) induced ˜P4. +For a pseudo-spider, let d be the duplicate and d0 the original node. The P ′ +4s are induced +by the partial injections I whose domain is {k, k′, f(k), f(k′)} for every (k, k′) ∈ K2 with +k ̸= k′, or by the partial injections I whose domain is {d, k′, f(d0), f(k′)} (resp. {f −1(d0), k′, d, f(k′)}) +for every k′ ∈ K with k′ ̸= d0 (resp. k′ ̸= f −1(d0)) if d0 is in K (resp. in S). In the 24 such +partial injections, only 2 are such that the graph induced is ˜P4. Since every induced ˜P4 +not containing d is counted twice, we have |K|(|K| − 1) + 2(|K| − 1) = (|K| + 2)(|K| − 1) +induced ˜P4. +□ +Remark. Note that this lemma implies that Occ ˜ +P4,a,P• = 0 for all the graph classes men- +tionned in Section 3. +Theorem 6.11. For each graph class introduced in Section 3, we have the following ex- +pressions for Occ ˜ +P4,P and Occ ˜ +P4,P•: + +40 +TH´EO LENOIR +P4-tidy +Occ ˜ +P4,P• +tidy(z) = (2 + 16z + 4z3) exp(z2) − 1 − 8z +Occ ˜ +P4,Ptidy(z) = Occ ˜ +P4,P• +tidy(z) + 5z +P4-lite +Occ ˜ +P4,P• +lite(z) = (2 + 16z + 4z3) exp(z2) − 1 − 8z +Occ ˜ +P4,Plite(z) = Occ ˜ +P4,P• +lite(z) + 4z +P4-extendible +Occ ˜ +P4,P• +ext(z) = 1 + 8z +Occ ˜ +P4,Pext = Occ ˜ +P4,P• +ext(z) + 5z +P4-sparse +Occ ˜ +P4,P•spa(z) = Occ ˜ +P4,Pspa(z) = 2 exp(z2) − 1 +P4-reducible +Occ ˜ +P4,P• +red(z) = Occ ˜ +P4,Pred = 1 +P4-free +Occ ˜ +P4,P•cog(z) = Occ ˜ +P4,Pcog(z) = 0 +Proof. We only detail the computation of Occ ˜ +P4,P• +tidy and Occ ˜ +P4,Ptidy for P4-tidy graphs as +this is the most involved case. Note that, with the notations of Section 4.1, +Occ ˜ +P4,P(z) = +� +n∈N +� +H∈RPn +� +H′∼H +Occ ˜ +P4(H)zN(H)−4 +N(H)! += +� +n∈N +� +H∈RPn +Occ ˜ +P4(H)zN(H)−4 +|Aut(H)| +and similarly +Occ ˜ +P4,P•(z) = +� +n∈N +� +H∈RP•n +� +H′∼H +Occ ˜ +P4(H)zN(H)−4 +N(H)! += +� +n∈N +� +H∈RPn +Occ ˜ +P4(H)zN(H)−4 +|Aut(H)| +According to Theorem 3.7, Ptidy is composed of one C5 that has 10 automorphisms and +10 induced ˜P4 and all its relabelings, one P5, and one P5 that both have 2 automorphisms +and 4 induced ˜P4’s and all their relabelings. +For k ≥ 3 (resp. k = 2), there are thin and fat spiders corresponding to the 2 (resp. 1) +different orbits of the action Φ2k over prime spiders of size 2k, each having k! automorphisms +and k(k − 1) ˜P4’s. +For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can +come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4) +cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over pseudo-spiders +of size 2k + 1, each having 2(k − 1)! automorphisms and (k + 2)(k − 1) ˜P4’s. +Thus we have +Occ ˜ +P4,P(z) = z + 4z +2 + 4z +2 + 2 +2 + 2 +� +k≥3 +k(k − 1)z2k−4 +k! ++ 44z +2 + 8 +� +k≥3 +(k + 2)(k − 1)z2k−3 +2(k − 1)! += 5z + 1 + 2 +� +k≥1 +z2k +k! + 8z + 4 +� +k≥1 +(k + 4)z2k+1 +k! += 5z + 1 + 2 exp(z2) − 2 + 8z + 4 +� +k≥0 +z2k+3 +k! ++ 16 +� +k≥1 +z2k+1 +k! + +GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS +41 += 5z + 2 exp(z2) − 1 + 4z3 exp(z2) + 16z exp(z2) − 8z += 5z + (2 + 16z + 4z3) exp(z2) − 1 − 8z +Now let’s compute Occ ˜ +P4,P•(z). For k ≥ 3 (resp. k = 2), there are thin and fat spiders +with blossom corresponding to the 2 (resp. 1) different orbits of the action Φ2k over blos- +somed prime spiders G with 2k non blossomed vertices, each having k! automorphisms and +k(k − 1) ˜P4’s. +For k ≥ 3 (resp. k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can +come from K or S, and can be connected or not to the initial vertex. These 8 (resp. 4) +cases correspond to the 8 (resp. 4) different orbits of the action Φ2k+1 over blossomed +pseudo-spiders with 2k + 1 non blossomed vertices, each having 2(k − 1)! automorphisms +and (k + 2)(k − 1) ˜P4’s. +Hence +Occ ˜ +P4,P•(z) = 2 +2 + 2 +� +k≥3 +k(k − 1)z2k−4 +k! ++ 44z +2 + 8 +� +k≥3 +(k + 2)(k − 1)z2k−3 +2(k − 1)! +Thus Occ ˜ +P4,P•(z) + 5z = Occ ˜ +P4,P(z) which gives the announced result. +□ +Combining Theorem 6.9, Theorem 6.11 and the remark above, we get that ˜P4 does not +verify (A), thus ˜P4 belongs to the linear case of Theorem 6.9: +Corollary 6.12. Let G(n) be a uniform graph of size n taken uniformly at random in one +of the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible, P4-reducible or P4-free. +Then E[Occ ˜ +P4(G(n))] ∼ K ˜ +P4n where K ˜ +P4 is defined in Theorem 6.9. +Here are the numerical approximations of K ˜ +P4 in the different cases: +class of graph +K ˜ +P4 +P4-tidy +0.29200322 +P4-lite +0.28507010 +P4-extendible +0.24959979 +P4-sparse +0.10280703 +P4-reducible +0.08249263 +P4-free +0 +Acknowledgements. I would like to thank Lucas Gerin and Fr´ed´erique Bassino for +useful discussions and for carefully reading many earlier versions of this manuscript. +References +[1] F. Bassino, M. Bouvel, V. F´eray, L. Gerin, M. Maazoun, and A. Pierrot. Random cographs: Brownian +graphon limit and asymptotic degree distribution. Random Struct. Algor., 60(2):166–200, 2022. +[2] C. Borgs, J. T. Chayes, L. Lov´asz, V. T. S´os, and K. Vesztergombi. Convergent sequences of dense +graphs I: Subgraph frequencies, metric properties and testing. Adv. Math., 219(6):1801–1851, 2008. +[3] A. Brandst¨adt, V. B. Le, and J. P. Spinrad. Graph Classes: A Survey. Society for Industrial and +Applied Mathematics, 1999. + +42 +TH´EO LENOIR +[4] A. Bretscher, D. Corneil, M. Habib, and C. Paul. A simple linear time LexBFS cograph recognition +algorithm. SIAM J. Discrete Math., 22(4):1277–1296, 2008. +[5] D. G. Corneil, H. Lerchs, and L. Stewart Burlingham. Complement reducible graphs. Discrete Appl. +Math., 3(3):163–174, 1981. +[6] D. G. Corneil, Y. Perl, and L. K. Stewart. A linear recognition algorithm for cographs. SIAM J. +Comput., 14(4):926–934, 1985. +[7] P. Diaconis and S. Janson. Graph limits and exchangeable random graphs. Rendiconti di Matematica, +28(1):33–61, 2008. +[8] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. +[9] T. Gallai. Transitiv orientierbare graphen. Acta Mathematica Academiae Scientiarum Hungarica, +18:25–66, 1967. +[10] V. Giakoumakis, F. Roussel, and H. Thuillier. On P4-tidy graphs. Discrete Math. Theor. Comput. +Sci., 1:17–41, 1997. +[11] V. Giakoumakis and J.-M. Vanherpe. On extended P4-reducible and extended P4-sparse graphs. The- +oretical Computer Science, 180(1):269–286, 1997. +[12] M. Habib and C. Paul. A simple linear time algorithm for cograph recognition. Discrete Appl. Math., +145(2):183–197, 2005. +[13] B. Jamison. A tree-representation for P4-sparse graphs. Discrete Appl. Math., 35(2):115–129, 1992. +[14] B. Jamison and S. Olariu. A new class of brittle graphs. Stud. Appl. Math., 81(1):89–92, 1989. +[15] B. Jamison and S. Olariu. P4-reducible graphs—class of uniquely tree-representable graphs. Stud. +Appl. Math., 81(1):79–87, 1989. +[16] B. Jamison and S. Olariu. On a unique tree representation for P4-extendible graphs. Discrete Appl. +Math., 34(1-3):151–164, 1991. +[17] B. Jamison and S. Olariu. Recognizing P4 sparse graphs in linear time. SIAM J. Comput., 21(2):381– +406, 1992. +[18] B. Jamison and S. Olariu. A linear-time recognition algorithm for P4-reducible graphs. Theoret. Com- +put. Sc., 145(1):329–344, 1995. +[19] L. Lov´asz. Large Networks and Graph Limits. Colloquium Publications. American Mathematical So- +ciety, 2012. +[20] R. H. M¨ohring. Algorithmic Aspects of Comparability Graphs and Interval Graphs, pages 41–101. +Springer, 1985. +[21] B. Stufler. Graphon convergence of random cographs. Random Struct. & Algor., 59:464 – 491, 2019. +Th´eo Lenoir theo.lenoir@polytechnique.fr +Cmap, Cnrs, ´Ecole polytechnique, +Institut Polytechnique de Paris, +91120 Palaiseau, France + diff --git a/3NFRT4oBgHgl3EQfoDdP/content/tmp_files/load_file.txt b/3NFRT4oBgHgl3EQfoDdP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e7d85d7f34a60cbc3286ef2e88f514fa22849c1 --- /dev/null +++ b/3NFRT4oBgHgl3EQfoDdP/content/tmp_files/load_file.txt @@ -0,0 +1,1587 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf,len=1586 +page_content='GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS TH´EO LENOIR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We consider large uniform labeled random graphs in different classes with few induced P4 (P4 is the graph consisting of a single line of 4 vertices), which generalize the case of cographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Our main result is the convergence to a Brownian limit object in the space of graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We also obtain an equivalent of the number of graphs of size n in the different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Finally we estimate the expected number of induced graphs isomorphic to a fixed graph H for a large variety of graphs H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Our proofs rely on tree encoding of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We then use mainly combinatorial argu- ments, including the symbolic method and singularity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Random graphs are one of the most studied objects in probability theory and in combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A natural question is to investigate the scaling limits of a uniformly chosen graph in a given family (an important example for this paper are the cographs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Cographs have been studied since the seventies by various authors, especially for their algorithmic properties: recognizing cographs can be solved in linear time [4, 6, 12], and many hard problems can be solved in polynomial time for cographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Several equivalent definitions exists of the class of cographs exists, here are two important ones: A graph is a cograph if and only if it has no induced P4 (a line of 4 vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The class of cograph is the smallest class containing every graph reduced to a single vertex, and stable by union and by join1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Simultaneously in [1] and [21], the authors exhibit a Brownian limiting object for a uniform cograph, called the Brownian cographon, which can be explicitly constructed from the Brownian excursion and a parameter p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The convergence holds in distribution in the sense of graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Introduced in [2], graphon is a well-established topic in graph theory but their probabilistic counterpart is more recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Graphon convergence can be seen as the convergence of the renormalized adjacency matrix for the so-called cut metric (a good reference on graphon theory is [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' One natural question to go further than the case of cographs is to study more complicated classes with, in some specific sense, few P4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A natural question is to study classes of graphs to which some algorithmic properties of cographs extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Several classes characterized by properties of their induced P4’s have thus been considered in the graph theory literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1the join of two graphs (G, H) is the graph obtained by adding an edge between every pair of vertices (g, h) ∈ G × H 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='13607v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='PR] 31 Jan 2023 2 TH´EO LENOIR The classes we will focus on here are the following: P4-reducible graphs [15,18], P4-sparse graphs [13,17] P4-lite graphs [14], P4-extendible graphs [16] and P4-tidy graphs [10] which can all be seen as classes defined by some constraints on the induced P4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' All these classes will be defined precisely in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The inclusion relations between these classes are sketched in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-tidy P4-lite P4-sparse P4-extendible P4-reducible P4-free (=cographs) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Inclusion relations between the different classes of graphs To our knowledge, these different classes have not been studied from a probabilistic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The main aim of this paper is to prove a result of universality of the Brownian cographon: for every class previously mentioned, a random graph will converge towards the Brownian cographon of parameter 1 2 (the rigorous construction is given by [1, Definition 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' An intermediate result is the asymptotic enumeration of each of these classes, which was unknown up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For a finite graph G, let WG be the embedding of the finite graph G in the set of graphon (the formal construction will be recalled in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Our main result is: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G(n) be a uniform graph of size n taken uniformly at random in one of the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible or P4-reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The following convergence in distribution holds in the sense of graphons: WG(n) n→∞ −→ W 1 2 where W 1 2 is the Brownian cographon of parameter 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Graphon convergence is equivalent to the joint convergence of subgraphs density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Di- aconis and Janson extended this criterion in [7] to random graphs: the convergence of a family (H(n))n≥1 of random graphs is characterized by the convergence in distribution of OccH(n)(H) nk for every positive integer k and for every finite graph H of size k, where OccG(H) is the number of induced subgraphs of G isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' All the necessary material on graphon will be recalled at the beginning of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 3 Figure 2 shows an example of the adjacency matrix of a random P4-extensible graph of size 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This picture gives an idea of what a realization of the Brownian cographon could look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The adjacency matrix of a random P4-extensible graph of size 200, simulation by Micka¨el Maazoun In the course of proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1, we get an equivalent of the number of graphs in the different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The number of labeled P4-sparse, P4-tidy, P4-lite, P4-extendible, P4-reducible or the number of P4-free graphs of size n is asymptotically equivalent to C n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Rnn 3 2 , for some R, C > 0, depending on the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We can compute with arbitrary precision the numerical values of R and C (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' All the numerical values of R and C vary according to each class which confirms that all these classes are significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 provides a precise estimation of OccH(G(n)) for every cograph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' But for every graph H which is not a cograph, the only information given by the convergence in the sense of graphon is that the number of induced H in G(n) is typically o(n|H|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Quite unexpectedly, thanks to the tools developed to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1, we are able to estimate the expected number of induced subgraphs isomorphic to a specific class of graphs H in G(n): the graphs that are called ”prime” for the modular decomposition (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G(n) be a uniform graph of size n taken uniformly at random in one of the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible or P4-reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let H be a prime graph, denote by OccH(G(n)) the number of labeled subgraphs of G(n) isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 回4 TH´EO LENOIR Then there exists KH ≥ 0 such that: E[OccH(G(n))] ∼ � � � KHn 3 2 if H verifies condition (A) KHn otherwise where (A) is defined p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='38 and constant KH is given in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This results follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9 which is stated in a more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The condition (A) depends on the class of graphs, checking if H verifies condition (A) and if KH is positive is quite straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' To make things more concrete, let us apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3 to the example of H = P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We can check that for each class P4 does not verify condition (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus a uniform random graph contains in average a linear number of induced P4, while Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 only implies that this number is o(n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The different numerical values of KP4 are explicitly computed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='41, and happen to take different values for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For each class, the graph called ”bull” (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 7) does not verify condition (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus a uniform random graph contains in average a number of induced bulls growing as n3/2, while Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 only implies that this number is o(n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' However, for non prime graphs H, the behavior of the expected value of induced subgraphs of G(n) isomorphic to H is not well-understood, which leads to interesting open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The proof is essentially combinatorial and is based on modular de- composition, which allows to encode a graph with a decorated tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Modular decomposition is a standard tool in graph theory (it was introduced in the 60’s by Gallai [9]) but to our knowledge it has been very little used in the context of random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In this paper we introduce an enriched modular decomposition which enables us to obtain exact enumer- ations for a large family of graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The five classes mentioned before fit in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We exploit those enumerative results with tools from analytic combinatorics to get asymptotic estimates in order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The more technical part of the proof is, for every finite graph H, to estimate the number of induced subgraphs of G(n) isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The enriched modular decomposition allows us to count the number of graphs with a specific induced subgraph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Again asymptotics are derived with tools from combinatorics to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In Section 2 we define the encoding of graphs with trees, the modular decomposition and the enriched modular decomposition which will be used throughout the different proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Section 3 presents the necessary material on the different classes of graphs studied: results are already widely known, most of them are quoted from the litterature and reformulated to suit our enriched modular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Sections 4 and 5 are about calculating generating series related to our graph classes: in Section 4 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2 and Section 5 deals with the generating series of graphs with a given induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 5 Section 6 presents the necessary material on graphons, and the proofs of Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Modular decomposition of graphs: old and new 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Labeled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following all the graphs considered are simple and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Each time a graph G is defined, we denote by V its set of vertices and E its set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Whenever there is an ambiguity, we denote by VG (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' EG) the set of vertices (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' edges) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We say that G = (V, E) is a weakly-labeled graph if every element of V has a distinct label in N and that G = (V, E) is a labeled graph if every element of V has a distinct label in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |V |}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The size of a graph G, denoted by |G|, is its number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The minimum of a graph G, denoted min(G), is the minimal label of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, every graph will be labeled, otherwise we will mention explicitly that the graph is weakly-labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We do not identify a vertex with its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A vertex of label i will be denoted vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The label of a vertex v will be denoted ℓ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For any weakly-labeled object (graph or tree) of size n, we call reduction the operation that reduces its labels to the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , n} while preserving the relative order of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For example if G labels 2, 4, 12, 63 then the reduced version of G is a copy of G in which 2, 4, 12, 63 are respectively replaced by 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Encoding graphs with trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph of size n and H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn be weakly-labeled graphs such that no label is given to two distinct vertices of �n i=1 Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The graph G[H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn] = (V, E) is the graph whose set of vertices is V = �n i=1 VHi and such that: for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , n} and every pair (v, v′) ∈ V 2 Hi, {v, v′} ∈ E if and only if {v, v′} ∈ EHi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every (i, j) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , n} with i ̸= j, and every pair (v, v′) ∈ VHi ×VHj, {v, v′} ∈ E if and only if {vi, vj} ∈ EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3 we will use the shortcut ⊕ for the complete graph of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus ⊕[H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn] is the graph obtained from copies of H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn in which for every i ̸= j every vertex of Hi is connected to every vertex of Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This graph is called the join of H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn We use the shortcut ⊖ for the empty graph of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus ⊖[H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn] is the graph given by the disjoint union of H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn This graph is called the union of H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This construction allows us to transform non-plane labeled trees with internal nodes decorated with graphs, ⊕ and ⊖ into graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 6 TH´EO LENOIR Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let T0 be the set of rooted non-plane trees whose leaves have distinct labels in N and whose internal nodes carry decorations satisfying the following constraints: internal nodes are decorated with ⊕, ⊖ or a graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node is decorated with some graph G then |G| ≥ 2 and this node has |G| children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node is decorated with ⊕ or ⊖ then it has at least 2 children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A tree t ∈ T0 is called a substitution tree if the labels of its leaves are in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |t|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We call linear the internal nodes decorated with ⊕ or ⊖ and non-linear the other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For a non-plane rooted tree t, and an internal node v of t, let tv be the multiset of trees attached to v and let t[v] be the non-plane tree rooted at v containing only the descendants of v in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We only consider non-plane trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' However it is sometimes convenient to order the subtrees of a given node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The convention is that for some v in a tree t the trees of tv are ordered according to their minimal leaf labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be an element of T0, the weakly-labeled graph Graph(t) is inductively defined as follows: if t is reduced to a single leaf labeled j, Graph(t) is the graph reduced to a single vertex labeled j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' otherwise, the root r of t is decorated with a graph H, and Graph(t) = H[Graph(t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Graph(t|H|)] where ti is the i-th tree of tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1 2 7 5 8 3 4 6 9 Root 1 9 6 2 3 8 7 5 4 t0 Graph(t0) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A substitution tree t0 and the corresponding graph Graph(t0) Note that if t is a substitution tree then Graph(t) is a labeled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 7 The following simple Lemma is essential to the study of the enriched decomposition of graphs introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a substitution tree such that the decoration of the root of t (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' its complementary) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then Graph(t) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' its complementary) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since both cases are similar, we only deal with the case of a connected decoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let r be the root of t, H its decoration and k the size of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , wk be vertices of Graph(t) such that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , k} there is a leaf labeled ℓ(wi) in the i-th tree of tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since the unlabeled graph induced by {wi | 1 ≤ i ≤ k} is isomorphic to H, it is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let C be the connected component of Graph(t) containing all wi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that for every vertex v of Graph(t), there exists p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k} such that the leaf labeled ℓ(v) belongs to the p-th tree of tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since H is connected and of size at least 2, there exists q ̸= p such that the vertices of label q and p are connected by an edge in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus v and wq are connected by an edge in Graph(t), which means that v ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This implies that C = V , thus Graph(t) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Modular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In this short section we gather the main definitions and properties of modular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The historical reference is [9], the interested reader may also look at [3] or [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The next definitions and theorems allows to get a unique recursive decomposition of any graph in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5, the modular decomposition, and to encode it by a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph (labeled or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A module M of G is a subset of V such that for every (x, y) ∈ M 2, and every z ∈ V \\M, {x, z} ∈ E if and only if {y, z} ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that ∅, V and {v} for v ∈ V are always modules of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Those sets are called the trivial modules of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is prime if it has at least 3 vertices and its only modules are the trivial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph is called ⊖-indecomposable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊕-indecomposable) if it cannot be written as ⊖[G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Gk] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊕[G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Gk]) for some k ≥ 2 and weakly-labeled graphs G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that a graph is ⊖-indecomposable if and only if it is connected, and ⊕-indecomposable if and only if its complementary is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10 (Modular decomposition, [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph with at least 2 vertices, there exists a unique partition M = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk} for some k ≥ 2, where each Mi is a module of G and such that either G = ⊕[M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk] and the (Mi)1≤i≤k are ⊕-indecomposable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' G = ⊖[M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk] and the (Mi)1≤i≤k are ⊖-indecomposable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' G = P[M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk] for some prime graph P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover, only one of the possibilities occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 8 TH´EO LENOIR This decomposition can be used to encode graphs by specific trees to get a one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a substitution tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We say that t is a canonical tree if its internal nodes are either ⊕, ⊖ or prime graphs, and if there is no child of a node decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖) which is decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' To a graph G we associate a canonical tree by recursively applying the decomposition of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10 to the modules (Mi)1≤i≤k, until they are of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' First of all, at each step, we order the different modules increasingly according to their minimal vertex labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Doing so, a labeled graph G can be encoded by a canonical tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The internal nodes are decorated with the different graphs that are encountered along the recursive decomposition process (⊕ if G = ⊕[M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk], ⊖ if G = ⊖[M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk], P if G = P[M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Mk]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' At the end, every module of size 1 is converted into a leaf labeled by the label of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This construction provides a one-to-one correspondence between labeled graphs and canonical trees that maps the size of a graph to the size of the corresponding tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph, and t its canonical tree, then t is the only canonical tree such that Graph(t) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is crucial to consider canonical trees as non-plane: otherwise, since prime graphs can have several labelings, there would be several canonical trees associated with the same graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Enriched modular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Unfortunately the modular decomposition alone does not provide usable decompositions for the graph classes that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The aim of this section is to solve this issue: we will state and prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='18 which provides in a very general settings a one-to-one encoding of graphs with substitution trees with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In Section 3 we will show that P4-reducible graphs, P4-sparse graphs, P4-lite graphs, P4-extendible graphs, P4-tidy graphs fit in the settings of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We say that G is a graph with blossoms if there exists k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |V |} such that exactly k vertices of G are labeled ∗, and the others ones have a distinct label in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |V | − k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The vertices labeled ∗ are called the blossoms of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let BG the set of vertices that are blossoms of G and N(G) := |V | − |BG| the number of vertices that are not a blossom of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the above definition, we allow k = 0, then the definition reduces to the one of a labeled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph with blossoms and π be a permutation of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , N(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The π-relabeling of G is the graph G′ such that: VG′ = VG and BG′ = BG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' for every vertex v in VG′\\BG′, we replace the label of the leaf v by π(ℓ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We write G ∼ G′ if there exists a permutation π of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , N(G)} such that G is iso- morphic to the π-relabeling of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 9 Note that ∼ is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph with blossoms, a permutation π of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , N(G)} is an automorphism of G if the π-relabeling of G is G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A module of a graph with blossoms is called flowerless if it does not contain any blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph with blossoms and M a non-empty flowerless module of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define bloM(G) to be the labeled graph obtained after the following transformations: M is replaced by a new vertex v, that is now labeled ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' for every vertex w ∈ G\\M, {w, v} is an edge if and only if {w, m} is an edge of G for every m ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' the graph obtained is replaced by its reduction as defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If G is a graph with one blossom and M is a non-empty flowerless module of G, we define bloM,0(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' bloM,1(G)) to be the graph bloM(G) where the label of the initial blossom of G is replaced by ∗0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ∗1) and the label of the new blossom is replaced by ∗1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ∗0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1 2 3 4 5 ∗ 1 2 4 5 6 3 7 8 M = {v3, v7, v8} bloM(G) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Illustration of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='16 Left: A graph G in which we have highlighted the module M = {v3, v7, v8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Right: The corresponding bloM(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In this paper, we only consider the construction bloM(G) for graphs with 0 or 1 blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We are now ready to precise the general framework of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' One of the key ingredient is the following recursive definition of families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P be a set of graphs with no blossom and P• be a set of graphs with one blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A tree t ∈ T0 is called (P, P•)-consistent if one of the following conditions holds: (D1) The tree t is a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (D2) The root r of t is decorated with a graph H ∈ P and tr (the multiset of trees attached to r) is a union of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (D3) The root r of t is decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖) and all the elements of tr are (P, P•)- consistent and their roots are not decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 10 TH´EO LENOIR (D4) The root r of t is decorated with a graph H /∈ {⊕, ⊖} and there exists at least one index i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |H|} such that the i-th tree of tr is (P, P•)-consistent, the remaining subtrees of tr are reduced to a single leaf and blo{vi}(H) ∈ P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define TP,P• to be the set of trees t that are (P, P•)-consistent and such that each leaf has a distinct label in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |t|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1 2 5 7 4 9 8 3 6 10 12 11 (D2) (D3) (D4) (D3) (D3) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' An example of tree in some TP,P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The different colours illustrate the different cases of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The subtree with leaves {5, 6} on the top-right is attached to the vertex which is circled in red inside the vertex of case (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This corresponds to the i-th subtree of case (D4) A graph G is called (P, P•)-consistent if there exists a (P, P•)-consistent tree t such that G = Graph(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We let GP,P• be the set of Graph(t) for t ∈ TP,P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The map t �→ Graph(t) from TP,P• to GP,P• is surjective, but without conditions on (P, P•) this map is not one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' To solve this issue, we introduce the following additional constraints on the set P, P•: Condition (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (C1) P and P• do not contain a graph of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (C2) For every F ∈ P and every module M of F, either bloM(F) ̸∈ P• or the subgraph of F induced by M is not (P, P•)-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (C3) For every F and F ′ in P•, and every flowerless modules M and M ′ of respectively F and F ′ one of the following conditions is verified: bloM,0(F) ̸= bloM′,1(F ′) The subgraph of F induced by M is not (P, P•)-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The subgraph of F ′ induced by M ′ is not (P, P•)-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (C4) Every element of P and P• is ⊕-indecomposable and ⊖-indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 11 (C5) For every G ∈ P•, the only modules of G containing the blossom are {∗} and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We say that (P, P•) verifies condition (C) if (C1) − (C5) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The last two constraints are not necessary to ensure that the map is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' However, giving necessary and sufficient conditions to have unicity that can be checked easily is quite complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that if condition (C) is satisfied for a pair of sets (P, P•) and Q ⊂ P and Q• ⊂ P•, it is also verified by (Q, Q•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P be a set of graphs with no blossom and P• a set of graphs with one blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Assume that (P, P•) verifies condition (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For any G ∈ GP,P•, there exists a unique t ∈ TP,P• such that G = Graph(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover, for any element of TP,P• satifying case (D4) in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17, the index i such that case (D4) holds is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Existence is guaranted by definition of GP,P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We proceed by contradiction to prove the uniqueness of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a smallest tree in TP,P• such that there exists another t′ in TP,P• verifying Graph(t) = Graph(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G = Graph(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The graph G cannot be reduced to a single vertex due to (C1), otherwise t and t′ would be a single leaf with label 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus we can assume that t and t′ are not in case (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6 and (C4), G is ⊕-indecomposable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖-indecomposable) if and only if t is not in case (D3) with a root decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus either t and t′ are both in case (D3) and their roots are both decorated ⊕ or ⊖, or they are both in case (D2) or (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Case (i): t, t′ are both in case (D3) and their are both decorated ⊕ or ⊖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let r and r′ be the roots of respectively t and t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Assume that both decorations are ⊖, the other case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The elements of tr induce connected graphs by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6 as their roots are either decorated with ⊕, or ⊖-indecomposable by (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since the roots of t and t′ are decorated with ⊖, we have a one-to-one correspondence between trees of tr and connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The same is true for t′ r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Assume that two trees corresponding to the same connected component of G are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since their set of labels are the same (they correspond to the labels of the vertices in the connected component) after reduction, one would obtain two trees t1, t2 that are different, (P, P•)-consistent and such that Graph(t1) = Graph(t2) since both are equal to the reduction of the corresponding connected component of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This contradicts the minimality of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Therefore tr = t′ r′ and t = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Case (ii): t, t′ are both in case (D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The graph G is simply the decoration of the root of t so t = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Case (iii): t is in case (D4), t′ is in case (D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let r be the root of t and H its decoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let i be one of the elements of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' |VH|} such that (D4) holds for t, H and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let M be the set of vertices of G whose labels are labels of leaves that belong to the i-th tree of tr: M is a module of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then bloM(G) is equal to blo{vi}(H) and thus belongs to P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover the subgraph of G induced by M is (P, P•)-consistent as the i-th subtree of t is also (P, P•)-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This contradicts (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 12 TH´EO LENOIR Case (iv): t, t′ are both in case (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let r and r′ be the roots of respectively t and t′ and H and H′ be their decorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let i be an element of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |VH|} such that (D4) is true for t, H and i, and i′ be an element of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , |VH′|} such that (D4) is true for t′, H′ and i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Consider M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' M ′) the set of vertices of G whose labels are labels of leaves that belong to the i-th tree of tr (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' i′-th tree of t′ r′): M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' M ′) is a module of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since the i-th tree of tr (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' the i′-th tree of t′ r′) is (P, P•)-consistent the subgraph of G induced by M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' M ′) is (P, P•)-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We now prove by contradiction that M = M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By symmetry we can assume that M ′ ̸⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' First assume that M ∩ M ′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that bloM,1(bloM′(G)) = bloM′,0(bloM(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since bloM(G) = blo{vi}(H) and bloM′(G) = blo{vi′}(H′), we get that bloM′,0(blo{vi}(H)) = bloM,1(blo{vi′}(H′)) which contradicts (C3) as both subgraphs of G induced by M and M ′ are (P, P•)-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Now assume that M ∩ M ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let L be the subset of VH such that v ∈ L if and only if the ℓ(v)-th tree of tr contains a leaf labeled with the label of an element of M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since M ′ is a module of G and M ∩ M ′ ̸= ∅, L is a module of blo{vi}(H) containing the blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since M ′ is not included in M, by (C5), L = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since M ′ ̸= G, there exists a vertex w in G such that w ̸∈ M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let w′ be the vertex of H such that w is in the ℓ(w′)-th tree of tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since M ′ is a module, every vertex of M ′ is either connected or not to w, thus w′ is connected to every vertex of H (except w′) or to none of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This means that H is either ⊕-decomposable or ⊖-decomposable, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus M = M ′ and blo{vi}(H) = bloM(G) = bloM′(G) = blo{vi′}(H′), and we get that H = H′, and that i = i′: thus i is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We know that the i-th tree of tr and the i-th tree of t′ r′ are (P, P•)-consistent and the associated graph is the one induced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By taking the reduction of the trees and the graph, we get by minimality of t that the reductions of both trees are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since M = M ′, it implies that both subtrees are the same: thus t = t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Zoology of graph classes with few P4’s Several classes have been defined as generalizations of the class of P4-free graphs, the cographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Here the classes we will focus on are the following: P4-reducible graphs [15,18], P4-sparse graphs [13,17] P4-lite graphs [14], P4-extendible graphs [16], P4-tidy graphs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The aim of this section is to give explicit sets P and P• such that GP,P• is one of the previously mentioned classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The following results and definitions are from [3, Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is a Pk if it is a path of k vertices, and a Ck if it is a cycle of k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The two vertices of degree one of a P4 are called the endpoints, the two vertices of degree two are called the midpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For a graph G, we denote by G its complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 13 The modular decompositions of classes of graphs we consider are already well-known [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' To explain the different properties, we need the notion of spider and bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A spider is a graph G, such that there exists a partition of VG in three parts, K, S, R, verifying: |K| ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' K induces a clique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' S induces a graph without edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' every element of R is connected to every element of K but to none of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' there exists a bijection f from K to S such that for every k ∈ K, k is only connected to f(k) in S, or such that for every k ∈ K, k is connected to every element of S except f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the first case the spider is called thin, in the second one it is called fat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' K S K S R R 1 1 2 2 3 3 4 4 5 5 6 6 7 7 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Left: a thin spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Right: a fat spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Both with |K| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every spider G, the partition (K, S, R) is uniquely determined by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover, the bijection f given by the definition is unique, except in the case |K| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In this case, since there is no difference between a thin and a fat spider, a spider with |K| = 2 is called thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A spider with |K| = 2 and |R| = 1 is called a bull, and a spider with |K| = 2 and |R| = 0 is simply a P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1 2 3 4 1 2 3 4 5 1 2 3 4 5 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' From left to right: a P4, a bull, a C5 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A spider is prime if and only if |R| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 14 TH´EO LENOIR In the following, if |R| = 1, the vertex belonging to R will be a blossom of the spider, and it will be its only blossom: such spiders will be called blossomed spiders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If |R| = 0, the spider will have no blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This also applies for bulls and P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We call a graph H a pseudo-spider if there exists a prime spider G such that, if we duplicate a vertex that is not a blossom of G (his label is the new number of vertices), and if either by adding or not an edge between the vertex and its duplicate, the graph obtained is a relabeling of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If |K| = 2, we also call H a pseudo-P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover, we say that H is a blossomed pseudo-spider if G is a blossomed spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If |K| = 2, we also call H a pseudo-bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 5 ∗ K S R 1 2 ∗ 3 4 5 6 Duplicate 7 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A blossomed pseudo-spider, a pseudo-bull, a pseudo P4 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A prime spider with 0 or 1 blossom has |K|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms (as there is a natural bijection between the automorphisms of the spider and the automorphisms of K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A pseudo-spider with 0 or 1 blossom has 2 × (|K| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-tidy graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is said to be a P4-tidy graph if, for every subgraph H of G inducing a P4, there exists at most one vertex y ∈ VG\\VH such that y is connected to at least one element of H but not all, and y is not connected to exactly both midpoints of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let Ptidy be the set containing all C5, P5, P5, all prime spiders without blossom and all pseudo-spiders without blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P• tidy be the set of all blossomed prime spiders and all blossomed pseudo-spiders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the set of graphs that are P4-tidy is GPtidy,P• tidy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is simply a reformulation in our setting of [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3] that states that a graph G is P4-tidy if and only if its canonical tree t verifies the following conditions: Every node in t is labeled with ⊕, ⊖, C5, P5, P5 or a prime spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with C5, P5 or P5, every element of tw is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 15 If a node w in t is decorated with a prime spider with |R| = 0, every element of tw is a tree of size at most two, and at most one is of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a prime spider H with |R| = 1, let v be the vertex of H in R, and t′ the ℓ(v)-th tree of tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Every element of tw\\{t′} is a tree of size at most two, and at most one is of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The pair (Ptidy, P• tidy) verifies (C) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that all the graph in Ptidy or P• tidy are prime except the pseudo-spiders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The only modules of the pseudo-spiders are the trivial ones, and the module formed by the vertex that was duplicated and its duplicate, which implies (C5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (C2) is also verified with the previous observation, as the modules of every graph in Ptidy are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (C1) is clearly verified and (C4) can be checked easily as all the graphs in P ∪ P• are connected, and their complementary is also connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For (C3), assume that for (F, F ′)2 ∈ P• tidy and M, M ′ are respectively flowerless modules of F and F ′, bloM,0(G) = bloM′,1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By cardinality argument, F and F ′ are either both spiders, or both pseudo-spiders of same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If both are spiders, as R is uniquely determined by the spiders, and the only element of R does not have the same label in bloM,0(G) and in bloM,1(G), we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If both are pseudo-spiders, note that the original node and its duplicate form the only module of size 2 of bloM,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus the only element of R (in the original spiders) is uniquely determined by the pseudo-spiders, and the only element of R does not have the same label in bloM,0(G) and in bloM,1(G), we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-lite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is said to be a P4-lite graph if every subgraph of G of size at most 6 does not contain three induced P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let Plite be the set containing all P5, P5, all prime spiders without blossom and all pseudo-spiders without blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P• lite to be the set containing all blossomed prime spiders and all blossomed pseudo-spiders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the set of graphs that are P4-lite is GPlite,P• lite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is simply a reformulation in our setting of [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8] that states that a graph G is P4-lite if and only if its canonical tree t verifies the following conditions: Every node in t is labeled with ⊕, ⊖, P5, P5 or a prime spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with P5 or P5, every element of tw is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a prime spider with |R| = 0, every element of tw is a tree of size at most two, and at most one is of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a prime spider H with |R| = 1, let v be the vertex of H in R, and t′ the ℓ(v)-th tree of tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Every element of tw\\{t′} is a tree of size at most two, and at most one is of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ 16 TH´EO LENOIR By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8 since Plite ⊂ Ptidy, P• lite ⊂ P• tidy we get that the pair (Plite, P• lite) verifies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-extendible graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is said to be a P4-extendible graph if, for every subgraph H of G inducing a P4, there exists at most one vertex y ∈ VG\\VH such that y belongs to an induced P4 sharing at least one vertex with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let Pext be the set containing all C5, P5, P5, P4 and all pseudo-P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P• ext be the set containing all bulls and all pseudo-bulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the set of graphs that are P4-extendible is GPext,P• ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is simply a reformulation in our setting of [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7] that states that a graph G is P4-extendible if and only if its canonical tree t verifies the following conditions: Every node in t is labeled with ⊕, ⊖, C5, P5, P5, P4 or a bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with C5, P5 or P5, every element of tw is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with P4, every element of tw is a tree of size at most two, and at most one is of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a bull G, let v be the vertex of G in R, and t′ the ℓ(v)-th tree of tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Every element of tw\\{t′} is a tree of size at most two, and at most one is of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8 since Pext ⊂ Ptidy, P• ext ⊂ P• tidy we get that the pair (Pext, P• ext) verifies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is said to be a P4-sparse graph if every subgraph of G of size 5 does not contain two induced P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P be the set containing all prime spiders without blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P• be the set containing all blossomed prime spiders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the set of graphs that are P4-sparse is GP,P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is simply a reformulation in our setting of [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4] that states that a graph G is P4-sparse if and only if its canonical tree t verifies the following conditions: Every node in t is labeled with ⊕, ⊖ or a prime spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a prime spider with |R| = 0, every element of tw is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a prime spider h with |R| = 1, let v be the vertex of H in R, and t′ the ℓ(v)-th tree of tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Every element of tw\\{t′} is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8 since Pspa ⊂ Ptidy, P• spa ⊂ P• tidy we get that the pair (Pspa, P• spa) verifies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-reducible graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is said to be a P4-reducible graph if every vertex of G belongs to at most one induced P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let Pred be the set containing all P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P• red be the set containing all bulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the set of graphs that are P4-reducible is GPred,P• red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is simply a reformulation in our setting of [11, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2] that states that a graph G is P4-reducible if and only if its canonical tree t verifies the following conditions: Every node in t is labeled with ⊕, ⊖, P4 or a bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a P4, every element of tw is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If a node w in t is decorated with a bull H, let v be the vertex of H in R, and t′ the ℓ(v)-th tree of tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Every element of tw\\{t′} is reduced to a single leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8 since Pred ⊂ Ptidy, P• red ⊂ P• tidy we get that the pair (Pred, P• red) verifies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-free graphs (cographs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graph G is said to be a cograph if no subgraph of G induces a P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Set Pcog = ∅ and P• cog = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the set of graphs that are cographs is GPcog,P•cog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It is simply a reformulation in our setting of [5, Theorem 7] that states that a graph G is a cograph if and only if its canonical tree t has no internal node decorated with a prime graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Clearly the pair (Pcog, P• cog) verifies (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Enriched modular decomposition: enumerative results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Exact enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, we establish combinatorial identities between formal power series involving subsets of P and P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Throughout this section, we consider generic pairs (P, P•) where P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P•) is a set of graphs with no blossom (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' with one blossom) verifying condition (C) defined p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Recall that for a graph G with blossoms, N(G) is the number of vertices that are not a blossom: this will be the crucial parameter in the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let P •(z) := � s∈P• zN(s) N(s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' and P(z) := � s∈P zN(s) N(s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='. For n ∈ N, let Pn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P• n) be the set of graphs G in P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P•) such that N(G) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that, if both P and P• are stable under relabeling (which is the case for the classes of graphs mentioned in Section 3), for each n ∈ N, there is a natural action Φn of the 18 TH´EO LENOIR permutations of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , n} over Pn and P• n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let RPn and RP•n be a system of representants of every orbit under this action, then P •(z) = � n∈N |P• n|zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = � n∈N � s∈RPn |RP•n| n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' |Aut(s)| zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = � n∈N � s∈RPn |RP•n| zn |Aut(s)| Similarly, we have: P(z) = � n∈N � s∈RPn |RPn| zn |Aut(s)| Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For each graph class introduced in Section 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' we have the following expres- sions for P and P •: P4-tidy P • tidy(z) = (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4 2 − 2z5 Ptidy(z) = P • tidy(z) + z5 + z5 10 P4-lite P • lite(z) = (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4 2 − 2z5 Plite(z) = P • lite(z) + z5 P4-extendible P • ext(z) = z4 2 + 2z5 Pext(z) = P • ext(z) + z5 + z5 10 P4-sparse P • spa(z) = Pspa(z) = 2(exp(z2) − 1 − z2 − z4 4 ) P4-reducible P • red(z) = Pred(z) = z4 2 P4-free P • cog(z) = Pcog(z) = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We only detail the computation of Ptidy and P • tidy for P4-tidy graphs as this is the most involved case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7, Ptidy is composed of one C5 that has 10 automorphisms and all its relabelings, one P5, and one P5 that both have 2 automorphisms and all their relabelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat spiders corresponding to the 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1) different orbits of the action Φ2k over prime spiders of size 2k, each having k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can come from K or S, and can be connected or not to the initial vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' These 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) cases correspond to the 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) different orbits of the action Φ2k+1 over pseudo-spiders of size 2k + 1, each having 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus we have Ptidy(z) = z5 10 + 2z5 2 + z4 2 + 2 � k≥3 z2k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' + 4z5 2 + 8 � k≥3 z2k+1 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Hence Ptidy(z) = z5 + z5 10 + (2 + 4z3) exp(z2) − 2 − 2z2 − 4z3 − z4 2 − 2z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Now let’s compute P • tidy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat spiders with blossom corresponding to the 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1) different orbits of the action Φ2k over blossomed prime spiders G with 2k non blossomed vertices, each having k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 19 For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can come from K or S, and can be connected or not to the initial vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' These 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) cases correspond to the 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) different orbits of the action Φ2k+1 over blossomed pseudo-spiders with 2k + 1 non blossomed vertices, each having 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Hence P • tidy(z) = z4 2 +2 � k≥3 z2k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' +4z5 2 +8 � k≥3 z2k+1 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = (2+4z3) exp(z2)−2−2z2−4z3− z4 2 −2z5, which gives the announced result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Let T be the exponential generating function of TP,P•, the set of trees defined in Def- inition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17 counted by their number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by Tnot⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Tnot⊖) the set of all t ∈ TP,P• whose root is not decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖) and by Tnot⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Tnot⊖) the corresponding exponential generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Together with Tnot⊕ = 0, the exponential generating function Tnot⊕ is de- termined (as a formal series) by the following equation: Tnot⊕ = z + P + (exp(Tnot⊕) − 1)P • + exp(Tnot⊕) − 1 − Tnot⊕, (1) and the series T and Tnot⊖ are simply given by the following equations: T = exp(Tnot⊕) − 1 (2) Tnot⊖ = Tnot⊕ (3) Moreover, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (1) with Tnot⊕(0) = 0 determines uniquely the generating function Tnot⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that there is a natural involution on TP,P•: the decoration of every linear node can be changed to its opposite: ⊕ to ⊖, and ⊖ to ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Therefore Tnot⊕ = Tnot⊖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' First, we prove that T = z + T × P • + P + 2 × (exp(Tnot⊕) − 1 − Tnot⊕) (4) We split the enumeration of the trees t ∈ TP,P• according to the different cases of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (D1) The tree t is a single leaf (which gives the z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (D2) The tree t has a root decorated with a graph H belonging to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The exponential generating function for a fixed H is zN(H) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='. Summing over all H and all n gives the term P in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (D3) The tree t has a root r decorated with ⊕ and having k children with k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In this case, the generating function of the set of the k subtrees of tr is T k not⊕ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Summing over all k implies that the exponential generating function of all trees in case (D3) with a root labeled ⊕ is exp(Tnot⊕) − 1 − Tnot⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The tree t can also have a root r decorated with ⊖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since Tnot⊕ = Tnot⊖, the exponential generating function of all trees in case (D3) with a root labeled ⊖ is exp(Tnot⊕) − 1 − Tnot⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 20 TH´EO LENOIR (D4) The tree t has a root r decorated with a graph H and there exists v ∈ VH such that blov(H) = W where W ∈ P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote t′ the ℓ(v)-th tree of tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The exponential generating function corresponding to the set of leaves in t\\t′ is zN(W ) N(W)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=', and the exponential generating function corresponding to t′ is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that the tree t is uniquely determined by W, the labeled product of t′ and the set of leaves of t\\t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus the corresponding generating function for a fixed W is T × zN(W ) N(W)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='. Summing over all W and all n gives the term T × P • in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Summing all terms gives Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Similarly, we get Tnot⊕ = z + T × P • + P + exp(Tnot⊕) − 1 − Tnot⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (5) Substracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (5) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (4) gives Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (1) is an easy consequence from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (2) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (1) can be rewritten as: Tnot⊕ = z + P + (exp(Tnot⊕) − 1)P • + � k≥2 T k not⊕ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (6) For every n ≥ 1, the coefficient of degree n of Tnot⊕ only depends on coefficients of lower degree as P •(z) has no term of degree 0 or 1 and Tnot⊕(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (1) combined with Tnot⊕(0) = 0 determines uniquely Tnot⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ We are going to define the notions of trees with marked leaves, and of blossomed trees, which will be crucial in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We insist on the fact that the size parameter will count the number of leaves including the marked ones but not the blossoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A marked tree is a pair (t, I) where t is a tree and I a partial injection from the set of labels of leaves of t to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The number of marked leaves is the size of the domain of I denoted by |(t, I)|, and a leaf is marked if its label j is in the domain, its mark being I(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, we will consider marked trees (t, I), and subtrees t′ of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The marked tree (t′, I) will refer to the marked tree (t′, I′) where I′ is the restriction of I to the set of labels of leaves of t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let F ∈ {TP,P•, Tnot⊖, Tnot⊕}, and F be its generating exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The exponential generating function of trees in F with a marked leaf is zF ′(z): if there are fn trees of size n in F, there are nfn trees with a marked leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus the generating exponential function is � n≥1 nfn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' zn = zF ′(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Blossoming transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a tree not reduced to a leaf in TP,P•, ℓ a leaf of t and n the parent of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If n is a linear node, we replace the label of ℓ by ∗, and do the reduction on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If v is a non-linear node, and ℓ is in the i-th tree of tv (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17), we replace the label of ℓ by ∗ and i by ∗ in the decoration of v, and do the reduction on both t and the decoration of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If t is GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 21 reduced to a leaf, we replace the leaf by a blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We call such this transformation the blossoming of (t, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We extend this operation to internal node: if n is a internal node, we replace t[n] by its leaf of smallest label, and do the blossoming operation on the tree obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The resulting tree is still called the blossoming of (t, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4 (Blossomed tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A blossomed tree is a tree that can be obtained by the blossoming of a tree in TP,P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Its size is its number of leaves without blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A blossom is ⊕-replaceable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖-replaceable) if its parent is not decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Similarly to a tree, a blossomed tree can be marked by a partial injection I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We will denote T b and T b a with a = not⊕, not⊖, and b = ⊕, ⊖ or blo the set of trees whose root is not ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖) if a = not⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' a = not⊖), and with one blossom that is b-replaceable if b = ⊕ or ⊖, or just with one blossom if b = blo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define T b a to be the corresponding exponential generating function of trees, counted by the number of non blossomed leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' However, we take the convention that T ⊕ not⊕(0) = 0 = T ⊖ not⊖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In other words, a single leaf is neither in T ⊕ not⊕ nor in T ⊖ not⊖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The other series have constant coefficient 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' From the previously defined involution, it follows that T ⊖ not⊕ = T ⊕ not⊖, T ⊕ not⊕ = T ⊖ not⊖ et T ⊕ = T ⊖ and T blo not⊕ = T blo not⊖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The functions T ⊕, T ⊕ not⊕, T ⊕ not⊖ are given by the following equations: T ⊕ = 1 2 − exp(Tnot⊕) − P • exp(Tnot⊕) (7) T ⊖ not⊕ = T ⊕ exp(Tnot⊕) (8) T ⊕ not⊕ = T ⊕ − 1 exp(Tnot⊕) (9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a tree in T ⊕ not⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that it cannot be reduced to a single leaf, have a root decorated with ⊕ or be in case (D2) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (D3) The tree t can have a root r decorated with ⊖ and having k children with k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' There are k−1 subtrees without blossom, and 1 with a blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus the generating function of the set of the k subtrees of tr is T k−1 not⊖ (k−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='T ⊖ not⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Summing over all k gives that the exponential generating function of all trees in case (D3) with a root labeled ⊖ is � k≥2 T k−1 not⊖ (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='T ⊕ not⊖ = (exp(Tnot⊖) − 1)T ⊕ not⊖ 22 TH´EO LENOIR ⊕ or (D4) ⊖ or (D4) ⊖ or (D4) not (D2) (D4) 2 9 6 or At least one tree, each does not have a root decorated with ⊕ If the previous node is in (D4) the marked leaf must be in the i-th tree Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Illustration of both cases in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5 (D4) The tree t can have a root r decorated with H and v ∈ VH such that blov(H) = W with W ∈ P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then the blossom must be in the ℓ(v)-th tree of tr that will be denoted t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The exponential generating function corresponding to the set of leaves in t\\t′ is zN(W ) N(W)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=', and the exponential generating function corresponding to t′ is T ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that the tree t is uniquely determined by W, the labeled product of t′ and the set of leaves of t\\t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus the corresponding generating function for a fixed W is T ⊕ × zN(W ) N(W)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='. Summing over all W and all n gives the exponential generating function T ⊕ × P •.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This implies the following equation: T ⊕ not⊕ = (exp(Tnot⊖) − 1)T ⊕ not⊖ + P •T ⊕ = (exp(Tnot⊕) − 1)T ⊖ not⊕ + P •T ⊕ (10) We have similarly: T ⊖ not⊕ = 1 + (exp(Tnot⊖) − 1)T ⊖ not⊖ + P •T ⊖ = 1 + (exp(Tnot⊕) − 1)T ⊕ not⊕ + P •T ⊕ (11) T ⊕ = 1 + (exp(Tnot⊖) − 1)T ⊕ not⊖ + (exp(Tnot⊕) − 1)T ⊕ not⊕ + P •T ⊕ (12) Thus: T ⊕ = 1 + (exp(Tnot⊕) − 1)(T ⊕ not⊕ + T ⊖ not⊕) + P •T ⊕ (13) By substracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (11) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (13), we get T ⊕ − T ⊖ not⊕ = (exp(Tnot⊕) − 1)T ⊖ not⊕ which implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (10) and (13), we get T ⊕ = 1 + (exp(Tnot⊕) − 1)T ⊕ not⊕ + T ⊕ not⊕ = 1 + exp(Tnot⊕)T ⊕ not⊕ which implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 23 Substituting T ⊕ not⊕ and T ⊖ not⊕ with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (9) and (10) ins Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (8), it follows that: T ⊕ − 1 = (exp(Tnot⊕) − 1)T ⊕ + exp(Tnot⊕)P •T ⊕ and T ⊕(2 − exp(Tnot⊕) − P • exp(Tnot⊕)) = 1 which implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We also have the following equations: T blo = exp(Tnot⊕) 2 − exp(Tnot⊕) − P • exp(Tnot⊕) (14) T blo not⊕ = 1 exp(Tnot⊕)T blo (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By the same techniques used as those of the previous proof, we establish that: T blo = 1 + 2(exp(Tnot⊕) − 1)T blo not⊕ + P •T blo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (16) T blo not⊕ = 1 + (exp(Tnot⊕) − 1)T blo not⊕ + P •T blo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (17) By substracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (17) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (16), we get that: T blo − T blo not⊕ = (exp(Tnot⊕) − 1)T blo not⊕ which implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (17) by exp(Tnot⊕) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (15) we get that: T blo (2 − P • exp(Tnot⊕) − exp(Tnot⊕)) = exp(Tnot⊕) which implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Combining Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6 we obtain: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We have the following equations: T blo = exp(Tnot⊕)T ⊕;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (18) T blo not⊕ = T ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (19) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Asymptotic enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, we derive from the previously obtained equations the radii of the different series introduced, the asymptotic behavior of the dif- ferent series in R and an equivalent of the number of graphs in GP,P• From now on, we assume that P and P • have a positive radius of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let R0 be the minimum of their radii of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by P(R0) and P •(R0) the limit in [0, +∞] of P and P • at R− 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, we assume that one of the conditions below is verified: P •(R0) ≥ 1 R0 + P(R0) + 2 ln(1 + P •(R0)) − P •(R0) > 2 ln(2) − 1 24 TH´EO LENOIR Note that one of these conditions is verified in the different classes of graphs we study, as R0 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by R the only solution in [0, R0) of the equation: R + P(R) + 2 ln(1 + P •(R)) − P •(R) = 2 ln(2) − 1 (20) such that P •(R) < 1 (unicity comes from the fact that z �→ 2 ln(1 + z) − z is increasing in [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that by definition, 0 < R < R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Recall that a formal series A is aperiodic if there does not exist two integers r ≥ 0 and d ≥ 2 and B a formal series such that A(z) = zrB(zd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The functions T, Tnot⊕, T ⊕, T ⊖ not⊕, T ⊕ not⊕, T blo, T blo not⊕ are aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' One can easily check that for each of the previous series, the coefficients of degree 3 and 4 are positive, and thus all the series are aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A set ∆ is a ∆-domain at 1 if there exist two positive numbers R and π 2 < φ < π such that ∆ = {z ∈ C||z| ≤ R, z ̸= 1, |arg(1 − z)| < φ} For every w ∈ C∗, a set is a ∆-domain at w if it is the image of a ∆-domain by the mapping z �→ zw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A power series U is said to be ∆-analytic if it has a positive radius of convergence ρ and there exists a ∆-domain D at ρ such that U has an analytic continuation on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Both T and Tnot⊕ have R as radius of convergence and a unique dominant singularity at R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' They are ∆-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Their asymptotic expansions near R are: Tnot⊕(z) = ln � 2 1 + P •(R) � − κ � 1 − z R + o �� 1 − z R � (21) T(z) = 2 1 + P •(R) − 1 − 2 1 + P •(R)κ � 1 − z R + o �� 1 − z R � (22) where κ is the constant given by: κ = � � � �R � 1 + P ′(R) + (1 − P •(R))(P •)′(R) 1 + P •(R) � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We begin with the expansion of Tnot⊕ for which we apply the smooth implicit the- orem [8, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='467].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Following [8, Sec VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1] we claim that Tnot⊕ satifies the settings of the so-called smooth implicit-function schema: Tnot⊕ is solution of T = G(z, T), where G(z, w) = z + P(z) + (exp(w) − 1)P •(z) + (exp(w) − 1 − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 25 The singularity analysis of Tnot⊕ will go through the study of the characteristic system: � � � G(r, s) = s, Gw(r, s) = 1 with 0 < r < R, s > 0 where Fx = ∂F ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that (r, s) = � R, ln � 2 1+P •(R) �� is a solution of the characteristic system of G since Gw(r, s) = exp(s)(1 + P •(R)) − 1 = 2 − 1 = 1 G(r, s) = R+P(R)−P •(R)+∂wG(r, s)−s = 2 ln(2)−1−2 ln(1+P •(R))+1−s = 2s − s = s Moreover Gz(r, s) = 1 + P ′(R) + (exp(s) − 1)(P •)′(R) = 1 + P ′(R) + (1−P •(R))(P •)′(R) (1+P •(R)) Gw,w(r, s) = exp(s)(1 + P •(r)) = 2 The expansion of T is then a consequence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (2) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='19 and of the expansion of Tnot⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The radius of convergence of T ⊕, T ⊖ not⊕, T ⊕ not⊕, T blo, and T blo not⊕ is R and R is the unique dominant singularity of these series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' They are ∆-analytic and their asymptotic expansions near R are: T ⊕ = 1 2κ � 1 − z R �− 1 2 + o �� 1 − z R �− 1 2 � (23) T ⊖ not⊕ = (1 + P•(R)) 4κ � 1 − z R �− 1 2 + o �� 1 − z R �− 1 2 � (24) T ⊕ not⊕ = (1 + P•(R)) 4κ � 1 − z R �− 1 2 + o �� 1 − z R �− 1 2 � (25) T blo = 1 (1 + P•(R))κ � 1 − z R �− 1 2 + o �� 1 − z R �− 1 2 � (26) T blo not⊕ = 1 2κ � 1 − z R �− 1 2 + o �� 1 − z R �− 1 2 � (27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' note that, if |z| ≤ R, |(1 + P •(z)) exp(Tnot⊕(z))| ≤ (1 + P •(|z|)) exp(|Tnot⊕(z)|) ≤ (1 + P •(R)) exp(Tnot⊕(R)) = 2 with equality if and only if z = R by aperiodicity from Daffodil lemma [8, Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1] and since Tnot⊕(R) ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Hence, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11 and by compacity, 2−(1+P •(z)) exp(Tnot⊕(z)) can be extended to a ∆-domain D at R with 2 − (1 + P •(z)) exp(Tnot⊕)(z) ̸= 0 for every z ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (7) shows that T ⊕ can be extended to D and yields the announced expansions when z tends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' These expansions show that all these series have a radius of convergence exactly equal to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ 26 TH´EO LENOIR Applying the Transfer Theorem [8, Corollary VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='392] to the results of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11, we obtain an equivalent of the number of trees of size n in TP,P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since there is a one-to-one correspondence between graphs in GP,P• and trees in TP,P•, we get the following result: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The number of graphs in GP,P• of size n is asymptotically equivalent to C n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Rnn 3 2 where C = κ √π(1 + P •(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Here are the numerical approximations of R and C in the different cases: class of graph R−1 R C P4-tidy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='90405818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='34434572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='40883495 P4-lite 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='90146936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='34465296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='40833239 P4-extendible 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='88492066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='34662998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='40351731 P4-sparse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='72743550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='36664478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='37405701 P4-reducible 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='71715531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='36803196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='37115484 P4-free 1 2 ln(2)−1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='58869945 2 ln(2) − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='38629436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='35065840 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Enumeration of graphs with a given induced subgraph 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Induced subtrees and subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We recall that the size of a graph is its number of vertices, and the size of a tree is its number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 (Induced subgraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a graph, k a positive integer and I a partial injection from the set of labels of G to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The labeled subgraph GI of G induced by I is defined as: The vertices of GI are the vertices of G whose label ℓ is in the domain of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every such vertex, we replace the label ℓ of the vertex by I(ℓ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For two vertices v and v′ of GI, (v, v′) is an edge of GI if and only if it is an edge of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2 (First common ancestor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a rooted tree and let ℓ1, ℓ2 be two distinct leaves of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The first common ancestor of ℓ1 and ℓ2 is the internal node of t that is the furthest from the root and that belongs to the shortest path from the root to ℓ1, and the shortest path from the root to ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3 (Induced subtree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let (t, I) be a marked tree in T0 (T0 is defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4, and the notion of marked tree in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The induced subtree tI of t induced by I is defined as: The leaves of tI are the leaves of t that are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every such leaf labeled with an integer ℓ, the new label of ℓ is I(ℓ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The internal nodes of tI are the internal nodes of t that are first common ancestors of two or more leaves of tI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The ancestor-descendent relation in tI is inherited from the one in t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 27 For every internal node v of t that appears in tI, let H be its decoration in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by J the set of positive integers k such that the k-th tree of tv contains a leaf of tI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every k in J, we define L(k) as the smallest image by I of a marked leaf label in the k-th tree of tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The decoration of v in tI is the reduction of HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every internal node v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' leaf ℓ) of tI, we also define φ(v) to be the only internal node (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' leaf) of t corresponding to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' When (t, I) is a marked tree and t′ is a subtree of t, we will denote t′ I the tree induced by the restriction of I to the set of labels of leaves of t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' As a consequence of Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3, we obtain: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let (t, I) be a marked tree in T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then Graph(t)I = Graph(tI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 8 3 7 2 6 4 5 1 4 1 3 2 Graph(t) 3 7 5 6 1 2 4 1 3 4 2 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Relations between induced subgraph and induced subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every pair of graphs (G, H) such that G has no blossom and H has at most one blossom, let OccG(H) be the number of partial injection I from the vertex labels of G to N such that no blossom is marked and HI is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every pair of graphs (G, H) and a ∈ N such that G has no blossom, H has exactly one blossom and a is the label of a vertex of G, let OccG,a(H) be the number (6) = 1, J(7) = 2, J(3) = 1, J(4) = 3tGraph(ts)Graph(t)>J(6) = 1, (7) = 2, J(1) = 3, J(3) = 428 TH´EO LENOIR of partial injection I from the vertex labels of G to N such that the image of the blossom by I is a and HI is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 2 3 1 ∗ 8 5 7 9 6 6 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Two occurences of a P4 in a blossomed graph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If G is a P4, the blue one is counted twice in OccG(H), the red one in counted once in OccG,a(H) iff a is the label of an extremity of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every graph G without blossom, and every a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , N(G) = |G|}, set: OccG,P(z) := � H∈P OccG(H)zN(H)−N(G) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' OccG,P•(z) := � H∈P• OccG(H)zN(H)−N(G) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' OccG,a,P•(z) := � H∈P• OccG,a(H)zN(H)−N(G)+1 N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' OccG,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' will only be used for graphs G with no blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every k ≥ 1 and every a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , k}: � G: N(G)=k OccG,P(z) = P (k)(z) (28) � G: N(G)=k OccG,P•(z) = (P •)(k)(z) (29) � G: N(G)=k OccG,a,P•(z) = (P •)(k−1)(z) (30) Thus for every graph G with no blossom and every a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , N(G)}, OccG,P, OccG,P• and OccG,a,P• have a radius of convergence strictly greater than R, the radius of convergence of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let H be an element of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since there are N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (N(H)−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' choices of partial injection whose image is {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , k}, we have: � G: N(G)=k OccG,P(z) = � H∈P � G: N(G)=k OccG(H)zN(H)−k N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = � H∈P zN(H)−k (N(H) − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = P (k)(z) GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 29 The proofs of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (29) and (30) are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (30), since I−1(a) must be ∗, there are exactly N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (N(H)−(k−1))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' choices for the partial injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every graph G, OccG,P has nonnegative coefficients and for every k ≥ 0, as mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2, P (k) has a radius of convergence at least R0, the minimum of the radii of convergence P and P •, which is greater than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' This implies that OccG,P has a radius of convergence greater than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The proof for the other series is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Enumerations of trees with a given induced subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The key step in the proof of our main theorem is to compute the limiting probability (when n → ∞) that a uniform induced subtree of a uniform tree in TP,P• with n leaves is a given substitution tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, let τ ∈ T0 be a fixed substitution tree of size at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define Tτ to be the set of marked trees (t, I) where t ∈ TP,P• and I is such that tI is isomorphic to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We also define Tτ to be the corresponding exponential generating function (where the size parameter is the total number of leaves, including the marked ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The aim now is to decompose a tree admitting τ as a subtree in smaller trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let (t, I) be in Tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A prime node v of τ is such that t[φ(v)] is either in case (D2) or (D4) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17: in other word, φ(v) must be a prime node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In constrast, knowing that an internal node v′ of τ is decorated with ⊕ or ⊖ does not give any information about the decoration of φ(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In order to state Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11 below, we need to partition the internal nodes of τ: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let (t, I) be in Tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We denote by V(t, I) the set of internal nodes v of τ such that φ(v) is non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The set V(t, I) can be partitioned in 4 subsets: V0(t, I) the set of internal nodes v such that t[φ(v)] is in case (D2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' V1(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and no marked leaf is in the i-th tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' V2(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and exactly one marked leaf is in the i-th tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' V3(t, I) the set of internal nodes v such that t[φ(v)] is in case (D4) and at least two marked leaves are in the i-th tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that the set of non-linear nodes of τ must be included in V(t, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since for every element v of V(t, I) at most one element of tφ(v) is non trivial, at most one element of τv is non trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus if τ has some non-linear nodes v such that two or more elements of τv are not reduced to a single leaf, Tτ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the following, we assume that it is not the case for τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If τv has exactly one non trivial subtree, then v ∈ V3(t, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Otherwise, τv is a union of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We denote by U0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' U1) the set of internal nodes v of τ such that no tree (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' exactly one tree) of τv has size greater or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 30 TH´EO LENOIR Note that by definition V0(t, I) ∪ V1(t, I) ∪ V2(t, I) ⊂ U0 and V3(t, I) ⊂ U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We also define rkt,I : V2(t, I) �→ N as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let v ∈ V2(t, I), we define rkt,I(v) to be the only integer k such that, if ℓ is the label of the k-th leaf of τv then the leaf of label I−1(ℓ) in t belongs to the i-th tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every v ∈ V2(t, I), we have 1 ≤ rkt,I(v) ≤ |τv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let τ be a substitution tree of size at least 2 such that every non-linear node of τ is in U0 ∪ U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let V0, V1 and V2 be three disjoint subsets of U0 and let V3 be a subset of U1 such that every non-linear node of τ is in V := V0 ∪ V1 ∪ V2 ∪ V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let rk: V2 → N be such that 1 ≤ rk(w) ≤ |τw| for every w ∈ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let Tτ,V0,V1,V2,V3,rk be the set of marked trees (t, I) in Tτ such that V0(t, I) = V0, V1(t, I) = V1, V2(t, I) = V2, V3(t, I) = V3, rkt,I = rk, and let Tτ,V0,V1,V2,V3,rk be its exponential gener- ating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then Tτ,V0,V1,V2,V3,rk = z|t|T root � T ⊕ not⊕ �d= � T ⊖ not⊕ �d̸= � T blo not⊕ �dV →V � T ′ not⊕ �dV →ℓ exp(nLTnot⊕) × T |V1|T ′|V2|(T ⊕)n1(T blo)n2F where F := � v∈V0 Occdec(v),P � v∈V3 Occdec(v),br(v),P• � v∈V1 Occdec(v),P• � v∈V2 Occdec(v),rk(v),P• and: d= is the number of edges between two internal nodes not in V with the same decoration (⊕ and ⊕, or ⊖ and ⊖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' d̸= is the number of edges between two internal nodes not in V decorated with different decorations (⊕ and ⊖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' dV →V is the number of edges between an internal node not belonging to V and one of its children belonging to V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' dV →ℓ is the number of edges between an internal node not in V and a leaf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' nL is the number of internal nodes not in V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' dec(v) is the decoration of v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' for every v ∈ V3, br(v) is the position of the subtree of τv not reduced to a leaf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' n1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' n2) is the number of internal nodes v in V3 such that the root of the br(v)-th tree of τv is not in V (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' is in V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' T root = T ⊕ if the root of τ is not in V , T root = T blo otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be a tree in Tτ,V0,V1,V2,V3,rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We decompose t into several disjoints subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The blossoms are nodes where (the root of) an other tree will be glued (and thus they are not counted in the generating series, to avoid counting them twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define t→root to be the tree t blossomed at φ(r0), where r0 is the root of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define the tree tv→ in the following way: If v is not in V , tv→ is the subtree of t containing φ(v) and all the subtrees of tφ(v) that do not contain a marked leaf of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 31 V0 V0 V1 V2 V3 V V Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A possible τ and choices of V0, V1, V2, V3 If v is in V0 ∪ V1 ∪ V2, tv→ is the tree t[φ(v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If v is in V3, tv→ is the tree t[φ(v)] obtained after blossoming the root of the non trivial tree of tφ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The blossom is marked with the smallest mark in the non trivial tree of tφ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every internal nodes v, v′ in τ such that v is not in V and v′ is a child of v, let tv→v′ be the unique tree of tφ(v) containing φ(v′), blossomed at φ(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every internal node v in τ not in V , and every leaf f which is a child of v in τ, we define tv→f to be the subtree of tφ(v) containing φ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every internal node v in V3, we define tv→br(v) to be the non trivial tree of tφ(v) blossomed at φ(v′), where v′ is the root of the br(v)-th tree of τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Now we need to analyze the properties of the trees that appear in this decomposition and compute the corresponding exponential generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the rest of the proof, we will say abusively that every blossomed tree belongs to TP,P•, and that two nodes both decorated with ⊕ or ⊖ have the same decoration, even if they do not have the same number of children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (i): analysis of t→root where v ̸∈ V The tree t→root is a tree in TP,P•, it has no marked leaf and a unique blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If the root is not in V and decorated with ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖), the blossom is ⊕-replaceable (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖-replaceable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If the root is in V , the blossom is replaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 32 TH´EO LENOIR root ii iii iv iv iv iv v v vii vii vii vi ix tv,v′ i viii viii x xi tv,f tv→ t→root tv→br(v) Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The decomposition of a tree admitting the graph τ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 12 as an induced tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The different notations correspond to the different cases of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is equal to T ⊕ if the root is not in V and equal to T blo otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (ii): analysis of tv→v′ where v ̸∈ V and v′ is a child of v not in V with the same decoration The tree tv→v′ is a tree in TP,P• whose root is not decorated with the same decoration as v and with one blossom ⊕-replaceable if v′ is decorated with ⊕, ⊖-replaceable otherwise and no marked leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The exponential generating function of such trees is either T ⊕ not⊕ if both nodes are deco- rated with ⊕ or T ⊖ not⊖ if both nodes are decorated with ⊖, which are both equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (iii): analysis of tv→v′ where v ̸∈ V and v′ is a child of v not in V with a different decoration The tree tv→v′ is a tree in TP,P• whose root is not decorated with the same decoration as v and with one blossom ⊕-replaceable if v′ is decorated with ⊕, ⊖-replaceable otherwise and no marked leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 33 The exponential generating function of such trees is either T ⊖ not⊕ if v is decorated with ⊕ and v′ with ⊖ or T ⊖ not⊖ if v is decorated with ⊖ and v′ with ⊕, which are both equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (iv): analysis of tv→v′ where v ̸∈ V and v′ is a child of v in V The tree tv→v′ is a tree in TP,P• whose root is not decorated with the decoration of v with one blossom and no marked leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is T blo not⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (v): analysis of tv→f where v ̸∈ V and f is a leaf which is a child of v The tree tv→f is a tree in TP,P• whose root is not decorated with the decoration of v with one marked leaf and no blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is zT ′ not⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (vi): analysis of tv→br(v) where v ∈ V3 The tree tv→br(v) is a tree with a blossom that is replaceable if the root of the br(v)-th subtree of t[v] is in V , ⊕-replaceable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖-replaceable) if the root is not in V and labeled ⊕ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ⊖), with no marked leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is equal to T ⊕ if the root of the br(v)-th tree of τv is not in V and equal to T blo otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (vii): analysis of tv→ where v ̸∈ V The tree tv→ is a tree whose root denoted is decorated with the same decoration as v, who has no marked leaf and no blossom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' It verifies all the conditions of being (P, P•)-consistent, except that the root can have 0 or 1 child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is � k≥0 T k not⊕ = exp(Tnot⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (viii): analysis of tv→ where v ∈ V0 The tree tv→ is a tree in TP,P• whose root is decorated with an element of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The subtree induced by the marked leaves of tv→ is τ[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover tv→ has only one internal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is � H∈P Occdec(v)(H)zN(H) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = zN(dec(v))Occdec(v),P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Indeed, for a given H ∈ P, the term zN(H) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' correspond to the set of leaves and the term Occdec(v)(H) to the possible markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (ix): analysis of tv→ where v ∈ V3 The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The subtree induced by the marked leaves of tv→ is τ[v], where the non-trivial tree of τv is replaced by a blossom, marked with the smallest mark in the non-trivial tree of τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover tv→ has only one internal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Similarly to case (viii), the corresponding exponential generating function is: � H∈P• Occdec(v),br(v)(H)zN(H) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = zN(dec(v))−1Occdec(v),rk(v),P•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (x): analysis of tv→ where v ∈ V1 34 TH´EO LENOIR The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The subtree induced by the marked leaves of tv→ is τ[v] and no marked leaf belongs to the i-th tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The corresponding exponential generating function is: � H∈P• Occdec(v)(H)zN(H) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' × T = zN(dec(v))Occdec(v),P• × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The sum corresponds to the choice of the root (as in the previous cases), and the factor T to the potential non trivial tree of tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' ( ¯ xi): analysis of tv→ where v ∈ V2 The tree tv→ is a tree (P, P•)-consistent in case (D4) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The subtree induced by the marked leaves of tv→ is τ[v] and there is only one marked leaf ℓ in the i-th tree of tφ(v) (where i is the element such that (D4) holds in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Moreover, if we denote by j the label of ℓ, the label of the rk(v)-th leaf of τv is I(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Similarly to case (x), the corresponding exponential generating function is: � H∈P• Occdec(v),rk(v)(H)zN(H) N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' × zT ′ = zN(dec(v))Occdec(v),rk(v),P• × T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' All these conditions ensure that we can recover t by gluing all the different trees and that the subtree of t induced by I is τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus, Tτ,V0,V1,V2,V3,rk is the product of the generating functions corresponding to labeled such trees and concludes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The series Tτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='rk has radius at least R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' is ∆-analytic and its asymptotic expansion near R is: Tτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='rk = Cτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='rk � 1 − z R �β (1 + o(1)) where Cτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='V3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='rk := ακγ(1 + P •(R))θ(1 − P •(R))|V1|2λRµ × F(R) with β = −1 + d= + d̸= + dV →V + dV →ℓ + |V2| + |V3| 2 γ = dV →ℓ + |V2| − d= − d̸= − dV →V − |V3| − 1 θ = d= + d̸= − |V1| − |V2| − n2 − nL λ = −dV →ℓ − n1 − 2d= − 2d̸= + dV →V + nL µ = −dV →ℓ − |V2| + l and α = 1 2 if the root is not in V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1 1+P •(κ) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof of the main theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Background on graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We now review the necessary material on graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We refer the reader to [19] for a comprehensive presentation of deterministic graphons, while [7] studies specifically the convergence of random graphs in the sens of graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Here we will only recall the properties needed to prove the convergence of random graphs toward the Brownian cographon (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A graphon is an equivalence class of symmetric functions f : [0, 1]2 �→ [0, 1], under the equivalence relation ∼, where f ∼ g if there exists a measurable function φ : [0, 1] �→ [0, 1] that is invertible and measure preserving such that, for almost every (x, y) ∈ [0, 1]2, f(φ(x), φ(y)) = g(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We denote by ˜ W the set of graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Intuitively graphons can be seen as continuous analogous of graph adjacency matrices, where graphs are considered up to relabeling (hence the quotient by ∼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' There is a natural way to embed a finite graph into graphons: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G be a (random) graph of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We define the (random) graphon WG to be the equivalence class of wG : [0, 1]2 �→ [0, 1] defined by: ∀(x, y) ∈ [0, 1]2 wG(x, y) := 1⌈nx⌉connected to⌈ny⌉ There exists a metric δ□ on the set of graphons ˜ W such that ( ˜ W, δ□) is compact [19, Chapter 8], thus we can define for δ□ the convergence in distribution of a random graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If (G(n))n≥1 is a sequence of random graphs, there exists a simple criterion [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1] characterizing the convergence in distribution of (WG(n)) with respect to δ□: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3 (Rephrasing of [7], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For any n, let G(n) be a random graph of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by WG(n) the random graphon associated to G(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The following assertions are equivalent: (a) The sequence of random graphons (WG(n))n≥1 converges in distribution to some random graphon W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (b) The random infinite vector � OccG(n)(H) n(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='(n−|H|+1) � H finite graph converges in distribution in the product topology to some random infinite vector (ΛH)H finite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For a finite graph H, the random variable ΛH can be seen as the density of the pattern H in the graphon W: the variables (ΛH)H play the roles of margins of W in the space of graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 1 and W a random graphon, we denote by Samplek(W) the unlabeled random graph built as follows: Samplek(W) has vertex set {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , vk} and, letting (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , Xk) be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' uniform random variables in [0, 1], we connect vertices vi and vj with probability w(Xi, Xj) (these events being independent, conditionally on (X1, · · · , Xk) and W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The construction does not depend on the representation of the graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' With the notations of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3, we have for any finite graph H ΛH = P(Sample|H|(W) = H | W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 36 TH´EO LENOIR The article [1] introduces a random graphon W1/2 called the Brownian cographon which can be explicitly constructed as a function of a realization of a Brownian excursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Besides, [1, Proposition 5] states that the distribution of the Brownian cographon is characterized2 by the fact that for every k ≥ 2, Samplek(W1/2) has the same law as the unlabeled version of Graph(bk) with bk a uniform labeled binary tree with k leaves and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' uniform decorations in {⊕, ⊖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A consequence of this characterization is a simple criterion for convergence to the Brow- nian cographon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4 (Rephrasing of [1] Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every positive integer n, let T(n) be a uniform random tree in TP,P• with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For every positive integer ℓ, Iℓ (n) be a uniform partial injection from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , n} to N whose image is {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , ℓ} and independent of T(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by T(n) Iℓ(n) the subtree induced by Iℓ (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Suppose that for every ℓ and for every binary tree τ with ℓ leaves, (31) P(T(n) I(n) = τ) −−−→ n→∞ (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (2ℓ − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='. Then WGraph(T(n)) converges as a graphon to the Brownian cographon W1/2 of parameter 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Conclusion of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let τ be a binary tree with ℓ ≥ 2 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The series Tτ has radius of convergence R, is ∆-analytic and its asymptotic expansion near R is: Tτ = κ (1 + P •(R))22ℓ−2 � 1 − z R �− 2ℓ−1 2 (1 + o(1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (32) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' As Tτ = � τ,V0,V1,V2,V3,rk Tτ,V0,V1,V2,V3,rk, the asymptotic expansions of the different series Tτ,V0,V1,V2,V3,rk yield the ∆-analyticity of Tτ, its asymptotic expansion and its radius of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that β ≤ 1+e 2 where e is the number of edge of τ, with equality if and only if V0, V1, V2 and V3 are all empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Therefore, only the series Tτ,∅,∅,∅,∅,rk contributes to the leading term of the asymptotic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In this case, dV →ℓ = ℓ, d= + d̸= = ℓ − 2 and nL = ℓ − 1 which gives the announced expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let τ be a binary tree with ℓ ≥ 2 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For n ≥ ℓ and T(n) be a uniform random tree in TP,P• with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let Iℓ (n) be a uniform partial injection from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , n} to N whose image is {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , ℓ} and independent of T(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Denote by T(n) Iℓ(n) the subtree induced by Iℓ (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 2This characterization is strongly linked to the remarkable property that k uniform leaves in the CRT induce a uniform binary tree with k leaves, see again [1, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 37 Then P(T(n) Iℓ(n) = τ) −−−→ n→∞ (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (2(ℓ − 1))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='. Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since Iℓ (n) is independent of T(n), P(T(n) Iℓ(n) = τ) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [zn]Tτ n(n − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (n − ℓ + 1)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [zn]T = [zn]Tτ n(n − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (n − ℓ + 1)[zn]T By applying the Transfer Theorem [8, Corollary VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='392] to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (32), we get [zn]Tτ ∼ κ (1 + P •(R))22ℓ−2 n 2ℓ−3 2 Γ � 2ℓ−1 2 � Rn and by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='13 we obtain n × · · · × (n − ℓ + 1)[zn]T ∼ nℓ κ √π(1 + P•(R)) 1 Rnn 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus when n goes to infinity P(T(n) Iℓ(n) = τ) → √π 22ℓ−2Γ � 2ℓ−1 2 � = (ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (2(ℓ − 1))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Combining Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='4 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='6 prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7 of which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 is a particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G(n) be a uniform random graph in GP,P• with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We have the following convergence in distribution in the sense of graphons: WG(n) n→∞ −→ W 1 2 where W 1 2 is the Brownian cographon of parameter 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Number of induced prime subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We now estimate for a prime graph H the number OccH(G(n)) of induced occurences of H in G(n) and show that in average it is null, linear or of order n 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We first observe that substitution trees encoding prime graphs have a very simple struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let H be a prime graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' If t is a substitution tree such that H = Graph(t), t is reduced to a single internal node decorated with a relabeling of H with |H| leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let t be such a tree and r its root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' To every element t′ of tr we can associate a module of H by taking the vertices whose labels are the labels of the leaves of t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus tr is a union of leaves, and the decoration of the root is a relabeling of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ We say that H verifies (A) if there exists a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' , ℓ} such that OccG,a,P•(R) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 38 TH´EO LENOIR Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let H be a prime graph and let ℓ be its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For n ≥ ℓ, let G(n) be a uniform random graph in GP,P• with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then if H verifies (A), E[OccH(G(n))] ∼ KHn 3 2 with KH = Rℓ−1√π � a∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=',ℓ} OccH,a,P•(R) κ(1 + P •(R)) otherwise, E[OccH(G(n))] ∼ KHn with KH = �1 − P •(R) 1 + P •(R)OccH,P•(R) + OccH,P(R) � Rℓ κ2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let T(n) be a uniform random tree in TP,P• with n vertices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let τ be the canonical tree of H and NT(n),τ the number of induced subtrees of Tn isomorphic to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since τ is the unique substitution tree of G, E[OccH(G(n))] = E[NT(n),τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By independence E[OccH(G(n))] = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [zn]Tτ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [zn]T = [zn]Tτ [zn]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11, since in this case the only node of τ is either in V0, V1 or V2, we have that: Tτ = zℓT blo � �T ′ � � � a∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=',ℓ} OccG,a,P• � � + TOccG,P• + OccG,P � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus in case (A), with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' (22) and (26) Tτ ∼ Rℓ R(1 + P•(R))2 � � � a∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=',ℓ} OccH,a,P•(R) � � � 1 − z R �−1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Otherwise, Tτ ∼ � 1−P •(R) 1+P •(R)OccH,P•(R) + OccH,P(R) � Rℓ κ(1+P •(R)) � 1 − z R �− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' By applying the Transfer Theorem [8, Corollary VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 392], In case (A), [zn]Tτ ∼ Rℓ Rn+1(1 + P•(R))2 � a∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=',ℓ} OccH,a,P•(R) Otherwise, [zn]Tτ ∼ �1 − P •(R) 1 + P •(R)OccH,P•(R) + OccH,P(R) � Rℓ √πκ(1 + P•(R)) 1 Rnn 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 39 By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='13, [zn]T ∼ κ √π(1 + P•(R)) 1 Rnn 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus: In case (A), E[OccG(G(n))] ∼ Rℓ−1√π � a∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=',ℓ} OccG,a,P•(R) κ(1 + P •(R)) n 3 2, Otherwise, E[OccG(G(n))] ∼ �1 − P •(R) 1 + P •(R)OccG,P•(R) + OccG,P(R) � Rℓ κ2 n, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ An interesting application of this theorem is the computation of the asymptotic number of ˜P4’s in a random uniform graph of each of the graph classes of Section 3, where ˜P4 is the only labeling of P4 with endpoints 1 and 4 and 2 connected to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A prime spider has exactly |K|(|K| − 1) induced ˜P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' A pseudo-spider of size k has exactly (|K| + 2)(|K| − 1) induced ˜P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' One can check that for a prime spider, the P ′ 4s are induced by the partial injections I whose domain is {k, k′, f(k), f(k′)} for every (k, k′) ∈ K2 with k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the 24 such partial injections, only 2 are such that the graph induced is ˜P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since every induced ˜P4 is counted twice, we have |K|(|K| − 1) induced ˜P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For a pseudo-spider, let d be the duplicate and d0 the original node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' The P ′ 4s are induced by the partial injections I whose domain is {k, k′, f(k), f(k′)} for every (k, k′) ∈ K2 with k ̸= k′, or by the partial injections I whose domain is {d, k′, f(d0), f(k′)} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' {f −1(d0), k′, d, f(k′)}) for every k′ ∈ K with k′ ̸= d0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k′ ̸= f −1(d0)) if d0 is in K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' in S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' In the 24 such partial injections, only 2 are such that the graph induced is ˜P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Since every induced ˜P4 not containing d is counted twice, we have |K|(|K| − 1) + 2(|K| − 1) = (|K| + 2)(|K| − 1) induced ˜P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that this lemma implies that Occ ˜ P4,a,P• = 0 for all the graph classes men- tionned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For each graph class introduced in Section 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' we have the following ex- pressions for Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P and Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P•: 40 TH´EO LENOIR P4-tidy Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• tidy(z) = (2 + 16z + 4z3) exp(z2) − 1 − 8z Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='Ptidy(z) = Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• tidy(z) + 5z P4-lite Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• lite(z) = (2 + 16z + 4z3) exp(z2) − 1 − 8z Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='Plite(z) = Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• lite(z) + 4z P4-extendible Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• ext(z) = 1 + 8z Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='Pext = Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• ext(z) + 5z P4-sparse Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P•spa(z) = Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='Pspa(z) = 2 exp(z2) − 1 P4-reducible Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P• red(z) = Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='Pred = 1 P4-free Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='P•cog(z) = Occ ˜ P4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='Pcog(z) = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' We only detail the computation of Occ ˜ P4,P• tidy and Occ ˜ P4,Ptidy for P4-tidy graphs as this is the most involved case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Note that, with the notations of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='1, Occ ˜ P4,P(z) = � n∈N � H∈RPn � H′∼H Occ ˜ P4(H)zN(H)−4 N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = � n∈N � H∈RPn Occ ˜ P4(H)zN(H)−4 |Aut(H)| and similarly Occ ˜ P4,P•(z) = � n∈N � H∈RP•n � H′∼H Occ ˜ P4(H)zN(H)−4 N(H)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = � n∈N � H∈RPn Occ ˜ P4(H)zN(H)−4 |Aut(H)| According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='7, Ptidy is composed of one C5 that has 10 automorphisms and 10 induced ˜P4 and all its relabelings, one P5, and one P5 that both have 2 automorphisms and 4 induced ˜P4’s and all their relabelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat spiders corresponding to the 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1) different orbits of the action Φ2k over prime spiders of size 2k, each having k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms and k(k − 1) ˜P4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can come from K or S, and can be connected or not to the initial vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' These 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) cases correspond to the 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) different orbits of the action Φ2k+1 over pseudo-spiders of size 2k + 1, each having 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms and (k + 2)(k − 1) ˜P4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus we have Occ ˜ P4,P(z) = z + 4z 2 + 4z 2 + 2 2 + 2 � k≥3 k(k − 1)z2k−4 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' + 44z 2 + 8 � k≥3 (k + 2)(k − 1)z2k−3 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = 5z + 1 + 2 � k≥1 z2k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' + 8z + 4 � k≥1 (k + 4)z2k+1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' = 5z + 1 + 2 exp(z2) − 2 + 8z + 4 � k≥0 z2k+3 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' + 16 � k≥1 z2k+1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' GRAPH CLASSES WITH FEW P4’S: UNIVERSALITY AND BROWNIAN GRAPHON LIMITS 41 = 5z + 2 exp(z2) − 1 + 4z3 exp(z2) + 16z exp(z2) − 8z = 5z + (2 + 16z + 4z3) exp(z2) − 1 − 8z Now let’s compute Occ ˜ P4,P•(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat spiders with blossom corresponding to the 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 1) different orbits of the action Φ2k over blos- somed prime spiders G with 2k non blossomed vertices, each having k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms and k(k − 1) ˜P4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' For k ≥ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' k = 2), there are thin and fat pseudo-spiders, the duplicated vertex can come from K or S, and can be connected or not to the initial vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' These 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) cases correspond to the 8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' 4) different orbits of the action Φ2k+1 over blossomed pseudo-spiders with 2k + 1 non blossomed vertices, each having 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' automorphisms and (k + 2)(k − 1) ˜P4’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Hence Occ ˜ P4,P•(z) = 2 2 + 2 � k≥3 k(k − 1)z2k−4 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' + 44z 2 + 8 � k≥3 (k + 2)(k − 1)z2k−3 2(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Thus Occ ˜ P4,P•(z) + 5z = Occ ˜ P4,P(z) which gives the announced result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' □ Combining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='11 and the remark above, we get that ˜P4 does not verify (A), thus ˜P4 belongs to the linear case of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Let G(n) be a uniform graph of size n taken uniformly at random in one of the following families: P4-sparse, P4-tidy, P4-lite, P4-extendible, P4-reducible or P4-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Then E[Occ ˜ P4(G(n))] ∼ K ˜ P4n where K ˜ P4 is defined in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Here are the numerical approximations of K ˜ P4 in the different cases: class of graph K ˜ P4 P4-tidy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='29200322 P4-lite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='28507010 P4-extendible 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='24959979 P4-sparse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='10280703 P4-reducible 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='08249263 P4-free 0 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' I would like to thank Lucas Gerin and Fr´ed´erique Bassino for useful discussions and for carefully reading many earlier versions of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Bassino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Bouvel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' F´eray, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Gerin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Maazoun, and A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' P4-reducible graphs—class of uniquely tree-representable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=', 81(1):79–87, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Jamison and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Olariu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' On a unique tree representation for P4-extendible graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=', 34(1-3):151–164, 1991.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=', 145(1):329–344, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Lov´asz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Large Networks and Graph Limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Colloquium Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' American Mathematical So- ciety, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' M¨ohring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Algorithmic Aspects of Comparability Graphs and Interval Graphs, pages 41–101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Springer, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Stufler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Graphon convergence of random cographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Random Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' & Algor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=', 59:464 – 491, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content=' Th´eo Lenoir theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='lenoir@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} +page_content='fr Cmap, Cnrs, ´Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFRT4oBgHgl3EQfoDdP/content/2301.13607v1.pdf'} diff --git a/3tAzT4oBgHgl3EQfuf3H/vector_store/index.faiss b/3tAzT4oBgHgl3EQfuf3H/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..009edec76b0d3b026c44de3d2141aca2ed9d0e65 --- /dev/null +++ b/3tAzT4oBgHgl3EQfuf3H/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5f699f66e91e9e1ad05b0783c98c789f03c8b91dcd8d941e4d74142402c80ce3 +size 3145773 diff --git a/49E4T4oBgHgl3EQf1A2T/content/tmp_files/2301.05287v1.pdf.txt b/49E4T4oBgHgl3EQf1A2T/content/tmp_files/2301.05287v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..895563210204d0b2ea1854a74b481a69ebf8cd4f --- /dev/null +++ b/49E4T4oBgHgl3EQf1A2T/content/tmp_files/2301.05287v1.pdf.txt @@ -0,0 +1,2572 @@ +Model selection in atomistic simulation +Jonathan E. Moussa +Molecular Sciences Software Institute, Virginia Tech, Blacksburg, Virginia 24060, USA +(*Electronic mail: godotalgorithm@gmail.com) +There are many atomistic simulation methods with very different costs, accuracies, transferabilities, and numbers of empirical parameters. +I show how statistical model selection can compare these methods fairly, even when they are very different. These comparisons are also +useful for developing new methods that balance cost and accuracy. As an example, I build a semiempirical model for hydrogen clusters. +I. +INTRODUCTION +Scientists have been building quantitative atomistic models for +over a century1. In that time, many atomistic models have evolved +into sophisticated computer simulations2. +While there are now +models based on a wide variety of atomistic simulation methods, +most development has focused on two contradictory goals. Classi- +cal molecular mechanics (MM) methods focus on minimizing cost +to access phenomena at large length scales and long time scales3. +However, the use of MM methods is limited by the availability and +accuracy of system-specific interatomic potentials4. In contrast, +first-principles quantum mechanics (QM) methods focus on mini- +mizing error for general-purpose simulations5, which can get very +expensive. MM methods can achieve simulation costs of less than +10−5 CPU-seconds per atom6, while high-accuracy QM methods +have asymptotic costs greater than 104 CPU-seconds per atom7. +Because of the large gaps in cost and utility, there are many +atomistic simulation tasks for which QM methods are too expensive +and MM methods have no suitable interatomic potential. In this +situation, a scientist needs an affordable model and must either +develop their own or use an existing one such as a semiempirical +QM (SQM) model8,9. In either case, they need to collect evidence +to support their model. They must either gather enough reference +data to fit a new model, or find enough examples of scientists +using an existing model for similar tasks to be confident that it will +work for them. This type of model selection process is a common +occurrence in atomistic science, and yet it remains rather informal +and subjective much of the time. +In this paper, I advocate for using statistical model selection10 +to develop and compare models for atomistic simulation. All else +being equal, a scientist should fit or choose a model to maximize +the probability that they will succeed at their simulation task. Since +the exact probability will be more expensive to compute than the +simulation task itself, they must rely on a proxy probability based +on related but simpler simulation tasks. Assumptions about the +transferability of a method’s accuracy between related simulation +tasks are unavoidable in atomistic science. Also, when considering +methods with different numbers of fitting parameters or costs, extra +penalties are needed to avoid overfitting or exceeding computa- +tional budgets. These same principles apply to the development +of general-purpose models that are intended to be used by many +scientists over a broad distribution of simulation tasks. +As an example, I apply statistical model selection to the task +of simulating random hydrogen clusters. First, I generate high- +accuracy QM reference data. Second, I compare the accuracy of +some popular SQM models and density functionals from density +functional theory (DFT)11. Third, I build new SQM models by +correcting this SQM and QM data with atomic pair potentials. Here, +model selection determines the optimal number of parameters in +the pair potentials and the computational budget thresholds for +switching between models. +II. +STATISTICAL MODEL SELECTION +The standard practice in fitting atomistic models with param- +eters is to minimize a distance between model predictions and +reference data. I consider vectors of 𝑚 reference data points x and +model predictions y(λ), which are determined by 𝑛 real parame- +ters λ. The value of λ is usually chosen by minimizing the mean +absolute error (MAE), +∥x − y(λ)∥1 = +𝑚 +∑︁ +𝑖=1 +|𝑥𝑖 − 𝑦𝑖(λ)|, +(1) +or the root-mean-square deviation (RMSD), +∥x − y(λ)∥2 = +� +� 𝑚 +∑︁ +𝑖=1 +[𝑥𝑖 − 𝑦𝑖(λ)]2. +(2) +The general expectation is that smaller distances correspond to bet- +ter accuracy and thus a higher chance of success when these models +are used for other simulation tasks. However, this relationship is +indirect because these distances are not operational measures of +success. An operational measure would describe the application of +a model by scientists in a more explicit and direct way, including +how successful they are. Directly optimizing an operational mea- +sure should produce a more successful model if the operational +measure itself is sufficiently accurate. +To use statistical model +selection as an operational measure in this context, I must first +introduce two distinct sources of randomness. +The first source of randomness is in the model predictions. +I consider a generalization of the reference information from data +points x to simulation tasks X. Each reference simulation task 𝑋𝑖 +defines one or more physical systems and calculations to perform, +together with reference output data and success criteria. The con- +ditional probability of success, 𝑝(λ|𝑋𝑖), after choosing a task 𝑋𝑖 +and using a model with parameters λ replaces a distance between +𝑥𝑖 and 𝑦𝑖(λ). The only constraint on the success criteria is that the +success probability for the method used to generate the reference +data must be one. +Viable models must always have a nonzero +success probability, which requires the model output or success +criteria to have a random component. +The second source of randomness is in the choice of reference +simulation tasks. I relate a set of reference simulation tasks to +the actual simulation task that a scientist wants to succeed at by +arXiv:2301.05287v1 [physics.chem-ph] 12 Jan 2023 + +2 +considering them to be randomly drawn from a common distribution +of simulation tasks. The probability of choosing a simulation task +𝑋 is 𝑝(𝑋), and the probability of choosing this task and then +succeeding with the model is +𝑝(λ, 𝑋) = 𝑝(λ|𝑋)𝑝(𝑋). +(3) +It is not strictly necessary for the simulation tasks to have been +randomly drawn from this distribution. Such a distribution is still +formally useful even when it is an artificial context and not even +precisely defined. It is simply the mathematical representation of +a computational scientist as a distribution over simulation tasks. +A. +Maximum likelihood estimation +I now apply the framework of maximum likelihood estimation +(MLE)10 to determine the best model in this randomized setting. +The operational measure of modeling success is the probability of +succeeding at all 𝑚 reference simulation tasks, +𝑃(λ) = +𝑚 +� +𝑖=1 +𝑝(λ|𝑋𝑖). +(4) +It is related to a statistical likelihood function, +𝐿(λ) = +𝑚 +� +𝑖=1 +𝑝(λ, 𝑋𝑖) = 𝑃(λ) +𝑚 +� +𝑖=1 +𝑝(𝑋𝑖), +(5) +over the joint distribution of simulation tasks and modeling success +or failure events. I follow the common convention of considering +the negative logarithm of the probability or likelihood, +− log 𝑃(λ) = − +𝑚 +∑︁ +𝑖=1 +log 𝑝(λ|𝑋𝑖) += − log 𝐿(λ) + +𝑚 +∑︁ +𝑖=1 +log 𝑝(𝑋𝑖), +(6) +which replaces the product over reference simulation tasks with a +more convenient sum. The negative logarithm is a strictly mono- +tonically decreasing function, and maximizing it corresponds to +maximizing the likelihood. Since 𝑝(𝑋𝑖) has no dependence on λ, +𝑃(λ) and 𝐿(λ) are maximized by the same value of λ. +The familiar case of minimizing RMSD follows from a simple +success criterion and error model. I assume that each simulation +task 𝑋𝑖 produces a single model output 𝑦𝑖(λ) that must be within 𝜖 +of a reference value 𝑥𝑖 for success. I further adjust each model output +by a Gaussian error model with mean 𝜇 and standard deviation 𝜎 +to guarantee a finite success probability. Each success probability +reduces to a quadratic penalty for small 𝜖 values, +− log 𝑝(λ|𝑋𝑖) = − log +∫ +𝑥𝑖+𝜖 +𝑥𝑖−𝜖 +𝑒−0.5[𝑧−𝜇−𝑦𝑖 (λ)]2/𝜎2 +𝜎 +√ +2𝜋 +𝑑𝑧 +≈ [𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2 +2𝜎2 ++ 1 +2 log 𝜋𝜎2 +2𝜖2 + 𝑂(𝜖). +(7) +While not clear from this notation, error model parameters such as +𝜇 and 𝜎 are also considered to be part of the parameter vector λ. +In the small-𝜖 limit, the operational measure of success reduces to +an RMSD-like formula, +− log 𝑃(λ) ≈ 𝑚 +2 log 𝜋𝜎2 +2𝜖2 + +𝑚 +∑︁ +𝑖=1 +[𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2 +2𝜎2 +. +(8) +When minimizing this formula over 𝜇 and 𝜎, the minimizers are +the mean and standard deviation of the model error distribution, +𝜇 = +𝑚 +∑︁ +𝑖=1 +𝑥𝑖 − 𝑦𝑖(λ) +𝑚 +, +𝜎 = +� +� 𝑚 +∑︁ +𝑖=1 +[𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2 +𝑚 +. +(9) +The remaining minimization over λ is equivalent to minimizing +the RMSD with a model bias correction of 𝜇. The minimum value +of the small-𝜖 success measure is +− log 𝑃(λ) ≈ 𝑚 +2 + 𝑚 +2 log 𝜋𝜎2 +2𝜖2 +(10) +for 𝜎 in Eq. (9), which is a monotonically increasing function of the +bias-corrected RMSD. In the absence of model bias, the RMSD +and success measure thus produce the same minimizing models +and rank them in the same order. +Using a Gaussian distribution to approximate model errors is +justified when they come from an accumulation of many small, +independent errors. A non-zero mean suggests that these small +errors are biased on average. The same small-𝜖 analysis can relate +a similar success measure to the MAE if the underlying error model +is a Laplace distribution, +𝜌(𝑥) = 𝑒− +√ +2|𝑥−𝜇|/𝜎 +𝜎 +√ +2 +. +(11) +However, non-Gaussian error distributions suggest a small number +of dominant, independent error sources that avoid the inevitable +consequences of the central limit theorem. +Also, the Laplace +distribution has a fatter tail than a Gaussian distribution, which +implies an increased tolerance of large error outliers. Ultimately, +the choice of distributions in an error model should be informed +by the observed distribution of errors between model and data. +A more sophisticated MLE example is a multi-Gaussian error +model. Here, we partition the reference simulation tasks into 𝑟 +groups of similar tasks, each with their model errors described +by a different Gaussian distribution. Such grouping is appropriate +when different groups of tasks are observed to have different error +statistics for models under consideration12. The small-𝜖 limit of +the success measure generalizes from Eq. (8) to +− log 𝑃(λ) ≈ +𝑟∑︁ +𝑖=1 +𝑚𝑖 +2 log 𝜋𝜎2 +𝑖 +2𝜖2 ++ +𝑟∑︁ +𝑖=1 +𝑚𝑖 +∑︁ +𝑗=1 +[𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) − 𝜇𝑖]2 +2𝜎2 +𝑖 +, +(12) +where the extra index is for the groups. The minimizing 𝜇𝑖 and 𝜎𝑖 +values generalize from Eq. (9) to +𝜇𝑖 = +𝑚𝑖 +∑︁ +𝑗=1 +𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) +𝑚𝑖 +, +𝜎𝑖 = +� +� 𝑚𝑖 +∑︁ +𝑖=1 +[𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) − 𝜇𝑖]2 +𝑚𝑖 +. +(13) + +3 +The minimization over λ is now equivalent to a weighted, bias- +corrected RMSD with weights proportional to the inverse error +variance. However, the minimum value of the success measure, +− log 𝑃(λ) ≈ 𝑚 +2 + +𝑟∑︁ +𝑖=1 +𝑚𝑖 +2 log 𝜋𝜎2 +𝑖 +2𝜖2 , +(14) +no longer ranks minimizing models in the same order as the cor- +responding weighted RMSD. Thus MLE rapidly deviates from +minimizing simple distances between model and reference data as +success criteria and error models get more complicated. +Beyond these simple examples, MLE can provide a lot of +flexibility to the model-fitting process. It is possible to fit low- +cost models that are designed to have only qualitative accuracy +by choosing success criteria that tolerate large but well-shaped +errors. For example, conformer searches only need to preserve the +order of conformer energies, which can be tested by the Spearman +rank correlation coefficient13. When fitting very accurate models, +many reference simulation tasks may have success probabilities +very close to one and effectively vanish from log 𝑃(λ). In this +highly successful regime, error outliers in a model will have a +greatly enhanced influence on the success measure and MLE may +become functionally equivalent to minimax optimization. +B. +Information criteria +Simple MLE is capable of selecting the best model from one +family of models parameterized by λ, but it cannot reliably compare +models from different families. Adding more free parameters to an +existing model and optimizing them can only improve the success +measure, and nested models with more parameters will always be +preferred. This can eventually cause the modeling phenomenon +of fitting noise rather than data, and there needs to be additional +modeling criteria for eliminating parameters that are not useful. +The most common approach is to introduce a penalty for adding +model parameters that is overcome by useful parameters. Such +measures of model accuracy with penalties for parameters are +called information criteria (IC), the oldest and most famous of +which is the Akaike information criterion (AIC)14. The Takeuchi +information criterion (TIC)15 is a more complicated generalization +of the AIC that does not assume model accuracy. Here, I provide +a minimal motivation and derivation of the TIC and AIC to justify +their use in fitting models for atomistic simulation. +An implicit assumption about both the IC derivations and +MLE itself is that 𝑃(λ) can be optimized over λ effectively in +practice. The mathematical structure of 𝑃(λ) depends on both +the model family and the success criteria of simulation tasks. I +specifically assume that 𝑃(λ) is twice differentiable with respect to +λ and that derivative information is used to find local minimizers. +I also assume that it is possible to choose initial values for λ in +the basin of convergence for the global minimizer. While there is +not enough structure here to guarantee or verify global minima, +there are often physical considerations to guide reasonable choices +of initial λ values. +Both the AIC and TIC come from attempting to change the +modeling success measure from Eq. (6) to +𝐷(λ) = −𝑚 +∑︁ +𝑋 +𝑝(𝑋) log 𝑝(λ|𝑋), +(15) +which is 𝑚 times the Kullback-Leibler divergence16 of the always +successful reference distribution from the model distribution that +can fail at simulation tasks. Minimizing this divergence maximizes +the asymptotic success probability for any large number of simula- +tion tasks drawn from the model distribution16. While this is more +reliable than only maximizing the success probability for a specific +set of 𝑚 simulation tasks, 𝐷(λ) and its minimizer ˆλ cannot be +calculated efficiently in general. The practical alternative is to use +− log 𝑃(λ) and its minimizer ˆλX to approximate these inaccessi- +ble quantities. To clarify their relationship, I use two convenient +intermediates, +𝐷X(λ) = − +𝑛 +∑︁ +𝑖=1 +log 𝑝(λ|𝑋𝑖), +∑︁ +X += +∑︁ +𝑋1 +𝑝(𝑋1) · · · +∑︁ +𝑋𝑛 +𝑝(𝑋𝑛), +(16) +to simplify the notation during the IC derivations. +For a constant value of λ, 𝐷X(λ) is an unbiased estimator of +𝐷(λ) when averaged over sets of 𝑚 simulation tasks X, +𝐷(λ) = +∑︁ +X +𝐷X(λ). +(17) +Since I cannot efficiently calculate ˆλ, I would like to evaluate 𝐷(λ) +at one λ = ˆλX value that I can calculate. If this was repeated and +averaged over sets of 𝑚 simulation tasks, it would be an unbiased +estimator of +𝐷min−ave = +∑︁ +X +𝐷(ˆλX). +(18) +However, with a single X, I can only evaluate 𝐷X(λ) at its own +minimum, λ = ˆλX, which is an unbiased estimator of the average +minimum, +𝐷ave−min = +∑︁ +X +𝐷X(ˆλX). +(19) +This has a negative bias relative to 𝐷(ˆλX) because each 𝐷(λ) is +evaluated at its own minimum instead of a common λ. A single +𝐷X(ˆλX) can be unbiased as an estimator of 𝐷(ˆλX) by adding a +bias correction, +Δ = 𝐷min−ave − 𝐷ave−min = +∑︁ +X +[𝐷(ˆλX) − 𝐷X(ˆλX)]. +(20) +I approximate Δ with several simplifying assumptions. +The first IC assumption is that 𝐷(λ) and 𝐷X(λ) are both +slowly changing in a region containing ˆλ and ˆλX. Both functions +can be extrapolated from their minimum to the other function’s +minimum with a second-order Taylor expansion, +𝐷(ˆλX) ≈ 𝐷(ˆλ) + 1 +2 (ˆλX − ˆλ)𝑇 F(ˆλX − ˆλ), +𝐷X(ˆλ) ≈ 𝐷X(ˆλX) + 1 +2 (ˆλ − ˆλX)𝑇 FX(ˆλ − ˆλX), +[F]𝑖, 𝑗 = +𝜕2𝐷 +𝜕𝜆𝑖𝜕𝜆 𝑗 +(ˆλ), +[FX]𝑖, 𝑗 = 𝜕2𝐷X +𝜕𝜆𝑖𝜕𝜆 𝑗 +(ˆλX). +(21) + +4 +These extrapolations can be combined using Eq. (17) to simplify +the bias correction in Eq. (20) to +Δ ≈ 1 +2 +∑︁ +X +(ˆλ − ˆλX)𝑇 (F + FX)(ˆλ − ˆλX). +(22) +Similarly, I can extrapolate 𝐷X(λ) from λ = ˆλ to λ = ˆλX, +𝐷X(ˆλX) ≈ 𝐷X(ˆλ) + (ˆλX − ˆλ)𝑇 𝜕𝐷X +𝜕λ (ˆλ) ++ 1 +2 (ˆλX − ˆλ)𝑇 F′ +X(ˆλX − ˆλ), +[F′ +X]𝑖, 𝑗 = 𝜕2𝐷X +𝜕𝜆𝑖𝜕𝜆 𝑗 +(ˆλ), +(23) +and minimize the quadratic form for the parameter variations, +ˆλ − ˆλX ≈ (F′ +X)−1 𝜕𝐷X +𝜕λ (ˆλ). +(24) +The second IC assumption is that F ≈ FX ≈ F′ +X, which allows for +the removal of FX and F′ +X from Eq. (22) after substituting Eq. (24), +Δ ≈ tr[ ˜FF−1], +[ ˜F]𝑖, 𝑗 = +∑︁ +X +𝜕𝐷X +𝜕𝜆𝑖 +(ˆλ) 𝜕𝐷X +𝜕𝜆 𝑗 +(ˆλ). +(25) +The validity of these two assumptions can be increased by adding +more reference data to reduce finite-sample effects until 𝐷X(λ) +and 𝐷(λ) have small differences in their gradients and negligible +differences in their Hessians at λ = ˆλX. +The TIC follows from a related assumption about small finite- +sampling effects. As a useful reference, I rearrange F and ˜F into a +similar form by rewriting ˜F as a sum over simulation tasks rather +than over groups of 𝑚 simulation tasks, +[ ˜F]𝑖, 𝑗 = 𝑚 +∑︁ +𝑋 +𝑝(𝑋) +� 𝜕 log 𝑝(λ|𝑋) +𝜕𝜆𝑖 +𝜕 log 𝑝(λ|𝑋) +𝜕𝜆 𝑗 +� +λ=ˆλ +, +[F]𝑖, 𝑗 = −𝑚 +∑︁ +𝑋 +𝑝(𝑋) +� 𝜕2 log 𝑝(λ|𝑋) +𝜕𝜆𝑖𝜕𝜆 𝑗 +� +λ=ˆλ +. +(26) +The TIC bias correction is a direct approximation of Eq. (25) by +Δ ≈ ΔTIC = tr[ ˜FXF−1 +X ], +[ ˜FX]𝑖, 𝑗 = +𝑚 +∑︁ +𝑘=1 +� 𝜕 log 𝑝(λ|𝑋𝑘) +𝜕𝜆𝑖 +𝜕 log 𝑝(λ|𝑋𝑘) +𝜕𝜆 𝑗 +� +λ=ˆλX +, +(27) +which again assumes that the 𝑚 samples in X are sufficient to +converge expectation values so that ˜F ≈ ˜FX and F ≈ FX. +The AIC follows from additional assumptions about model +accuracy. I can simplify the difference between F and ˜F in Eq. (26) +by rearranging and combining the logarithmic derivatives into +[ ˜F − F]𝑖, 𝑗 = 𝑚 +∑︁ +𝑋 +𝑝(𝑋) +𝑝(ˆλ|𝑋) +� 𝜕2𝑝(λ|𝑋) +𝜕𝜆𝑖𝜕𝜆 𝑗 +� +λ=ˆλ +. +(28) +Next, I consider a modified form of 𝐷(λ) from Eq. (15) in which +the reference simulation tasks are assigned a failure rate 𝛿, +𝐷(λ) = −𝑚 +∑︁ +𝑋 +(1 − 𝛿)𝑝(𝑋) log 𝑝(λ|𝑋) +− 𝑚 +∑︁ +𝑋 +𝛿𝑝(𝑋) log(1 − 𝑝(λ|𝑋)). +(29) +The original form is recovered in the 𝛿 → 0 limit. +If the IC +derivation is repeated for the modified form, Eq. (28) becomes +[ ˜F − F]𝑖, 𝑗 = 𝑚 +∑︁ +𝑋 +(1 − 𝛿)𝑝(𝑋) +𝑝(ˆλ|𝑋) +� 𝜕2𝑝(λ|𝑋) +𝜕𝜆𝑖𝜕𝜆 𝑗 +� +λ=ˆλ ++ 𝑚 +∑︁ +𝑋 +𝛿𝑝(𝑋) +1 − 𝑝(ˆλ|𝑋) +� 𝜕2[1 − 𝑝(λ|𝑋)] +𝜕𝜆𝑖𝜕𝜆 𝑗 +� +λ=ˆλ +. +(30) +The final AIC assumption is that the optimized model can recover +the reference distribution, resulting in 𝑝(ˆλ|𝑋) ≈ 1 − 𝛿 here. I can +then cancel the 𝛿 factors and combine the two terms in Eq. (30), +[ ˜F − F]𝑖, 𝑗 ≈ 𝑚 +� +𝜕2 +𝜕𝜆𝑖𝜕𝜆 𝑗 +∑︁ +𝑋 +𝑝(𝑋) +� +λ=ˆλ += 0. +(31) +The difference between ˜F and F disappears for any value of 𝛿. In +this scenario, ˜F and F are 𝑚 times the Fisher information matrix16 +of 𝑝(λ, 𝑋). The AIC bias correction corresponds to ignoring this +difference and keeping only the trace of the identity matrix over +the 𝑛-dimensional parameter space, +Δ = 𝑛 + tr[( ˜F − F)F−1] ≈ ΔAIC = 𝑛. +(32) +The validity of the good model assumption can be increased by +improving the model family and relaxing the success criteria to +increase all optimized success probabilities towards one. +C. +Transferability +A statistical framework for model selection can also sup- +port more precise statistical statements about model transferability. +Here, I briefly contrast a notion of statistical transferability from +that of physical transferability, which is frequently discussed when +building models for atomistic simulation17. I argue that while sta- +tistical transferability is the more desirable goal of model building, +it is often impractical to avoid physical transferability assumptions +given the present state of atomistic simulation methods. +Statistical transferability can directly predict the average future +success of a model when simulation tasks can be interpreted as +being drawn from the same distribution that was used to fit the +model. This is a form of model transferability to future simulation +tasks that were not part of the reference data. For a model fit with +𝑚 reference simulation tasks to a minimum divergence 𝐷(ˆλ) in +Eq. (15), the asymptotic fraction of successful simulations will be +exp(−𝐷(ˆλ)/𝑚). +(33) +If the task distribution is designed to predict or approximate typical +workloads of typical users of a model, then the model fitting process +provides a direct operational statement about how effective the +model should be for its users. +Statistical transferability can also be used to recycle refer- +ence data by transferring it between task distributions. Reference +data sampled from a second distribution 𝑝′(𝑋) over a superset of +simulation tasks can be reused to estimate 𝐷(λ) for 𝑝(𝑋), +𝐷(λ) ≈ − +� +min +𝑖 +𝑝′(𝑋𝑖) +𝑝(𝑋𝑖) +� +𝑚 +∑︁ +𝑖=1 +𝑝(𝑋𝑖) +𝑝′(𝑋𝑖) log 𝑝(λ|𝑋𝑖). +(34) + +5 +This is an implicit form of rejection sampling, and it requires the +ability to calculate probability ratios between two task distribu- +tions. +It can also be used heuristically to reduce the influence +of data that is necessary to fit a model but not representative of +its typical applications. Operationally, this can be interpreted as +rare instances when users validate the model for themselves on the +original reference data. The effective sample size associated with +this resampling procedure is +𝑚′ = +� +min +𝑖 +𝑝′(𝑋𝑖) +𝑝(𝑋𝑖) +� +𝑚 +∑︁ +𝑖=1 +𝑝(𝑋𝑖) +𝑝′(𝑋𝑖) , +(35) +which can be small if 𝑝(𝑋) and 𝑝′(𝑋) are very different. +Physical transferability is a set of observations and assumptions +about the spatial locality of physics at an atomistic length scale. It +assumes that some model details and parameters describing short- +range interatomic effects will be insensitive to distant changes in +a large system with many atoms and then observes the varying +degrees to which this is true. The underlying first-principles QM +equations are completely local and transferable when long-range +interactions are mediated by local fields. Unfortunately, locality +and transferability are both degraded when encapsulating many- +body effects and non-essential degrees of freedom to build simpler +models. Physical transferability assumptions are essential for justi- +fying the use of methods that decompose large systems into a set of +small fragments and simulate them individually, often embedded +in simpler model environments. Such methods include implicit +solvation models18, QM/MM embedding19, and the use of periodic +supercells20. However, the effectiveness of these methods can be +highly system dependent, an important example being the reduced +locality of electronic effects in metallic systems that complicate +efforts to develop low-cost methods21. +In the context of statistical model selection, physical transfer- +ability assumptions are unavoidable when generating reference data +for task distributions containing large systems. Reliable methods +for reference data generation generally have large cost prefactors or +poor cost scaling with system size that prevent their direct use on +the task distribution. Physical transferability can be used to justify +the use of more accessible reference data corresponding to a proxy +task distribution over small embedded fragments. Tasks from the +original distribution can be decomposed into sets of proxy tasks +on fragments to generate the proxy distribution. While these small +proxy tasks may all be contained within the original task distri- +bution, the proxy distribution is over a strict subset of simulation +tasks. It is statistically impossible to sample from a distribution +by weighting samples from a second distribution over a subset of +events, but this is avoided by the physical fragmentation process. +While rigorous error analysis of this process is difficult, the general +expectation is that the use of larger system fragments increases the +validity of physical transferability assumptions. +D. +Cost penalties +The primary purpose of fitting models in statistics is to explain +data in the absence of a prior explanation. In contrast, the purpose +of fitting models for atomistic simulation is to avoid the large cost of +evaluating a known first-principles model. Statistics is concerned +with efficiency, but its main consideration is in getting the most +value out of limited data to avoid the potentially high cost of +collecting or generating data. Without some penalty for the cost +of models, the inevitable conclusion of statistical model selection +in atomistic simulation is to choose the expensive model that was +used to generate the reference data. The IC already add penalties +to the success measure that limits the number of model parameters, +and the simplest approach is to introduce a cost penalty with a +similar form. The linear parameter penalty in Eq. (32) looks like a +Lagrange multiplier, except that the coefficient is not adjustable and +the number of parameters is a trivial function of parameter values. +The average model evaluation cost can be a non-trivial function of +model parameter values, and it can be controlled using a Lagrange +multiplier that penalizes excessive cost. +With both cost and parameter penalties added, the operational +measure of modeling success is +˜𝐷(λ) = 𝛾(𝑡 − 𝑡0) + Δ − +𝑚 +∑︁ +𝑖=1 +log 𝑝(λ|𝑋𝑖). +(36) +Here, 𝛾 is a Lagrange multiplier, 𝑡 is the total cost of applying +the model to the 𝑚 simulation tasks, 𝑡0 is the target computational +budget, and Δ is an IC penalty approximating Eq. (20). Between +multiple model families with different costs and parameters, the +family that produces the minimum value of ˜𝐷(λ) for a common 𝛾 +value should be selected. The stationary condition of the Lagrange +multiplier, +𝜕 +𝜕𝛾 +˜𝐷(λ) = 𝑡 − 𝑡0 = 0, +(37) +should be applied to the model family with the smallest minimum +˜𝐷(λ) value. If this best model family has a parameter-invariant 𝑡 +value, then 𝛾 should be adjusted until the minimum cost-penalized +˜𝐷(λ) is equal for two different models. +In this scenario, the +cost of the two best models, 𝑡1 and 𝑡2, should bracket 𝑡0 as 𝑡1 ≤ +𝑡0 ≤ 𝑡2. A new, hybrid model can then achieve the target cost by +randomly switching tasks between the two bracketing models with +probabilities (𝑡2 − 𝑡0)/(𝑡2 − 𝑡1) and (𝑡0 − 𝑡1)/(𝑡2 − 𝑡1). It is often +more practical to minimize Eq. (36) over λ without any penalties +and then add in the penalties with no further optimization of λ. +The use of cost penalties may be more complicated if applied +to proxy distributions of fragmented simulation tasks as described +in the previous subsection. If sets of fragmented simulation tasks +are meant to represent a larger simulation task, then the model eval- +uation cost for the larger simulation task may not be approximated +well by the sum of costs for the proxy tasks. In this situation, a +proxy cost penalty could be constructed from resource estimates +that approximate the unknown cost of the larger simulation task +from the known costs of the proxy tasks and other task-specific +data. Models usually have a well-understood scaling with system +size and cost prefactors can be estimated from the proxy calcu- +lations. More detailed, model-specific resource estimation is also +possible22. The estimated total simulation cost of the model on the +large simulation tasks could then be used as 𝑡 in Eq. (36) instead +of the total proxy simulation cost that is directly observed. + +6 +III. +HYDROGEN CLUSTER EXAMPLE +To demonstrate the principles of statistical model selection, I +consider a simple set of simulation tasks on randomly generated +hydrogen clusters. By only considering hydrogen atoms, I keep +the elemental diversity at a minimum to simplify the process of +fitting SQM models with element-specific parameters. I keep the +phenomenological diversity high by considering two distributions +of clusters. A “dense” distribution of clusters forces the minimum +interatomic distance between hydrogen atoms to be less than the +Coulson-Fischer point23 near 1 Å, while a “sparse” distribution +allows larger minimum separations. +Molecular orbitals tend to +remain grouped into pairs with opposite spin and similar spatial +character in the dense distribution, while the sparse distribution +generates many clusters that favor spin-polarized, atom-localized +orbitals. +Because of limitations in methods and software that +generate accurate and reliable reference data, the only observable +that I consider is the total energy of clusters. +I consider three +simulation tasks to calculate energies for three cluster modifications: +removal of an atom, removal of an electron, and addition of an +electron. A success is defined as the calculation of one such energy +with an error of 1 kcal/mol or less. While these distributions and +tasks are artificial and not directly motivated by any application, +there is some experimental interest in positively24 and negatively25 +charged hydrogen clusters. +I generate the dense and sparse distribution of hydrogen clus- +ters by sequential rejection sampling. Atoms are assigned uniformly +random positions in a box containing the valid domain, and the +atom is rejected and repositioned if it violates a distance constraint. +The minimum allowed interatomic distance for both distributions +is 0.3 Å, near the classical turning point of the H2 potential energy +surface. The maximum allowed value for the minimum interatomic +distance is 1 Å for the dense distribution and 4 Å for the sparse +distribution. The sparse distribution is a strict superset of the dense +distribution, and some sparse clusters could be recycled as dense +clusters. However, the recycling rate of two-atom clusters is only +0.016, and it decreases rapidly with increasing cluster size. This is +an example of low recycling efficiency between distributions that +are very different. For the reference data set, I generate 10,000 +nested sequences of clusters between two and seven atoms for each +distribution, resulting in 120,000 distinct structures. The three sim- +ulation tasks require calculations of three different charge states – +0, 1, and -1 – corresponding to 360,003 total energy calculations +including an isolated hydrogen atom. +I briefly compare this reference data set with the MGCDB84 +data set that is popular for testing DFT functionals26. Both data +sets are restricted to total energies of small, isolated groups of +atoms. +MGCDB84 corresponds to 5,931 total energy calcula- +tions of structures that are 52.6% hydrogen, 29.2% carbon, 8.8% +oxygen, 5.5% nitrogen, and less than 1% each of main-group ele- +ments from the first four rows of the periodic table. Thus, while +it is not restricted to only hydrogen atoms, hydrogen is the most +well-represented element in MGCDB84. MGCDB84 is organized +into 84 subsets of data corresponding to different simulation tasks, +including non-covalent binding energies, isomerization energies, +formation energies, and barrier heights. +However, this data set +lacks diversity by some measures, such as 95.2% of the structures +being closed-shell singlets and 93.0% being charge neutral. Also, +0 +1 +2 +3 +4 +5 +101 +103 +105 +MGCDB84 +0 +1 +2 +3 +4 +5 +101 +103 +105 +dense +0 +1 +2 +3 +4 +5 +H-H distance +101 +103 +105 +sparse +FIG. 1. Histograms of interatomic distances between hydrogen atoms in +the structures from three reference data sets. +MGCDB84 mostly contains structures and properties of interest to +organic chemistry with structures at equilibrium or saddle points. +MGCDB84 is thus a reasonable proxy for the interests of organic +chemists, while the hydrogen cluster data sets broadly sample from +the potential energy surface of many hydrogen atoms. Of partic- +ular interest when fitting distance-dependent parameters such as +pair potentials and one-body matrix elements is the distribution +of interatomic distances between hydrogen atoms. These distance +distributions are shown in Fig. 1 for MGCDB84 and the two distri- +butions of hydrogen clusters considered here. MGCDB84 has poor +coverage at distances less than 1.4 Å and is not a good reference +to fit distance-dependent parameters for hydrogen interactions. +A. +Reference data +I gather high-level reference data for the hydrogen clusters at +the CCSD(T) level of theory27 with the def2-QZVPP basis set28. I +also record data at the Hartree-Fock (HF), MP2, and CCSD levels +of theory during the CCSD(T) calculations. In addition to the high- +level reference data, I also gather data using several popular SQM +models and DFT functionals to test their transferability. There are +too many SQM models and DFT functionals to test all of them, and +this study is limited to a few important representative examples. +AM129 was the most popular SQM thermochemistry model of the +last century, and PM730 is the most recent model from that family +of MNDO-like models31. +GFN132 and GFN233 are two recent +SQM models from the density functional tight-binding (DFTB) +framework34. PBE35 is the most popular DFT functional in solid- +state physics and materials science. B3LYP36 is the most popular + +7 +DFT functional in chemistry. 𝜔B97M-V37 is claimed to be the most +accurate DFT functional without including terms from many-body +perturbation theory. While a smaller basis set might be sufficient, +I perform all DFT calculations using the def2-QZVPP basis set +for consistency. In total, I gather data from eleven QM and SQM +models, which corresponds to 3,960,033 total energy calculations. +All QM calculations use a post-2.1.1 development version of +PySCF38–40. All calculations use spin-unrestricted orbitals. For +HF theory and every DFT functional, the large-basis calculations +are initialized by projecting a converged density matrix from a +calculation in the smaller def2-SVP basis set28. +The def2-SVP +density matrix is taken from the calculation with the lowest total +energy from a systematic ground-state search for each structure +and charge state. First, a def2-SVP calculation is performed for +every spin state from the standard spin-averaged independent-atom +density matrix guess. Second, a custom density matrix guess is +constructed from spin-polarized independent-atom density matrices +with every combination of atomic charges and spin orientations. +Third, after performing all of these small-basis calculations with the +default DIIS algorithm41, they are all repeated with an alternative +ADIIS algorithm42. +The large-basis calculation uses the same +algorithm, either DIIS or ADIIS, as the small-basis calculation that +is used to initialize it. Even with all of this redundancy, it is not +possible to converge a self-consistent field (SCF) cycle for every +charge and spin state of every structure. +While the variational +nature of SCF calculations guarantees the existence of stable local +energy minima, DIIS-based algorithms provide no guarantees of +convergence. All large-basis DFT calculations use a (99,590) local +grid and a SG-1 nonlocal grid, following the recommendations for +the 𝜔B97M-V functional37. +For SQM calculations, MOPAC 22.0.543 is used for AM1 and +PM7 calculations, and xTB 6.5.19 is used for GFN1 and GFN2 +calculations. MOPAC calculations follow the same ground-state +search procedure as the PySCF calculations except with only DIIS +and without any projection into a larger basis. There are fewer +points of failure in minimal-basis calculations, and MOPAC is able +to converge an SCF calculation for every structure and charge state. +xTB calculations do not contain Fock exchange and depend on an +initial electronic density guess rather than a density matrix guess. +I only use the default spin-averaged density guess and restrict the +ground-state search to total spin values. There is a high failure rate +for SCF convergence in xTB with the default options for this data +set. However, it is possible to converge every structure and charge +state in xTB with calculations at elevated electronic temperatures +of 3000 K and then 1500 K followed by linear extrapolation of the +total energies to zero temperature. +At this level of automation and scale of data generation, it is not +possible to converge every iterative solve for HF, DFT, and CCSD +calculations in PySCF. The choice of solver options is important +as it changes success statistics and average run times. I did not +try to optimize these choices in a systematic way, but they were +adjusted during the implementation of the workflow to improve +success rates44. In addition to convergence failures, a DFT or HF +calculation is considered to fail if the def2-QZVPP total energy +is more than 10 kcal/mol larger than the smallest def2-SVP total +energy. Total energies tend to be lower for larger basis sets because +they have more variational degrees of freedom. I attribute these +energy increases to the DIIS phenomenon of escaping from the +0.00 +0.01 +0.02 +0.03 +0.04 +101 +103 +105 +dense +0.00 +0.01 +0.02 +0.03 +0.04 +101 +103 +105 +sparse +FIG. 2. Histograms of the maximum deviation from zero and one of the +unrelaxed MP2 1RDM eigenvalues for all structures and charge states from +the reference data sets. +basin of convergence of a ground state and converging to a very +different stationary state with a larger energy. The failure rate of +DFT calculations is 3.4%, the failure rate of CCSD(T) calculations +is 1.0%, and the overall failure rate of the simulation tasks is 4.3%. +If at least one model fails to produce an output for a simulation +task, then that task is omitted from the final data set and statistical +analysis. Such failures distort the distribution of simulation tasks +because they act as a form of rejection sampling. +I also validate the CCSD(T)/def2-QZVPP level of theory for +this data set while gathering data. The main validity concern is +strong electron correlation effects, which are known to occur in +hydrogen clusters45. These effects are caused by multi-reference +ground states that come from a superposition of many electronic +spin configurations with nearly degenerate energies in the atomic +limit. Randomly generated hydrogen clusters are unlikely to have +many degenerate spin configurations, and they are expected to be +more weakly correlated on average. The most direct validity test +would be the overlap between the normalized HF and CCSD many- +body wave-functions, but this quantity is not efficiently computable. +Instead, I use the eigenvalues of the one-particle density matrix +(1RDM) at the unrelaxed MP2 level of theory as an accessible +proxy for this overlap. The maximum deviation of the eigenvalues +from zero and one is strictly zero when the overlap is one, and +the deviation increases as the overlap is reduced. This deviation +is plotted for every structure in every charge state in Fig. 2. The +sparse distribution that is expected to be more susceptible to multi- +reference effects because of spin symmetry breaking does not have +larger deviations than the dense distribution. +The other major validity concern is the basis-set convergence +of CCSD(T)/def2-QZVPP. A quadruple-zeta basis such as def2- +QZVPP does not typically converge absolute post-HF energies to +chemical accuracy of 1 kcal/mol or less without basis-set extrapo- +lation or explicit correlation corrections. However, the simulation +tasks considered here only require energy differences between struc- + +8 +−2.5 +0.0 +2.5 +101 +103 +105 +CCSD +–0.5 +±0.5 +remove atom +dense +−5 +0 +5 +–0.0 +±0.1 +sparse +−2.5 +0.0 +2.5 +–0.5 +±0.5 +remove electron +dense +0 +25 +0.1 +±0.7 +sparse +0.0 +2.5 +0.1 +±0.5 +add electron +dense +0 +20 +0.7 +±1.0 +sparse +−20 +0 +101 +103 +105 +MP2 +–2.7 +±3.1 +−20 +0 +–2.3 +±4.7 +−20 +0 +20 +–2.0 +±3.4 +−25 +0 +25 +–3.0 +±5.7 +−10 +0 +10 +0.0 +±0.9 +−10 +0 +10 +6.6 +±4.2 +−50 +0 +101 +103 +105 +HF +–18.8 +±8.9 +−25 +0 +–4.1 +±8.1 +−25 +0 +–17.1 +±9.7 +−50 +0 +–5.0 +±8.3 +0 +25 +1.2 +±4.2 +0 +25 +17.8 +±6.3 +−25 +0 +25 +101 +103 +105 +ωB97M-V +0.7 +±2.3 +−10 +0 +10 +–1.3 +±2.4 +−25 +0 +25 +0.5 +±2.6 +−50 +0 +50 +–29.1 +±8.8 +0 +25 +–0.2 +±1.1 +−20 +0 +–7.6 +±3.4 +−25 +0 +25 +101 +103 +105 +B3LYP +2.2 +±2.8 +0 +10 +–0.1 +±1.6 +0 +25 +4.2 +±3.3 +−50 +0 +–36.9 +±13.4 +0 +20 +–0.4 +±1.4 +−50 +0 +–21.5 +±6.1 +−25 +0 +25 +101 +103 +105 +PBE +0.9 +±4.5 +−10 +0 +10 +–0.8 +±2.0 +−25 +0 +25 +0.7 +±4.6 +0 +200 +–47.6 +±19.0 +0 +25 +–0.2 +±1.1 +−25 +0 +25 +–15.6 +±5.7 +−1000 +0 +1000 +101 +103 +105 +GFN2 +–32.7 +±96.5 +−100 +0 +6.8 +±9.3 +−200 +0 +21.0 +±54.0 +0 +100 +21.4 +±17.8 +−100 +0 +–19.1 +±33.2 +−200 +0 +–157.1 +±19.7 +−250 +0 +101 +103 +105 +GFN1 +–7.7 +±54.3 +−200 +0 +9.4 +±13.2 +0 +100 +68.5 +±26.5 +0 +100 +36.1 +±21.5 +−100 +0 +–10.7 +±21.6 +−200 +0 +–153.3 +±23.4 +−2000 +0 +101 +103 +105 +PM7 +–23.5 +±150.1 +−1000 +0 +–1.3 +±20.6 +−100 +0 +–42.1 +±15.9 +−100 +0 +–50.4 +±7.7 +0 +50 +1.3 +±4.6 +0 +50 +17.9 +±6.4 +0 +250 +101 +103 +105 +AM1 +15.6 +±42.9 +0 +200 +–2.0 +±7.5 +−100 +0 +–21.8 +±22.2 +−100 +0 +–41.6 +±8.5 +0 +50 +1.3 +±4.6 +0 +50 +17.9 +±6.4 +FIG. 3. Error histograms in kcal/mol for all models and tasks along with their means, standard deviations, and moment-matching Gaussian model fits. + +9 +−25 +0 +25 +101 +103 +105 +ωB97M-V +0.7 +±1.6 +remove atom +dense +−10 +0 +10 +–1.3 +±2.4 +sparse +−25 +0 +25 +0.5 +±2.0 +remove electron +dense +−50 +0 +50 +–27.7 +±8.9 +sparse +0 +25 +–0.2 +±0.7 +add electron +dense +−20 +0 +–6.9 +±2.9 +sparse +−25 +0 +25 +101 +103 +105 +B3LYP +2.2 +±2.3 +0 +10 +–0.1 +±1.6 +0 +25 +4.1 +±3.0 +−50 +0 +–33.3 +±14.0 +0 +20 +–0.5 +±1.2 +−50 +0 +–19.5 +±6.2 +−25 +0 +25 +101 +103 +105 +PBE +0.8 +±4.2 +−10 +0 +10 +–0.8 +±2.1 +−25 +0 +25 +0.6 +±4.4 +0 +200 +–42.8 +±14.9 +0 +25 +–0.2 +±0.9 +−25 +0 +25 +–14.0 +±5.8 +FIG. 4. Error histograms in kcal/mol for DFT models and all tasks along with the means, standard deviations, and moment-matching Gaussian model +fits of the marked data with consistent total spin values between HF and DFT. +tures that differ by at most one atom, which should be less sensitive +to finite-basis errors. The most basis-set sensitive structures are +correlation-bound anions, which account for 5.3% of the anions +in the dense distribution and 45.2% in the sparse distribution. +Correlation-bound anions do not have a proper complete basis set +(CBS) limit with HF orbitals because the overlap between the HF +and CCSD wave-functions tends to zero as the unbound HF orbital +delocalizes. A formally correct treatment of correlation-bound an- +ions in the CBS limit requires Brueckner orbitals46. However, I +do not expect the def2-QZVPP basis set to be large enough for the +pathological CBS limit to have a substantial effect on this data set. +B. +Anomaly detection +Anomaly detection is a natural part of error analysis when +gathering large amounts of data within a statistical framework. +The basic expectation of a good model is that its errors are an +accumulation of a large number of small, independent errors, which +tend to induce Gaussian distributions of model errors. Errors in the +hydrogen cluster data organized by model and task are shown in +Fig. 3 with moment-matching Gaussian fits. While many errors are +effectively described by the Gaussian model, there are also several +fat error tails, many of which are rare enough to be unlikely to +appear in data generation at smaller scales. What is not shown are +some even larger error tails that were present in earlier versions of +the data set as the workflow was being refined to detect and avoid +more failure events and silent errors. +This statistical overview +of error distributions along with metadata collected during the +primary data generation are essential for detecting and correcting +rare failures. Unfortunately, sufficiently rare failures are unlikely +to occur in small-scale preliminary testing of a workflow precisely +because of how rare they are. +There is not necessarily a clean partition between model, +algorithm, and software errors in large-scale data generation. For +example, the lack of reliability in DIIS-based SCF solvers causes +enough gaps in the ground-state searches that the wrong total +spin is assigned in some DFT calculations. +As a result, some +DFT calculations produce total energies that are too high, which +are likely a source of some rare error outliers. +However, there +is no guarantee that the DFT and HF ground states for a given +structure and charge state will have the same total spin. There is +not enough information to distinguish model from algorithm errors +here without more reliable SCF solver algorithms to fill gaps in +data. Similarly, software bugs may cause failures in one algorithm +implementation that are not reproduced by other implementations, +and custom improvements to algorithms may cause successes that +are also not reproducible in other software. To see the impact of +spin inconsistency, the DFT data is shown in Fig. 4 with spin- +consistent calculations marked and fit to Gaussian error models. +The spin-inconsistent data contains most of the error outliers but +does not substantially change the overall error statistics since the +spin-consistent data has similar means and standard deviations. +The broadest error distributions in Fig. 3 are in the SQM atom +removal data from the dense distribution. It is likely that errors in +short-range pair potentials and matrix elements account for much +of this error since these SQM models are mostly fit to data from +near-equilibrium structures. I test this hypothesis by separating data +in Fig. 5 based on the minimum interatomic distance in a structure +being greater than or less than 0.74 Å, the equilibrium bond length +of H2. There is a clear narrowing of the error distributions for the +structures without short interatomic distances, which supports the +error hypothesis. +It may not be possible to detect or explain all error outliers. +The CCSD error tails from the sparse distribution in Fig. 3 imply +rare instances of large perturbative triples corrections to the total +energy. In these cases, the exact ground-state wave-function may +have strong multi-reference character. However, the multi-reference +test in Fig. 2 has no corresponding outliers, and a variety of multi- +reference tests may be needed to increase detection reliability47. + +10 +−1000 +0 +1000 +101 +103 +105 +GFN2 +–1.6 +±37.2 +remove atom +dense +−100 +0 +6.9 +±9.3 +sparse +−200 +0 +51.3 +±21.6 +remove electron +dense +0 +100 +21.3 +±17.7 +sparse +−100 +0 +–25.4 +±38.8 +add electron +dense +−200 +0 +–157.3 +±19.0 +sparse +−250 +0 +101 +103 +105 +GFN1 +15.1 +±35.6 +−200 +0 +9.5 +±12.6 +0 +100 +88.0 +±19.0 +0 +100 +35.9 +±21.4 +−100 +0 +–16.7 +±29.4 +−200 +0 +–153.6 +±22.8 +−2000 +0 +101 +103 +105 +PM7 +26.5 +±14.5 +−1000 +0 +–1.0 +±6.2 +−100 +0 +–37.5 +±9.7 +−100 +0 +–50.4 +±7.6 +0 +50 +0.7 +±3.4 +0 +50 +17.9 +±6.4 +0 +250 +101 +103 +105 +AM1 +5.3 +±20.3 +0 +200 +–2.4 +±5.7 +−100 +0 +–28.8 +±17.3 +−100 +0 +–41.6 +±8.4 +0 +50 +0.7 +±3.4 +0 +50 +17.9 +±6.4 +FIG. 5. Error histograms in kcal/mol for SQM models and all tasks along with the means, standard deviations, and moment-matching Gaussian model +fits of the marked data from structures with minimum interatomic distances greater than 0.74 Å. +The failures that statistical model selection in Sec. II seeks to +avoid are silent failures. Anomaly detection implies an ability to +detect and herald some types of failures. For the example data set in +this paper, I havechosen to removesome heralded failuresassociated +with algorithm-specific SCF convergence problems to increase the +emphasis on errors in the physical models. This formally changes +the underlying task distributions by a small amount. To be faithful to +the original task distributions, a more complete model would always +produce a viable output by branching to less accurate but more +reliable calculations and eventually resorting to a random guess. +When trying to increase a model’s overall success probability, +improving the ability to detect and respond to rare failures and +error outliers can be just as important as improving the average +model accuracy for typical inputs. +C. +Model fitting +I now consider a minimal representative example of using +model selection to fit SQM models. First, I highlight the benefits of +using more complicated error models to improve success measures. +Second, I fit an atomic pair potential to all QM and SQM data, +primarily to correct the large error outliers in the SQM data. Pair +potentials are one of the most common and basic elements of both +interatomic potentials and SQM models. While pair potentials are +often restricted to a simple form before fitting them, I consider a +general form and rely on model selection to limit the number of +parameters that define the pair potential. +Because some models being considered are near chemical +accuracy, the small-𝜖 approximation used in Eq. (7) is not always +accurate. Instead, I use the exact success probability, +𝑝(λ|𝑋𝑖) = +∫ +𝑥𝑖+𝜖 +𝑥𝑖−𝜖 +𝑒−0.5[𝑧−𝜇−𝑦𝑖 (λ)]2/𝜎2 +𝜎 +√ +2𝜋 +𝑑𝑧 += erf +� +𝑥𝑖−𝑦𝑖 (λ)−𝜇+𝜖 +√ +2𝜋𝜎 +� +− erf +� +𝑥𝑖−𝑦𝑖 (λ)−𝜇−𝜖 +√ +2𝜋𝜎 +� +, +(38) +for a success interval [𝑥𝑖 − 𝜖, 𝑥𝑖 + 𝜖] around a reference data value +𝑥𝑖. For chemical accuracy, 𝜖 = 1 kcal/mol. This interval needs +to be adjusted for electron addition and removal energies that are +near their vacuum-limited values. The energy to add an electron +cannot be greater than zero, and the energy to remove an electron +cannot be less than zero. If the success interval crosses into this +physically forbidden region, then I ignore the unphysical end point +and consider a semi-infinite success interval in Eq. (38). The form +of the pair potential is a polynomial at short range that goes to +zero at an adjustable cutoff 𝑅 and strictly zero beyond that. The +success measure in Eq. (36) and its analytical first and second +derivatives with respect to λ are tedious but straightforward to +evaluate. I minimize the success measure with a sequence of line +searches that use this derivative information to achieve asymptotic +quadratic convergence. +As I increase the polynomial degree, I +use the minimizing model with one fewer degree as the initial +guess for minimization. For degree one, I use the moment-based +approximations of the error model in Eq. (13) and a zero pair +potential with 𝑅 = 4 Å as the initial guess. The TIC bias correction +in Eq. (27) is calculated at the penalty-free minimum of the success +measure instead of being included in the minimization process. +The models that minimize the success measure are summarized +in Table I. There is a clear benefit to using a richer error model +with a separate Gaussian error model for each type of simulation + +11 +model +˜𝐷1g +˜𝐷6g +𝜇 +𝜎 +𝜇rad +𝜎rad +𝜇ras +𝜎ras +𝜇red 𝜎red +𝜇res +𝜎res +𝜇aed 𝜎aed +𝜇aes 𝜎aes 𝑡 +CCSD+PP 1.57 × 105 9.20 × 104 +0.2 +0.7 +0.1 +0.4 -0.1 +0.3 +-0.6 +0.3 +0.1 +0.8 +1.3 +0.4 +0.8 +0.9 3.64 × 108 +CCSD +1.72 × 105 9.57 × 104 +0.0 +0.7 +-0.6 +0.4 -0.2 +0.3 +-0.6 +0.3 +0.1 +0.8 +1.3 +0.4 +0.8 +0.9 3.64 × 108 +MP2 +7.41 × 105 6.17 × 105 +0.0 +5.5 +-2.7 +3.0 -2.3 +4.7 +-2.0 +3.4 +-3.0 +5.6 +2.0 +1.8 +6.9 +3.8 2.10 × 108 +HF +1.06 × 106 8.17 × 105 +-2.9 +16.0 +-18.9 +8.8 -4.1 +8.1 +-1.7 +9.7 +-5.0 +8.3 +12.5 +6.0 +18.3 +5.5 5.42 × 107 +𝜔B97M-V 9.82 × 105 5.63 × 105 +-4.5 +12.3 +0.7 +2.2 -1.3 +2.3 +0.5 +2.5 +-29.1 +8.8 +2.8 +2.8 +-7.5 +3.2 2.57 × 108 +B3LYP +1.09 × 106 6.28 × 105 +-6.2 +17.8 +2.2 +2.7 -0.1 +1.5 +4.2 +3.3 +-36.9 13.4 +4.3 +4.3 +-21.4 +6.0 8.51 × 107 +PBE +1.14 × 106 7.00 × 105 +-7.5 +21.1 +0.9 +4.5 -0.8 +2.0 +0.7 +4.6 +-47.6 19.0 +2.9 +2.8 +-15.4 +5.6 1.14 × 108 +GFN2+PP +1.53 × 106 1.17 × 106 +-9.0 +82.3 +-52.5 +92.0 +2.2 +5.2 +21.0 54.0 +21.4 17.8 111.2 98.9 -156.8 20.0 9.31 × 104 +GFN2 +1.54 × 106 1.21 × 106 +-15.1 +85.1 +-32.7 +96.5 +6.8 +9.3 +21.0 54.0 +21.4 17.8 111.2 98.9 -156.8 20.0 9.31 × 104 +GFN1+PP +1.52 × 106 1.13 × 106 +5.2 +79.4 +-23.2 +42.7 +4.7 +8.4 +68.5 26.5 +36.1 21.5 +69.4 60.7 -153.0 23.9 9.08 × 104 +GFN1 +1.52 × 106 1.17 × 106 +2.2 +81.2 +-7.7 +54.3 +9.4 13.2 +68.5 26.5 +36.1 21.5 +69.4 60.7 -153.0 23.9 9.08 × 104 +PM7+PP +1.27 × 106 9.11 × 105 +-6.5 +33.2 +13.8 +29.7 -0.7 +7.3 +-42.1 15.9 +-50.4 +7.7 +13.7 +6.8 +18.4 +5.7 1.22 × 106 +PM7 +1.50 × 106 1.07 × 106 +-7.3 +75.0 +-23.5 150.1 -1.3 20.6 +-42.1 15.9 +-50.4 +7.7 +13.7 +6.8 +18.4 +5.7 1.22 × 106 +AM1+PP +1.21 × 106 9.00 × 105 +-5.0 +26.8 +15.1 +25.1 -0.9 +4.6 +-21.8 22.2 +-41.6 +8.5 +13.7 +6.8 +18.4 +5.7 1.18 × 106 +AM1 +1.26 × 106 9.60 × 105 +-0.8 +32.1 +15.6 +42.9 -2.0 +7.5 +-21.8 22.2 +-41.6 +8.5 +13.7 +6.8 +18.4 +5.7 1.18 × 106 +PP +1.69 × 106 1.15 × 106 -100.8 135.5 143.8 121.8 -2.4 +9.7 -228.1 76.2 -293.1 17.7 +13.7 +6.8 +18.4 +5.7 5.25 × 10−2 +none +1.72 × 106 1.18 × 106 +-62.7 154.5 +20.8 100.3 -7.5 19.4 -228.1 76.2 -293.1 17.7 +13.7 +6.8 +18.4 +5.7 3.38 × 10−2 +TABLE I. Comparison of minimized success measures over 𝑚 = 344, 513 simulation tasks for various models, including a pair potential (PP) correction +when the improvement is greater than one percent. This comparison includes one-Gaussian (1g) error models (𝜇, 𝜎) and six-Gaussian (6g) error models +fit to atom removal (ra), electron removal (re), and electron addition (ae) on both dense (d) and sparse (s) distributions. The success measures do not +include parameter or cost penalties. The error model parameters are in kcal/mol and the total model evaluation times 𝑡 are in CPU-seconds. The cost of +generating the reference data is 𝑡 = 4.13 × 108. +task. Many of the large standard deviations in the overall error +are better explained as biases in a specific task type with a smaller +standard deviation per type. Some of these biases are obvious and +expected, but it is still useful to quantify them. The GFN1 and +GFN2 models predict a very large electron binding energy for most +hydrogen clusters, while AM1 and PM7 do not predict any binding +of excess electrons to any hydrogen cluster. The HF model has +biases associated with the absence of electron correlation energy, +which is always negative and usually proportional to the number of +electrons. The DFT models are known to have large delocalization +errors48 that are likely to be biasing the electron removal energies of +the sparse distribution. If an error model is used to improve success +probabilities by adding random numbers to a model’s outputs, then +an improvement to the error model is an improvement to the model +as a whole. +The effects of the IC penalties on the selection of the pair +potential are shown for two representative model families in Fig. 6. +CCSD is a more accurate model than PM7, and the AIC is likewise +a better approximation of the TIC for CCSD. For PM7, the AIC +is unable to compensate for the parameter bias enough to create +a local minimum in the success measure. For CCSD, the AIC is +able to create a local minimum, but its location is different than for +the TIC. In this example, the TIC correction introduces significant +numerical noise, which appear as values above the smoother trend +line. +The TIC is a response property that depends sensitively +on the numerical quality of the success measure minimum. The +derivative discontinuity that I allow at the large-distance cutoff +point 𝑅 of the pair potential introduces derivative discontinuities +in the 𝑅 dependence of the success measure that complicates the +minimization. Even under such non-ideal conditions, the TIC is +0 +20 +40 +60 +80 +100 +−3700 +−3690 +−3680 +−3670 +−3660 +−3650 +−3640 +˜D6g change for CCSD +AIC +TIC +0 +20 +40 +60 +80 +100 +polynomial degree +−159500 +−159000 +−158500 +−158000 +−157500 +−157000 +−156500 +˜D6g change for PM7 +AIC +TIC +FIG. 6. Reduction of the success measure ˜𝐷6g with a six-Gaussian error +model as the polynomial degree of the pair potential is increased. The +TIC is regularized by replacing small and negative eigenvalues of the ˜𝐷6g +Hessian with 10−9 times the largest eigenvalue when that value is greater. + +12 +0.5 +1.0 +1.5 +2.0 +2.5 +−0.4 +−0.2 +0.0 +0.2 +PP for CCSD (kcal/mol) +no TIC +TIC +0.3 +0.4 +0.5 +0.6 +0.7 +distance (Å) +−200 +−150 +−100 +−50 +0 +PP for PM7 (kcal/mol) +no TIC +TIC +FIG. 7. Short-range polynomial pair potential corrections for the CCSD +and PM7 models. With the TIC penalty, the minimizing polynomial has +degree 22 for CCSD and degree 14 for PM7. Without an IC penalty, there +is no local minimum in polynomial degree and best degree 100 polynomial +is shown as an example of overfitting. Outside of the plotted range, the +PM7 pair potential decreases to -1790 kcal/mol at 0.3 Å. +still functional for model selection with appropriate regularization +of the success measure Hessian. +The TIC is more challenging +to calculate for parameterized QM calculations that involve QM +response properties in the parameter derivatives of the success +measure. +The benefits of a pair potential correction are not uniform +over models or tasks. Since the pair potential only depends on +the atomic structure and not electronic structure, it cannot correct +the electron addition and removal tasks. For many models, the +overall reduction of the success measure is one percent or less, and +these minor improvements are omitted from Table I. The largest +improvement comes from the PM7 pair potential, shown in Fig. +7. Apparently, the short-range hydrogen-hydrogen pair potential +in PM7 is much too repulsive at distances just below the bond +length of H2. In contrast, the CCSD pair potential is much longer +in range and much smaller in magnitude. It is not surprising that +the largest correction occurs near the Coulson-Fischer point around +1 Å. However, it is surprising that something as complicated as +the CCSD(T) triples correction can be partially approximated by a +pair potential. The IC penalties succeed in suppressing the high- +frequency oscillations typically attributed to overfitting noise, but +there are still some artifacts near the edges of the polynomial’s +domain. There are other ways to reduce unphysical oscillations +in pair potentials, such as considering reference tasks that depend +directly on derivatives of a pair potential or explicit functional +regularization49. As shown in Fig. 8, the pair potential corrections +eliminate most of the large error outliers in SQM models except for +GFN2 on the dense distribution. I expect that the persistent error +in GFN2 is from the short-range part of either a 3-body potential +term or a Hamiltonian matrix element, neither of which can be +repaired by a pair potential. +This example demonstrates the benefits of having an excessive +amount of data available when fitting models. As the amount of +data increases, the utility and reliability of statistical tools and con- +cepts increases. The abundance of data creates a comfortable safety +buffer between the number of parameters needed to fit a model ac- +curately and the maximum number of parameters that can be fit with +statistical significance. The model selection process then enables +an accurate model to be carved from an accessible set of redundant, +overfit models. Such large amounts of data are accessible because +of the massive scale of modern high-performance computing, an +ability to generate data sets procedurally, and careful use of phys- +ical transferability assumptions. This strongly contrasts with how +SQM models such as PM730 and GFN233 have been developed. +They prescribe simple model forms with a few tens of parameters +per element and collect enough reference data to fit those forms +specifically. They do not gather enough data to consider or rule out +more complicated models with more parameters, and many SQM +model design choices have remained frozen for decades. +PM7 +still uses the MNDO model form31 proposed in 1977, just with +the addition of more complicated classical correction terms. De- +spite being from a much newer model family, GFN2 also contains +old model forms such as the Wolfsberg–Helmholz approximation50 +from 1952 relating Hamiltonian and overlap off-diagonal matrix +elements. With an increasing amount of data, model forms can +−2.5 +0.0 +2.5 +101 +103 +105 +CCSD+PP +0.1 +±0.4 +remove atom +dense +−5 +0 +5 +0.0 +±0.1 +sparse +−1000 +0 +1000 +101 +103 +105 +GFN2+PP +–52.3 +±92.0 +−50 +0 +2.2 +±5.2 +−100 +0 +100 +101 +103 +105 +GFN1+PP +–22.9 +±42.7 +−50 +0 +4.7 +±8.5 +−200 +−100 +0 +100 +101 +103 +105 +PM7+PP +13.7 +±29.7 +−50 +0 +50 +–0.7 +±7.4 +−100 +0 +100 +101 +103 +105 +AM1+PP +15.0 +±25.2 +−50 +0 +50 +–0.9 +±4.7 +FIG. 8. Revisions of error histograms from Fig. 3 in kcal/mol for the +models and tasks that benefit from a pair potential correction. + +13 +shift more towards what is objectively supported by the data and +farther from the subjective technical opinions of specific model +builders. +D. +Cost budgeting +Considerations of model cost are always more complicated +than model accuracy because they are much more sensitive to +software, hardware, and fine details of a workflow. All calculations +reported in Table I are performed on the same computing cluster, +with AMD EPYC 7702 CPU cores and two gigabytes of memory +per core. Except for MP2, CCSD, and CCSD(T) calculations, all +calculations are performed on a single CPU core for maximum +throughput. Some of the MP2, CCSD, and CCSD(T) calculations +exceed the memory budget of a single CPU core, and they are run +with four cores per calculation for a safety buffer of memory usage. +Parts of the calculation are threaded and make use of multiple +cores, but the thread scaling is limited. This complicates some +cost comparisons. For example, the HF and MP2 calculations have +very similar run times under similar conditions, and the large cost +difference reported in Table I is caused by the different number +of cores required. +Also, the AM1 and PM7 calculations have +similar run times as GFN1 and GFN2 calculations for an individual +calculation, but their workflow requires a combinatorial search over +atomic spin configurations. The limited sensitivity of GFN1 and +GFN2 calculations to spin order makes this search unnecessary and +reduces their overall run time per simulation task, but may also be +related to their relatively poor accuracy here. +A visual way to compare success measures with varying cost +penalties is to plot them versus cost as in Fig. 9 and draw the +convex hull connecting minimum-cost models with various rates +of success. Models on the convex hull are optimal for a range +of computational budgets, and models above the convex hull are +not worth using for these simulation tasks according to this cost +analysis. In this example, the convex hull connects PP, AM1+PP, +B3LYP, and CCSD+PP, with GFN1+PP also just on the convex +hull. As noted in Sec. II D, any model cost versus accuracy along +the convex hull can be achieved by randomly switching between the +models on the end points with a probability varying linearly between +zero and one along the facet. Thus, there is a natural continuum of +hybrid models between the cheapest and most expensive models. +The practice of randomly mixing models with different accu- +racies as suggested here is usually avoided in atomistic simulation. +Models often rely on some form of error cancellation or a study +of qualitative trends rather than precise quantitative predictions +that can be disrupted by comparing data between different models. +These behaviors can be described within a framework of statisti- +cally independent simulation tasks by carefully defining tasks as +groups of simulations with success based on comparisons rather +than the absolute value of outputs. In the simple example of a +ranking, random pairs of simulations might be performed with the +output being the decision of which system had the larger value for +a specific output. Such a grouping forces comparisons to remain +within a specific model while still allowing for the use of a different +model for each independent ranking decision. A more common +practice of mixing models is to filter a larger number of systems +with a cheap, inaccurate model and then filter the remains systems +10−2 +100 +102 +104 +106 +108 +1010 +1012 +t (CPU-seconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +˜D6g +×106 +CCSD+PP +MP2 +HF +ωB97M-V +B3LYP +PBE +GFN2+PP +GFN1+PP +PM7+PP +AM1+PP +PP +FIG. 9. Success measure versus total cost of models from Table I with the +convex hull denoting the most cost-effective models in the example. +that pass the first filter with a more expensive and accurate model. +The intention of this practice is to approximate the effect of ap- +plying the more expensive filter to all of the systems with a lower +overall cost. However, this requires the cheaper model to have a +sufficiently low false positive rate that the overall cost is actually +reduced while maintaining a very low false negative rate to avoid +distorting the outcome. +Simultaneous considerations of model cost and accuracy at +a large enough scale that reliability also matters as in Fig. 9 is a +very challenging test for models. It is much easier to show cost +benchmarks of a model or software under ideal conditions, accuracy +benchmarks under a different set of ideal conditions, and ignore +problematic cases altogether. Even the hydrogen cluster example +considered here is artificially generous because a small fraction +of structures that caused SCF convergence failures were omitted +from the set of simulation tasks. While the models are depicted +as points on the plot, they are more generally going to be regions +corresponding to the set of possible changes in a workflow that alter +both cost and accuracy. For example, the combinatorial search over +atomic spin configurations for the hydrogen cluster example could +have been avoided, which would have substantially reduced the +cost of many models. However, many of the calculations would +have failed to find the lowest energy ground state, and the overall +accuracy would have been reduced as a result. Cost and accuracy +could have been balanced more carefully by randomly sampling a +limited set of spin configurations rather than using an exhaustive +combinatorial search. Adjusting details of a model workflow to +improve the convex hull of optimal models requires a careful balance +of these cost, accuracy, and reliability considerations. + +14 +IV. +CONCLUSION +As scientists continue to develop more diverse and sophisti- +cated models for atomistic simulation, how models are compared +and how their successes are judged become increasingly important. +Progress in method development can slow down or stop if scientists +have different, incompatible definitions for what success is51. This +paper has presented an operational success measure for judging +atomistic models that is based on statistical model selection. Using +simple simulation tasks on hydrogen clusters as an example, I have +shown how this measure can be used to compare the cost and accu- +racy of a diverse set of QM and SQM models. I have also used it to +fit a minimal SQM model that applies a pair potential correction to +this QM and SQM data and select the potential form that best fits +the data. The TIC provides a reliable parameter penalty to avoid +selecting over-complicated models, while the AIC is not a reliable +penalty because some atomistic models are too inaccurate for its +assumptions to hold. For a computational budget that is too small +for a high-accuracy model but excessive for a low-accuracy model, +the success measure predicts the efficacy of splitting a workload +between models to match the budget. By adjusting the operational +definition of success for simulation tasks, this success measure +can be equally good for designing expensive models to succeed at +difficult tasks and cheap models to succeed at easy tasks. +An essential aspect of model building in atomistic simulation +is the availability of high-quality reference data for fitting and test- +ing. While models have historically relied on reference data from +experiments, it is now possible to generate accurate data using ex- +pensive QM models. As shown in the hydrogen cluster example, +CCSD(T) data is affordable for small molecular fragments, and +less accurate DFT data remains affordable for larger molecules and +periodic systems. For data generation at scales larger than what +has been presented in this paper, reliability issues will become +increasingly important alongside cost and accuracy considerations. +SCF convergence problems can cause heralded failures, while SCF +convergence to excited states can cause silent failures. Without +more fundamentally reliable algorithms to reduce failure rates, a +fixed rate of failure means an increasing number of failure events +as data sets grow larger in size. There are increasingly sophisti- +cated tools52 for remote, automated computing of large workloads +and organizing large data sets with modern database standards. +However, limitations in the reliability of the underlying tasks being +automated may have a strongly negative influence on the cost and +accuracy of generating large data sets as failures persist against +increasing computational redundancy. +The hydrogen cluster example considered here is sufficiently +different from typical reference data sets that it serves as a challeng- +ing test of physical transferability. There is a significant difference +in the apparent progress that DFT and SQM models have made in +developing transferable models. The improvement in transferability +from PBE to B3LYP to 𝜔B97M-V is consistent with the develop- +ment roadmap of DFT functionals with increasing complexity53. +While likely a coincidence, the SQM models considered here have +systematically degrading performance in chronological order of +their development. A simple explanation of this difference might +be that DFT functionals are fundamentally more transferable than +minimal-basis SQM models. However, it is also important to con- +sider the vastly differing amounts of technical effort that have been +invested in these two approaches. +The development path from +B3LYP to 𝜔B97M-V includes the development of hundreds of +DFT functionals from numerous research groups over more than +three decades26. In contrast, the development path from AM1 to +PM7 consists of only a few other models developed by a single +scientist – Dr. James J. P. Stewart – working mostly in isolation +outside of academia for more than three decades. GFN1 and GFN2 +were developed much more recently by a single academic group +– the research group of Prof. Stefan Grimme at the University of +Bonn. While there are other SQM models outside of the GFN and +MNDO-like model families, these are the two most widely used +families and the only non-commerical models54 to be fit for com- +binations of elements over most of the periodic table. The GFN +models incorporate ideas from both MNDO-like models (multipole +expansions of electrostatics, avoidance of diatomic parameters) and +DFTB models (expansion around an atomic limit, DFT-like cor- +relation models). All of the SQM models considered here have +similar superficial complexity, similar numbers of parameters per +element, and are fit to similar amounts of reference data. Except +for a belief in the superiority of DFT-like models, there is no com- +pelling theoretical reason why any SQM model from this set should +perform any better than any other on systems that are very different +from their training data. +The concepts presented in this paper are meant to inform the +process of designing, fitting, and selecting models for atomistic +simulation tasks. If a simulation task is not going to be repeated a +very large number of times, then the formal process of gathering +reference data and calculating a success measure might not be +worth the amount of human effort required. However, the statistical +model selection process can still be useful as a conceptual guide +even when it is not worthwhile to perform it carefully or explicitly. +For tasks that are performed frequently by many scientists, it may +be worthwhile to capture that activity as a distribution of tasks +and a representative sampling from that distribution. +Quantum +chemistry has a tradition of curating reference data sets to guide +method development26. Expanding that tradition to accommodate +larger data sets, statistical interpretations, and success measures that +capture the real needs of applied scientists could create an even +better guide for method development. It is difficult for a scientist +to characterize real application needs while also developing novel +simulation methods to satisfy those needs, and it would be helpful +to decouple those important research activities from each other. +ACKNOWLEDGMENTS +J. E. M. thanks Jimmy Stewart for helpful discussions. The +Molecular Sciences Software Institute is supported by NSF Grant +No. ACI-1547580. The computational resources used in this work +were provided by Advanced Research Computing at Virginia Tech. +AUTHOR DECLARATIONS +Conflict of Interest +The author has no conflicts to disclose. + +15 +Author Contributions +Jonathan E. Moussa: Conceptualization (equal); Data curation +(equal); Formal analysis (equal); Investigation (equal); Method- +ology (equal); Resources (equal); Software (equal); Validation +(equal); Visualization (equal); Writing – original draft (equal); +Writing – review & editing (equal). +DATA AVAILABILITY +The data and software that support the findings of this study +are available on Zenodo at the DOI 10.5281/zenodo.7530231. +REFERENCES +1M. Born and A. Landé, “Die Abstände der Atome im Molekül und im Kristalle,” +Die Naturwissenschaften 6, 496 (1918). +2L. Talirz, L. M. Ghiringhelli, and B. Smit, “Trends in Atomistic Simulation +Software Usage [Article v1.0],” Living J. 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Theory Comput. 9, 4006–4017 (2013). + diff --git a/49E4T4oBgHgl3EQf1A2T/content/tmp_files/load_file.txt b/49E4T4oBgHgl3EQf1A2T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1bd4caa00ca2c5bf669384d3303b8cb6f4b9d05 --- /dev/null +++ b/49E4T4oBgHgl3EQf1A2T/content/tmp_files/load_file.txt @@ -0,0 +1,1623 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf,len=1622 +page_content='Model selection in atomistic simulation Jonathan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Moussa Molecular Sciences Software Institute, Virginia Tech, Blacksburg, Virginia 24060, USA (*Electronic mail: godotalgorithm@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='com) There are many atomistic simulation methods with very different costs, accuracies, transferabilities, and numbers of empirical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I show how statistical model selection can compare these methods fairly, even when they are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' These comparisons are also useful for developing new methods that balance cost and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As an example, I build a semiempirical model for hydrogen clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' INTRODUCTION Scientists have been building quantitative atomistic models for over a century1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In that time, many atomistic models have evolved into sophisticated computer simulations2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While there are now models based on a wide variety of atomistic simulation methods, most development has focused on two contradictory goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Classi- cal molecular mechanics (MM) methods focus on minimizing cost to access phenomena at large length scales and long time scales3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the use of MM methods is limited by the availability and accuracy of system-specific interatomic potentials4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In contrast, first-principles quantum mechanics (QM) methods focus on mini- mizing error for general-purpose simulations5, which can get very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MM methods can achieve simulation costs of less than 10−5 CPU-seconds per atom6, while high-accuracy QM methods have asymptotic costs greater than 104 CPU-seconds per atom7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Because of the large gaps in cost and utility, there are many atomistic simulation tasks for which QM methods are too expensive and MM methods have no suitable interatomic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this situation, a scientist needs an affordable model and must either develop their own or use an existing one such as a semiempirical QM (SQM) model8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In either case, they need to collect evidence to support their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' They must either gather enough reference data to fit a new model, or find enough examples of scientists using an existing model for similar tasks to be confident that it will work for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This type of model selection process is a common occurrence in atomistic science, and yet it remains rather informal and subjective much of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this paper, I advocate for using statistical model selection10 to develop and compare models for atomistic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' All else being equal, a scientist should fit or choose a model to maximize the probability that they will succeed at their simulation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Since the exact probability will be more expensive to compute than the simulation task itself, they must rely on a proxy probability based on related but simpler simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Assumptions about the transferability of a method’s accuracy between related simulation tasks are unavoidable in atomistic science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Also, when considering methods with different numbers of fitting parameters or costs, extra penalties are needed to avoid overfitting or exceeding computa- tional budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' These same principles apply to the development of general-purpose models that are intended to be used by many scientists over a broad distribution of simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As an example, I apply statistical model selection to the task of simulating random hydrogen clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' First, I generate high- accuracy QM reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Second, I compare the accuracy of some popular SQM models and density functionals from density functional theory (DFT)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Third, I build new SQM models by correcting this SQM and QM data with atomic pair potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Here, model selection determines the optimal number of parameters in the pair potentials and the computational budget thresholds for switching between models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' STATISTICAL MODEL SELECTION The standard practice in fitting atomistic models with param- eters is to minimize a distance between model predictions and reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I consider vectors of 𝑚 reference data points x and model predictions y(λ), which are determined by 𝑛 real parame- ters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The value of λ is usually chosen by minimizing the mean absolute error (MAE), ∥x − y(λ)∥1 = 𝑚 ∑︁ 𝑖=1 |𝑥𝑖 − 𝑦𝑖(λ)|, (1) or the root-mean-square deviation (RMSD), ∥x − y(λ)∥2 = � � 𝑚 ∑︁ 𝑖=1 [𝑥𝑖 − 𝑦𝑖(λ)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (2) The general expectation is that smaller distances correspond to bet- ter accuracy and thus a higher chance of success when these models are used for other simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, this relationship is indirect because these distances are not operational measures of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' An operational measure would describe the application of a model by scientists in a more explicit and direct way, including how successful they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Directly optimizing an operational mea- sure should produce a more successful model if the operational measure itself is sufficiently accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' To use statistical model selection as an operational measure in this context, I must first introduce two distinct sources of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The first source of randomness is in the model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I consider a generalization of the reference information from data points x to simulation tasks X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Each reference simulation task 𝑋𝑖 defines one or more physical systems and calculations to perform, together with reference output data and success criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The con- ditional probability of success, 𝑝(λ|𝑋𝑖), after choosing a task 𝑋𝑖 and using a model with parameters λ replaces a distance between 𝑥𝑖 and 𝑦𝑖(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The only constraint on the success criteria is that the success probability for the method used to generate the reference data must be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Viable models must always have a nonzero success probability, which requires the model output or success criteria to have a random component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The second source of randomness is in the choice of reference simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I relate a set of reference simulation tasks to the actual simulation task that a scientist wants to succeed at by arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='05287v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='chem-ph] 12 Jan 2023 2 considering them to be randomly drawn from a common distribution of simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The probability of choosing a simulation task 𝑋 is 𝑝(𝑋), and the probability of choosing this task and then succeeding with the model is 𝑝(λ, 𝑋) = 𝑝(λ|𝑋)𝑝(𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (3) It is not strictly necessary for the simulation tasks to have been randomly drawn from this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such a distribution is still formally useful even when it is an artificial context and not even precisely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is simply the mathematical representation of a computational scientist as a distribution over simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Maximum likelihood estimation I now apply the framework of maximum likelihood estimation (MLE)10 to determine the best model in this randomized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The operational measure of modeling success is the probability of succeeding at all 𝑚 reference simulation tasks, 𝑃(λ) = 𝑚 � 𝑖=1 𝑝(λ|𝑋𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (4) It is related to a statistical likelihood function, 𝐿(λ) = 𝑚 � 𝑖=1 𝑝(λ, 𝑋𝑖) = 𝑃(λ) 𝑚 � 𝑖=1 𝑝(𝑋𝑖), (5) over the joint distribution of simulation tasks and modeling success or failure events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I follow the common convention of considering the negative logarithm of the probability or likelihood, − log 𝑃(λ) = − 𝑚 ∑︁ 𝑖=1 log 𝑝(λ|𝑋𝑖) = − log 𝐿(λ) + 𝑚 ∑︁ 𝑖=1 log 𝑝(𝑋𝑖), (6) which replaces the product over reference simulation tasks with a more convenient sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The negative logarithm is a strictly mono- tonically decreasing function, and maximizing it corresponds to maximizing the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Since 𝑝(𝑋𝑖) has no dependence on λ, 𝑃(λ) and 𝐿(λ) are maximized by the same value of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The familiar case of minimizing RMSD follows from a simple success criterion and error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I assume that each simulation task 𝑋𝑖 produces a single model output 𝑦𝑖(λ) that must be within 𝜖 of a reference value 𝑥𝑖 for success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I further adjust each model output by a Gaussian error model with mean 𝜇 and standard deviation 𝜎 to guarantee a finite success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Each success probability reduces to a quadratic penalty for small 𝜖 values, − log 𝑝(λ|𝑋𝑖) = − log ∫ 𝑥𝑖+𝜖 𝑥𝑖−𝜖 𝑒−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5[𝑧−𝜇−𝑦𝑖 (λ)]2/𝜎2 𝜎 √ 2𝜋 𝑑𝑧 ≈ [𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2 2𝜎2 + 1 2 log 𝜋𝜎2 2𝜖2 + 𝑂(𝜖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (7) While not clear from this notation, error model parameters such as 𝜇 and 𝜎 are also considered to be part of the parameter vector λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In the small-𝜖 limit, the operational measure of success reduces to an RMSD-like formula, − log 𝑃(λ) ≈ 𝑚 2 log 𝜋𝜎2 2𝜖2 + 𝑚 ∑︁ 𝑖=1 [𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2 2𝜎2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (8) When minimizing this formula over 𝜇 and 𝜎, the minimizers are the mean and standard deviation of the model error distribution, 𝜇 = 𝑚 ∑︁ 𝑖=1 𝑥𝑖 − 𝑦𝑖(λ) 𝑚 , 𝜎 = � � 𝑚 ∑︁ 𝑖=1 [𝑥𝑖 − 𝑦𝑖(λ) − 𝜇]2 𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (9) The remaining minimization over λ is equivalent to minimizing the RMSD with a model bias correction of 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The minimum value of the small-𝜖 success measure is − log 𝑃(λ) ≈ 𝑚 2 + 𝑚 2 log 𝜋𝜎2 2𝜖2 (10) for 𝜎 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (9), which is a monotonically increasing function of the bias-corrected RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In the absence of model bias, the RMSD and success measure thus produce the same minimizing models and rank them in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Using a Gaussian distribution to approximate model errors is justified when they come from an accumulation of many small, independent errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A non-zero mean suggests that these small errors are biased on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The same small-𝜖 analysis can relate a similar success measure to the MAE if the underlying error model is a Laplace distribution, 𝜌(𝑥) = 𝑒− √ 2|𝑥−𝜇|/𝜎 𝜎 √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (11) However, non-Gaussian error distributions suggest a small number of dominant, independent error sources that avoid the inevitable consequences of the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Also, the Laplace distribution has a fatter tail than a Gaussian distribution, which implies an increased tolerance of large error outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Ultimately, the choice of distributions in an error model should be informed by the observed distribution of errors between model and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A more sophisticated MLE example is a multi-Gaussian error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Here, we partition the reference simulation tasks into 𝑟 groups of similar tasks, each with their model errors described by a different Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such grouping is appropriate when different groups of tasks are observed to have different error statistics for models under consideration12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The small-𝜖 limit of the success measure generalizes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (8) to − log 𝑃(λ) ≈ 𝑟∑︁ 𝑖=1 𝑚𝑖 2 log 𝜋𝜎2 𝑖 2𝜖2 + 𝑟∑︁ 𝑖=1 𝑚𝑖 ∑︁ 𝑗=1 [𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) − 𝜇𝑖]2 2𝜎2 𝑖 , (12) where the extra index is for the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The minimizing 𝜇𝑖 and 𝜎𝑖 values generalize from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (9) to 𝜇𝑖 = 𝑚𝑖 ∑︁ 𝑗=1 𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) 𝑚𝑖 , 𝜎𝑖 = � � 𝑚𝑖 ∑︁ 𝑖=1 [𝑥𝑖, 𝑗 − 𝑦𝑖, 𝑗 (λ) − 𝜇𝑖]2 𝑚𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (13) 3 The minimization over λ is now equivalent to a weighted, bias- corrected RMSD with weights proportional to the inverse error variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the minimum value of the success measure, − log 𝑃(λ) ≈ 𝑚 2 + 𝑟∑︁ 𝑖=1 𝑚𝑖 2 log 𝜋𝜎2 𝑖 2𝜖2 , (14) no longer ranks minimizing models in the same order as the cor- responding weighted RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Thus MLE rapidly deviates from minimizing simple distances between model and reference data as success criteria and error models get more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Beyond these simple examples, MLE can provide a lot of flexibility to the model-fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is possible to fit low- cost models that are designed to have only qualitative accuracy by choosing success criteria that tolerate large but well-shaped errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For example, conformer searches only need to preserve the order of conformer energies, which can be tested by the Spearman rank correlation coefficient13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' When fitting very accurate models, many reference simulation tasks may have success probabilities very close to one and effectively vanish from log 𝑃(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this highly successful regime, error outliers in a model will have a greatly enhanced influence on the success measure and MLE may become functionally equivalent to minimax optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Information criteria Simple MLE is capable of selecting the best model from one family of models parameterized by λ, but it cannot reliably compare models from different families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Adding more free parameters to an existing model and optimizing them can only improve the success measure, and nested models with more parameters will always be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This can eventually cause the modeling phenomenon of fitting noise rather than data, and there needs to be additional modeling criteria for eliminating parameters that are not useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The most common approach is to introduce a penalty for adding model parameters that is overcome by useful parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such measures of model accuracy with penalties for parameters are called information criteria (IC), the oldest and most famous of which is the Akaike information criterion (AIC)14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The Takeuchi information criterion (TIC)15 is a more complicated generalization of the AIC that does not assume model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Here, I provide a minimal motivation and derivation of the TIC and AIC to justify their use in fitting models for atomistic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' An implicit assumption about both the IC derivations and MLE itself is that 𝑃(λ) can be optimized over λ effectively in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The mathematical structure of 𝑃(λ) depends on both the model family and the success criteria of simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I specifically assume that 𝑃(λ) is twice differentiable with respect to λ and that derivative information is used to find local minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I also assume that it is possible to choose initial values for λ in the basin of convergence for the global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While there is not enough structure here to guarantee or verify global minima, there are often physical considerations to guide reasonable choices of initial λ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Both the AIC and TIC come from attempting to change the modeling success measure from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (6) to 𝐷(λ) = −𝑚 ∑︁ 𝑋 𝑝(𝑋) log 𝑝(λ|𝑋), (15) which is 𝑚 times the Kullback-Leibler divergence16 of the always successful reference distribution from the model distribution that can fail at simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Minimizing this divergence maximizes the asymptotic success probability for any large number of simula- tion tasks drawn from the model distribution16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While this is more reliable than only maximizing the success probability for a specific set of 𝑚 simulation tasks, 𝐷(λ) and its minimizer ˆλ cannot be calculated efficiently in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The practical alternative is to use − log 𝑃(λ) and its minimizer ˆλX to approximate these inaccessi- ble quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' To clarify their relationship, I use two convenient intermediates, 𝐷X(λ) = − 𝑛 ∑︁ 𝑖=1 log 𝑝(λ|𝑋𝑖), ∑︁ X = ∑︁ 𝑋1 𝑝(𝑋1) · · · ∑︁ 𝑋𝑛 𝑝(𝑋𝑛), (16) to simplify the notation during the IC derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For a constant value of λ, 𝐷X(λ) is an unbiased estimator of 𝐷(λ) when averaged over sets of 𝑚 simulation tasks X, 𝐷(λ) = ∑︁ X 𝐷X(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (17) Since I cannot efficiently calculate ˆλ, I would like to evaluate 𝐷(λ) at one λ = ˆλX value that I can calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If this was repeated and averaged over sets of 𝑚 simulation tasks, it would be an unbiased estimator of 𝐷min−ave = ∑︁ X 𝐷(ˆλX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (18) However, with a single X, I can only evaluate 𝐷X(λ) at its own minimum, λ = ˆλX, which is an unbiased estimator of the average minimum, 𝐷ave−min = ∑︁ X 𝐷X(ˆλX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (19) This has a negative bias relative to 𝐷(ˆλX) because each 𝐷(λ) is evaluated at its own minimum instead of a common λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A single 𝐷X(ˆλX) can be unbiased as an estimator of 𝐷(ˆλX) by adding a bias correction, Δ = 𝐷min−ave − 𝐷ave−min = ∑︁ X [𝐷(ˆλX) − 𝐷X(ˆλX)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (20) I approximate Δ with several simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The first IC assumption is that 𝐷(λ) and 𝐷X(λ) are both slowly changing in a region containing ˆλ and ˆλX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Both functions can be extrapolated from their minimum to the other function’s minimum with a second-order Taylor expansion, 𝐷(ˆλX) ≈ 𝐷(ˆλ) + 1 2 (ˆλX − ˆλ)𝑇 F(ˆλX − ˆλ), 𝐷X(ˆλ) ≈ 𝐷X(ˆλX) + 1 2 (ˆλ − ˆλX)𝑇 FX(ˆλ − ˆλX), [F]𝑖, 𝑗 = 𝜕2𝐷 𝜕𝜆𝑖𝜕𝜆 𝑗 (ˆλ), [FX]𝑖, 𝑗 = 𝜕2𝐷X 𝜕𝜆𝑖𝜕𝜆 𝑗 (ˆλX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (21) 4 These extrapolations can be combined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (17) to simplify the bias correction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (20) to Δ ≈ 1 2 ∑︁ X (ˆλ − ˆλX)𝑇 (F + FX)(ˆλ − ˆλX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (22) Similarly, I can extrapolate 𝐷X(λ) from λ = ˆλ to λ = ˆλX, 𝐷X(ˆλX) ≈ 𝐷X(ˆλ) + (ˆλX − ˆλ)𝑇 𝜕𝐷X 𝜕λ (ˆλ) + 1 2 (ˆλX − ˆλ)𝑇 F′ X(ˆλX − ˆλ), [F′ X]𝑖, 𝑗 = 𝜕2𝐷X 𝜕𝜆𝑖𝜕𝜆 𝑗 (ˆλ), (23) and minimize the quadratic form for the parameter variations, ˆλ − ˆλX ≈ (F′ X)−1 𝜕𝐷X 𝜕λ (ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (24) The second IC assumption is that F ≈ FX ≈ F′ X, which allows for the removal of FX and F′ X from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (22) after substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (24), Δ ≈ tr[ ˜FF−1], [ ˜F]𝑖, 𝑗 = ∑︁ X 𝜕𝐷X 𝜕𝜆𝑖 (ˆλ) 𝜕𝐷X 𝜕𝜆 𝑗 (ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (25) The validity of these two assumptions can be increased by adding more reference data to reduce finite-sample effects until 𝐷X(λ) and 𝐷(λ) have small differences in their gradients and negligible differences in their Hessians at λ = ˆλX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The TIC follows from a related assumption about small finite- sampling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As a useful reference, I rearrange F and ˜F into a similar form by rewriting ˜F as a sum over simulation tasks rather than over groups of 𝑚 simulation tasks, [ ˜F]𝑖, 𝑗 = 𝑚 ∑︁ 𝑋 𝑝(𝑋) � 𝜕 log 𝑝(λ|𝑋) 𝜕𝜆𝑖 𝜕 log 𝑝(λ|𝑋) 𝜕𝜆 𝑗 � λ=ˆλ , [F]𝑖, 𝑗 = −𝑚 ∑︁ 𝑋 𝑝(𝑋) � 𝜕2 log 𝑝(λ|𝑋) 𝜕𝜆𝑖𝜕𝜆 𝑗 � λ=ˆλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (26) The TIC bias correction is a direct approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (25) by Δ ≈ ΔTIC = tr[ ˜FXF−1 X ], [ ˜FX]𝑖, 𝑗 = 𝑚 ∑︁ 𝑘=1 � 𝜕 log 𝑝(λ|𝑋𝑘) 𝜕𝜆𝑖 𝜕 log 𝑝(λ|𝑋𝑘) 𝜕𝜆 𝑗 � λ=ˆλX , (27) which again assumes that the 𝑚 samples in X are sufficient to converge expectation values so that ˜F ≈ ˜FX and F ≈ FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The AIC follows from additional assumptions about model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I can simplify the difference between F and ˜F in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (26) by rearranging and combining the logarithmic derivatives into [ ˜F − F]𝑖, 𝑗 = 𝑚 ∑︁ 𝑋 𝑝(𝑋) 𝑝(ˆλ|𝑋) � 𝜕2𝑝(λ|𝑋) 𝜕𝜆𝑖𝜕𝜆 𝑗 � λ=ˆλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (28) Next, I consider a modified form of 𝐷(λ) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (15) in which the reference simulation tasks are assigned a failure rate 𝛿, 𝐷(λ) = −𝑚 ∑︁ 𝑋 (1 − 𝛿)𝑝(𝑋) log 𝑝(λ|𝑋) − 𝑚 ∑︁ 𝑋 𝛿𝑝(𝑋) log(1 − 𝑝(λ|𝑋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (29) The original form is recovered in the 𝛿 → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If the IC derivation is repeated for the modified form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (28) becomes [ ˜F − F]𝑖, 𝑗 = 𝑚 ∑︁ 𝑋 (1 − 𝛿)𝑝(𝑋) 𝑝(ˆλ|𝑋) � 𝜕2𝑝(λ|𝑋) 𝜕𝜆𝑖𝜕𝜆 𝑗 � λ=ˆλ + 𝑚 ∑︁ 𝑋 𝛿𝑝(𝑋) 1 − 𝑝(ˆλ|𝑋) � 𝜕2[1 − 𝑝(λ|𝑋)] 𝜕𝜆𝑖𝜕𝜆 𝑗 � λ=ˆλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (30) The final AIC assumption is that the optimized model can recover the reference distribution, resulting in 𝑝(ˆλ|𝑋) ≈ 1 − 𝛿 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I can then cancel the 𝛿 factors and combine the two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (30), [ ˜F − F]𝑖, 𝑗 ≈ 𝑚 � 𝜕2 𝜕𝜆𝑖𝜕𝜆 𝑗 ∑︁ 𝑋 𝑝(𝑋) � λ=ˆλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (31) The difference between ˜F and F disappears for any value of 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this scenario, ˜F and F are 𝑚 times the Fisher information matrix16 of 𝑝(λ, 𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The AIC bias correction corresponds to ignoring this difference and keeping only the trace of the identity matrix over the 𝑛-dimensional parameter space, Δ = 𝑛 + tr[( ˜F − F)F−1] ≈ ΔAIC = 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (32) The validity of the good model assumption can be increased by improving the model family and relaxing the success criteria to increase all optimized success probabilities towards one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Transferability A statistical framework for model selection can also sup- port more precise statistical statements about model transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Here, I briefly contrast a notion of statistical transferability from that of physical transferability, which is frequently discussed when building models for atomistic simulation17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I argue that while sta- tistical transferability is the more desirable goal of model building, it is often impractical to avoid physical transferability assumptions given the present state of atomistic simulation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Statistical transferability can directly predict the average future success of a model when simulation tasks can be interpreted as being drawn from the same distribution that was used to fit the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This is a form of model transferability to future simulation tasks that were not part of the reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For a model fit with 𝑚 reference simulation tasks to a minimum divergence 𝐷(ˆλ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (15), the asymptotic fraction of successful simulations will be exp(−𝐷(ˆλ)/𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (33) If the task distribution is designed to predict or approximate typical workloads of typical users of a model, then the model fitting process provides a direct operational statement about how effective the model should be for its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Statistical transferability can also be used to recycle refer- ence data by transferring it between task distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Reference data sampled from a second distribution 𝑝′(𝑋) over a superset of simulation tasks can be reused to estimate 𝐷(λ) for 𝑝(𝑋), 𝐷(λ) ≈ − � min 𝑖 𝑝′(𝑋𝑖) 𝑝(𝑋𝑖) � 𝑚 ∑︁ 𝑖=1 𝑝(𝑋𝑖) 𝑝′(𝑋𝑖) log 𝑝(λ|𝑋𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (34) 5 This is an implicit form of rejection sampling, and it requires the ability to calculate probability ratios between two task distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It can also be used heuristically to reduce the influence of data that is necessary to fit a model but not representative of its typical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Operationally, this can be interpreted as rare instances when users validate the model for themselves on the original reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The effective sample size associated with this resampling procedure is 𝑚′ = � min 𝑖 𝑝′(𝑋𝑖) 𝑝(𝑋𝑖) � 𝑚 ∑︁ 𝑖=1 𝑝(𝑋𝑖) 𝑝′(𝑋𝑖) , (35) which can be small if 𝑝(𝑋) and 𝑝′(𝑋) are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Physical transferability is a set of observations and assumptions about the spatial locality of physics at an atomistic length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It assumes that some model details and parameters describing short- range interatomic effects will be insensitive to distant changes in a large system with many atoms and then observes the varying degrees to which this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The underlying first-principles QM equations are completely local and transferable when long-range interactions are mediated by local fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Unfortunately, locality and transferability are both degraded when encapsulating many- body effects and non-essential degrees of freedom to build simpler models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Physical transferability assumptions are essential for justi- fying the use of methods that decompose large systems into a set of small fragments and simulate them individually, often embedded in simpler model environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such methods include implicit solvation models18, QM/MM embedding19, and the use of periodic supercells20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the effectiveness of these methods can be highly system dependent, an important example being the reduced locality of electronic effects in metallic systems that complicate efforts to develop low-cost methods21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In the context of statistical model selection, physical transfer- ability assumptions are unavoidable when generating reference data for task distributions containing large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Reliable methods for reference data generation generally have large cost prefactors or poor cost scaling with system size that prevent their direct use on the task distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Physical transferability can be used to justify the use of more accessible reference data corresponding to a proxy task distribution over small embedded fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Tasks from the original distribution can be decomposed into sets of proxy tasks on fragments to generate the proxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While these small proxy tasks may all be contained within the original task distri- bution, the proxy distribution is over a strict subset of simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is statistically impossible to sample from a distribution by weighting samples from a second distribution over a subset of events, but this is avoided by the physical fragmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While rigorous error analysis of this process is difficult, the general expectation is that the use of larger system fragments increases the validity of physical transferability assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Cost penalties The primary purpose of fitting models in statistics is to explain data in the absence of a prior explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In contrast, the purpose of fitting models for atomistic simulation is to avoid the large cost of evaluating a known first-principles model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Statistics is concerned with efficiency, but its main consideration is in getting the most value out of limited data to avoid the potentially high cost of collecting or generating data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Without some penalty for the cost of models, the inevitable conclusion of statistical model selection in atomistic simulation is to choose the expensive model that was used to generate the reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The IC already add penalties to the success measure that limits the number of model parameters, and the simplest approach is to introduce a cost penalty with a similar form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The linear parameter penalty in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (32) looks like a Lagrange multiplier, except that the coefficient is not adjustable and the number of parameters is a trivial function of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The average model evaluation cost can be a non-trivial function of model parameter values, and it can be controlled using a Lagrange multiplier that penalizes excessive cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' With both cost and parameter penalties added, the operational measure of modeling success is ˜𝐷(λ) = 𝛾(𝑡 − 𝑡0) + Δ − 𝑚 ∑︁ 𝑖=1 log 𝑝(λ|𝑋𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (36) Here, 𝛾 is a Lagrange multiplier, 𝑡 is the total cost of applying the model to the 𝑚 simulation tasks, 𝑡0 is the target computational budget, and Δ is an IC penalty approximating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Between multiple model families with different costs and parameters, the family that produces the minimum value of ˜𝐷(λ) for a common 𝛾 value should be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The stationary condition of the Lagrange multiplier, 𝜕 𝜕𝛾 ˜𝐷(λ) = 𝑡 − 𝑡0 = 0, (37) should be applied to the model family with the smallest minimum ˜𝐷(λ) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If this best model family has a parameter-invariant 𝑡 value, then 𝛾 should be adjusted until the minimum cost-penalized ˜𝐷(λ) is equal for two different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this scenario, the cost of the two best models, 𝑡1 and 𝑡2, should bracket 𝑡0 as 𝑡1 ≤ 𝑡0 ≤ 𝑡2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A new, hybrid model can then achieve the target cost by randomly switching tasks between the two bracketing models with probabilities (𝑡2 − 𝑡0)/(𝑡2 − 𝑡1) and (𝑡0 − 𝑡1)/(𝑡2 − 𝑡1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is often more practical to minimize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (36) over λ without any penalties and then add in the penalties with no further optimization of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The use of cost penalties may be more complicated if applied to proxy distributions of fragmented simulation tasks as described in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If sets of fragmented simulation tasks are meant to represent a larger simulation task, then the model eval- uation cost for the larger simulation task may not be approximated well by the sum of costs for the proxy tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this situation, a proxy cost penalty could be constructed from resource estimates that approximate the unknown cost of the larger simulation task from the known costs of the proxy tasks and other task-specific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Models usually have a well-understood scaling with system size and cost prefactors can be estimated from the proxy calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' More detailed, model-specific resource estimation is also possible22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The estimated total simulation cost of the model on the large simulation tasks could then be used as 𝑡 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (36) instead of the total proxy simulation cost that is directly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 6 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' HYDROGEN CLUSTER EXAMPLE To demonstrate the principles of statistical model selection, I consider a simple set of simulation tasks on randomly generated hydrogen clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' By only considering hydrogen atoms, I keep the elemental diversity at a minimum to simplify the process of fitting SQM models with element-specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I keep the phenomenological diversity high by considering two distributions of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A “dense” distribution of clusters forces the minimum interatomic distance between hydrogen atoms to be less than the Coulson-Fischer point23 near 1 Å, while a “sparse” distribution allows larger minimum separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Molecular orbitals tend to remain grouped into pairs with opposite spin and similar spatial character in the dense distribution, while the sparse distribution generates many clusters that favor spin-polarized, atom-localized orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Because of limitations in methods and software that generate accurate and reliable reference data, the only observable that I consider is the total energy of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I consider three simulation tasks to calculate energies for three cluster modifications: removal of an atom, removal of an electron, and addition of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A success is defined as the calculation of one such energy with an error of 1 kcal/mol or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While these distributions and tasks are artificial and not directly motivated by any application, there is some experimental interest in positively24 and negatively25 charged hydrogen clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I generate the dense and sparse distribution of hydrogen clus- ters by sequential rejection sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Atoms are assigned uniformly random positions in a box containing the valid domain, and the atom is rejected and repositioned if it violates a distance constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The minimum allowed interatomic distance for both distributions is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 Å, near the classical turning point of the H2 potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The maximum allowed value for the minimum interatomic distance is 1 Å for the dense distribution and 4 Å for the sparse distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The sparse distribution is a strict superset of the dense distribution, and some sparse clusters could be recycled as dense clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the recycling rate of two-atom clusters is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='016, and it decreases rapidly with increasing cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This is an example of low recycling efficiency between distributions that are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For the reference data set, I generate 10,000 nested sequences of clusters between two and seven atoms for each distribution, resulting in 120,000 distinct structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The three sim- ulation tasks require calculations of three different charge states – 0, 1, and -1 – corresponding to 360,003 total energy calculations including an isolated hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I briefly compare this reference data set with the MGCDB84 data set that is popular for testing DFT functionals26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Both data sets are restricted to total energies of small, isolated groups of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MGCDB84 corresponds to 5,931 total energy calcula- tions of structures that are 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6% hydrogen, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2% carbon, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8% oxygen, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5% nitrogen, and less than 1% each of main-group ele- ments from the first four rows of the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Thus, while it is not restricted to only hydrogen atoms, hydrogen is the most well-represented element in MGCDB84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MGCDB84 is organized into 84 subsets of data corresponding to different simulation tasks, including non-covalent binding energies, isomerization energies, formation energies, and barrier heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, this data set lacks diversity by some measures, such as 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2% of the structures being closed-shell singlets and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0% being charge neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Also, 0 1 2 3 4 5 101 103 105 MGCDB84 0 1 2 3 4 5 101 103 105 dense 0 1 2 3 4 5 H-H distance 101 103 105 sparse FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Histograms of interatomic distances between hydrogen atoms in the structures from three reference data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MGCDB84 mostly contains structures and properties of interest to organic chemistry with structures at equilibrium or saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MGCDB84 is thus a reasonable proxy for the interests of organic chemists, while the hydrogen cluster data sets broadly sample from the potential energy surface of many hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Of partic- ular interest when fitting distance-dependent parameters such as pair potentials and one-body matrix elements is the distribution of interatomic distances between hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' These distance distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 1 for MGCDB84 and the two distri- butions of hydrogen clusters considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MGCDB84 has poor coverage at distances less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 Å and is not a good reference to fit distance-dependent parameters for hydrogen interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Reference data I gather high-level reference data for the hydrogen clusters at the CCSD(T) level of theory27 with the def2-QZVPP basis set28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I also record data at the Hartree-Fock (HF), MP2, and CCSD levels of theory during the CCSD(T) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In addition to the high- level reference data, I also gather data using several popular SQM models and DFT functionals to test their transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There are too many SQM models and DFT functionals to test all of them, and this study is limited to a few important representative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' AM129 was the most popular SQM thermochemistry model of the last century, and PM730 is the most recent model from that family of MNDO-like models31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' GFN132 and GFN233 are two recent SQM models from the density functional tight-binding (DFTB) framework34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' PBE35 is the most popular DFT functional in solid- state physics and materials science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' B3LYP36 is the most popular 7 DFT functional in chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 𝜔B97M-V37 is claimed to be the most accurate DFT functional without including terms from many-body perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While a smaller basis set might be sufficient, I perform all DFT calculations using the def2-QZVPP basis set for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In total, I gather data from eleven QM and SQM models, which corresponds to 3,960,033 total energy calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' All QM calculations use a post-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 development version of PySCF38–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' All calculations use spin-unrestricted orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For HF theory and every DFT functional, the large-basis calculations are initialized by projecting a converged density matrix from a calculation in the smaller def2-SVP basis set28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The def2-SVP density matrix is taken from the calculation with the lowest total energy from a systematic ground-state search for each structure and charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' First, a def2-SVP calculation is performed for every spin state from the standard spin-averaged independent-atom density matrix guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Second, a custom density matrix guess is constructed from spin-polarized independent-atom density matrices with every combination of atomic charges and spin orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Third, after performing all of these small-basis calculations with the default DIIS algorithm41, they are all repeated with an alternative ADIIS algorithm42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The large-basis calculation uses the same algorithm, either DIIS or ADIIS, as the small-basis calculation that is used to initialize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Even with all of this redundancy, it is not possible to converge a self-consistent field (SCF) cycle for every charge and spin state of every structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While the variational nature of SCF calculations guarantees the existence of stable local energy minima, DIIS-based algorithms provide no guarantees of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' All large-basis DFT calculations use a (99,590) local grid and a SG-1 nonlocal grid, following the recommendations for the 𝜔B97M-V functional37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For SQM calculations, MOPAC 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='543 is used for AM1 and PM7 calculations, and xTB 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='19 is used for GFN1 and GFN2 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' MOPAC calculations follow the same ground-state search procedure as the PySCF calculations except with only DIIS and without any projection into a larger basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There are fewer points of failure in minimal-basis calculations, and MOPAC is able to converge an SCF calculation for every structure and charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' xTB calculations do not contain Fock exchange and depend on an initial electronic density guess rather than a density matrix guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I only use the default spin-averaged density guess and restrict the ground-state search to total spin values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There is a high failure rate for SCF convergence in xTB with the default options for this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, it is possible to converge every structure and charge state in xTB with calculations at elevated electronic temperatures of 3000 K and then 1500 K followed by linear extrapolation of the total energies to zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' At this level of automation and scale of data generation, it is not possible to converge every iterative solve for HF, DFT, and CCSD calculations in PySCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The choice of solver options is important as it changes success statistics and average run times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I did not try to optimize these choices in a systematic way, but they were adjusted during the implementation of the workflow to improve success rates44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In addition to convergence failures, a DFT or HF calculation is considered to fail if the def2-QZVPP total energy is more than 10 kcal/mol larger than the smallest def2-SVP total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Total energies tend to be lower for larger basis sets because they have more variational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I attribute these energy increases to the DIIS phenomenon of escaping from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='04 101 103 105 dense 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='04 101 103 105 sparse FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Histograms of the maximum deviation from zero and one of the unrelaxed MP2 1RDM eigenvalues for all structures and charge states from the reference data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' basin of convergence of a ground state and converging to a very different stationary state with a larger energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The failure rate of DFT calculations is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4%, the failure rate of CCSD(T) calculations is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0%, and the overall failure rate of the simulation tasks is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If at least one model fails to produce an output for a simulation task, then that task is omitted from the final data set and statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such failures distort the distribution of simulation tasks because they act as a form of rejection sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I also validate the CCSD(T)/def2-QZVPP level of theory for this data set while gathering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The main validity concern is strong electron correlation effects, which are known to occur in hydrogen clusters45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' These effects are caused by multi-reference ground states that come from a superposition of many electronic spin configurations with nearly degenerate energies in the atomic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Randomly generated hydrogen clusters are unlikely to have many degenerate spin configurations, and they are expected to be more weakly correlated on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The most direct validity test would be the overlap between the normalized HF and CCSD many- body wave-functions, but this quantity is not efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Instead, I use the eigenvalues of the one-particle density matrix (1RDM) at the unrelaxed MP2 level of theory as an accessible proxy for this overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The maximum deviation of the eigenvalues from zero and one is strictly zero when the overlap is one, and the deviation increases as the overlap is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This deviation is plotted for every structure in every charge state in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The sparse distribution that is expected to be more susceptible to multi- reference effects because of spin symmetry breaking does not have larger deviations than the dense distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The other major validity concern is the basis-set convergence of CCSD(T)/def2-QZVPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A quadruple-zeta basis such as def2- QZVPP does not typically converge absolute post-HF energies to chemical accuracy of 1 kcal/mol or less without basis-set extrapo- lation or explicit correlation corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the simulation tasks considered here only require energy differences between struc- 8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 101 103 105 CCSD –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 remove atom dense −5 0 5 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 ±0.' metadata={'source': 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+page_content='2 −100 0 –41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 0 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 0 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Error histograms in kcal/mol for all models and tasks along with their means, standard deviations, and moment-matching Gaussian model fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 9 −25 0 25 101 103 105 ωB97M-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 remove atom dense −10 0 10 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 sparse −25 0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 remove electron dense −50 0 50 –27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 sparse 0 25 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 add electron dense −20 0 –6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 sparse −25 0 25 101 103 105 B3LYP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 0 10 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 0 25 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 −25 0 25 101 103 105 PBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 −10 0 10 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 −25 0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0 200 –42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 0 25 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 −25 0 25 –14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Error histograms in kcal/mol for DFT models and all tasks along with the means, standard deviations, and moment-matching Gaussian model fits of the marked data with consistent total spin values between HF and DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' tures that differ by at most one atom, which should be less sensitive to finite-basis errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The most basis-set sensitive structures are correlation-bound anions, which account for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3% of the anions in the dense distribution and 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2% in the sparse distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Correlation-bound anions do not have a proper complete basis set (CBS) limit with HF orbitals because the overlap between the HF and CCSD wave-functions tends to zero as the unbound HF orbital delocalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A formally correct treatment of correlation-bound an- ions in the CBS limit requires Brueckner orbitals46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, I do not expect the def2-QZVPP basis set to be large enough for the pathological CBS limit to have a substantial effect on this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Anomaly detection Anomaly detection is a natural part of error analysis when gathering large amounts of data within a statistical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The basic expectation of a good model is that its errors are an accumulation of a large number of small, independent errors, which tend to induce Gaussian distributions of model errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Errors in the hydrogen cluster data organized by model and task are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 3 with moment-matching Gaussian fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While many errors are effectively described by the Gaussian model, there are also several fat error tails, many of which are rare enough to be unlikely to appear in data generation at smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' What is not shown are some even larger error tails that were present in earlier versions of the data set as the workflow was being refined to detect and avoid more failure events and silent errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This statistical overview of error distributions along with metadata collected during the primary data generation are essential for detecting and correcting rare failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Unfortunately, sufficiently rare failures are unlikely to occur in small-scale preliminary testing of a workflow precisely because of how rare they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There is not necessarily a clean partition between model, algorithm, and software errors in large-scale data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For example, the lack of reliability in DIIS-based SCF solvers causes enough gaps in the ground-state searches that the wrong total spin is assigned in some DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As a result, some DFT calculations produce total energies that are too high, which are likely a source of some rare error outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, there is no guarantee that the DFT and HF ground states for a given structure and charge state will have the same total spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There is not enough information to distinguish model from algorithm errors here without more reliable SCF solver algorithms to fill gaps in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Similarly, software bugs may cause failures in one algorithm implementation that are not reproduced by other implementations, and custom improvements to algorithms may cause successes that are also not reproducible in other software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' To see the impact of spin inconsistency, the DFT data is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 4 with spin- consistent calculations marked and fit to Gaussian error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The spin-inconsistent data contains most of the error outliers but does not substantially change the overall error statistics since the spin-consistent data has similar means and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The broadest error distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 3 are in the SQM atom removal data from the dense distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is likely that errors in short-range pair potentials and matrix elements account for much of this error since these SQM models are mostly fit to data from near-equilibrium structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I test this hypothesis by separating data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 5 based on the minimum interatomic distance in a structure being greater than or less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='74 Å, the equilibrium bond length of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There is a clear narrowing of the error distributions for the structures without short interatomic distances, which supports the error hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It may not be possible to detect or explain all error outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The CCSD error tails from the sparse distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 3 imply rare instances of large perturbative triples corrections to the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In these cases, the exact ground-state wave-function may have strong multi-reference character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the multi-reference test in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 2 has no corresponding outliers, and a variety of multi- reference tests may be needed to increase detection reliability47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 10 −1000 0 1000 101 103 105 GFN2 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 ±37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 remove atom dense −100 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 sparse −200 0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 remove electron dense 0 100 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 sparse −100 0 –25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 ±38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 add electron dense −200 0 –157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 sparse −250 0 101 103 105 GFN1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 ±35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 −200 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 ±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 0 100 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 ±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 0 100 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 −100 0 –16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 −200 0 –153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 ±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 −2000 0 101 103 105 PM7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 −1000 0 –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 −100 0 –37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 −100 0 –50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0 250 101 103 105 AM1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 0 200 –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 −100 0 –28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 ±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 −100 0 –41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Error histograms in kcal/mol for SQM models and all tasks along with the means, standard deviations, and moment-matching Gaussian model fits of the marked data from structures with minimum interatomic distances greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='74 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The failures that statistical model selection in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' II seeks to avoid are silent failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Anomaly detection implies an ability to detect and herald some types of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For the example data set in this paper, I havechosen to removesome heralded failuresassociated with algorithm-specific SCF convergence problems to increase the emphasis on errors in the physical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This formally changes the underlying task distributions by a small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' To be faithful to the original task distributions, a more complete model would always produce a viable output by branching to less accurate but more reliable calculations and eventually resorting to a random guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' When trying to increase a model’s overall success probability, improving the ability to detect and respond to rare failures and error outliers can be just as important as improving the average model accuracy for typical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Model fitting I now consider a minimal representative example of using model selection to fit SQM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' First, I highlight the benefits of using more complicated error models to improve success measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Second, I fit an atomic pair potential to all QM and SQM data, primarily to correct the large error outliers in the SQM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Pair potentials are one of the most common and basic elements of both interatomic potentials and SQM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While pair potentials are often restricted to a simple form before fitting them, I consider a general form and rely on model selection to limit the number of parameters that define the pair potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Because some models being considered are near chemical accuracy, the small-𝜖 approximation used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (7) is not always accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Instead, I use the exact success probability, 𝑝(λ|𝑋𝑖) = ∫ 𝑥𝑖+𝜖 𝑥𝑖−𝜖 𝑒−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5[𝑧−𝜇−𝑦𝑖 (λ)]2/𝜎2 𝜎 √ 2𝜋 𝑑𝑧 = erf � 𝑥𝑖−𝑦𝑖 (λ)−𝜇+𝜖 √ 2𝜋𝜎 � − erf � 𝑥𝑖−𝑦𝑖 (λ)−𝜇−𝜖 √ 2𝜋𝜎 � , (38) for a success interval [𝑥𝑖 − 𝜖, 𝑥𝑖 + 𝜖] around a reference data value 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For chemical accuracy, 𝜖 = 1 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This interval needs to be adjusted for electron addition and removal energies that are near their vacuum-limited values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The energy to add an electron cannot be greater than zero, and the energy to remove an electron cannot be less than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If the success interval crosses into this physically forbidden region, then I ignore the unphysical end point and consider a semi-infinite success interval in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The form of the pair potential is a polynomial at short range that goes to zero at an adjustable cutoff 𝑅 and strictly zero beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The success measure in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (36) and its analytical first and second derivatives with respect to λ are tedious but straightforward to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I minimize the success measure with a sequence of line searches that use this derivative information to achieve asymptotic quadratic convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As I increase the polynomial degree, I use the minimizing model with one fewer degree as the initial guess for minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For degree one, I use the moment-based approximations of the error model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (13) and a zero pair potential with 𝑅 = 4 Å as the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The TIC bias correction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' (27) is calculated at the penalty-free minimum of the success measure instead of being included in the minimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The models that minimize the success measure are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There is a clear benefit to using a richer error model with a separate Gaussian error model for each type of simulation 11 model ˜𝐷1g ˜𝐷6g 𝜇 𝜎 𝜇rad 𝜎rad 𝜇ras 𝜎ras 𝜇red 𝜎red 𝜇res 𝜎res 𝜇aed 𝜎aed 𝜇aes 𝜎aes 𝑡 CCSD+PP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='57 × 105 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='25 × 10−2 none 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='72 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='18 × 106 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 -228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 -293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='38 × 10−2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Comparison of minimized success measures over 𝑚 = 344, 513 simulation tasks for various models, including a pair potential (PP) correction when the improvement is greater than one percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This comparison includes one-Gaussian (1g) error models (𝜇, 𝜎) and six-Gaussian (6g) error models fit to atom removal (ra), electron removal (re), and electron addition (ae) on both dense (d) and sparse (s) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The success measures do not include parameter or cost penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The error model parameters are in kcal/mol and the total model evaluation times 𝑡 are in CPU-seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The cost of generating the reference data is 𝑡 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='13 × 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Many of the large standard deviations in the overall error are better explained as biases in a specific task type with a smaller standard deviation per type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Some of these biases are obvious and expected, but it is still useful to quantify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The GFN1 and GFN2 models predict a very large electron binding energy for most hydrogen clusters, while AM1 and PM7 do not predict any binding of excess electrons to any hydrogen cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The HF model has biases associated with the absence of electron correlation energy, which is always negative and usually proportional to the number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The DFT models are known to have large delocalization errors48 that are likely to be biasing the electron removal energies of the sparse distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If an error model is used to improve success probabilities by adding random numbers to a model’s outputs, then an improvement to the error model is an improvement to the model as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The effects of the IC penalties on the selection of the pair potential are shown for two representative model families in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' CCSD is a more accurate model than PM7, and the AIC is likewise a better approximation of the TIC for CCSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For PM7, the AIC is unable to compensate for the parameter bias enough to create a local minimum in the success measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For CCSD, the AIC is able to create a local minimum, but its location is different than for the TIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this example, the TIC correction introduces significant numerical noise, which appear as values above the smoother trend line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The TIC is a response property that depends sensitively on the numerical quality of the success measure minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The derivative discontinuity that I allow at the large-distance cutoff point 𝑅 of the pair potential introduces derivative discontinuities in the 𝑅 dependence of the success measure that complicates the minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Even under such non-ideal conditions, the TIC is 0 20 40 60 80 100 −3700 −3690 −3680 −3670 −3660 −3650 −3640 ˜D6g change for CCSD AIC TIC 0 20 40 60 80 100 polynomial degree −159500 −159000 −158500 −158000 −157500 −157000 −156500 ˜D6g change for PM7 AIC TIC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Reduction of the success measure ˜𝐷6g with a six-Gaussian error model as the polynomial degree of the pair potential is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The TIC is regularized by replacing small and negative eigenvalues of the ˜𝐷6g Hessian with 10−9 times the largest eigenvalue when that value is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 PP for CCSD (kcal/mol) no TIC TIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 distance (Å) −200 −150 −100 −50 0 PP for PM7 (kcal/mol) no TIC TIC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Short-range polynomial pair potential corrections for the CCSD and PM7 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' With the TIC penalty, the minimizing polynomial has degree 22 for CCSD and degree 14 for PM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Without an IC penalty, there is no local minimum in polynomial degree and best degree 100 polynomial is shown as an example of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Outside of the plotted range, the PM7 pair potential decreases to -1790 kcal/mol at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' still functional for model selection with appropriate regularization of the success measure Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The TIC is more challenging to calculate for parameterized QM calculations that involve QM response properties in the parameter derivatives of the success measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The benefits of a pair potential correction are not uniform over models or tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Since the pair potential only depends on the atomic structure and not electronic structure, it cannot correct the electron addition and removal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For many models, the overall reduction of the success measure is one percent or less, and these minor improvements are omitted from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The largest improvement comes from the PM7 pair potential, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Apparently, the short-range hydrogen-hydrogen pair potential in PM7 is much too repulsive at distances just below the bond length of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In contrast, the CCSD pair potential is much longer in range and much smaller in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is not surprising that the largest correction occurs near the Coulson-Fischer point around 1 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, it is surprising that something as complicated as the CCSD(T) triples correction can be partially approximated by a pair potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The IC penalties succeed in suppressing the high- frequency oscillations typically attributed to overfitting noise, but there are still some artifacts near the edges of the polynomial’s domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There are other ways to reduce unphysical oscillations in pair potentials, such as considering reference tasks that depend directly on derivatives of a pair potential or explicit functional regularization49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 8, the pair potential corrections eliminate most of the large error outliers in SQM models except for GFN2 on the dense distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I expect that the persistent error in GFN2 is from the short-range part of either a 3-body potential term or a Hamiltonian matrix element, neither of which can be repaired by a pair potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This example demonstrates the benefits of having an excessive amount of data available when fitting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As the amount of data increases, the utility and reliability of statistical tools and con- cepts increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The abundance of data creates a comfortable safety buffer between the number of parameters needed to fit a model ac- curately and the maximum number of parameters that can be fit with statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The model selection process then enables an accurate model to be carved from an accessible set of redundant, overfit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such large amounts of data are accessible because of the massive scale of modern high-performance computing, an ability to generate data sets procedurally, and careful use of phys- ical transferability assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This strongly contrasts with how SQM models such as PM730 and GFN233 have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' They prescribe simple model forms with a few tens of parameters per element and collect enough reference data to fit those forms specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' They do not gather enough data to consider or rule out more complicated models with more parameters, and many SQM model design choices have remained frozen for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' PM7 still uses the MNDO model form31 proposed in 1977, just with the addition of more complicated classical correction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' De- spite being from a much newer model family, GFN2 also contains old model forms such as the Wolfsberg–Helmholz approximation50 from 1952 relating Hamiltonian and overlap off-diagonal matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' With an increasing amount of data, model forms can −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 101 103 105 CCSD+PP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 remove atom dense −5 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='1 sparse −1000 0 1000 101 103 105 GFN2+PP –52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='3 ±92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 −50 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 −100 0 100 101 103 105 GFN1+PP –22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 −50 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5 −200 −100 0 100 101 103 105 PM7+PP 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 −50 0 50 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 ±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 −100 0 100 101 103 105 AM1+PP 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 ±25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 −50 0 50 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='9 ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Revisions of error histograms from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 3 in kcal/mol for the models and tasks that benefit from a pair potential correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 13 shift more towards what is objectively supported by the data and farther from the subjective technical opinions of specific model builders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Cost budgeting Considerations of model cost are always more complicated than model accuracy because they are much more sensitive to software, hardware, and fine details of a workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' All calculations reported in Table I are performed on the same computing cluster, with AMD EPYC 7702 CPU cores and two gigabytes of memory per core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Except for MP2, CCSD, and CCSD(T) calculations, all calculations are performed on a single CPU core for maximum throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Some of the MP2, CCSD, and CCSD(T) calculations exceed the memory budget of a single CPU core, and they are run with four cores per calculation for a safety buffer of memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Parts of the calculation are threaded and make use of multiple cores, but the thread scaling is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This complicates some cost comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For example, the HF and MP2 calculations have very similar run times under similar conditions, and the large cost difference reported in Table I is caused by the different number of cores required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Also, the AM1 and PM7 calculations have similar run times as GFN1 and GFN2 calculations for an individual calculation, but their workflow requires a combinatorial search over atomic spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The limited sensitivity of GFN1 and GFN2 calculations to spin order makes this search unnecessary and reduces their overall run time per simulation task, but may also be related to their relatively poor accuracy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A visual way to compare success measures with varying cost penalties is to plot them versus cost as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 9 and draw the convex hull connecting minimum-cost models with various rates of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Models on the convex hull are optimal for a range of computational budgets, and models above the convex hull are not worth using for these simulation tasks according to this cost analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In this example, the convex hull connects PP, AM1+PP, B3LYP, and CCSD+PP, with GFN1+PP also just on the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' II D, any model cost versus accuracy along the convex hull can be achieved by randomly switching between the models on the end points with a probability varying linearly between zero and one along the facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Thus, there is a natural continuum of hybrid models between the cheapest and most expensive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The practice of randomly mixing models with different accu- racies as suggested here is usually avoided in atomistic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Models often rely on some form of error cancellation or a study of qualitative trends rather than precise quantitative predictions that can be disrupted by comparing data between different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' These behaviors can be described within a framework of statisti- cally independent simulation tasks by carefully defining tasks as groups of simulations with success based on comparisons rather than the absolute value of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In the simple example of a ranking, random pairs of simulations might be performed with the output being the decision of which system had the larger value for a specific output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Such a grouping forces comparisons to remain within a specific model while still allowing for the use of a different model for each independent ranking decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A more common practice of mixing models is to filter a larger number of systems with a cheap, inaccurate model and then filter the remains systems 10−2 100 102 104 106 108 1010 1012 t (CPU-seconds) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='2 ˜D6g ×106 CCSD+PP MP2 HF ωB97M-V B3LYP PBE GFN2+PP GFN1+PP PM7+PP AM1+PP PP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Success measure versus total cost of models from Table I with the convex hull denoting the most cost-effective models in the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' that pass the first filter with a more expensive and accurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The intention of this practice is to approximate the effect of ap- plying the more expensive filter to all of the systems with a lower overall cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, this requires the cheaper model to have a sufficiently low false positive rate that the overall cost is actually reduced while maintaining a very low false negative rate to avoid distorting the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Simultaneous considerations of model cost and accuracy at a large enough scale that reliability also matters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 9 is a very challenging test for models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is much easier to show cost benchmarks of a model or software under ideal conditions, accuracy benchmarks under a different set of ideal conditions, and ignore problematic cases altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Even the hydrogen cluster example considered here is artificially generous because a small fraction of structures that caused SCF convergence failures were omitted from the set of simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While the models are depicted as points on the plot, they are more generally going to be regions corresponding to the set of possible changes in a workflow that alter both cost and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For example, the combinatorial search over atomic spin configurations for the hydrogen cluster example could have been avoided, which would have substantially reduced the cost of many models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, many of the calculations would have failed to find the lowest energy ground state, and the overall accuracy would have been reduced as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Cost and accuracy could have been balanced more carefully by randomly sampling a limited set of spin configurations rather than using an exhaustive combinatorial search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Adjusting details of a model workflow to improve the convex hull of optimal models requires a careful balance of these cost, accuracy, and reliability considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 14 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' CONCLUSION As scientists continue to develop more diverse and sophisti- cated models for atomistic simulation, how models are compared and how their successes are judged become increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Progress in method development can slow down or stop if scientists have different, incompatible definitions for what success is51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' This paper has presented an operational success measure for judging atomistic models that is based on statistical model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Using simple simulation tasks on hydrogen clusters as an example, I have shown how this measure can be used to compare the cost and accu- racy of a diverse set of QM and SQM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' I have also used it to fit a minimal SQM model that applies a pair potential correction to this QM and SQM data and select the potential form that best fits the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The TIC provides a reliable parameter penalty to avoid selecting over-complicated models, while the AIC is not a reliable penalty because some atomistic models are too inaccurate for its assumptions to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For a computational budget that is too small for a high-accuracy model but excessive for a low-accuracy model, the success measure predicts the efficacy of splitting a workload between models to match the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' By adjusting the operational definition of success for simulation tasks, this success measure can be equally good for designing expensive models to succeed at difficult tasks and cheap models to succeed at easy tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' An essential aspect of model building in atomistic simulation is the availability of high-quality reference data for fitting and test- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While models have historically relied on reference data from experiments, it is now possible to generate accurate data using ex- pensive QM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' As shown in the hydrogen cluster example, CCSD(T) data is affordable for small molecular fragments, and less accurate DFT data remains affordable for larger molecules and periodic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For data generation at scales larger than what has been presented in this paper, reliability issues will become increasingly important alongside cost and accuracy considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' SCF convergence problems can cause heralded failures, while SCF convergence to excited states can cause silent failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Without more fundamentally reliable algorithms to reduce failure rates, a fixed rate of failure means an increasing number of failure events as data sets grow larger in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There are increasingly sophisti- cated tools52 for remote, automated computing of large workloads and organizing large data sets with modern database standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, limitations in the reliability of the underlying tasks being automated may have a strongly negative influence on the cost and accuracy of generating large data sets as failures persist against increasing computational redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The hydrogen cluster example considered here is sufficiently different from typical reference data sets that it serves as a challeng- ing test of physical transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' There is a significant difference in the apparent progress that DFT and SQM models have made in developing transferable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The improvement in transferability from PBE to B3LYP to 𝜔B97M-V is consistent with the develop- ment roadmap of DFT functionals with increasing complexity53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While likely a coincidence, the SQM models considered here have systematically degrading performance in chronological order of their development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' A simple explanation of this difference might be that DFT functionals are fundamentally more transferable than minimal-basis SQM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, it is also important to con- sider the vastly differing amounts of technical effort that have been invested in these two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The development path from B3LYP to 𝜔B97M-V includes the development of hundreds of DFT functionals from numerous research groups over more than three decades26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' In contrast, the development path from AM1 to PM7 consists of only a few other models developed by a single scientist – Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' James J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Stewart – working mostly in isolation outside of academia for more than three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' GFN1 and GFN2 were developed much more recently by a single academic group – the research group of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Stefan Grimme at the University of Bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' While there are other SQM models outside of the GFN and MNDO-like model families, these are the two most widely used families and the only non-commerical models54 to be fit for com- binations of elements over most of the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The GFN models incorporate ideas from both MNDO-like models (multipole expansions of electrostatics, avoidance of diatomic parameters) and DFTB models (expansion around an atomic limit, DFT-like cor- relation models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' All of the SQM models considered here have similar superficial complexity, similar numbers of parameters per element, and are fit to similar amounts of reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Except for a belief in the superiority of DFT-like models, there is no com- pelling theoretical reason why any SQM model from this set should perform any better than any other on systems that are very different from their training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The concepts presented in this paper are meant to inform the process of designing, fitting, and selecting models for atomistic simulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' If a simulation task is not going to be repeated a very large number of times, then the formal process of gathering reference data and calculating a success measure might not be worth the amount of human effort required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' However, the statistical model selection process can still be useful as a conceptual guide even when it is not worthwhile to perform it carefully or explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' For tasks that are performed frequently by many scientists, it may be worthwhile to capture that activity as a distribution of tasks and a representative sampling from that distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Quantum chemistry has a tradition of curating reference data sets to guide method development26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Expanding that tradition to accommodate larger data sets, statistical interpretations, and success measures that capture the real needs of applied scientists could create an even better guide for method development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' It is difficult for a scientist to characterize real application needs while also developing novel simulation methods to satisfy those needs, and it would be helpful to decouple those important research activities from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' ACKNOWLEDGMENTS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' thanks Jimmy Stewart for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The Molecular Sciences Software Institute is supported by NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' ACI-1547580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' The computational resources used in this work were provided by Advanced Research Computing at Virginia Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' AUTHOR DECLARATIONS Conflict of Interest The author has no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' 15 Author Contributions Jonathan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Moussa: Conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Data curation (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Formal analysis (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Investigation (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Method- ology (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Resources (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Software (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Validation (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Visualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Writing – original draft (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' Writing – review & editing (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content=' DATA AVAILABILITY The data and software that support the findings of this study are available on Zenodo at the DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E4T4oBgHgl3EQf1A2T/content/2301.05287v1.pdf'} 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Yunnan, 653100, P. R. China +Abstract +Motivated by the Higgs boson decaying to gg at one loop approximation, the amplitude of scalar one +loop three-point diagram with two different internal masses are evaluated and fully analytic results are +obtained. The main ingredient of the evaluation is a integral in which the integrand is product of the +reciprocal of the integral variable and a logarithm, where the argument of the logarithm is a quadratic +function of the general form. The results depend on the choice of the masses of the propagators and +the massive external line. In the first case the amplitude contains an infinite series in which each +term is a hypergeometric function, in the second case the result is expressed through dilogarithms. In +particular, if the three internal lines are taking the same mass, the results will reduce to the known +functions in one loop evaluation of Higgs decaying to gg or γγ. +PACS numbers: +∗ jinzhang@yxnu.edu.cn +1 + +I. +INTRODUCTION +In framework of the Standard Model(SM) and its minimal supersymmetric extension, the +evaluation of scalar one loop three-point amplitudes play a fundamental role in deciphering the +property of the Higgs boson[1, 2] though its decaying to gg(or γγ) and the inverse process, i.e., +production of the Higgs boson by gluon fusion[3–7]. Owing to the coupling of the Higgs boson +to the fermions gHf ¯f and the coupling of fermions to gluon, the three propagators take the +same mass at the leading order of perturbative theory. The evaluated amplitude, if the high +energy approximation[9, 10] is not exploited, will be expressed as function of mf/mH, where +mf and mH are the masses of the internal fermion and the Higgs boson, respectively. At the +final stage of the evaluation, a integral of the following form must be handled carefully +I1 = +� 1 +0 +dx ln(ax2 − ax + 1 − iε) +x +, +a > 0 +(1) +where ε is positive infinitesimal. The integral in Eq.(1) necessarily arises both in the evaluation +of the amplitude of Higgs boson decaying to gg and its production via gluon fusion at one loop. +In addition to the top quark, there is considerable mass hierarchy between the Higgs boson +and other quarks, an economic way to compute Eq.(1) is taking the limit that the masses of +the propagators in the triangle are negligible compared with mass of the Higgs boson, then +the result of Eq.(1) tends to constant. However, in this manner we can not tell the different +contributions from various competing processes to the amplitude. Thus, fully analytic results +is essential to analyze H → gg, then results will be distinguished the cases 0 < a < 4 from the +case a > 4 as detailed in the later works[5, 11–17]. +As a natural generalization of Eq.(1), let us consider the following integral +I2 = +� 1 +0 +dx ln(ax2 − bx + 1 − iε) +x +, +(2) +the parameters a and b satisfy +a > 0, +b > 0, +a ̸= b +(3) +It is obvious that if a = b, Eq.(2) reduce to Eq.(1). This integral can be derived from the +evaluation of the scalar one loop three-point diagram depicted in fig.1, in this case a and b will +be functions of the masses of the propagators ω1 and ω1 as well as the mass of external line m1. +Unfortunately, a close inspection to fig.1 indicates that it does not connect with real decaying +2 + +p1 +p2 +p3 +k +ω2 +ω2 +ω1 +FIG. 1: Massive triangle with two massless external lines. The solid lines and dashed lines denote +massive and massless particles, respectively. +processes of the Higgs boson even though the contribution from Higgs-Kibble ghosts associated +with the W ± and Z bosons are taken into account. Maybe it is the reason that rarely can +we look up the evaluation of integral displayed in Eq.(2) in the one loop evaluation of Higgs +boson decaying to gg1. An thorough investigation of the integral in Eq.(2) on the footing of +perturbative theory is necessary. Therefore, in this paper we will present a systematic study on +Eq.(2) based on the evaluation of scalar one loop three-point amplitude, the complete analytic +results are derived. We hope that the results can be applied to some decaying process under +reasonable approximation addition to H → gg, but also enrich the results of scalar one loop +three-point diagram from the viewpoint of analytic evaluation. +The paper is organized as follows. +In section II we introduce the integral in Eq.(2) by +evaluation the amplitude depicted in fig.1 in a scalar field theory, some general results are +derived. In section III the analytic results of fig.1 are obtained and the special case a = b +are discussed. A short summary are presented in IV. Some useful formulas are listed in the +appendix. +1 Integral of this type has been computed in Eq.(23) of Ref.[8], but only the case b2 − 4a > 0 is considered. +3 + +II. +THE FORMULAS +A. +The massive triangle with two massless external lines +To start with, we write down the amplitude corresponding to fig.1 +I = +� +d4k +(2π)4 +1 +A1A2A3 +, +(4) +where the three denominators are defined by +A1 = k2 − ω2 +1 + iε +A2 = (p1 − k)2 − ω2 +2 + iε +A3 = (p1 − p2 − k)2 − ω2 +2 + iε, +(5) +and ε is real positive infinitesimal, the three external momentum satisfy +p2 +1 = m2 +1, +p2 +2 = p2 +3 = 0. +(6) +Using the Feynman’s trick, Eq.(4) can be written as +I = +� +d4k +(2π)4 +� +dxdydz2! δ(1 − x − y − z) +[D(x, y, z)]3 +, +(7) +where +D(x, y, z) = x(k2 − ω2 +1 + iε) + y[(p1 − k)2 − ω2 +2 + iε] + z[(p1 − p2 − k)2 − ω2 +2 + iε], +(8) +Since the amplitude given by Eq.(4) is both ultraviolet and infrared finite, thus regularization +is unnecessary, the evaluation can be carried out in the four-dimensional space-time. We first +perform the integral over k and z, obtaining +I = − +i +16π2 +� 1 +0 +dx +� 1−x +0 +dy +1 +−yxm2 +1 + x(ω2 +1 − ω2 +2) + ω2 +2 − iε +(9) +The integral over y in Eq.(9) is trivial, combining with Eq.(A1), we arrive at the following +intermediate result +I = +i +16π2m2 +1 +� 1 +0 +dx1 +x +� +ln +�m2 +1 +ω2 +2 +x2 − m2 +1 − ω2 +1 + ω2 +2 +ω2 +2 +x + 1 − iε +� +− ln +�ω2 +1 − ω2 +2 +ω2 +2 +x + 1 − iε +�� += +i +16π2m2 +1 +� +Li2 +� +1 − ω2 +1 +ω2 +2 +� ++ +� 1 +0 +dx1 +x ln +�m2 +1 +ω2 +2 +x2 − m2 +1 − ω2 +1 + ω2 +2 +ω2 +2 +x + 1 − iε +�� +. +(10) +The remaining work is the evaluation of the last integral in Eq.(10). For brevity, in the forth- +coming sections the pre-factor i/(16π2) will be suppressed while 1/m2 +1 will be preserved so as +to maintain the correct dimension of the primitive amplitude displayed in Eq.(4). +4 + +B. +evaluation of integral with logarithms +The evaluation of the last term in Eq.(10) motivates a general investigation on the integral +of the following type +F = +� 1 +0 +dx ln(ax2 − bx + 1 − iε) +x +, +a > 0, +b > 0 +(11) +Since the argument of the logarithm is quadratic in x, we first consider the case b2 − 4a < 0, +in this case the argument of the logarithm is positive definite thus the iε term can be safely +dropped. A feasible way to calculate the integral of Eq.(11) turns out to be expressing it as +F = +� 1 +0 +dx +� 1 +0 +dz +ax − b +1 + zx(ax − b) += a +� 1 +0 +dz +� 1 +0 +dx +x +1 + zx(ax − b) − b +� 1 +0 +dz +� 1 +0 +dx +1 +1 + zx(ax − b) += 1 +2 +� 1 +0 +dzln[1 + z(a − b)] +z +− b +2 +� 1 +0 +dz +� 1 +0 +dx +1 +1 + zx(ax − b)) += −1 +2Li2(b − a) − b +2 +� 1 +0 +dz +� 1 +0 +dx +1 +1 + zx(ax − b). +(12) +Now we concentrate on the last integral in Eq.(12), for later convenience we label it as A, the +integral over x can be calculated[18] +A = +� 1 +0 +dz +� 1 +0 +dx +1 +1 + zx(ax − b) += +� 1 +0 +dz +2 +√ +4az − b2z2 +� +arctan +bz +√ +4az − b2z2 + arctan +2az − bz +√ +4az − b2z2 +� +, +(13) +In deriving Eq.(13) we employ the property that arctan(x) is odd +arctan(−x) = − arctan(x), +(14) +To proceed we separate the integral in Eq.(13) into two parts +A = 2(A1 + A2), +(15) +where +A1 = +� 1 +0 +dz +1 +√ +4az − b2z2 arctan +bz +√ +4az − b2z2, +5 + +A2 = +� 1 +0 +dz +1 +√ +4az − b2z2 arctan +2az − bz +√ +4az − b2z2. +(16) +It is not difficult to demonstrate that +� +arcsin +� +b +� z +4a +��′ = +� +arctan +bz +√ +4az − b2z2 +�′ = b +2 +1 +√ +4az − b2z2, +(17) +Hence, the integral in A1 is easy to calculate +A1 = 1 +b +� +arctan +b +√ +4a − b2 +�2, +(18) +Using the identity[21] +arctan x = arcsin +x +√ +1 + x2, +(19) +leads to the final result for A1 +A1 = 1 +b +� +arcsin +b +2√a +�2. +(20) +Next, we evaluate A2, by making use integration by parts, getting +A2 = 2 +b arcsin( +b +2√a) arctan +2a − b +√ +4a − b2 − 2a − b +b +� 1 +0 +arcsin +� +b� z +4a +� +[1 + (a − b)z] +√ +4az − b2z2 dz, +(21) +We label the second term in Eq.(21) as B and define +u = b +� z +4a, +(22) +such that this term casts into the following form +B = 2a − b +b2 +� b/(2√a) +0 +arcsin u +(1 + αu2) +√ +1 − u2 du, +α = 4a(a − b) +b2 +(23) +Since 0 < u < 1, we employ the following expansion[22] +arcsin u +√ +1 − u2 = ++∞ +� +n=0 +2nn! +(2n + 1)!!u2n+1, +(24) +Plugging Eq.(24) into Eq.(23), interchanging the order of integration and summation, yields +B = 2a − b +b2 ++∞ +� +n=0 +2nn! +(2n + 1)!! +� b/(2√a) +0 +u2n+1 +1 + αu2 du, +(25) +It is convenient to make variable transformation for the second time +ξ = u2, +(26) +6 + +Then we obtain +B = 2a − b +b2 ++∞ +� +n=0 +2nn! +(2n + 1)!! +� b2/(4a) +0 +ξn +1 + αξdξ, +(27) +Changing the variable further +ξ = b2 +4aρ, +(28) +Eq.(27) can be written as +B = 2a − b +b2 ++∞ +� +n=0 +2nn! +(2n + 1)!! +� b2 +4a +�n+1 � 1 +0 +ρn� +1 + αb2 +4a ρ +�−1 +dρ, +(29) +Comparing with Eq.(A4), we identify +α = 1, +β = n + 1, +γ = n + 2, +z = b − a, +(30) +which generates the following result for B +B = 2a − b +2a ++∞ +� +n=0 +2nn! +(n + 1)(2n + 1)!! +� b2 +4a +�n +2F1(1, n + 1; n + 2; b − a), +(31) +Combining Eq.(20), Eq.(21) and Eq.(31), we arrive at the final result of F in the case b2−4a < 0 +F = −1 +2Li2(b − a) − +� +arcsin +b +2√a +�2 − 2 arcsin( +b +2√a) arctan +2a − b +√ +4a − b2 ++ (2a − b)b +2a ++∞ +� +n=0 +2nn! +(n + 1)(2n + 1)!! +� b2 +4a +�n +2F1(1, n + 1; n + 2; b − a). +(32) +Next, we consider the case b2 − 4a > 0. In this case there are two zeros of the logarithm in +the range [0, 1], the iε prescription must be retained appropriately. By exploiting integration +by parts, it is easy to get +F = − +� 1 +0 +(2ax − b) ln x +a(x − x1)(x − x2) dx, +(33) +where x1 and x2 are the two roots of the argument of the logarithm +x1 = x+ + iε, +x+ = b + +√ +b2 − 4a +2a +, +x2 = x− − iε, +x− = b − +√ +b2 − 4a +2a +, +(34) +Making use partial fraction expansion and Eq.(A3), we obtain the following result +F = +1 +x1 − x2 +( b +a − 2x1) +� 1 +0 +ln x +x − x1 +dx + +1 +x1 − x2 +(2x2 − b +a) +� 1 +0 +ln x +x − x2 +dx += +1 +x1 − x2 +� +( b +a − 2x1)Li2[ 1 +x+ +− iε sgn(x+)] − ( b +a − 2x2)Li2[ 1 +x− ++ iε sgn(x−)] +� +. +(35) +with the function sgn(x) defined in Eq.(A3). +7 + +III. +RESULTS AND DISCUSSIONS +We shall now apply the results obtained in Section 2 to calculate the integral left in Eq.(10). +Comparing Eq.(10) and Eq.(11), we identify that +a = m2 +1 +ω2 +2 +, +b = m2 +1 − ω2 +1 + ω2 +2 +ω2 +2 +, +(36) +In the case b2 − 4a < 0 which implies that λ(m2 +1, ω2 +1, ω2 +2) < 0, where λ(x, y, z) is the well-known +K¨allen function +λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz, +(37) +By employing Eq.(32), yields the following explicit result +I = +1 +m2 +1 +�1 +2 Li2 +� +1 − ω2 +1 +ω2 +2 +� +− +� +arcsin m2 +1 − ω2 +1 + ω2 +2 +2m1ω2 +�2 +− 2 arcsin m2 +1 − ω2 +1 + ω2 +2 +2m1ω2 +arctan m2 +1 + ω2 +1 − ω2 +2 +� +λ(m2 +1, ω2 +1, ω2 +2) ++ m4 +1 − (ω2 +1 − ω2 +2)2 +8m2 +1ω2 +2 +� +∞ +� +n=0 +2nn! +(n + 1)(2n + 1)!! +�(m2 +1 − ω2 +1 + ω2 +2)2 +4m2 +1ω2 +1 +�n +× 2F1 +� +1, n + 1; n + 2; 1 − ω2 +1 +ω2 +2 +��� +. +(38) +In order to apply the result in Eq.(38) correctly, the following comments are necessary. First, +from the conditions of a and b declared in Eq.(11), to guarantee Eq.(38) holds true for the +evaluation, in addition to λ(m2 +1, ω2 +1, ω2 +2) < 0, the masses must obey +m2 +1 > ω2 +1 − ω2 +2, +(39) +Second, since Eq.(38) is summed over hypergeometric functions, a crucial issue is that if the +summation of the infinite series is convergent. Due to λ(m2 +1, ω2 +1, ω2 +2) < 0, it is obvious that +0 < (m2 +1 − ω2 +1 + ω2 +2)2 +4m2 +1ω2 +1 +< 1. +(40) +and the hypergeometric functions are always taking finite value, thus the summation is conver- +gent. Finally, in considering the analytic property of the dilogarithm in Eq.(A2), a question is +that if Eq.(38) can develop imaginary part. In other words, if the argument of the dilogarithm +can be greater than 1. But it is impossible since both ω1 and ω2 are assumed to be real, thus +0 < 1 − (ω2 +1/ω2 +2) < 1 is always satisfied, therefore there is no imaginary part can be developed. +8 + +In the case b2 − 4a > 0, this implies that λ(m2 +1, ω2 +1, ω2 +2) > 0, exploiting Eq.(35) we get +I = 1 +m2 +1 +�1 +2 Li2(1 − ω2 +1 +ω2 +2 +) − Li2[ 1 +x+ +− iε sgn(x+)] − Li2[ 1 +x− ++ iε sgn(x−)] +� +, +(41) +where +x+ = (m2 +1 − ω2 +1 + ω2 +2) + λ1/2(m2 +1, ω2 +1, ω2 +2) +2m2 +1 +, +x− = (m2 +1 − ω2 +1 + ω2 +2) − λ1/2(m2 +1, ω2 +1, ω2 +2) +2m2 +1 +, +(42) +Since m2 +1 > ω2 +1 − ω2 +2 as presented in Eq.(39), both x+ and x− are positive definite, thus Eq.(41) +simplified to +I = 1 +m2 +1 +�1 +2 Li2(1 − ω2 +1 +ω2 +2 +) − Li2( 1 +x+ +− iε) − Li2( 1 +x− ++ iε) +� +. +(43) +In order to explore phenomenological implications of the results presented in Eq.(38) and +Eq.(43), it is instructive to consider the special case that the coefficients in Eq.(11) are con- +strained by the following conditions +a = b > 0, +(44) +This is the integral indispensable in the evaluation of H → gg decay. studied in past. Supposing +the mass of each propagator of the triangle is ω1, setting +a = b = m2 +1 +ω2 +1 +, +(45) +Hence, Eq.(10) reduces to +I = 1 +m2 +1 +� 1 +0 +dxln(ax2 − ax + 1 − iε) +x +, +a = m2 +1 +ω2 +1 +(46) +First consider the case 0 < a < 4, i.e., 0 < m1 < 2ω1. To get the correct results we must trace +back to Eq.(12) and Eq.(13), other than taking advantage of Eq.(32) and simply setting a = b, +otherwise it will make some mistakes. By exploiting Eq.(17), we immediately obtain +I = − a +2m2 +1 +� 1 +0 +dz +� 1 +0 +dx +1 +1 + zax(x − 1) += − a +2m2 +1 +� 1 +0 +dz +4 +√ +4az − a2z2 arctan +az +√ +4az − a2z2 += − 2 +m2 +1 +� +arctan +a +√ +4a − a2 +�2 +, +(47) +9 + +By using Eq.(19), we get the well-known function appeared in one loop evaluation of H → gg +I = − 2 +m2 +1 +� +arcsin m1 +2ω1 +�2 +, +0 < m1 < 2ω1. +(48) +Next, we consider the case of a > 4, i.e., m1 > 2ω1. From Eq.(35) we get +I = 1 +m2 +1 +� +− Li2( 1 +x− ++ iε) − Li2( 1 +x+ +− iε) +� +, +(49) +x+ and x− are given by +x+ = 1 +2 +� +1 + +� +1 − 4 +a +� +, +x− = 1 +2 +� +1 − +� +1 − 4 +a +� +, +(50) +Since both x+ and x− are less than 1, in order to simplify the Eq.(49), defining +α = 1 +x− +> 1, +1 +x+ += +1 +1 − x− += +α +α − 1 > 1, +(51) +Then Eq.(49) can be written as +I = 1 +m2 +1 +� +− Li2(α + iε) − Li2( +α +α − 1 − iε) +� +, +(52) +Combining Eq.(A2) we obtain +I = +1 +m2 +1 +� +− +� +Re Li2(α) + iπ ln α +� +− +� +Re Li2( +α +α − 1) − iπ ln +α +α − 1 +� += +1 +m2 +1 +� +− Re +� +Li2(α) + Li2( +α +α − 1) +� +− iπ ln(α − 1) +� +, +(53) +In considering the property od dilogarithm +Re +� +Li2(x) + Li2( +x +x − 1) +� += π2 +2 − ln2(x − 1) +2 +, +x > 1 +(54) +which leads to +I = 1 +m2 +1 +� +− π2 +2 + ln2(α − 1) +2 +− iπ ln(α − 1) +� +, +(55) +Substituting Eq.(51) into Eq.(55) and noticing that +α − 1 = x+ +x− +, +(56) +we get +I = 1 +m2 +1 +� +− π2 +2 + 1 +2 ln2 x+ +x− +− iπ ln x+ +x− +� +, +(57) +Plugging Eq.(50) into Eq.(57) we arrive at the well-known function in the one loop evaluation +of H → gg decay +I = − 1 +2m2 +1 +� +π + i ln +1 + +� +1 − 4ω2 +1 +m2 +1 +1 − +� +1 − 4ω2 +1 +m2 +1 +�2 +. +m1 > 2ω1 +(58) +10 + +IV. +SUMMARY +In this paper, the amplitude of scalar one loop three-point diagram in which the masses of +the three internal propagators are assigned two different masses is evaluated. A general type +integral is extracted and analytic results are obtained. Conditions of the validity of the results +are discussed in details. As a check to the results, we consider the case that each propagator of +the triangle taking the same mass, we find that the general results will reduce to the functions +obtained in the lowest order evaluation of H → gg decay. We also notice that the results are +mathematically preferred in that the diagram does not corresponds to real process in H → gg +in the SM. However, we still hope that the results and techniques may be found its applications +in triangle mediated decays addition to H → gg. +Appendix A: The dilogarithm function and integral representation of Gauss hyperge- +ometric function +In this section we list some necessary formula in our evaluation. The dilogarithm is defined +as[19] +Li2(x) = ++∞ +� +k=1 +xk +k2 = − +� 1 +0 +ln(1 − xt) +t +dt, +|x| < 1 +(A1) +There is a branch cut from 1 to ∞, for ε → 0 +Li2(x + iε) = Re Li2(x) + iπ sgn(ε)Θ(x − 1) ln x, +(A2) +where Θ is the step function, the sgn(x) is +sgn(x) = + + + + + +1 +x > 0 +−1 +x < 0 +Another two formulas we need are the following equation of dilogarithm[20] +� 1 +0 +ln x +a + bx dx = 1 +bLi2 +� +− b +a +� +, +(A3) +and the integral representation of the Gauss hypergeometric function[23, 24] +2F1(α, β; γ; z) = +Γ(γ) +Γ(β)Γ(γ − β) +� 1 +0 +tβ−1(1 − t)γ−β−1(1 − zt)−αdt, +(A4) +11 + +where +Re(γ) > Re(β) > 0, +|arg(1 − z)| < π. +(A5) +[1] G. Aad et al.(ATLAS), PhysLett. B 716, 1(2012). +[2] S. Chatrchyan et al.(CMS), B 716, 30(2012). +[3] J. F. Gunion, H. E. Haber, G. Kane, S. Dawson, The Higgs Hunter’s Guide (Perseus Publishing, +Cambridge, Massachusetts, 1990). +[4] B. A. Kniehl, Phys. Rept. 240, 211(1994). +[5] A. Djouadi, Phys. Rept. 457, 1(2008), Phys. Rept. 459, 1(2008). +[6] M. Spira, Prog. Part. Nucl. Phys. 95, 98(2017). +[7] M. Carena, C. Grojean, M. Kado et al, “Status of Higgs Boson Physics”, R. L. Workman et +al.(Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01(2022). +[8] M. Roth, A. Denner, Nucl. Phys. B 479, 495(1996). +[9] J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B 106, 292(1976). +[10] T. G. Rizzo, Phys. Rev. D 22, 178(1980), Phys. Rev. D 22, 1824(1980). +[11] A. I. Vainshtein, M. B. Voloshin, V. I. Zakharov et al, Sov. J. Nucl. Phys. 30, 711(1979). +[12] L. B. Okun, Leptons and Quarks(North-Holland, Amsterdam, 1982). +[13] J. F. Gunion and H. E. Haber, Nucl. Phys. B 278, 449(1986), Erratum: Nucl. Phys. B 402, +569(1993). +[14] R. K. Ellis, I. Hinchliffe, M. Soldate et al, Nucl. Phys. B 297, 221(1988). +[15] D. Huang, Y. Tang and Y-L. Wu, Commun. Theor. Phys. 57, 427(2012). +[16] M. Shifman, A. Vainshtein, M. B. Voloshin, et al, Phys. Rev. D 85, 013015(2012). +[17] W. J. Marciano, C. Zhang and S. Willenbrock, Phys. Rev. D 85, 013002(2012). +[18] H. B. Dwight, Tables of Integrals and Other Mathematical Data(The Macmillan Company, New +York, 1957), Third edition. +[19] L. Lewin, Polylogarithms and Associated Functions(North Holland, New York, 1981), Second +Edition. +[20] A. Devoto and D. W. Duke, Riv. Nuovo Cim. 7N6, 1(1984). +12 + +[21] I. S. Gradshteyn I. M. Ryzhik, Table of Integrals, Series, and Products, Eighth Edition(Academic +Press, London, 2014). +[22] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with formulas, Graphs and +Mathematical Tables(Dover Publications, New York,1972). +[23] A. Erdelyi, Higher Transcendental Functions, Vol.I (McGrill-Hall Book Company, New York, +1953). +[24] Z. X. Wang, D. R. Guo, Special Functions(World Scientific, Singapore, 1989). +13 + diff --git a/5tE4T4oBgHgl3EQfBgtg/content/tmp_files/load_file.txt b/5tE4T4oBgHgl3EQfBgtg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..510e96b03b2f460591cd121247135f347e9b9f30 --- /dev/null +++ b/5tE4T4oBgHgl3EQfBgtg/content/tmp_files/load_file.txt @@ -0,0 +1,333 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf,len=332 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='04852v1 [hep-ph] 12 Jan 2023 Evaluation of one type scalar one loop three-point amplitude inspired by H → gg decay in the standard model Jin Zhang∗ School of Physics and Engineering, Yuxi Normal University, Yuxi, Yunnan, 653100, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' China Abstract Motivated by the Higgs boson decaying to gg at one loop approximation, the amplitude of scalar one loop three-point diagram with two different internal masses are evaluated and fully analytic results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The main ingredient of the evaluation is a integral in which the integrand is product of the reciprocal of the integral variable and a logarithm, where the argument of the logarithm is a quadratic function of the general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The results depend on the choice of the masses of the propagators and the massive external line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In the first case the amplitude contains an infinite series in which each term is a hypergeometric function, in the second case the result is expressed through dilogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In particular, if the three internal lines are taking the same mass, the results will reduce to the known functions in one loop evaluation of Higgs decaying to gg or γγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' PACS numbers: ∗ jinzhang@yxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='cn 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' INTRODUCTION In framework of the Standard Model(SM) and its minimal supersymmetric extension, the evaluation of scalar one loop three-point amplitudes play a fundamental role in deciphering the property of the Higgs boson[1, 2] though its decaying to gg(or γγ) and the inverse process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=', production of the Higgs boson by gluon fusion[3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Owing to the coupling of the Higgs boson to the fermions gHf ¯f and the coupling of fermions to gluon, the three propagators take the same mass at the leading order of perturbative theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The evaluated amplitude, if the high energy approximation[9, 10] is not exploited, will be expressed as function of mf/mH, where mf and mH are the masses of the internal fermion and the Higgs boson, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' At the final stage of the evaluation, a integral of the following form must be handled carefully I1 = � 1 0 dx ln(ax2 − ax + 1 − iε) x , a > 0 (1) where ε is positive infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (1) necessarily arises both in the evaluation of the amplitude of Higgs boson decaying to gg and its production via gluon fusion at one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In addition to the top quark, there is considerable mass hierarchy between the Higgs boson and other quarks, an economic way to compute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (1) is taking the limit that the masses of the propagators in the triangle are negligible compared with mass of the Higgs boson, then the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (1) tends to constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' However, in this manner we can not tell the different contributions from various competing processes to the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Thus, fully analytic results is essential to analyze H → gg, then results will be distinguished the cases 0 < a < 4 from the case a > 4 as detailed in the later works[5, 11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' As a natural generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (1), let us consider the following integral I2 = � 1 0 dx ln(ax2 − bx + 1 − iε) x , (2) the parameters a and b satisfy a > 0, b > 0, a ̸= b (3) It is obvious that if a = b, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2) reduce to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' This integral can be derived from the evaluation of the scalar one loop three-point diagram depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='1, in this case a and b will be functions of the masses of the propagators ω1 and ω1 as well as the mass of external line m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Unfortunately, a close inspection to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='1 indicates that it does not connect with real decaying 2 p1 p2 p3 k ω2 ω2 ω1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 1: Massive triangle with two massless external lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The solid lines and dashed lines denote massive and massless particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' processes of the Higgs boson even though the contribution from Higgs-Kibble ghosts associated with the W ± and Z bosons are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Maybe it is the reason that rarely can we look up the evaluation of integral displayed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2) in the one loop evaluation of Higgs boson decaying to gg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' An thorough investigation of the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2) on the footing of perturbative theory is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Therefore, in this paper we will present a systematic study on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2) based on the evaluation of scalar one loop three-point amplitude, the complete analytic results are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' We hope that the results can be applied to some decaying process under reasonable approximation addition to H → gg, but also enrich the results of scalar one loop three-point diagram from the viewpoint of analytic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In section II we introduce the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2) by evaluation the amplitude depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='1 in a scalar field theory, some general results are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In section III the analytic results of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='1 are obtained and the special case a = b are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' A short summary are presented in IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Some useful formulas are listed in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 1 Integral of this type has been computed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (23) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' [8], but only the case b2 − 4a > 0 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' THE FORMULAS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The massive triangle with two massless external lines To start with, we write down the amplitude corresponding to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='1 I = � d4k (2π)4 1 A1A2A3 , (4) where the three denominators are defined by A1 = k2 − ω2 1 + iε A2 = (p1 − k)2 − ω2 2 + iε A3 = (p1 − p2 − k)2 − ω2 2 + iε, (5) and ε is real positive infinitesimal, the three external momentum satisfy p2 1 = m2 1, p2 2 = p2 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (6) Using the Feynman’s trick, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (4) can be written as I = � d4k (2π)4 � dxdydz2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' δ(1 − x − y − z) [D(x, y, z)]3 , (7) where D(x, y, z) = x(k2 − ω2 1 + iε) + y[(p1 − k)2 − ω2 2 + iε] + z[(p1 − p2 − k)2 − ω2 2 + iε], (8) Since the amplitude given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (4) is both ultraviolet and infrared finite, thus regularization is unnecessary, the evaluation can be carried out in the four-dimensional space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' We first perform the integral over k and z, obtaining I = − i 16π2 � 1 0 dx � 1−x 0 dy 1 −yxm2 1 + x(ω2 1 − ω2 2) + ω2 2 − iε (9) The integral over y in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (9) is trivial, combining with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A1), we arrive at the following intermediate result I = i 16π2m2 1 � 1 0 dx1 x � ln �m2 1 ω2 2 x2 − m2 1 − ω2 1 + ω2 2 ω2 2 x + 1 − iε � − ln �ω2 1 − ω2 2 ω2 2 x + 1 − iε �� = i 16π2m2 1 � Li2 � 1 − ω2 1 ω2 2 � + � 1 0 dx1 x ln �m2 1 ω2 2 x2 − m2 1 − ω2 1 + ω2 2 ω2 2 x + 1 − iε �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (10) The remaining work is the evaluation of the last integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' For brevity, in the forth- coming sections the pre-factor i/(16π2) will be suppressed while 1/m2 1 will be preserved so as to maintain the correct dimension of the primitive amplitude displayed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' evaluation of integral with logarithms The evaluation of the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (10) motivates a general investigation on the integral of the following type F = � 1 0 dx ln(ax2 − bx + 1 − iε) x , a > 0, b > 0 (11) Since the argument of the logarithm is quadratic in x, we first consider the case b2 − 4a < 0, in this case the argument of the logarithm is positive definite thus the iε term can be safely dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' A feasible way to calculate the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (11) turns out to be expressing it as F = � 1 0 dx � 1 0 dz ax − b 1 + zx(ax − b) = a � 1 0 dz � 1 0 dx x 1 + zx(ax − b) − b � 1 0 dz � 1 0 dx 1 1 + zx(ax − b) = 1 2 � 1 0 dzln[1 + z(a − b)] z − b 2 � 1 0 dz � 1 0 dx 1 1 + zx(ax − b)) = −1 2Li2(b − a) − b 2 � 1 0 dz � 1 0 dx 1 1 + zx(ax − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (12) Now we concentrate on the last integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (12), for later convenience we label it as A, the integral over x can be calculated[18] A = � 1 0 dz � 1 0 dx 1 1 + zx(ax − b) = � 1 0 dz 2 √ 4az − b2z2 � arctan bz √ 4az − b2z2 + arctan 2az − bz √ 4az − b2z2 � , (13) In deriving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (13) we employ the property that arctan(x) is odd arctan(−x) = − arctan(x), (14) To proceed we separate the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (13) into two parts A = 2(A1 + A2), (15) where A1 = � 1 0 dz 1 √ 4az − b2z2 arctan bz √ 4az − b2z2, 5 A2 = � 1 0 dz 1 √ 4az − b2z2 arctan 2az − bz √ 4az − b2z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (16) It is not difficult to demonstrate that � arcsin � b � z 4a ��′ = � arctan bz √ 4az − b2z2 �′ = b 2 1 √ 4az − b2z2, (17) Hence, the integral in A1 is easy to calculate A1 = 1 b � arctan b √ 4a − b2 �2, (18) Using the identity[21] arctan x = arcsin x √ 1 + x2, (19) leads to the final result for A1 A1 = 1 b � arcsin b 2√a �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (20) Next, we evaluate A2, by making use integration by parts, getting A2 = 2 b arcsin( b 2√a) arctan 2a − b √ 4a − b2 − 2a − b b � 1 0 arcsin � b� z 4a � [1 + (a − b)z] √ 4az − b2z2 dz, (21) We label the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (21) as B and define u = b � z 4a, (22) such that this term casts into the following form B = 2a − b b2 � b/(2√a) 0 arcsin u (1 + αu2) √ 1 − u2 du, α = 4a(a − b) b2 (23) Since 0 < u < 1, we employ the following expansion[22] arcsin u √ 1 − u2 = +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='u2n+1, (24) Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (24) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (23), interchanging the order of integration and summation, yields B = 2a − b b2 +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' � b/(2√a) 0 u2n+1 1 + αu2 du, (25) It is convenient to make variable transformation for the second time ξ = u2, (26) 6 Then we obtain B = 2a − b b2 +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' � b2/(4a) 0 ξn 1 + αξdξ, (27) Changing the variable further ξ = b2 4aρ, (28) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (27) can be written as B = 2a − b b2 +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' � b2 4a �n+1 � 1 0 ρn� 1 + αb2 4a ρ �−1 dρ, (29) Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A4), we identify α = 1, β = n + 1, γ = n + 2, z = b − a, (30) which generates the following result for B B = 2a − b 2a +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (n + 1)(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' � b2 4a �n 2F1(1, n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' b − a), (31) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (20), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (31), we arrive at the final result of F in the case b2−4a < 0 F = −1 2Li2(b − a) − � arcsin b 2√a �2 − 2 arcsin( b 2√a) arctan 2a − b √ 4a − b2 + (2a − b)b 2a +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (n + 1)(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' � b2 4a �n 2F1(1, n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' b − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (32) Next, we consider the case b2 − 4a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In this case there are two zeros of the logarithm in the range [0, 1], the iε prescription must be retained appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' By exploiting integration by parts, it is easy to get F = − � 1 0 (2ax − b) ln x a(x − x1)(x − x2) dx, (33) where x1 and x2 are the two roots of the argument of the logarithm x1 = x+ + iε, x+ = b + √ b2 − 4a 2a , x2 = x− − iε, x− = b − √ b2 − 4a 2a , (34) Making use partial fraction expansion and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A3), we obtain the following result F = 1 x1 − x2 ( b a − 2x1) � 1 0 ln x x − x1 dx + 1 x1 − x2 (2x2 − b a) � 1 0 ln x x − x2 dx = 1 x1 − x2 � ( b a − 2x1)Li2[ 1 x+ − iε sgn(x+)] − ( b a − 2x2)Li2[ 1 x− + iε sgn(x−)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (35) with the function sgn(x) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 7 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS We shall now apply the results obtained in Section 2 to calculate the integral left in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (11), we identify that a = m2 1 ω2 2 , b = m2 1 − ω2 1 + ω2 2 ω2 2 , (36) In the case b2 − 4a < 0 which implies that λ(m2 1, ω2 1, ω2 2) < 0, where λ(x, y, z) is the well-known K¨allen function λ(x, y, z) = x2 + y2 + z2 − 2xy − 2xz − 2yz, (37) By employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (32), yields the following explicit result I = 1 m2 1 �1 2 Li2 � 1 − ω2 1 ω2 2 � − � arcsin m2 1 − ω2 1 + ω2 2 2m1ω2 �2 − 2 arcsin m2 1 − ω2 1 + ω2 2 2m1ω2 arctan m2 1 + ω2 1 − ω2 2 � λ(m2 1, ω2 1, ω2 2) + m4 1 − (ω2 1 − ω2 2)2 8m2 1ω2 2 � +∞ � n=0 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (n + 1)(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' �(m2 1 − ω2 1 + ω2 2)2 4m2 1ω2 1 �n × 2F1 � 1, n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' n + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 1 − ω2 1 ω2 2 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (38) In order to apply the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (38) correctly, the following comments are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' First, from the conditions of a and b declared in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (11), to guarantee Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (38) holds true for the evaluation, in addition to λ(m2 1, ω2 1, ω2 2) < 0, the masses must obey m2 1 > ω2 1 − ω2 2, (39) Second, since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (38) is summed over hypergeometric functions, a crucial issue is that if the summation of the infinite series is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Due to λ(m2 1, ω2 1, ω2 2) < 0, it is obvious that 0 < (m2 1 − ω2 1 + ω2 2)2 4m2 1ω2 1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (40) and the hypergeometric functions are always taking finite value, thus the summation is conver- gent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Finally, in considering the analytic property of the dilogarithm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A2), a question is that if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (38) can develop imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' In other words, if the argument of the dilogarithm can be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' But it is impossible since both ω1 and ω2 are assumed to be real, thus 0 < 1 − (ω2 1/ω2 2) < 1 is always satisfied, therefore there is no imaginary part can be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 8 In the case b2 − 4a > 0, this implies that λ(m2 1, ω2 1, ω2 2) > 0, exploiting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (35) we get I = 1 m2 1 �1 2 Li2(1 − ω2 1 ω2 2 ) − Li2[ 1 x+ − iε sgn(x+)] − Li2[ 1 x− + iε sgn(x−)] � , (41) where x+ = (m2 1 − ω2 1 + ω2 2) + λ1/2(m2 1, ω2 1, ω2 2) 2m2 1 , x− = (m2 1 − ω2 1 + ω2 2) − λ1/2(m2 1, ω2 1, ω2 2) 2m2 1 , (42) Since m2 1 > ω2 1 − ω2 2 as presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (39), both x+ and x− are positive definite, thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (41) simplified to I = 1 m2 1 �1 2 Li2(1 − ω2 1 ω2 2 ) − Li2( 1 x+ − iε) − Li2( 1 x− + iε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (43) In order to explore phenomenological implications of the results presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (38) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (43), it is instructive to consider the special case that the coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (11) are con- strained by the following conditions a = b > 0, (44) This is the integral indispensable in the evaluation of H → gg decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' studied in past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Supposing the mass of each propagator of the triangle is ω1, setting a = b = m2 1 ω2 1 , (45) Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (10) reduces to I = 1 m2 1 � 1 0 dxln(ax2 − ax + 1 − iε) x , a = m2 1 ω2 1 (46) First consider the case 0 < a < 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=', 0 < m1 < 2ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' To get the correct results we must trace back to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (12) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (13), other than taking advantage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (32) and simply setting a = b, otherwise it will make some mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' By exploiting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (17), we immediately obtain I = − a 2m2 1 � 1 0 dz � 1 0 dx 1 1 + zax(x − 1) = − a 2m2 1 � 1 0 dz 4 √ 4az − a2z2 arctan az √ 4az − a2z2 = − 2 m2 1 � arctan a √ 4a − a2 �2 , (47) 9 By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (19), we get the well-known function appeared in one loop evaluation of H → gg I = − 2 m2 1 � arcsin m1 2ω1 �2 , 0 < m1 < 2ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (48) Next, we consider the case of a > 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=', m1 > 2ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (35) we get I = 1 m2 1 � − Li2( 1 x− + iε) − Li2( 1 x+ − iε) � , (49) x+ and x− are given by x+ = 1 2 � 1 + � 1 − 4 a � , x− = 1 2 � 1 − � 1 − 4 a � , (50) Since both x+ and x− are less than 1, in order to simplify the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (49), defining α = 1 x− > 1, 1 x+ = 1 1 − x− = α α − 1 > 1, (51) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (49) can be written as I = 1 m2 1 � − Li2(α + iε) − Li2( α α − 1 − iε) � , (52) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A2) we obtain I = 1 m2 1 � − � Re Li2(α) + iπ ln α � − � Re Li2( α α − 1) − iπ ln α α − 1 � = 1 m2 1 � − Re � Li2(α) + Li2( α α − 1) � − iπ ln(α − 1) � , (53) In considering the property od dilogarithm Re � Li2(x) + Li2( x x − 1) � = π2 2 − ln2(x − 1) 2 , x > 1 (54) which leads to I = 1 m2 1 � − π2 2 + ln2(α − 1) 2 − iπ ln(α − 1) � , (55) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (51) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (55) and noticing that α − 1 = x+ x− , (56) we get I = 1 m2 1 � − π2 2 + 1 2 ln2 x+ x− − iπ ln x+ x− � , (57) Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (50) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (57) we arrive at the well-known function in the one loop evaluation of H → gg decay I = − 1 2m2 1 � π + i ln 1 + � 1 − 4ω2 1 m2 1 1 − � 1 − 4ω2 1 m2 1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' m1 > 2ω1 (58) 10 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' SUMMARY In this paper, the amplitude of scalar one loop three-point diagram in which the masses of the three internal propagators are assigned two different masses is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' A general type integral is extracted and analytic results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Conditions of the validity of the results are discussed in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' As a check to the results, we consider the case that each propagator of the triangle taking the same mass, we find that the general results will reduce to the functions obtained in the lowest order evaluation of H → gg decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' We also notice that the results are mathematically preferred in that the diagram does not corresponds to real process in H → gg in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' However, we still hope that the results and techniques may be found its applications in triangle mediated decays addition to H → gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Appendix A: The dilogarithm function and integral representation of Gauss hyperge- ometric function In this section we list some necessary formula in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' The dilogarithm is defined as[19] Li2(x) = +∞ � k=1 xk k2 = − � 1 0 ln(1 − xt) t dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' |x| < 1 (A1) There is a branch cut from 1 to ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' for ε → 0 Li2(x + iε) = Re Li2(x) + iπ sgn(ε)Θ(x − 1) ln x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A2) where Θ is the step function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' the sgn(x) is sgn(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 x > 0 −1 x < 0 Another two formulas we need are the following equation of dilogarithm[20] � 1 0 ln x a + bx dx = 1 bLi2 � − b a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A3) and the integral representation of the Gauss hypergeometric function[23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' 24] 2F1(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' z) = Γ(γ) Γ(β)Γ(γ − β) � 1 0 tβ−1(1 − t)γ−β−1(1 − zt)−αdt, (A4) 11 where Re(γ) > Re(β) > 0, |arg(1 − z)| < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (A5) [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' (ATLAS), PhysLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfBgtg/content/2301.04852v1.pdf'} +page_content=' B 716, 1(2012).' metadata={'source': 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Quantum teleportation has enabled the transfer of quantum +information, but teleportation of quantum physical quantities has not yet been realized. +Here +we report the first realization and observation of quantum energy teleportation on real quantum +hardware. +We achieve this by using several IBM’s superconducting quantum computers. +The +results are consistent with the exact solution of the theory and are improved by the mitigation +of measurement error. Quantum energy teleportation requires only local operations and classical +communication. Therefore our results provide a realistic benchmark that is fully achievable with +current quantum computing and communication technologies. +I. +QUANTUM ENERGY TELEPORTATION +While it is fairly widely known that information about +quantum states can be transported to remote loca- +tions [1–4], it is less well known that quantum state +energy can be similarly transmitted, despite its impact +and potential for future applications. Quantum informa- +tion transferred by quantum teleportation is not a phys- +ical quantity, but energy is a distinct physical quantity. +Transferring physical quantities to remote locations is an +unexplored area of technology. Quantum Energy Tele- +portation (QET) was proposed by Hotta about 15 years +ago and has been studied theoretically for spin chains [5– +7], an ion trap system [8], a quantum Hall system [9], and +other various theoretical systems [10, 11]. It is surprising +that (to the best of knowledge of the author) QET has +never been confirmed by any experiment on any system +before, even though it can be achieved with a very simple +quantum system. The purpose of this paper is to make +the first experimental realization of QET in actual quan- +tum hardware and to establish the quantum circuits that +make it possible. We achieved the realization of QET us- +ing some IBM quantum computers by applying quantum +error mitigation [12–14]. The methods we have estab- +lished can be applied to any system capable of QET. +In what follows, we explain that QET is a universal +means of quantum energy transfer, just as quantum tele- +portation is a universal means of quantum information +transfer. Any non-trivial local operations, including mea- +surements on the ground state of a quantum many-body +system give rise to excited states, which in turn increase +the energy expectation value. Note that the increase in +energy is supplied by the experimental devices. An im- +portant property of the ground state of a quantum many- +body system is that it has entanglement, which brings +about quantum fluctuations in the global ground state +energy. In other words, quantum fluctuations in the en- +ergy of the local systems are entangled. Local measure- +∗kazuki7131@gmail.com +ment of the quantum state at a subsystem A partially de- +stroys this ground state entanglement. At the same time, +energy EA from the device making the measurement is +injected into the entire system. The injected energy EA +stays around the subsystem A in the very early stages +of time evolution, but operations around A alone cannot +extract EA from the system. This is because informa- +tion about EA is also stored in remote locations other +than A due to the entanglement that exists prior to the +measurement. In other words, the locally injected energy +EA can be partially extracted at any location other than +A [15]. QET is the protocol that makes this possible. Up +to this point, no special assumptions about the system +have been used. +The crucial property of QET is that +it can be realized entirely by the general nature of the +ground state of the quantum many-body system and the +universal fact of measurement. +We work on the minimal QET model given in [16]. +One of the purposes of this paper is to give a quantum +circuit that utilizes QETs with real quantum computers +and quantum networks. The complete form of quantum +circuits we used for QET is displayed in Fig. 1. The max- +imum circuit depth is 10 and the number of qubits used +is 2. +Hence, current quantum computers are powerful +enough to implement QET. +Let k, h be positive real numbers. The Hamiltonian of +the minimal model is +Htot = H0 + H1 + V, +(1) +Hn = hZn + +h2 +√ +h2 + k2 , (n = 0, 1) +(2) +V = 2kX0X1 + +2k2 +√ +h2 + k2 . +(3) +The ground state of Htot is +|g⟩ = +1 +√ +2 +� +1 − +h +√ +h2 + k2 |00⟩− 1 +√ +2 +� +1 + +h +√ +h2 + k2 |11⟩ , +(4) +The constant terms in the Hamiltonians are added so +that the ground state |g⟩ of Htot returns the zero mean +arXiv:2301.02666v1 [quant-ph] 7 Jan 2023 + +2 +FIG. 1: Quantum gate operations used for quantum energy teleportation. (A) preparation of ground state and Alice’s X0 +measurement to deposit her energy. She tells Bob via classical communication whether µ = −1 or µ = +1 was observed. (B) +Bob’s conditional operations to receive energy. He selects an operation U1(+1) or U1(−1) based on µ = +1 or −1, corresponding +to the Maxwell demon operation. (C) Equivalent implementation of Bob’s operations on a quantum computer. +energy for all local and global Hamiltonians: +⟨g| Htot |g⟩ = ⟨g| H0 |g⟩ = ⟨g| H1 |g⟩ = ⟨g| V |g⟩ = 0. (5) +However it should be noted that |g⟩ is neither a ground +state nor an eigenstate of Hn, V, Hn + V (n = 0, 1). The +essence of QET is to extract negative ground state energy +of those local and semi-local Hamiltonians. +The QET protocol is as follows. First, Alice makes a +measurement on her Pauli operator X0 by P0(µ) = 1 +2(1+ +(−1)µX0) and then she obtains either µ = −1 or +1. At +this point, Alice’s expectation energy is E0 = +h2 +√ +h2+k2 . +Via a classical channel, Alice then sends her measure- +ment result µ to Bob, who applies an operation U1(µ) to +his qubit and measures H1 and V . The density matrix +ρQET after Bob operates U1(µ) to P0(µ) |g⟩ is +ρQET = +� +µ∈{−1,1} +U1(µ)P0(µ) |g⟩ ⟨g| P0(µ)U † +1(µ). +(6) +Using ρQET, the expected local energy at Bob’s subsys- +tem is evaluated as ⟨E1⟩ = Tr[ρQET(H1 + V )], which +is negative in general. +Due to the conservation of en- +ergy, EB = −⟨E1⟩(> 0) is extracted from the system by +the device that operates U1(µ) [17]. In this way, Alice +and Bob can transfer the energy of the quantum system +only by operations on their own local system and classical +communication (LOCC). +II. +QUANTUM CIRCUIT IMPLEMENTATION +OF QUANTUM ENERGY TELEPORTATION +A. +Preparation of Ground State +Here we explain how to construct a quantum circuit +(Fig. 1 (A)) that generates the exact ground state |g⟩. +Let us begin with a Bell state |Φ−⟩ = |00⟩−|11⟩ +√ +2 +since |g⟩ +is resemble to it. |Φ−⟩ can be prepared by +|00⟩ − |11⟩ +√ +2 += (Z ⊗ I)CNOT(H ⊗ I |00⟩ +(7) +where CNOT= |0⟩ ⟨0|⊗I +|1⟩ ⟨1|⊗X. Using Y = SXS†, +we can perform a gate operation that maps |Φ−⟩ to the +ground state |g⟩ (eq. (4)) by a combination of one- and +two-qubit operators +|g⟩ = exp(−iαX ⊗ Y ) |Φ−⟩ += +1 +√ +2(cos α + sin α) |00⟩ − 1 +√ +2(cos α − sin α) |11⟩ . +(8) +where +α +is +designed +to +satisfy +cos α + sin α += +� +1 − +h +√ +h2+k2 and cos α − sin α = +� +1 + +h +√ +h2+k2 . +Those quantum operations are implemented by the +quantum circuit in Fig. 1 (A). +B. +Step 1: Deposit Energy +We use the following projective measurement operator +P0(µ) = 1 +2(1 + (−1)µX0). +(9) + +H +H +st +Rx(2α) +s +H +Ui(+1) +Ui(-1) +Ui(+1) +Ui(-1) +H +Ui(+1) +Ui(-1) +Ui(+1) +Ui(-1)3 +We measure Alice’s X operator, by which we obtain a +state |+⟩ or |−⟩. This operation does not affect Bob’s +energy since [X0, V ] = [X0, H1] = 0. Using [P0(µ), V ] = +0 and ⟨+| Z |+⟩ = ⟨−| Z |−⟩ = 0, we find that Alice’s +mean energy to deposit is +⟨E0⟩ = +� +µ∈{−1,1} +⟨g| P0(µ)HtotP0(µ) |g⟩ = +h2 +√ +h2 + k2 . +(10) +Alice’s operation can be implemented on a quantum +circuit in Fig 1 (A). ⟨E0⟩ can be calculated with the out- +put bit-strings 00, 01, 10, 11. Analytical values ⟨E0⟩ and +results with quantum computers for different pairs of k +and h are summarized in Table I. +C. +Step 2: Receive Energy +As soon as Alice observes µ ∈ {0, 1}, she tells her result +to Bob who operates UB(µ) to his qubit and measures his +energy. Here UB(µ) is +U1(µ) = cos θI − iµ sin θY1 = RY (2µθ), +(11) +where θ obeys +cos(2θ) = +h2 + k2 +� +(h2 + 2k2)2 + h2k2 +(12) +sin(2θ) = +hk +� +(h2 + 2k2)2 + h2k2 . +(13) +The average quantum state ρQET eq.(6) is obtained af- +ter Bob operates U1(µ) to P0(µ) |g⟩. Then the average +energy Bob measures is +⟨E1⟩ = Tr[ρQET(H1 + V )] = Tr[ρQETHtot] − ⟨E0⟩, (14) +where we used [U1(µ), H1] = 0. It is important that the +map � +µ∈{−1,1} P0(µ) |g⟩ ⟨g| P0(µ) → ρQET is not a uni- +tary transformation. Therefore eq. (14) can be negative. +This is in contrast to eq. (A7). +Now let us explain quantum circuits for the QET pro- +tocol. Since V and H1 do not commute, measurement on +those terms should be done separately. In other words, +Bob measures V and H1 independently and obtains ⟨V ⟩ +and ⟨H1⟩ statistically. As the figures show, ⟨V ⟩ is always +negative and ⟨H1⟩ is always positive. Therefore is suffi- +cient for Bob to measure only ⟨V ⟩ to receive energy by +QET. +We consider V (µ) = ⟨g| P0(µ)U † +1(µ)V U1(µ)P0(µ) |g⟩. +The quantum circuit to measure V (µ) is shown in the +right panel of Fig. 1 (B). It is important to note that, +since Bob knows µ which contains Alice’s information, he +can obtain VQET(µ) by local measurement only, although +V is not a local operator. Similarly we can measure H1 +in Z-basis as in the left panel of Fig. 1 (B). The corre- +sponding quantum circuit is obtained by removing the +second Hadamard gate from the previous circuit Fig. 1 +(C). On average the circuit generates +⟨E1⟩ = +� +µ∈{−1,1} +⟨g| P0(µ)U † +1(µ)(H1 + V )U1(µ)P0(µ) |g⟩ += − +1 +√ +h2 + k2 [hk sin(2θ) − (h2 + 2k2)(1 − cos(2θ))]. +(15) +If θ is small, ⟨E1⟩ is negative. +Bob receives energy +⟨EB⟩ = −⟨E1⟩ on average. +In Appendix B, we per- +formed measurement of V (µ) and H1 based on the quan- +tum circuit Fig. 1 (B) and summarized data in Table II, +where numerical values are compared with analytical val- +ues given in eq. (15). +D. +QET on Real Quantum Hardware +Here we describe how to implement the conditional +operations that may not be natively supported by many +quantum computers and quantum devices. In the QET +protocol, Bob’s operation must be selected according to +the results of Alice’s measurements, as shown in Fig. 1 +(B). Even in environments where conditional statements +are not supported, QET can be implemented without +problems through the technique of deferred measure- +ment. +We can postpone Alice’s measurement until the end +of the circuit, and obtain the same results. The condi- +tional operations can be created by a controlled U gate +Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U and an anti-controlled U +gate (X ⊗ I)Λ(U)(X ⊗ I). One would find the equiva- +lence between the following two circuits. We use the right +circuit enclosed by the orange dashed frame in Fig. 1 (C). +We performed quantum computation using 6 dif- +ferent types of IBM quantum hardware ibmq lima, +ibmq jakarta, ibmq hanoi, ibm cairo, ibm auckland +and ibmq montreal. +The properties of each quantum +computers can be seen from Fig. 2. +ibmq lima con- +sists of 5 qubits (Fig. 2 [Left]) and ibmq jakarta has +7 qubits (Fig. 2 [Middle]). +ibm cairo is a 27-qubit +hardware, and ibmq hanoi, ibm cairo, ibm auckland +and ibmq montreal have the same graph structure as +ibm cairo (Fig. 2 [Right]). A direct CNOT gate can be +applied to two qubits connected at the edge. +We can +choose two qubits placed on the graph of the hardware +to perform a quantum computation. We conducted the +experiment by choosing two qubits connected at the edge +with relatively small errors. +We also performed a simulation using a simulator +qasm simulator, which can classically emulate gate op- +erations on the same quantum circuits we used for +quantum computation. +We summarize results with +ibmq lima, ibmq jakarta and ibm cairo in Table I. The +results using the simulator agreed with the analytical so- +lution with high accuracy, confirming that the quantum +circuit was implemented correctly. +More experimental + +4 +FIG. 2: (A) properties of quantum computers we used. Each graph of qubits corresponds to the layout of the hardware. A +direct CNOT gate can be applied to two qubits connected at the edge. (B) Distribution of states compared with a simulator +qasm simulator and a quantum computer ibm cairo (raw results and mitigated results) +results are summarized in Table IV in Appendix D. We +describe details of machine properties and experimental +conditions in Table III in Appendix C. +The most significant achievement in this study is the +observation of negative energy ⟨E1⟩ < 0. The value of +⟨V ⟩ that was closest to the exact analysis value was - +0.1079 (h = 1.5, k = 1 with ibmq jakarta), which is +about 76% accurate. +As emphasised in Hotta’s origi- +nal works [5–11, 16], after Alice observes her X0, no +unitary operation can make ⟨E1⟩ negative (eq. (A7)). +In order for Bob to obtain the correct ⟨E1⟩, Alice and +Bob must repeat the experiment an enormous number of +times, and the correct value of ⟨V ⟩ and ⟨H1⟩ can be ob- +tained only when Alice and Bob communicate correctly +in the quantum circuit in Fig. 1 (C). Distributions of +states obtained by a quantum computer ibm cairo are +shown in Fig. 2 (B), where distributions of raw results +and error mitigated results are compared with a simu- +lator qasm simulator. We used a simple measurement +error mitigation to determine the effects of measurement +errors. +We prepared a list of 4 measurement calibra- +tion circuits for the full Hilbert space. Then we immedi- +ately measured them to obtain the probability distribu- +tions. Then we applied the calibration matrix to correct +the measured results. The average measurement fidelity +when using each quantum computer is summarized in +Table III in Appendix C. The histograms of the observed +states showed similar tendencies for all other quantum +computers we used. It can be seen that the histograms +obtained by the measurement of H agree with the sim- +ulator results with good accuracy. The improvement of +the values due to measurement error mitigation is also +confirmed by the results in Table I. The observation of V +is of utmost importance in this study. Although the raw +data from quantum computers deviated from the simula- +tor results, in some cases error mitigation improved them +enough to observe negative energy expectation values. +It should also be emphasized that we observed negative +⟨V ⟩ for all parameter (k, h) combinations in all quantum +computers used. As emphasized in Sec. II C, the amount +of energy available to Bob is greater if only V is observed, +since ⟨H1⟩ is always positive (Fig. 3). +This would be +enough for practical purposes.. Note that the energy that +Bob gains becomes smaller when he observes H1. + +iloa mal lima Error Map +lomi cairo Emor Map) +Keidaut: Hror tw +Heridoun: Hror ta? +Featour: :Error tw? +1 +2 +12 +2.41 +H arrar te ( [avg. : : DuE4s) +Cl aror hie h avg. +Distriloution of states when measuring V (ilom cairo, k = 1, h = 1) +Distriloution of states when measuring H (ilom cairo, k = 1, h = 1) +raw +raw +mitigated +0.4 +0.474 +mitigated +0.4702473 +0.350.349 +0.362 +0.330367362 +0.457 +simullator +simullator +0.45 +0.3 +Probabilities +0.30 +Probal +0.2 +0.139130139 +0.15 +0.1 +0.059 +04.8 +0.049 +.040 +0.027 +0.026 +0.00 +0.0 +15 +Backend +(h, k) = (1, 0.2) +(h, k) = (1, 0.5) +(h, k) = (1, 1) +(h, k) = (1.5, 1) +Analytical value +⟨E0⟩ +0.9806 +0.8944 +0.7071 +1.2481 +qasm simulator +0.9827 ± 0.0031 +0.8941 ± 0.0001 +0.7088 ± 0.0001 +1.2437 ± 0.0047 +ibmq lima +error mitigated +0.9423 ± 0.0032 +0.8169 ± 0.0032 +0.6560 ± 0.0031 +1.2480 ± 0.0047 +unmitigated +0.9049 ± 0.0017 +0.8550 ± 0.0032 +0.6874 ± 0.0031 +1.4066 ± 0.0047 +ibmq jakarta +error mitigated +0.9299 ± 0.0056 +0.8888 ± 0.0056 +0.7039 ± 0.0056 +1.2318 ± 0.0084 +unmitigated +0.9542 ± 0.0056 +0.9089 ± 0.0056 +0.7232 ± 0.0056 +1.2624 ± 0.0083 +ibm cairo +error mitigated +0.9571 ± 0.0032 +0.8626 ± 0.0031 +0.7277 ± 0.0031 +1.2072 ± 0.0047 +unmitigated +0.9578 ± 0.0031 +0.8735 ± 0.0031 +0.7362 ± 0.0031 +1.2236 ± 0.0047 +Analytical value +⟨H1⟩ +0.0521 +0.1873 +0.2598 +0.3480 +qasm simulator +0.0547 ± 0.0012 +0.1857 ± 0.0022 +0.2550 ± 0.0028 +0.3487 ± 0.0038 +ibmq lima +error mitigated +0.0733 ± 0.0032 +0.1934 ± 0.0032 +0.2526 ± 0.0032 +0.3590 ± 0.0047 +unmitigated +0.1295 ± 0.0053 +0.2422 ± 0.0024 +0.2949 ± 0.0028 +0.4302 ± 0.0039 +ibmq jakarta +error mitigated +0.0736 ± 0.0055 +0.2018 ± 0.0056 +0.2491 ± 0.0056 +0.3390 ± 0.0084 +unmitigated +0.0852 ± 0.0022 +0.2975 ± 0.0045 +0.3365 ± 0.0052 +0.4871 ± 0.0073 +ibm cairo +error mitigated +0.0674 ± 0.0032 +0.1653 ± 0.0031 +0.2579 ± 0.0031 +0.3559 ± 0.0047 +unmitigated +0.0905 ± 0.0014 +0.1825 ± 0.0022 +0.2630 ± 0.0027 +0.3737 ± 0.0037 +Analytical value +⟨V ⟩ +-0.0701 +-0.2598 +-0.3746 +-0.4905 +qasm simulator +−0.0708 ± 0.0012 −0.2608 ± 0.0032 −0.3729 ± 0.0063 −0.4921 ± 0.0038 +ibmq lima +error mitigated −0.0655 ± 0.0012 −0.2041 ± 0.0031 −0.2744 ± 0.0063 −0.4091 ± 0.0063 +unmitigated +−0.0538 ± 0.0011 −0.1471 ± 0.0025 −0.1233 ± 0.0041 −0.2737 ± 0.0046 +ibmq jakarta +error mitigated −0.0515 ± 0.0022 −0.2348 ± 0.0056 −0.3255 ± 0.0112 −0.4469 ± 0.0112 +unmitigated +−0.0338 ± 0.0021 −0.1371 ± 0.0046 −0.0750 ± 0.0075 −0.2229 ± 0.0083 +ibm cairo +error mitigated −0.0497 ± 0.0013 −0.1968 ± 0.0031 −0.2569 ± 0.0063 −0.3804 ± 0.0063 +unmitigated +−0.0471 ± 0.0012 −0.1682 ± 0.0026 −0.1733 ± 0.0038 −0.3089 ± 0.0045 +Analytical value +⟨E1⟩ +-0.0180 +-0.0726 +-0.1147 +-0.1425 +qasm simulator +−0.0161 ± 0.0017 −0.0751 ± 0.00398 −0.1179 ± 0.0069 −0.1433 ± 0.0054 +ibmq lima +error mitigated +0.0078 ± 0.0034 +−0.0107 ± 0.0045 −0.0217 ± 0.0071 −0.0501 ± 0.0079 +unmitigated +0.0757 ± 0.0054 +0.0950 ± 0.0035 +0.1715 ± 0.0050 +0.1565 ± 0.0060 +ibmq jakarta +error mitigated +0.0221 ± 0.0059 +−0.0330 ± 0.0079 −0.0764 ± 0.0125 −0.1079 ± 0.0140 +unmitigated +0.0514 ± 0.0030 +0.1604 ± 0.0064 +0.2615 ± 0.0091 +0.2642 ± 0.00111 +ibm cairo +error mitigated +0.0177 ± 0.0035 +−0.0315 ± 0.0044 +0.0010 ± 0.0070 +−0.0245 ± 0.0079 +unmitigated +0.0433 ± 0.0018 +0.0143 ± 0.0034 +0.0897 ± 0.0047 +0.0648 ± 0.0058 +TABLE I: Comparison between analytical values of ⟨E0⟩, ⟨H1⟩, ⟨V ⟩, ⟨E1⟩ and results from IBM’s real quantum computers, +ibmq lima, ibmq jakarta and ibm cairo. We evaluate ⟨E1⟩ = ⟨H1⟩ + ⟨V ⟩. ”error mitigated” means results using measurement +error mitigation and ”unmitigated” corresponds to results without measurement error mitigation. +III. +IMPLICATIONS FOR OUR REAL WORLD +Our results provide implications for new quantum com- +munication technologies with respect to different phases +in the short, medium and long term. It is important to +note that, like quantum teleportation, energy can also +be teleported only by LOCC. Reproducing the minimal +QET model we used in our demonstration in a labora- +tory system is something that can be tackled in the short +term with current quantum computing and communica- +tion technology. A quantum device with 2 qubits and a +gate depth of 10 would be ready for immediate experi- +mentation. This is expected to lead to new developments +in the use of quantum memory [18–20]. Furthermore, ver- +ifying QET in a variety of quantum systems and materi- +als beyond the minimal model is an important challenge +for future applications. +Quantum energy teleportation without limit of dis- +tance is also provided [21]. The ability to transfer quan- +tum energy over long distances will bring about a new +revolution in quantum communication technology. +In +other words, a world in which physical quantities are +freely and instantaneously transmitted to remote loca- +tions connected by a large-scale Quantum Internet (Net- +work) can be realized in the near future. For example +there is a long-distance (∼158km) SBU/BNL quantum +network in Long Island, New York [22]. Various quan- +tum networks have been developed [23–25]. +Realizing +QET on a quantum network, which is expected to be + +6 +in practical use around the 2030s, would be a milestone +toward realizing QET on a worldwide quantum network. +The realization of a long-range QET will have impor- +tant implications beyond the development of information +and communication technology and quantum physics. In- +formation and energy are physical, but also economic. +Allowing physical quantities to be traded concretely on +the quantum network means that a new economic mar- +ket will be born [26]. Quantum teleportation is an es- +tablished technology and is being developed for practical +use. In addition to this, if QET is put to practical use, it +will mean that various quantum resources will be at the +disposal of us. The expected value of the Hermite op- +erator is called energy, but it need not literally be used +only as energy. +Teleported energy can be used as en- +ergy, as well as for other uses. The ability to teleport a +concrete physical quantity, energy, means that quantum +information will have added value. In a quantum market +where Alice, Bob, and Charlie exist, if Bob can get more +energy from Charlie than from Alice, Bob may prefer to +do business with Charlie rather than Alice, and he may +prefer an entangle state with Charlie. However, depend- +ing on transaction costs, Bob may choose Alice. A lot +of such game-theoretic situations can be created [27–31]. +This implies that quantum information economics (which +does not yet exist) will become a meaningful idea in the +future. +Acknowledgement +I thank David Frenklakh, +Adrien Florio, +Sebas- +tian Grieninger, Fangcheng He, Dmitri Kharzeev, Yuta +Kikuchi, Vladimir Korepin, Qiang Li, Adam Lowe, +Shuzhe Shi, Hiroki Sukeno, Tzu-Chieh Wei, Kwangmin +Yu and Ismail Zahed for fruitful communication and col- +laboration. I thank Megumi Ikeda for providing the car- +toons. I acknowledge the use of IBM quantum comput- +ers. I was supported by the U.S. Department of Energy, +Office of Science, National Quantum Information Science +Research Centers, Co-design Center for Quantum Advan- +tage (C2QA) under Contract No.DESC0012704. +Author contribution +All work was performed by the author. +competing interests +The author declares that there is no competing finan- +cial interests. +References +[1] C. H. Bennett, G. Brassard, C. 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We use the following one- +qubit operators whose matrix representations are given +as +X = +� +0 1 +1 0 +� +, Y = +� +0 −i +i +0 +� +, Z = +� +1 +0 +0 −1 +� +, +H = +1 +√ +2 +� +1 +1 +1 −1 +� +, S = +� +1 0 +0 i +� +. +(A1) +We use |0⟩ = +�1 +0 +� +, |1⟩ = +�0 +1 +� +for the computational +basis states, which are eigenstates of Z: +Z |0⟩ += +|0⟩ , Z |1⟩ = − |1⟩. +We also work with another ba- +sis vectors |±⟩ = +|0⟩±|1⟩ +√ +2 +. +They are eignestates of X: +X |−⟩ = − |−⟩ , X |+⟩ = − |+⟩. Note that |±⟩ are created +by applying H to |0⟩ and |1⟩; H |0⟩ = |+⟩ , H |1⟩ = |−⟩. +Those are used for measuring Hn, V (n = 1, 2) in the +QET protocol. For example, Alice finds µ = ±1 by ob- +serving the eigenvalues ±1 of her local Pauli X operator. +The rotation of X, Y, Z is defined by +RX(α) = e−i α +2 X, RY (α) = e−i α +2 Y , RZ(α) = e−i α +2 Z. +(A2) +Note that X and Y gates are related in a way that Y = +SXS†. +Using those representations, it will be easy to +check the matrix representation +exp(−iαX ⊗ Y ) = (I ⊗ S) exp(−iαX ⊗ X)(I ⊗ S†) += +� +� +� +� +� +cos α +0 +0 +− sin α +0 +cos α +sin α +0 +0 +− sin α cos α +0 +sin α +0 +0 +cos α +� +� +� +� +� +(A3) +and the exact form of the ground state eq. (8): +|g⟩ = exp(−iαX ⊗ Y ) |Φ−⟩ += +1 +√ +2(cos α + sin α) |00⟩ − 1 +√ +2(cos α − sin α) |11⟩ . +We use two-qubit gate operations. In general, a control +U operation Λ(U) is defined by +Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U +(A4) +and the corresponding diagram is drwan as +control U= +U +One of the most frequently used controlled gates is a +CNOT gate CNOT = Λ(X), whose diagram is especially +drawn as +CNOT= +It is convenient to define an anti-control gate, which is +activated when the control bit is in state |0⟩: |1⟩ ⟨1|⊗I + +|0⟩ ⟨0| ⊗ U, whose diagram is drawn as +Anti-control U= +U += +X +X +U +Now we describe measurement of quantum operators. +We measure Z1 and X0X1. Measurement of Z1 is done +by the following circuit +The output of the measurement is a bit string b0b1 ∈ +{00, 01, 10, 11}. Since the eigenvalues of Z are −1, 1, we +convert the bit string into 1 − 2b1. +Let nshot be the +number of repetitions of the circuit, and countsb0b1 be +the number of times b0 and b1 are detected. Therefore +countsb0b1 +nshots +is the probability that a bit string b0b1 is ob- +tained. Then the expectation value of Z1 is computed by +the formula +⟨Z1⟩ = +� +b0,b1 +(1 − 2b1)countsb0b1 +nshots +. +(A5) +Measurement of X0X1 is done by the following circuit +H +H +As we described previously, +H +maps |0⟩ , |1⟩ to +|+⟩ , |−⟩, which are eigenvectors of X. The output of the +measurement is again a bit string b0b1 ∈ {00, 01, 10, 11}. +They are converted to the eigenvalues of X0X1 by (1 − +2b0)(1 − 2b1). +Then the expectation value of X0X1 is +computed by the formula +⟨X0X1⟩ = +� +b0,b1 +(1 − 2b0)(1 − 2b1)countsb0b1 +nshots +. +(A6) + +9 +FIG. 3: Heat maps visualizing expectation values ⟨V ⟩ = Tr[ρQETV ] and ⟨H1⟩ = Tr[ρQETH1] by (k, h). +2. +Some details of the model +Here we describe details of the model we used. For +more information please refer to Hotta’s original papers. +First it is important to note that the ground state of +the total Hamiltonian H is not the ground state of lo- +cal operators. +For example, V has three degenerated +ground states |−+⟩ , |+−⟩ , |−+⟩+|+−⟩ +√ +2 +, and the ground +state energy of V is −2k + +2k2 +√ +h2+k2 . +It is important +that V ’s ground state energy is negative for all k > 0. +This is also true for Hn, whose ground state energy is +−h + +h2 +√ +h2+k2 . The expected values of ⟨V ⟩ = Tr[ρQETV ] +and ⟨H1⟩ = Tr[ρQETH1] obtained by QET are shown in +Fig. 3. +To understand the non-triviality of the QET protocol, +it is important to note that after Alice’s measurement, +no matter what unitary operation W1 is performed on +Bob’s qubit, no energy can be extracted. This can be +confirmed by +Tr[ρW Htot] − ⟨E0⟩ = ⟨g| W † +1 HtotW1 |g⟩ ≥ 0, +(A7) +where +ρW = W † +1 +� +� +� +µ∈{−1,1} +P0(µ) |g⟩ ⟨g| P0(µ) +� +� W1. +(A8) +The inequality in eq. (A7) is guaranteed by eq. (5). +If Bob does not perform any operations on his own +system after Alice’s measurement, the time evolution of +Bob’s local system is as follows +⟨H1(t)⟩ = Tr[ρMeitHH1e−itH] = h2(1 − cos(4kt)) +2 +√ +h2 + k2 +⟨V (t)⟩ = Tr[ρMeitHV e−itH] = 0, +(A9) +where ρM = � +µ∈{±1} P0(µ) |g⟩ ⟨g| P0(µ). +As a result of the natural time evolution of the sys- +tem, energy is indeed transferred to Bob’s local system, +but this is no more than energy propagation in the usual +sense. In QET, energy is not obtained through the nat- +ural time evolution of the system, but instantaneously +as a result of communication. Since we consider a non- +relativistic quantum many-body system, the speed of en- +ergy propagation is sufficiently slower than the speed of +light. Classical communication, realized by optical com- +munication, can convey information to remote locations +much faster than the time evolution of physical systems. +Hence, QET can be described as a fast energy propaga- +tion protocol. +It is known that the change in entropy before and after +the measurement can be evaluated as follows +∆SAB = SAB − +� +µ∈{±1} +pµSAB(µ) +(A10) +≥ 1 + sin2 ξ +2 cos3 ξ +ln 1 + cos ξ +1 − cos ξ +EB +√ +h2 + k2 +(A11) +where pµ is the probability distribution of µ, SAB(µ) is +the entanglement entropy after the measurement, ξ = +arctan +� k +h +� +and EB is the amount of energy that Bob can +receive (EB = −⟨E1⟩ > 0) [16]. Moreover the maximal +energy that Bob would receive is bounded below by the +difference of entropy: +max +U1(µ) EB ≥ +2 +√ +h2 + k2( +� +4 − 3 cos2 ξ − 2 + cos2 ξ)∆SAB +(1 + cos ξ) ln +� +2 +1+cos ξ +� ++ (1 − cos ξ) ln +� +2 +1−cos ξ +�. +(A12) +Appendix B: Simulation of Hotta’s original QET +protocol +Hotta’s original QET protocol, which can be imple- +mented by Fig. 1 (B) in the main text, does require the +conditional operations based on a signal µ ∈ {−1, +1} + +(V) +(H1) + 0.0 +2.0 + 0.5 +18 +- +0.1 +18 +1.6 +- +16 + 0.4 +0.2 +E'O- + 0.3 +10 +0.4 +- +8°0 +0.8 +- + 0.2 +0.5 +0.6 +- +0.6 +0.4 + 0.1 +0.7 +- + 0.0 +00.0 +0.4 +0.6 +8:0 +10 +12 +141618 +00.0 +20 +0.4 +9'0 +8:0 +10 +12 +14 +16 +18 +2.0 +k +k10 +(h, k) +(1,0.1) +(1,0.2) +(1,0.5) +(1,1) +(1.5,1) +Analytical ⟨E0⟩ +0.9950 +0.9806 +0.8944 +0.7071 +1.2481 +qasm simulator ⟨E0⟩ +0.9929 ± 0.0010 +0.9807 ± 0.0010 +0.8948 ± 0.0010 +0.7067 ± 0.0010 +1.2492 ± 0.0015 +Analytical ⟨V ⟩ +-0.0193 +-0.0701 +-0.2598 +-0.3746 +-0.4905 +qasm simulator ⟨V ⟩ −0.0194 ± 0.0057 −0.0682 ± 0.0011 −0.2625 ± 0.0061 −0.3729 ± 0.0063 −0.4860 ± 0.0061 +Analytical ⟨H1⟩ +0.0144 +0.0521 +0.1873 +0.2598 +0.3480 +qasm simulator ⟨H1⟩ +0.0136 ± 0.0006 +0.0501 ± 0.0011 +0.1857 ± 0.0022 +0.2550 ± 0.0028 +0.3493 ± 0.0038 +Analytical ⟨E1⟩ +-0.0049 +-0.0180 +-0.0726 +−0.1147 +-0.1425 +qasm simulator ⟨E1⟩ −0.0058 ± 0.0057 −0.0181 ± 0.016 −0.0768 ± 0.0064 −0.1179 ± 0.0068 −0.1367 ± 0.0072 +TABLE II: Comparison between analytical values and numerical values from the quantum circuits with conditional opera- +tion (Fig. 1 (B)). Each error corresponds to statistical error of 105 shots. We evaluate ⟨E1⟩ = ⟨H1⟩ + ⟨V ⟩. +that Bob receives from Alice. We performed quantum +computation on the equivalent circuit (right quantum cir- +cuit in Fig. 1) (C) that yielded exactly the same results. +Let Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U be a controlled U +gate. Note that Λ(U(−1)) and (X ⊗ I)Λ(U(+1))(X ⊗ I) +commute: +U(−1) +U(+1) += +U(+1) +U(−1) +Of course, the equivalence of these circuits is theoreti- +cally trivial, we used qasm simulator and executed our +simulation based on the left quantum circuit in Fig. 1 +(C), in order to confirm the consistency between them. +Table II summarizes the numerical results and shows per- +fect agreement with the analytical results as well as re- +sults (Table IV) with the right circuit in Fig. 1 (C). +Appendix C: Properties of Quantum Hardware +Here we describe more on our experiments with IBM +quantum computers. Graphs of IBM quantum computers +we used are displayed in Fig 4. For example, the layout +of ibmq lima corresponds to (A) in Fig. 4 and we used +the pair of qubits in [0,1] that had the smallest readout +assignment error among all pairs (Fig. 2 (A) [Left]). We +can perform a direct CNOT operation between qubits +connected at the edge. For ibmq lima, the CNOT error +between [1,2] qubits were 0.00510 (Table. III). +Appendix D: Additional results with 6 different +quantum hardware +Here we describe additional results obtained by some +other IBM quantum computers. In the main text we fo- +cused on best results with ibmq lima and ibmq jakarta, +but in fact we also experimented with ibmq hanoi, +ibm cairo, ibm auckland, ibmq montreal. +Table IV +summarizes the complete lists of the best data we ob- +FIG. 4: +Configurations of qubits on graphs: +(A) the lay- +out of ibmq lima which has 5 qubits; (B) the layout of +ibmq jakarta which has 7 qubits; (C) the layout of 27-qubit +hardware including ibmq hanoi, ibm cairo, ibm auckland +and ibmq montreal. A direct CNOT gate can be applied to +two qubits connected at the edge. +tained and Table III summarizes the experimental con- +ditions used for each hardware. In the entire circuit, the +total number N of qubits is 2 and the circuit depth d(N) +that can be executed is 9 (excluding measurement of V ) +and 10 (including measurement of V ). The quantum vol- +ume is defined by QV = +� +arg maxn≤N min{n, d(n)} +�2. +Therefore quantum computers with QV += +128 are +enough for this work. Here QV is a metric that quantifies +the largest random circuit of equal width and depth that +a quantum computer can successfully implement. How- +ever, QV may not be a crucial metric in this study, since +we are only dealing with 2-qubit, relatively simple quan- +tum circuits. Errors in quantum computers result from a +combination of various factors, including readout error, +CNOT error, etc.. Table IV shows that Alice’s measure- +ments of X0 are relatively accurate in almost all cases. +With respect to the observation of V , there is a deviation + +0 +2 +0 +3 +3 +4 +4 +5 +(6) +17 +4 +10 +12 +15 +18 +21 +23 +13 +24 +5 +8 +11 +14 +16 +19 +22 +25 +26 +2011 +Backend +ibmq lima +ibmq jakarta +ibm cairo +ibm hanoi +ibmq auckland ibmq montreal +Ntot +5 +7 +27 +27 +27 +27 +Quantum Volume +8 +16 +64 +64 +64 +128 +shots +105 +3.2 × 104 +105 +105 +105 +3.2 × 104 +Measurement fidelity +0.961075 +0.924695 +0.961935 +0.979530 +0.979383 +0.957484 +qubits used +[0,1] +[3,5] +[13,14] +[14,16] +[14,16] +[14,16] +CNOT error +0.00510 +0.00665 +0.00439 +0.01996 +0.00570 +0.00739 +Gate time (ns) +305.778 +291.556 +220.444 +472.889 +355.556 +355.556 +First qubit +t1(µs) +75.67 +93.53 +146.43 +219.15 +60.97 +129.56 +t2(µs) +141.39 +41.09 +164.29 +25.75 +150.49 +168.53 +Frequency (GHz) +5.030 +5.178 +5.282 +5.047 +5.167 +4.961 +Anharmonicity (GHz) +-0.33574 +-0.34112 +-0.33874 +-0.34412 +-0.34196 +-0.32314 +Pauli X error +2.781 × 10−4 2.140 × 10−4 1.630 × 10−4 2.305 × 10−4 2.4842 × 10−4 +1.942 × 10−4 +Readout assignment error 1.960 × 10−2 2.440 × 10−2 8.500 × 10−3 7.400 × 10−3 +8.100 × 10−3 +1.310 × 10−2 +Second qubit +t1(µs) +58.03 +143.52 +94.28 +190.07 +73.16 +83.73 +t2(µs) +74.97 +59.33 +186.99 +253.46 +183.12 +39.92 +Frequency (GHz) +5.128 +5.063 +5.044 +4.883 +4.970 +5.086 +Anharmonicity (GHz) +-0.31835 +-0.34129 +-0.34289 +-0.34591 +-0.34389 +-0.33707 +Pauli X error +1.469 × 10−4 1.708 × 10−4 1.732 × 10−4 4.708 × 10−4 +2.052 × 10−4 +2.221 × 10−4 +Readout assignment error 1.300 × 10−2 2.400 × 10−2 8.000 × 10−3 9.600 × 10−3 +7.700 × 10−3 +9.800 × 10−3 +TABLE III: Machine properties of IBM quantum computers and parameters we used. +shots is the number of iterations +we performed for sampling. Average measurement fidelity was computed when preparing a calibration matrix and used for +measurement error mitigation. CNOT error corresponds to the direct CNOT error between two qubits [q0, q1] used. Gate time +corresponds to the gate time between [q0, q1]. First and second qubits corresponds to q0 and q1, respectively. t1 is relaxation +time and t2 is dephasing time. +from the analytical value. It was confirmed that the error +mitigation improved the results. In this study, what is +important is that negative expectation values ⟨V ⟩ were +observed for all cases. It is a noteworthy achievement +that negative energy expectation values ⟨E⟩ < 0 were +observed by error mitigation. In fact, the histograms of +states (Fig. 2 (B)) have improved to approach the ex- +act values, indicating that all operations were performed +correctly. + +12 +Backend +(h, k) = (1, 0.2) +(h, k) = (1, 0.5) +(h, k) = (1, 1) +(h, k) = (1.5, 1) +Analytical value +⟨E0⟩ +0.9806 +0.894 +0.7071 +1.2481 +ibmq lima +error mitigated +0.9423 ± 0.0032 +0.8169 ± 0.0032 +0.6560 ± 0.0031 +1.2480 ± 0.0047 +unmitigated +0.9049 ± 0.0017 +0.8550 ± 0.0032 +0.6874 ± 0.0031 +1.4066 ± 0.0047 +ibmq jakarta +error mitigated +0.9299 ± 0.0056 +0.8888 ± 0.0056 +0.7039 ± 0.0056 +1.2318 ± 0.0084 +unmitigated +0.9542 ± 0.0056 +0.9089 ± 0.0056 +0.7232 ± 0.0056 +1.2624 ± 0.0083 +ibm hanoi +error mitigated +1.0685 ± 0.0032 +0.9534 ± 0.0032 +0.7852 ± 0.0031 +1.3728 ± 0.0047 +unmitigated +1.0670 ± 0.0031 +0.9524 ± 0.0031 +0.7809 ± 0.0031 +1.3663 ± 0.0047 +ibm cairo +error mitigated +0.9571 ± 0.0032 +0.8626 ± 0.0031 +0.7277 ± 0.0031 +1.2072 ± 0.0047 +unmitigated +0.9578 ± 0.0031 +0.8735 ± 0.0031 +0.7362 ± 0.0031 +1.2236 ± 0.0047 +ibm auckland +error mitigated +0.9766 ± 0.0032 +0.8703 ± 0.0032 +0.6925 ± 0.0032 +1.2482 ± 0.0047 +unmitigated +0.9771 ± 0.0032 +0.8712 ± 0.0032 +0.6931 ± 0.0032 +1.2487 ± 0.0047 +ibmq montreal +error mitigated +0.8774 ± 0.0056 +0.8084 ± 0.0056 +0.6315 ± 0.0056 +1.1449 ± 0.0084 +unmitigated +0.9036 ± 0.0056 +0.8338 ± 0.0056 +0.6564 ± 0.0056 +1.1819 ± 0.0084 +Analytical value +⟨H1⟩ +0.0521 +0.1873 +0.2598 +0.3480 +ibmq lima +error mitigated +0.0733 ± 0.0032 +0.1934 ± 0.0032 +0.2526 ± 0.0032 +0.3590 ± 0.0047 +unmitigated +0.1295 ± 0.0053 +0.2422 ± 0.0024 +0.2949 ± 0.0028 +0.4302 ± 0.0039 +ibmq jakarta +error mitigated +0.0736 ± 0.0055 +0.2018 ± 0.0056 +0.2491 ± 0.0056 +0.3390 ± 0.0084 +unmitigated +0.0852 ± 0.0022 +0.2975 ± 0.0045 +0.3365 ± 0.0052 +0.4871 ± 0.0073 +ibm hanoi +error mitigated +0.1786 ± 0.0032 +0.3256 ± 0.0032 +0.4276 ± 0.0032 +0.5890 ± 0.0047 +unmitigated +0.2012 ± 0.0019 +0.3427 ± 0.0026 +0.4378 ± 0.0031 +0.6104 ± 0.0042 +ibm cairo +error mitigated +0.0674 ± 0.0032 +0.1653 ± 0.0031 +0.2579 ± 0.0031 +0.3559 ± 0.0047 +unmitigated +0.0905 ± 0.0014 +0.1825 ± 0.0022 +0.2630 ± 0.0027 +0.3737 ± 0.0037 +ibm auckland +error mitigated +0.1218 ± 0.0032 +0.2004 ± 0.0031 +0.2181 ± 0.0032 +0.3215 ± 0.0047 +unmitigated +0.1455 ± 0.0017 +0.2205 ± 0.0023 +0.2337 ± 0.0027 +0.3493 ± 0.0038 +ibmq montreal +error mitigated +0.0897 ± 0.0056 +0.1618 ± 0.0056 +0.1921 ± 0.0056 +0.2816 ± 0.0084 +unmitigated +0.1603 ± 0.0032 +0.2251 ± 0.0041 +0.2454 ± 0.0049 +0.3704 ± 0.0068 +Analytical value +⟨V ⟩ +-0.0701 +-0.2598 +-0.3746 +-0.4905 +ibmq lima +error mitigated −0.0655 ± 0.0012 −0.2041 ± 0.0031 −0.2744 ± 0.0063 −0.4091 ± 0.0063 +unmitigated +−0.0538 ± 0.0011 −0.1471 ± 0.0025 −0.1233 ± 0.0041 −0.2737 ± 0.0046 +ibmq jakarta +error mitigated −0.0515 ± 0.0022 −0.2348 ± 0.0056 −0.3255 ± 0.0112 −0.4469 ± 0.0112 +unmitigated +−0.0338 ± 0.0021 −0.1371 ± 0.0046 −0.0750 ± 0.0075 −0.2229 ± 0.0083 +ibm hanoi +error mitigated −0.1136 ± 0.0013 −0.2820 ± 0.0031 −0.3497 ± 0.0063 −0.5512 ± 0.0063 +unmitigated +−0.1061 ± 0.0011 −0.2494 ± 0.0022 −0.2704 ± 0.0034 −0.4767 ± 0.0038 +ibm cairo +error mitigated −0.0497 ± 0.0013 −0.1968 ± 0.0031 −0.2569 ± 0.0063 −0.3804 ± 0.0063 +unmitigated +−0.0471 ± 0.0012 −0.1682 ± 0.0026 −0.1733 ± 0.0038 −0.3089 ± 0.0045 +ibm auckland +error mitigated −0.0138 ± 0.0012 −0.0854 ± 0.0032 −0.0591 ± 0.0063 −0.1887 ± 0.0063 +unmitigated +−0.0113 ± 0.0012 −0.0665 ± 0.0027 −0.0046 ± 0.0044 −0.1412 ± 0.0049 +ibmq montreal error mitigated −0.0157 ± 0.0022 −0.1207 ± 0.0056 −0.1275 ± 0.0112 −0.1967 ± 0.0112 +unmitigated +−0.0091 ± 0.0021 −0.0764 ± 0.0048 −0.0043 ± 0.0079 −0.0926 ± 0.0089 +Analytical value +⟨E1⟩ +-0.0180 +-0.0726 +-0.1147 +-0.1425 +ibmq lima +error mitigated +0.0078 ± 0.0034 +−0.0107 ± 0.0045 −0.0217 ± 0.0071 −0.0501 ± 0.0079 +unmitigated +0.0757 ± 0.0054 +0.0950 ± 0.0035 +0.1715 ± 0.0050 +0.1565 ± 0.0060 +ibmq jakarta +error mitigated +0.0221 ± 0.0059 +−0.0330 ± 0.0079 −0.0764 ± 0.0125 −0.1079 ± 0.0140 +unmitigated +0.0514 ± 0.0030 +0.1604 ± 0.0064 +0.2615 ± 0.0091 +0.2642 ± 0.00111 +ibm hanoi +error mitigated +0.065 ± 0.0034 +0.0436 ± 0.0044 +0.0779 ± 0.0071 +1.2481 ± 0.015 +unmitigated +0.0950 ± 0.0022 +0.0933 ± 0.0021 +0.1674 ± 0.0046 +1.0566 ± 0.015 +ibm cairo +error mitigated +0.0177 ± 0.0035 +−0.0315 ± 0.0044 +0.0010 ± 0.0070 +−0.0245 ± 0.0079 +unmitigated +0.0433 ± 0.0018 +0.0143 ± 0.0034 +0.0897 ± 0.0047 +0.0648 ± 0.0058 +ibm auckland +error mitigated +0.1080 ± 0.0034 +0.1149 ± 0.0045 +0.5877 ± 0.0031 +1.2072 ± 0.0047 +unmitigated +0.1341 ± 0.0021 +0.154 ± 0.0035 +0.6364 ± 0.0031 +1.2236 ± 0.0047 +ibmq montreal +error mitigated +0.0740 ± 0.0060 +0.0411 ± 0.0079 +0.0645 ± 0.0057 +0.0849 ± 0.0140 +unmitigated +0.1512 ± 0.0038 +0.1487 ± 0.0063 +0.2411 ± 0.0093 +0.2778 ± 0.0112 +TABLE +IV: +Results +by +ibmq lima, +ibmq jakarta, +ibmq hanoi, ibm cairo, ibm auckland, ibmq montreal. + diff --git a/E9E0T4oBgHgl3EQfzAJD/content/tmp_files/load_file.txt b/E9E0T4oBgHgl3EQfzAJD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5f80cd34b6acb57c6c9d2a73be30738453f6c24 --- /dev/null +++ b/E9E0T4oBgHgl3EQfzAJD/content/tmp_files/load_file.txt @@ -0,0 +1,1374 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf,len=1373 +page_content='First Realization of Quantum Energy Teleportation on Quantum Hardware Kazuki Ikeda1, ∗ 1Co-design Center for Quantum Advantage & Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Teleporting physical quantities to remote locations is a remaining key challenge for quantum information science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum teleportation has enabled the transfer of quantum information, but teleportation of quantum physical quantities has not yet been realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Here we report the first realization and observation of quantum energy teleportation on real quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We achieve this by using several IBM’s superconducting quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The results are consistent with the exact solution of the theory and are improved by the mitigation of measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum energy teleportation requires only local operations and classical communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Therefore our results provide a realistic benchmark that is fully achievable with current quantum computing and communication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' QUANTUM ENERGY TELEPORTATION While it is fairly widely known that information about quantum states can be transported to remote loca- tions [1–4], it is less well known that quantum state energy can be similarly transmitted, despite its impact and potential for future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum informa- tion transferred by quantum teleportation is not a phys- ical quantity, but energy is a distinct physical quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Transferring physical quantities to remote locations is an unexplored area of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum Energy Tele- portation (QET) was proposed by Hotta about 15 years ago and has been studied theoretically for spin chains [5– 7], an ion trap system [8], a quantum Hall system [9], and other various theoretical systems [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is surprising that (to the best of knowledge of the author) QET has never been confirmed by any experiment on any system before, even though it can be achieved with a very simple quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The purpose of this paper is to make the first experimental realization of QET in actual quan- tum hardware and to establish the quantum circuits that make it possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We achieved the realization of QET us- ing some IBM quantum computers by applying quantum error mitigation [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The methods we have estab- lished can be applied to any system capable of QET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In what follows, we explain that QET is a universal means of quantum energy transfer, just as quantum tele- portation is a universal means of quantum information transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Any non-trivial local operations, including mea- surements on the ground state of a quantum many-body system give rise to excited states, which in turn increase the energy expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Note that the increase in energy is supplied by the experimental devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' An im- portant property of the ground state of a quantum many- body system is that it has entanglement, which brings about quantum fluctuations in the global ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In other words, quantum fluctuations in the en- ergy of the local systems are entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Local measure- ∗kazuki7131@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='com ment of the quantum state at a subsystem A partially de- stroys this ground state entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' At the same time, energy EA from the device making the measurement is injected into the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The injected energy EA stays around the subsystem A in the very early stages of time evolution, but operations around A alone cannot extract EA from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This is because informa- tion about EA is also stored in remote locations other than A due to the entanglement that exists prior to the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In other words, the locally injected energy EA can be partially extracted at any location other than A [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' QET is the protocol that makes this possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Up to this point, no special assumptions about the system have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The crucial property of QET is that it can be realized entirely by the general nature of the ground state of the quantum many-body system and the universal fact of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We work on the minimal QET model given in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' One of the purposes of this paper is to give a quantum circuit that utilizes QETs with real quantum computers and quantum networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The complete form of quantum circuits we used for QET is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The max- imum circuit depth is 10 and the number of qubits used is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Hence, current quantum computers are powerful enough to implement QET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Let k, h be positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The Hamiltonian of the minimal model is Htot = H0 + H1 + V, (1) Hn = hZn + h2 √ h2 + k2 , (n = 0, 1) (2) V = 2kX0X1 + 2k2 √ h2 + k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (3) The ground state of Htot is |g⟩ = 1 √ 2 � 1 − h √ h2 + k2 |00⟩− 1 √ 2 � 1 + h √ h2 + k2 |11⟩ , (4) The constant terms in the Hamiltonians are added so that the ground state |g⟩ of Htot returns the zero mean arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='02666v1 [quant-ph] 7 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1: Quantum gate operations used for quantum energy teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A) preparation of ground state and Alice’s X0 measurement to deposit her energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' She tells Bob via classical communication whether µ = −1 or µ = +1 was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (B) Bob’s conditional operations to receive energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' He selects an operation U1(+1) or U1(−1) based on µ = +1 or −1, corresponding to the Maxwell demon operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (C) Equivalent implementation of Bob’s operations on a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' energy for all local and global Hamiltonians: ⟨g| Htot |g⟩ = ⟨g| H0 |g⟩ = ⟨g| H1 |g⟩ = ⟨g| V |g⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (5) However it should be noted that |g⟩ is neither a ground state nor an eigenstate of Hn, V, Hn + V (n = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The essence of QET is to extract negative ground state energy of those local and semi-local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The QET protocol is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' First, Alice makes a measurement on her Pauli operator X0 by P0(µ) = 1 2(1+ (−1)µX0) and then she obtains either µ = −1 or +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' At this point, Alice’s expectation energy is E0 = h2 √ h2+k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Via a classical channel, Alice then sends her measure- ment result µ to Bob, who applies an operation U1(µ) to his qubit and measures H1 and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The density matrix ρQET after Bob operates U1(µ) to P0(µ) |g⟩ is ρQET = � µ∈{−1,1} U1(µ)P0(µ) |g⟩ ⟨g| P0(µ)U † 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (6) Using ρQET, the expected local energy at Bob’s subsys- tem is evaluated as ⟨E1⟩ = Tr[ρQET(H1 + V )], which is negative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Due to the conservation of en- ergy, EB = −⟨E1⟩(> 0) is extracted from the system by the device that operates U1(µ) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In this way, Alice and Bob can transfer the energy of the quantum system only by operations on their own local system and classical communication (LOCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' QUANTUM CIRCUIT IMPLEMENTATION OF QUANTUM ENERGY TELEPORTATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Preparation of Ground State Here we explain how to construct a quantum circuit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (A)) that generates the exact ground state |g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Let us begin with a Bell state |Φ−⟩ = |00⟩−|11⟩ √ 2 since |g⟩ is resemble to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' |Φ−⟩ can be prepared by |00⟩ − |11⟩ √ 2 = (Z ⊗ I)CNOT(H ⊗ I |00⟩ (7) where CNOT= |0⟩ ⟨0|⊗I +|1⟩ ⟨1|⊗X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Using Y = SXS†, we can perform a gate operation that maps |Φ−⟩ to the ground state |g⟩ (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (4)) by a combination of one- and two-qubit operators |g⟩ = exp(−iαX ⊗ Y ) |Φ−⟩ = 1 √ 2(cos α + sin α) |00⟩ − 1 √ 2(cos α − sin α) |11⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (8) where α is designed to satisfy cos α + sin α = � 1 − h √ h2+k2 and cos α − sin α = � 1 + h √ h2+k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Those quantum operations are implemented by the quantum circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Step 1: Deposit Energy We use the following projective measurement operator P0(µ) = 1 2(1 + (−1)µX0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (9) H H st Rx(2α) s H Ui(+1) Ui(-1) Ui(+1) Ui(-1) H Ui(+1) Ui(-1) Ui(+1) Ui(-1)3 We measure Alice’s X operator, by which we obtain a state |+⟩ or |−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This operation does not affect Bob’s energy since [X0, V ] = [X0, H1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Using [P0(µ), V ] = 0 and ⟨+| Z |+⟩ = ⟨−| Z |−⟩ = 0, we find that Alice’s mean energy to deposit is ⟨E0⟩ = � µ∈{−1,1} ⟨g| P0(µ)HtotP0(µ) |g⟩ = h2 √ h2 + k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (10) Alice’s operation can be implemented on a quantum circuit in Fig 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' ⟨E0⟩ can be calculated with the out- put bit-strings 00, 01, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Analytical values ⟨E0⟩ and results with quantum computers for different pairs of k and h are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Step 2: Receive Energy As soon as Alice observes µ ∈ {0, 1}, she tells her result to Bob who operates UB(µ) to his qubit and measures his energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Here UB(µ) is U1(µ) = cos θI − iµ sin θY1 = RY (2µθ), (11) where θ obeys cos(2θ) = h2 + k2 � (h2 + 2k2)2 + h2k2 (12) sin(2θ) = hk � (h2 + 2k2)2 + h2k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (13) The average quantum state ρQET eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (6) is obtained af- ter Bob operates U1(µ) to P0(µ) |g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Then the average energy Bob measures is ⟨E1⟩ = Tr[ρQET(H1 + V )] = Tr[ρQETHtot] − ⟨E0⟩, (14) where we used [U1(µ), H1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is important that the map � µ∈{−1,1} P0(µ) |g⟩ ⟨g| P0(µ) → ρQET is not a uni- tary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Therefore eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (14) can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This is in contrast to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Now let us explain quantum circuits for the QET pro- tocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Since V and H1 do not commute, measurement on those terms should be done separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In other words, Bob measures V and H1 independently and obtains ⟨V ⟩ and ⟨H1⟩ statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' As the figures show, ⟨V ⟩ is always negative and ⟨H1⟩ is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Therefore is suffi- cient for Bob to measure only ⟨V ⟩ to receive energy by QET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We consider V (µ) = ⟨g| P0(µ)U † 1(µ)V U1(µ)P0(µ) |g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The quantum circuit to measure V (µ) is shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is important to note that, since Bob knows µ which contains Alice’s information, he can obtain VQET(µ) by local measurement only, although V is not a local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Similarly we can measure H1 in Z-basis as in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The corre- sponding quantum circuit is obtained by removing the second Hadamard gate from the previous circuit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' On average the circuit generates ⟨E1⟩ = � µ∈{−1,1} ⟨g| P0(µ)U † 1(µ)(H1 + V )U1(µ)P0(µ) |g⟩ = − 1 √ h2 + k2 [hk sin(2θ) − (h2 + 2k2)(1 − cos(2θ))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (15) If θ is small, ⟨E1⟩ is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Bob receives energy ⟨EB⟩ = −⟨E1⟩ on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In Appendix B, we per- formed measurement of V (µ) and H1 based on the quan- tum circuit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (B) and summarized data in Table II, where numerical values are compared with analytical val- ues given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' QET on Real Quantum Hardware Here we describe how to implement the conditional operations that may not be natively supported by many quantum computers and quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In the QET protocol, Bob’s operation must be selected according to the results of Alice’s measurements, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Even in environments where conditional statements are not supported, QET can be implemented without problems through the technique of deferred measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We can postpone Alice’s measurement until the end of the circuit, and obtain the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The condi- tional operations can be created by a controlled U gate Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U and an anti-controlled U gate (X ⊗ I)Λ(U)(X ⊗ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' One would find the equiva- lence between the following two circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We use the right circuit enclosed by the orange dashed frame in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We performed quantum computation using 6 dif- ferent types of IBM quantum hardware ibmq lima, ibmq jakarta, ibmq hanoi, ibm cairo, ibm auckland and ibmq montreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The properties of each quantum computers can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' ibmq lima con- sists of 5 qubits (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2 [Left]) and ibmq jakarta has 7 qubits (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2 [Middle]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' ibm cairo is a 27-qubit hardware, and ibmq hanoi, ibm cairo, ibm auckland and ibmq montreal have the same graph structure as ibm cairo (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2 [Right]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' A direct CNOT gate can be applied to two qubits connected at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We can choose two qubits placed on the graph of the hardware to perform a quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We conducted the experiment by choosing two qubits connected at the edge with relatively small errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We also performed a simulation using a simulator qasm simulator, which can classically emulate gate op- erations on the same quantum circuits we used for quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We summarize results with ibmq lima, ibmq jakarta and ibm cairo in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The results using the simulator agreed with the analytical so- lution with high accuracy, confirming that the quantum circuit was implemented correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' More experimental 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2: (A) properties of quantum computers we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Each graph of qubits corresponds to the layout of the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' A direct CNOT gate can be applied to two qubits connected at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (B) Distribution of states compared with a simulator qasm simulator and a quantum computer ibm cairo (raw results and mitigated results) results are summarized in Table IV in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We describe details of machine properties and experimental conditions in Table III in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The most significant achievement in this study is the observation of negative energy ⟨E1⟩ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The value of ⟨V ⟩ that was closest to the exact analysis value was - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1079 (h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='5, k = 1 with ibmq jakarta), which is about 76% accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' As emphasised in Hotta’s origi- nal works [5–11, 16], after Alice observes her X0, no unitary operation can make ⟨E1⟩ negative (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In order for Bob to obtain the correct ⟨E1⟩, Alice and Bob must repeat the experiment an enormous number of times, and the correct value of ⟨V ⟩ and ⟨H1⟩ can be ob- tained only when Alice and Bob communicate correctly in the quantum circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Distributions of states obtained by a quantum computer ibm cairo are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2 (B), where distributions of raw results and error mitigated results are compared with a simu- lator qasm simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We used a simple measurement error mitigation to determine the effects of measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We prepared a list of 4 measurement calibra- tion circuits for the full Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Then we immedi- ately measured them to obtain the probability distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Then we applied the calibration matrix to correct the measured results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The average measurement fidelity when using each quantum computer is summarized in Table III in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The histograms of the observed states showed similar tendencies for all other quantum computers we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It can be seen that the histograms obtained by the measurement of H agree with the sim- ulator results with good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The improvement of the values due to measurement error mitigation is also confirmed by the results in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The observation of V is of utmost importance in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Although the raw data from quantum computers deviated from the simula- tor results, in some cases error mitigation improved them enough to observe negative energy expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It should also be emphasized that we observed negative ⟨V ⟩ for all parameter (k, h) combinations in all quantum computers used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' As emphasized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' II C, the amount of energy available to Bob is greater if only V is observed, since ⟨H1⟩ is always positive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This would be enough for practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='. Note that the energy that Bob gains becomes smaller when he observes H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' iloa mal lima Error Map lomi cairo Emor Map) Keidaut: Hror tw Heridoun: Hror ta?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Featour: :Error tw?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 2 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='41 H arrar te ( [avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' : : DuE4s) Cl aror hie h avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Distriloution of states when measuring V (ilom cairo, k = 1, h = 1) Distriloution of states when measuring H (ilom cairo, k = 1, h = 1) raw raw mitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='474 mitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='4702473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='349 0.' metadata={'source': 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jakarta error mitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0221 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0059 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0330 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0079 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0764 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0125 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1079 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0140 unmitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1604 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2615 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2642 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00111 ibm cairo error mitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0177 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0315 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0070 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0245 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0079 unmitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0433 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0143 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0897 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0648 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0058 TABLE I: Comparison between analytical values of ⟨E0⟩, ⟨H1⟩, ⟨V ⟩, ⟨E1⟩ and results from IBM’s real quantum computers, ibmq lima, ibmq jakarta and ibm cairo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We evaluate ⟨E1⟩ = ⟨H1⟩ + ⟨V ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' ”error mitigated” means results using measurement error mitigation and ”unmitigated” corresponds to results without measurement error mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' IMPLICATIONS FOR OUR REAL WORLD Our results provide implications for new quantum com- munication technologies with respect to different phases in the short, medium and long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is important to note that, like quantum teleportation, energy can also be teleported only by LOCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Reproducing the minimal QET model we used in our demonstration in a labora- tory system is something that can be tackled in the short term with current quantum computing and communica- tion technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' A quantum device with 2 qubits and a gate depth of 10 would be ready for immediate experi- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This is expected to lead to new developments in the use of quantum memory [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Furthermore, ver- ifying QET in a variety of quantum systems and materi- als beyond the minimal model is an important challenge for future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum energy teleportation without limit of dis- tance is also provided [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The ability to transfer quan- tum energy over long distances will bring about a new revolution in quantum communication technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In other words, a world in which physical quantities are freely and instantaneously transmitted to remote loca- tions connected by a large-scale Quantum Internet (Net- work) can be realized in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' For example there is a long-distance (∼158km) SBU/BNL quantum network in Long Island, New York [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Various quan- tum networks have been developed [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Realizing QET on a quantum network, which is expected to be 6 in practical use around the 2030s, would be a milestone toward realizing QET on a worldwide quantum network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The realization of a long-range QET will have impor- tant implications beyond the development of information and communication technology and quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In- formation and energy are physical, but also economic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Allowing physical quantities to be traded concretely on the quantum network means that a new economic mar- ket will be born [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum teleportation is an es- tablished technology and is being developed for practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In addition to this, if QET is put to practical use, it will mean that various quantum resources will be at the disposal of us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The expected value of the Hermite op- erator is called energy, but it need not literally be used only as energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Teleported energy can be used as en- ergy, as well as for other uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The ability to teleport a concrete physical quantity, energy, means that quantum information will have added value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In a quantum market where Alice, Bob, and Charlie exist, if Bob can get more energy from Charlie than from Alice, Bob may prefer to do business with Charlie rather than Alice, and he may prefer an entangle state with Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' However, depend- ing on transaction costs, Bob may choose Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' A lot of such game-theoretic situations can be created [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This implies that quantum information economics (which does not yet exist) will become a meaningful idea in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Acknowledgement I thank David Frenklakh, Adrien Florio, Sebas- tian Grieninger, Fangcheng He, Dmitri Kharzeev, Yuta Kikuchi, Vladimir Korepin, Qiang Li, Adam Lowe, Shuzhe Shi, Hiroki Sukeno, Tzu-Chieh Wei, Kwangmin Yu and Ismail Zahed for fruitful communication and col- laboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' I thank Megumi Ikeda for providing the car- toons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' I acknowledge the use of IBM quantum comput- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' I was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advan- tage (C2QA) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='DESC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Author contribution All work was performed by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' competing interests The author declares that there is no competing finan- cial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' References [1] C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Ikeda, Quantum Information Processing (to appear), arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='05435 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Ikeda and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Aoki, Quantum Information Processing 20, 387 (2021), arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='05588 [quant-ph] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Ikeda and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Lowe, arXiv e-prints , arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='02073 (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='02073 [quant-ph] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Ikeda, Quantum Information Processing 20, 313 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 8 Appendix A: Description of the Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Quantum Gates and Measurement Here we give a self-contained description of the back- ground knowledge of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We use the following one- qubit operators whose matrix representations are given as X = � 0 1 1 0 � , Y = � 0 −i i 0 � , Z = � 1 0 0 −1 � , H = 1 √ 2 � 1 1 1 −1 � , S = � 1 0 0 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A1) We use |0⟩ = �1 0 � , |1⟩ = �0 1 � for the computational basis states, which are eigenstates of Z: Z |0⟩ = |0⟩ , Z |1⟩ = − |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We also work with another ba- sis vectors |±⟩ = |0⟩±|1⟩ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' They are eignestates of X: X |−⟩ = − |−⟩ , X |+⟩ = − |+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Note that |±⟩ are created by applying H to |0⟩ and |1⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' H |0⟩ = |+⟩ , H |1⟩ = |−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Those are used for measuring Hn, V (n = 1, 2) in the QET protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' For example, Alice finds µ = ±1 by ob- serving the eigenvalues ±1 of her local Pauli X operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The rotation of X, Y, Z is defined by RX(α) = e−i α 2 X, RY (α) = e−i α 2 Y , RZ(α) = e−i α 2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A2) Note that X and Y gates are related in a way that Y = SXS†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Using those representations, it will be easy to check the matrix representation exp(−iαX ⊗ Y ) = (I ⊗ S) exp(−iαX ⊗ X)(I ⊗ S†) = � � � � � cos α 0 0 − sin α 0 cos α sin α 0 0 − sin α cos α 0 sin α 0 0 cos α � � � � � (A3) and the exact form of the ground state eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (8): |g⟩ = exp(−iαX ⊗ Y ) |Φ−⟩ = 1 √ 2(cos α + sin α) |00⟩ − 1 √ 2(cos α − sin α) |11⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We use two-qubit gate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' a control U operation Λ(U) is defined by Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U (A4) and the corresponding diagram is drwan as control U= U One of the most frequently used controlled gates is a CNOT gate CNOT = Λ(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' whose diagram is especially drawn as CNOT= It is convenient to define an anti-control gate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' which is activated when the control bit is in state |0⟩: |1⟩ ⟨1|⊗I + |0⟩ ⟨0| ⊗ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' whose diagram is drawn as Anti-control U= U = X X U Now we describe measurement of quantum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We measure Z1 and X0X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Measurement of Z1 is done by the following circuit The output of the measurement is a bit string b0b1 ∈ {00, 01, 10, 11}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Since the eigenvalues of Z are −1, 1, we convert the bit string into 1 − 2b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Let nshot be the number of repetitions of the circuit, and countsb0b1 be the number of times b0 and b1 are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Therefore countsb0b1 nshots is the probability that a bit string b0b1 is ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Then the expectation value of Z1 is computed by the formula ⟨Z1⟩ = � b0,b1 (1 − 2b1)countsb0b1 nshots .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A5) Measurement of X0X1 is done by the following circuit H H As we described previously, H maps |0⟩ , |1⟩ to |+⟩ , |−⟩, which are eigenvectors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The output of the measurement is again a bit string b0b1 ∈ {00, 01, 10, 11}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' They are converted to the eigenvalues of X0X1 by (1 − 2b0)(1 − 2b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Then the expectation value of X0X1 is computed by the formula ⟨X0X1⟩ = � b0,b1 (1 − 2b0)(1 − 2b1)countsb0b1 nshots .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A6) 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 3: Heat maps visualizing expectation values ⟨V ⟩ = Tr[ρQETV ] and ⟨H1⟩ = Tr[ρQETH1] by (k, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Some details of the model Here we describe details of the model we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' For more information please refer to Hotta’s original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' First it is important to note that the ground state of the total Hamiltonian H is not the ground state of lo- cal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' For example, V has three degenerated ground states |−+⟩ , |+−⟩ , |−+⟩+|+−⟩ √ 2 , and the ground state energy of V is −2k + 2k2 √ h2+k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is important that V ’s ground state energy is negative for all k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This is also true for Hn, whose ground state energy is −h + h2 √ h2+k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The expected values of ⟨V ⟩ = Tr[ρQETV ] and ⟨H1⟩ = Tr[ρQETH1] obtained by QET are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' To understand the non-triviality of the QET protocol, it is important to note that after Alice’s measurement, no matter what unitary operation W1 is performed on Bob’s qubit, no energy can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' This can be confirmed by Tr[ρW Htot] − ⟨E0⟩ = ⟨g| W † 1 HtotW1 |g⟩ ≥ 0, (A7) where ρW = W † 1 � � � µ∈{−1,1} P0(µ) |g⟩ ⟨g| P0(µ) � � W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A8) The inequality in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A7) is guaranteed by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' If Bob does not perform any operations on his own system after Alice’s measurement, the time evolution of Bob’s local system is as follows ⟨H1(t)⟩ = Tr[ρMeitHH1e−itH] = h2(1 − cos(4kt)) 2 √ h2 + k2 ⟨V (t)⟩ = Tr[ρMeitHV e−itH] = 0, (A9) where ρM = � µ∈{±1} P0(µ) |g⟩ ⟨g| P0(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' As a result of the natural time evolution of the sys- tem, energy is indeed transferred to Bob’s local system, but this is no more than energy propagation in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In QET, energy is not obtained through the nat- ural time evolution of the system, but instantaneously as a result of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Since we consider a non- relativistic quantum many-body system, the speed of en- ergy propagation is sufficiently slower than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Classical communication, realized by optical com- munication, can convey information to remote locations much faster than the time evolution of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Hence, QET can be described as a fast energy propaga- tion protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is known that the change in entropy before and after the measurement can be evaluated as follows ∆SAB = SAB − � µ∈{±1} pµSAB(µ) (A10) ≥ 1 + sin2 ξ 2 cos3 ξ ln 1 + cos ξ 1 − cos ξ EB √ h2 + k2 (A11) where pµ is the probability distribution of µ, SAB(µ) is the entanglement entropy after the measurement, ξ = arctan � k h � and EB is the amount of energy that Bob can receive (EB = −⟨E1⟩ > 0) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Moreover the maximal energy that Bob would receive is bounded below by the difference of entropy: max U1(µ) EB ≥ 2 √ h2 + k2( � 4 − 3 cos2 ξ − 2 + cos2 ξ)∆SAB (1 + cos ξ) ln � 2 1+cos ξ � + (1 − cos ξ) ln � 2 1−cos ξ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (A12) Appendix B: Simulation of Hotta’s original QET protocol Hotta’s original QET protocol, which can be imple- mented by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (B) in the main text, does require the conditional operations based on a signal µ ∈ {−1, +1} (V) (H1) 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0726 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1425 qasm simulator ⟨E1⟩ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0057 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0181 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0768 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0064 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1179 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0068 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1367 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0072 TABLE II: Comparison between analytical values and numerical values from the quantum circuits with conditional opera- tion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Each error corresponds to statistical error of 105 shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We evaluate ⟨E1⟩ = ⟨H1⟩ + ⟨V ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' that Bob receives from Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We performed quantum computation on the equivalent circuit (right quantum cir- cuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1) (C) that yielded exactly the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Let Λ(U) = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ U be a controlled U gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Note that Λ(U(−1)) and (X ⊗ I)Λ(U(+1))(X ⊗ I) commute: U(−1) U(+1) = U(+1) U(−1) Of course, the equivalence of these circuits is theoreti- cally trivial, we used qasm simulator and executed our simulation based on the left quantum circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (C), in order to confirm the consistency between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Table II summarizes the numerical results and shows per- fect agreement with the analytical results as well as re- sults (Table IV) with the right circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 1 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Appendix C: Properties of Quantum Hardware Here we describe more on our experiments with IBM quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Graphs of IBM quantum computers we used are displayed in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' For example, the layout of ibmq lima corresponds to (A) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 4 and we used the pair of qubits in [0,1] that had the smallest readout assignment error among all pairs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2 (A) [Left]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' We can perform a direct CNOT operation between qubits connected at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' For ibmq lima, the CNOT error between [1,2] qubits were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00510 (Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Appendix D: Additional results with 6 different quantum hardware Here we describe additional results obtained by some other IBM quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In the main text we fo- cused on best results with ibmq lima and ibmq jakarta, but in fact we also experimented with ibmq hanoi, ibm cairo, ibm auckland, ibmq montreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Table IV summarizes the complete lists of the best data we ob- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 4: Configurations of qubits on graphs: (A) the lay- out of ibmq lima which has 5 qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (B) the layout of ibmq jakarta which has 7 qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' (C) the layout of 27-qubit hardware including ibmq hanoi, ibm cairo, ibm auckland and ibmq montreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' A direct CNOT gate can be applied to two qubits connected at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' tained and Table III summarizes the experimental con- ditions used for each hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In the entire circuit, the total number N of qubits is 2 and the circuit depth d(N) that can be executed is 9 (excluding measurement of V ) and 10 (including measurement of V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' The quantum vol- ume is defined by QV = � arg maxn≤N min{n, d(n)} �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Therefore quantum computers with QV = 128 are enough for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Here QV is a metric that quantifies the largest random circuit of equal width and depth that a quantum computer can successfully implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' How- ever, QV may not be a crucial metric in this study, since we are only dealing with 2-qubit, relatively simple quan- tum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Errors in quantum computers result from a combination of various factors, including readout error, CNOT error, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='. Table IV shows that Alice’s measure- ments of X0 are relatively accurate in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' With respect to the observation of V , there is a deviation 0 2 0 3 3 4 4 5 (6) 17 4 10 12 15 18 21 23 13 24 5 8 11 14 16 19 22 25 26 2011 Backend ibmq lima ibmq jakarta ibm cairo ibm hanoi ibmq auckland ibmq montreal Ntot 5 7 27 27 27 27 Quantum Volume 8 16 64 64 64 128 shots 105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2 × 104 105 105 105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2 × 104 Measurement fidelity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='961075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='924695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='961935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='979530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='979383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='957484 qubits used [0,1] [3,5] [13,14] [14,16] [14,16] [14,16] CNOT error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='01996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='00739 Gate time (ns) 305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='778 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='556 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='444 472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='889 355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='556 355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='556 First qubit t1(µs) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='67 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='53 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='43 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='15 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='97 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='56 t2(µs) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='39 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='09 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='29 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='75 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='49 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='53 Frequency (GHz) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='030 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='178 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='282 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='047 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='167 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='961 Anharmonicity (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='33574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='33874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='32314 Pauli X error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='781 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='140 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='630 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='305 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='4842 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='942 × 10−4 Readout assignment error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='960 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='440 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='500 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='400 × 10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='100 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='310 × 10−2 Second qubit t1(µs) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='03 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='52 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='28 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='07 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='16 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='73 t2(µs) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='97 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='33 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='99 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='46 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='12 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='92 Frequency (GHz) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='128 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='063 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='044 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='883 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='970 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='086 Anharmonicity (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='31835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='34389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='33707 Pauli X error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='469 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='708 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='732 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='708 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='052 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='221 × 10−4 Readout assignment error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='300 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='400 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='000 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='600 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='700 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='800 × 10−3 TABLE III: Machine properties of IBM quantum computers and parameters we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' shots is the number of iterations we performed for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Average measurement fidelity was computed when preparing a calibration matrix and used for measurement error mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' CNOT error corresponds to the direct CNOT error between two qubits [q0, q1] used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' Gate time corresponds to the gate time between [q0, q1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' First and second qubits corresponds to q0 and q1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' t1 is relaxation time and t2 is dephasing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' from the analytical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It was confirmed that the error mitigation improved the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In this study, what is important is that negative expectation values ⟨V ⟩ were observed for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' It is a noteworthy achievement that negative energy expectation values ⟨E⟩ < 0 were observed by error mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' In fact, the histograms of states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 2 (B)) have improved to approach the ex- act values, indicating that all operations were performed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content=' 12 Backend (h, k) = (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2) (h, k) = (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='5) (h, k) = (1, 1) (h, k) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='5, 1) Analytical value ⟨E0⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='9806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='7071 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2481 ibmq lima error mitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='9423 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='8169 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='6560 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0031 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2480 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0047 unmitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='9049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='8550 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='6874 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0031 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='4066 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0047 ibmq jakarta error mitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='9299 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='8888 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='7039 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0056 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2318 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0084 unmitigated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='9542 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='9089 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='7232 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0056 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2624 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0083 ibm hanoi error mitigated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1512 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='1487 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2411 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='2778 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} +page_content='0112 TABLE IV: Results by ibmq lima, ibmq jakarta, ibmq hanoi, ibm cairo, ibm auckland, ibmq montreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfzAJD/content/2301.02666v1.pdf'} diff --git a/EdE2T4oBgHgl3EQfowg5/content/tmp_files/2301.04021v1.pdf.txt b/EdE2T4oBgHgl3EQfowg5/content/tmp_files/2301.04021v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..292aa56ea12ea1fb1e25e417a5352d695b5670d7 --- /dev/null +++ b/EdE2T4oBgHgl3EQfowg5/content/tmp_files/2301.04021v1.pdf.txt @@ -0,0 +1,218 @@ +Need for “special” states in a deterministic +theory of quantum mechanics +L. S. Schulman +Clarkson University, Potsdam, New York 13699-5820, USA +email: schulmanclarkson.edu +January 11, 2023 +Abstract +There are several theories or processes which may underlie quantum mechanics and +make it deterministic. Some references are given in the main text. Any such theory, plus +a number of reasonable assumptions, implies the existence of what I have called “special” +states. The assumptions are conservation laws, obedience (up to a point) of Schr¨odinger’s +equation, and a single world, in the sense of the many worlds interpretation (the last one a +consequence of any deterministic theory). This article also, for clarity, gives an example of +a “special” state. There is an experimental test of the “special” state theory. +1 +Introduction +Determinism is a loophole in Bell’s [1] ideas, which he was aware of. I unwittingly exploited it in +1984 [18] with what I will call the “special” state theory of measurement. (In Sec. 2 an example +of a “special” state is given.) The present article reports new motivation for “special” states. +There have been quite a few attempts to find an underlying process that would make the +Schr¨odinger equation deterministic. I am not referring to Bohm’s interpretation [2] or that of +his followers. Rather I have in mind those theories which would restore determinism, such as +(not exclusively) those of ’tHooft [22, 23], Palmer [16], De la Pena Auerbach and Cetto [13], +Cavalleri et al. [5], Cufaro-Petroni and Vigier [6] and Marshall [15]. For at least some of these +the Schr¨odinger equation is an approximation—a good approximation, but an approximation +nevertheless. There has also been discussion about the consequences of determinism [22, 16, 12, +4, 11]. +There is an experimental test of the “special” state theory, which, if successful, would lend +credence to some of the theories advanced. If negative, it would be challenging to maintain +determinism. +2 +“Special” states +Most of this section is review. It may be skipped by those familiar with the kind of “special” +states that I have in mind. +1 +arXiv:2301.04021v1 [quant-ph] 8 Jan 2023 + +We take, as an example of a “special” state, a spin, initially pointing in the positive z +direction with a 50% probability of overturning at some given time, say at 0.15 (since all is +determined the time of observation is also fixed). Moreover, we don’t deal with “registration” of +the measurement; that will be accomplished by additional degrees of freedom.1 The Hamiltonian +is +H = ε +2 (1 + σz) + ωa†a + βσx(a† + a). +(1) +The Pauli matrices σx and σz are the operators for the 2-state spin system, a and a† are the +boson operators and ε, β and ω are parameters. +“Special” states are particular initial conditions of the bath such that the microscopic final +state of the spin is (either) all up, (eiφ1 +0 ), or all down, ( +0 +eiφ2 ). “Final” refers to the time of +measurement, namely when (even) more degrees of freedom are involved (we use parameter +values ϵ = 0.5, ω = 0.1, β = 0.6 and a time of 0.15). +The system begins in all up and ordinarily at time 0.15 has probability of half up, half down. +As indicated, that is not the case for these “special” initial conditions. If the probabilities are +as in Fig. 1a—with fixed phases (not shown)—then the system will be found in an up state. If +the initial state is as shown in Fig. 1b (again with particular phases, not shown) then the system +will be found in a down state. +There are three points to be raised: the first is what about residual amplitudes? The ampli- +tude for (say) the up state is not perfect and for the given cutoff of the bosons at 250 is about +10−4; the same is true for the state which is “fully” decayed. The second question has to do with +Schr¨odinger’s cat. And the third issue is how do you find these states? +Now 10−4 is a big number, especially since the final state of one interaction is the initial +state for the next. I could improve that number if I had better computer power, but I doubt +if it could be zero. But it doesn’t have to be zero! It only needs to be accurate as far as the +Schr¨odinger equation has been checked. And I don’t think it has been checked to 10−12 (which I +am reasonably confident I could get the discrepancy down to). +The second issue I mentioned is, what about Schr¨odinger cats? The (possibly) decaying spin +could be the determinant of whether the poison is released.2 With “special” states the cat is +either alive or dead. It should be noticed that there are only what ’tHooft calls “ontological” +states [22, 23]. I believe “special” and “ontological” have the same meaning in this case. +Finally there is question of how “special” states are found. +You can define a projection +operator (cf. [19]) on the spin: P ≡ (ψupψ† +up)⊗1boson bath. Using this operator, the probability of +being all up at time t is +Pr(up) = ⟨ψup ⊗ ψbath∣U †PU∣ψup ⊗ ψbath⟩ = ⟨ψup ⊗ ψbath∣PU †PPUP∣ψup ⊗ ψbath⟩, +(2) +with U ≡ exp(−iHt/ℏ) and where PP = P is used. Defining A ≡ PUP and using P † = P, we have +Pr(up) = ⟨ψup⊗ψbath∣A†A∣ψup⊗ψbath⟩. Defining B ≡ A†A, it follows that the issue of whether any +initial state (of the bath) can lead to a measurement of up, using purely unitary time evolution +1The irreversible “registration” of the result of a measurement by the observer has been studied in many +contexts. For example, in [9] the “measurement” is accompanied by the bath’s (not the same as the bath in +the current Eq. (1)) changing in an irreversible fashion. Other models of measurement (e.g., [3, 10]) show the +same feature. As a result, our considerations in the present article do not pursue the registration issue once the +observer is coupled to the system, that coupling taking place (in our forthcoming example) at 0.15 time units. +2I assume familiarity with the Schr¨odinger cat paradox. +2 + +is the matter of whether B has eigenvalues equal to one. For any fully decayed states you must +have an eigenvalue (of B) be zero. (Of course A and B are functions of t, since U is.) +Remark: +It also is true that the number of decay states and non-decay (“special”) states is +roughly equal at time 0.15. +Figure 1: “Special” time-0 oscillator states. Figure (a) shows the (initial) probability of excitation +of oscillator states that contribute to the non-decay state. Only shown are even states, since there +is total amplitude zero for the odd states. Phases of the states are not shown, but are also fixed +by the non-decay condition. In Figure (b) are shown the probabilities for the state that decays; +in this case (and for the same reason) only even oscillator states are shown. As in image a, the +phases, though not shown are crucial to the “special” nature of the state. +This is the main idea of the “special” state theory: no macroscopic superpositions because +of particular initial conditions. There is also no entanglement. At time-0.15 the spin state is +wholly in one state or the other. +3 +Determinism implies “special” states +The title of this section needs a bit of enhancement: you need a few more concessions to reality. +Besides determinism you need conservation laws and Schr¨odinger’s equation, at least to the +extent that it’s been checked. It is also understood that there is just one world. These rules, +together with determinism, imply “special” states. +You start with a wave function describing some state, say a spin in a Stern-Gerlach experi- +ment. Then it must go to some particular outcome, say spin up. Presumably there were involved +other coordinates (such as the bosons in the above example) that fixed its outcome. The final +state is definite. But the Schr¨odinger equation holds also. Therefore it could only have evolved +to that final state. How can that be? There must have been a coordination of degrees of freedom +on the initial state that forced it to its final form. That is, there must have been a “special” +state. +4 +Experiment +Finally, there is the matter of experiment. In [20] and [21] we have described in detail experi- +mental tests of the “special” state theory. An example is the double Stern-Gerlach experiment +3 + +0.2 +a +0.15 +0.1 +0.05 +0 +0 +50 +100 +state label +200 +2500.07 +b +0.06 +0.05 +probability +0.04 +0.03 +0.02 +0.01 +0 +0 +50 +100 +200 +250 +state label([17, 8, 14, 7]) which requires the detection of a magnetic field of 5×10−8 tesla in an environment +of half a tesla, a challenging experiment. A firm absence of the small magnetic field would in my +opinion spell the end of efforts to find a deterministic theory (but no-go theorems are made to +be disproved). +5 +Conclusions +You don’t have to believe in any of the deterministic theories to reach the conclusion that +“special” states are needed in any theory which is deterministic, goes from one “special” state +into another, satisfies Schr¨odinger’s equations (as far as has been measured), has a single world +and satisfies conservation laws. You only have to believe that it’s possible. +Three points are worth mentioning. First—and this is new—you don’t need to eliminate +“incorrect” choices (by “special” states) at the level of (say) 10−12, since the Schr¨odinger equa- +tion has not been checked at that level. Second, there is an experimental test of the special +state theory. Failure would eliminate deterministic theories (or leave people struggling for an +explanation), while success would encourage attempts to find deterministic theories. Third, it +may be that ’tHooft is right, and one should look to extremely small times and distances for +theoretical support for determinism. However, given the fragmentary understanding of events at +10−17 cm I’d be reluctant to make predictions about what happens at 10−33 cm. +References +[1] J. S. Bell. +Speakable and unspeakable in quantum mechanics. +Cambridge Univ. Press, +Cambridge, 1987. +[2] David Bohm. Quntum Theory. Prentice Hall, New Jersey, 1961. +[3] P. B´ona. A solvable model of particle detection in quantum theory. Acta Facultatis Rerum +Naturalium Uuniversitatis Comenianae Physica, XX:65–95, 1980. +[4] C. H. Brans. Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theor. +Phys., 27:219, 1988. +[5] G. Cavalleri, E. Cesaroni, E. Tonni, and P. Di Sia. About the derivation of Planck’s black +body spectrum from classical mechanics. in “Physical Interpretations of Relativity Theory +VI” (Imperial College, London, 15-18 September 2000), Ed. by M.C.Duffy, School of Me- +chanical Engineering, Sunderland Polytecnic, Chester Road, Sunderland, SR1 3SD, England +(p. 78), 09 2000. +[6] N. Cufaro-Petroni and J-P. Vigier. Single-particle trajectories and interferences in quantum +mechanics. Found. Phys., 22:1–40, 1992. +[7] R. Frisch, T. E. Phipps, E. Segre, and O. Stern. Process of space quantisation. Nature, 130 +(no. 3293):892–3, 1932. +4 + +[8] R. Frisch and E. Segr`e. ¨Uber die Einstellung der Richtungsquantelung. II. Zeitschrift f¨ur +Physik, 80:610–616, 1933. +[9] B. Gaveau and L. S. Schulman. Model apparatus for quantum measurements. J. Stat. Phys., +58:1209–1230, 1990. +[10] H. S. Green. Observation in quantum mechanics. Nuov. Cim., 9:880–889, 1958. +[11] M. J. W. Hall. Local deterministic model of singlet state correlations based on relaxing +measurement independence. Phys. Rev. Lett., 105:250404, 2010. +[12] S. Hossenfelder and T. Palmer. Rethinking superdeterminism. Front. Phys., 8:139, 2020. +[13] L. De la Pe˜na Auerbach and A. M. Cetto. Stochastic electrodynamics as a foundation for +quantum mechanics. Phys. Lett., 56A:253–254, 1976. +[14] E. Majorana. Atomi orientati in campo magnetico variabile. Nuovo Cimento, 9:43–50, 1932. +[15] T. W. Marshall. Statistical electrodynamcs. Proc. Camb. Phil. Soc., 61:537–546, 1965. +[16] T. N. Palmer. The Invariant Set Postulate: a new geometric framework for the foundations +of quantum theory and the role played by gravity. Proc. Roy. Soc. A, 465:3165–3185, 2009. +[17] T. E. Phipps and O. Stern. ¨Uber die einstellung der richtungsquantelung. Zeitschrift f¨ur +Physik, 73:185–191, 1932. +[18] L. S. Schulman. Definite measurements and deterministic quantum evolution. Phys. Lett. +A, 102:396–400, 1984. +[19] L. S. Schulman. Time’s Arrows and Quantum Measurement. Cambridge Univ. Press, New +York, 1997. +[20] L. S. Schulman. Experimental test of the “special state” theory of quantum measurement. +Entropy, 14:665–686, 2012. +[21] L. S. Schulman and M. G. E. da Luz. Looking for the source of change. Found. Phys., +46:1495–1501, 2016. +[22] G. ’tHooft. The Cellular Automaton Interpretation of Quantum Mechanics. Springer, Berlin, +2016. +[23] G. ’tHooft. Deterministic quantum mechanics: The mathematical equations. Front. Phys., +8:253, 2020. +5 + diff --git a/EdE2T4oBgHgl3EQfowg5/content/tmp_files/load_file.txt b/EdE2T4oBgHgl3EQfowg5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4893306d019d975998f9c35e1382530c22827ede --- /dev/null +++ b/EdE2T4oBgHgl3EQfowg5/content/tmp_files/load_file.txt @@ -0,0 +1,287 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf,len=286 +page_content='Need for “special” states in a deterministic theory of quantum mechanics L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Schulman Clarkson University, Potsdam, New York 13699-5820, USA email: schulmanclarkson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='edu January 11, 2023 Abstract There are several theories or processes which may underlie quantum mechanics and make it deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Some references are given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Any such theory, plus a number of reasonable assumptions, implies the existence of what I have called “special” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The assumptions are conservation laws, obedience (up to a point) of Schr¨odinger’s equation, and a single world, in the sense of the many worlds interpretation (the last one a consequence of any deterministic theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' This article also, for clarity, gives an example of a “special” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There is an experimental test of the “special” state theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 1 Introduction Determinism is a loophole in Bell’s [1] ideas, which he was aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' I unwittingly exploited it in 1984 [18] with what I will call the “special” state theory of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' (In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 2 an example of a “special” state is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=') The present article reports new motivation for “special” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There have been quite a few attempts to find an underlying process that would make the Schr¨odinger equation deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' I am not referring to Bohm’s interpretation [2] or that of his followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Rather I have in mind those theories which would restore determinism, such as (not exclusively) those of ’tHooft [22, 23], Palmer [16], De la Pena Auerbach and Cetto [13], Cavalleri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' [5], Cufaro-Petroni and Vigier [6] and Marshall [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' For at least some of these the Schr¨odinger equation is an approximation—a good approximation, but an approximation nevertheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There has also been discussion about the consequences of determinism [22, 16, 12, 4, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There is an experimental test of the “special” state theory, which, if successful, would lend credence to some of the theories advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' If negative, it would be challenging to maintain determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 2 “Special” states Most of this section is review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' It may be skipped by those familiar with the kind of “special” states that I have in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='04021v1 [quant-ph] 8 Jan 2023 We take, as an example of a “special” state, a spin, initially pointing in the positive z direction with a 50% probability of overturning at some given time, say at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15 (since all is determined the time of observation is also fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Moreover, we don’t deal with “registration” of the measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' that will be accomplished by additional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='1 The Hamiltonian is H = ε 2 (1 + σz) + ωa†a + βσx(a† + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' (1) The Pauli matrices σx and σz are the operators for the 2-state spin system, a and a† are the boson operators and ε, β and ω are parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' “Special” states are particular initial conditions of the bath such that the microscopic final state of the spin is (either) all up, (eiφ1 0 ), or all down, ( 0 eiφ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' “Final” refers to the time of measurement, namely when (even) more degrees of freedom are involved (we use parameter values ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='5, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='6 and a time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The system begins in all up and ordinarily at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15 has probability of half up, half down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' As indicated, that is not the case for these “special” initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' If the probabilities are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 1a—with fixed phases (not shown)—then the system will be found in an up state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' If the initial state is as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 1b (again with particular phases, not shown) then the system will be found in a down state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There are three points to be raised: the first is what about residual amplitudes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The ampli- tude for (say) the up state is not perfect and for the given cutoff of the bosons at 250 is about 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' the same is true for the state which is “fully” decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The second question has to do with Schr¨odinger’s cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' And the third issue is how do you find these states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Now 10−4 is a big number, especially since the final state of one interaction is the initial state for the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' I could improve that number if I had better computer power, but I doubt if it could be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' But it doesn’t have to be zero!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' It only needs to be accurate as far as the Schr¨odinger equation has been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' And I don’t think it has been checked to 10−12 (which I am reasonably confident I could get the discrepancy down to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The second issue I mentioned is, what about Schr¨odinger cats?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The (possibly) decaying spin could be the determinant of whether the poison is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='2 With “special” states the cat is either alive or dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' It should be noticed that there are only what ’tHooft calls “ontological” states [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' I believe “special” and “ontological” have the same meaning in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Finally there is question of how “special” states are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' You can define a projection operator (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' [19]) on the spin: P ≡ (ψupψ† up)⊗1boson bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Using this operator, the probability of being all up at time t is Pr(up) = ⟨ψup ⊗ ψbath∣U †PU∣ψup ⊗ ψbath⟩ = ⟨ψup ⊗ ψbath∣PU †PPUP∣ψup ⊗ ψbath⟩, (2) with U ≡ exp(−iHt/ℏ) and where PP = P is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Defining A ≡ PUP and using P † = P, we have Pr(up) = ⟨ψup⊗ψbath∣A†A∣ψup⊗ψbath⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Defining B ≡ A†A, it follows that the issue of whether any initial state (of the bath) can lead to a measurement of up, using purely unitary time evolution 1The irreversible “registration” of the result of a measurement by the observer has been studied in many contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' For example, in [9] the “measurement” is accompanied by the bath’s (not the same as the bath in the current Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' (1)) changing in an irreversible fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Other models of measurement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=', [3, 10]) show the same feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' As a result, our considerations in the present article do not pursue the registration issue once the observer is coupled to the system, that coupling taking place (in our forthcoming example) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 2I assume familiarity with the Schr¨odinger cat paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 2 is the matter of whether B has eigenvalues equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' For any fully decayed states you must have an eigenvalue (of B) be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' (Of course A and B are functions of t, since U is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=') Remark: It also is true that the number of decay states and non-decay (“special”) states is roughly equal at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Figure 1: “Special” time-0 oscillator states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Figure (a) shows the (initial) probability of excitation of oscillator states that contribute to the non-decay state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Only shown are even states, since there is total amplitude zero for the odd states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Phases of the states are not shown, but are also fixed by the non-decay condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' In Figure (b) are shown the probabilities for the state that decays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' in this case (and for the same reason) only even oscillator states are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' As in image a, the phases, though not shown are crucial to the “special” nature of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' This is the main idea of the “special” state theory: no macroscopic superpositions because of particular initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There is also no entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' At time-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15 the spin state is wholly in one state or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 3 Determinism implies “special” states The title of this section needs a bit of enhancement: you need a few more concessions to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Besides determinism you need conservation laws and Schr¨odinger’s equation, at least to the extent that it’s been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' It is also understood that there is just one world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' These rules, together with determinism, imply “special” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' You start with a wave function describing some state, say a spin in a Stern-Gerlach experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Then it must go to some particular outcome, say spin up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Presumably there were involved other coordinates (such as the bosons in the above example) that fixed its outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' The final state is definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' But the Schr¨odinger equation holds also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Therefore it could only have evolved to that final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' How can that be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' There must have been a coordination of degrees of freedom on the initial state that forced it to its final form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' That is, there must have been a “special” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 4 Experiment Finally, there is the matter of experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' In [20] and [21] we have described in detail experi- mental tests of the “special” state theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' An example is the double Stern-Gerlach experiment 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='2 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='05 0 0 50 100 state label 200 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='07 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='05 probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content='01 0 0 50 100 200 250 state label([17, 8, 14, 7]) which requires the detection of a magnetic field of 5×10−8 tesla in an environment of half a tesla, a challenging experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' A firm absence of the small magnetic field would in my opinion spell the end of efforts to find a deterministic theory (but no-go theorems are made to be disproved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' 5 Conclusions You don’t have to believe in any of the deterministic theories to reach the conclusion that “special” states are needed in any theory which is deterministic, goes from one “special” state into another, satisfies Schr¨odinger’s equations (as far as has been measured), has a single world and satisfies conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' You only have to believe that it’s possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Three points are worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' First—and this is new—you don’t need to eliminate “incorrect” choices (by “special” states) at the level of (say) 10−12, since the Schr¨odinger equa- tion has not been checked at that level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Second, there is an experimental test of the special state theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Failure would eliminate deterministic theories (or leave people struggling for an explanation), while success would encourage attempts to find deterministic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Third, it may be that ’tHooft is right, and one should look to extremely small times and distances for theoretical support for determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' However, given the fragmentary understanding of events at 10−17 cm I’d be reluctant to make predictions about what happens at 10−33 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Bell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQfowg5/content/2301.04021v1.pdf'} +page_content=' Speakable and unspeakable in quantum mechanics.' metadata={'source': 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+ROWAN.MCALLISTER@TRI.GLOBAL +Toyota Research Institute, Los Altos, CA +Koushil Sreenath +KOUSHILS@BERKELEY.EDU +University of California, Berkeley +Adrien Gaidon +ADRIEN.GAIDON@TRI.GLOBAL +Toyota Research Institute, Los Altos, CA +Abstract +Learning-based control approaches have shown great promise in performing complex tasks di- +rectly from high-dimensional perception data for real robotic systems. Nonetheless, the learned +controllers can behave unexpectedly if the trajectories of the system divert from the training data +distribution, which can compromise safety. In this work, we propose a control filter that wraps +any reference policy and effectively encourages the system to stay in-distribution with respect to +offline-collected safe demonstrations. Our methodology is inspired by Control Barrier Functions +(CBFs), which are model-based tools from the nonlinear control literature that can be used to con- +struct minimally invasive safe policy filters. While existing methods based on CBFs require a +known low-dimensional state representation, our proposed approach is directly applicable to sys- +tems that rely solely on high-dimensional visual observations by learning in a latent state-space. We +demonstrate that our method is effective for two different visuomotor control tasks in simulation +environments, including both top-down and egocentric view settings. +Keywords: Distributional Shift, Control Barrier Functions, State Representation Learning +1. Introduction +The modern advances in the representation learning literature have been an enabling factor for +the recent surge of a wide variety of methods for robotic control directly from images or high- +dimensional sensory observations (Watter et al., 2015; Ebert et al., 2018; Hafner et al., 2019; Lenz +et al., 2015; Zhang et al., 2019; Van Hoof et al., 2016). These approaches for visuomotor planning +and control have the potential to solve challenging tasks in which the state of the system might +not be directly observable, or even not possible to model analytically. While promising, the high- +dimensionality of the problem make these methods susceptible to several open challenges. For +example, the exploration requirements of reinforcement learning (RL) algorithms are significantly +exacerbated for these tasks, due to the high-dimensionality of the observations. This means that +trying to learn safe control policies using RL often requires us to accept that abundant failures +will occur during training. On the other hand, supervised learning approaches for control, such as +behavioral cloning, would in principle seem less prone to exhibit unsafe behaviors. However, it is +© 2023 F. Casta˜neda, H. Nishimura, R. McAllister, K. Sreenath & A. Gaidon. +arXiv:2301.12012v1 [cs.RO] 27 Jan 2023 + +IN-DISTRIBUTION BARRIER FUNCTIONS +well known that simply because of their data-dependent nature, these methods are still susceptible to +a key challenge named distributional shift: if the trajectories of the system divert from the training +data distribution, the controller might take unexpected actions. +On the other hand, the control theory literature extensively covers the problem of long-horizon +constraint satisfaction. In particular, Control Barrier Functions (CBFs, Ames et al. (2014)) are a +popular model-based tool used to restrict the trajectories of the system from entering undesirable +regions of the state-space. One of the properties of CBFs that explain their recent popularity is that +they decouple the problem of constraint-satisfaction from any performance objective. Specifically, +if a CBF is available, then Ames et al. (2014) showed that we can construct a minimally invasive +safety filter that transforms into safety-preserving control actions any unsafe commands that an +arbitrary reference policy could output. +The main question we want to address in this work naturally emerges from the previous dis- +cussion: can we take inspiration from CBFs to avoid out-of-distribution (OOD) states when using +data-driven controllers for visuomotor tasks? Even though CBFs are model-based tools that, as +such, require knowledge of the state-space and dynamics of the system, the recent advances on +learning latent state-space representations and associated dynamics models clearly set a path for +linking data-driven visuomotor policy learning with the use of model-based control-theoretic tools +such as CBFs. +Contributions: We present an end-to-end self-supervised approach for learning a task-agnostic +policy filter which prevents the system from entering OOD states. We do not assume knowledge of +the state-space or system dynamics. In addition, our framework only requires an offline-collected +dataset of safe demonstrations (where the concept of safety is only linked to the demonstrator’s +subjectivity, as it is their responsibility to provide the dataset). We therefore do not require any un- +safe demonstrations to learn a safe policy filter, in contrast to most other works tackling constrained +policy learning. Furthermore, to the best of our knowledge, this is the first work that uses CBFs +for constructing policy filters in learned latent state-spaces. This endows our approach with the +flexibility of being applicable to systems with high-dimensional sensory observations, in contrast +to most prior CBF-based methods. We present simulation experiments on two different visuomotor +control tasks, which suggest that our framework, taking only raw RGB images as input, can learn +to significantly reduce the distributional shift from safe demonstrations and, consequently, critically +improve the safety of both systems. +2. Related Work +There exist some constructive procedures for synthesizing CBFs based on sum-of-squares program- +ming (Jarvis-Wloszek et al., 2003; Majumdar et al., 2013; Dai and Permenter, 2022; Wang et al., +2022) or Hamilton-Jacobi reachability (Choi et al., 2021). However, these methods require knowl- +edge of the dynamics of the system and typically suffer from scalability issues for high-dimensional +systems. More in line with this work, some recent results show that CBFs can be learned from +data (Jin et al., 2020; Dawson et al., 2022; Qin et al., 2022; Robey et al., 2020; Lindemann et al., +2021; Abate et al., 2021; Jagtap et al., 2020). None of these works, however, consider systems with +high-dimensional observations. Furthermore, the works Jin et al. (2020); Dawson et al. (2022); Qin +et al. (2022) assume a priori knowledge of a control-invariant safe set, and focus on building a CBF +for that particular set. The line of research of Robey et al. (2020); Lindemann et al. (2021) has the +most similar problem setup to our work, as they also consider learning from safe demonstrations. +2 + +IN-DISTRIBUTION BARRIER FUNCTIONS +Although, notably, the authors provide formal verification arguments for the learned CBFs, their +methods are not applicable to high-dimensional observations, assume a nominal dynamics model is +given, and use an algorithmic approach for the detection of the boundary of the dataset that does +not scale to large datasets. Other recent approaches build signed distance functions from sensory +measurements that are obtained from a LiDAR or stereo cameras (Long et al., 2021; Srinivasan +et al., 2020; Cosner et al., 2022). However, these functions are not encouraged to satisfy any set +invariance property. +Extensions of the CBF-based control filters to systems with dynamics or measurement-model +uncertainty have also been recently proposed (Nguyen and Sreenath, 2021; Casta˜neda et al., 2021; +Taylor et al., 2021; Dhiman et al., 2021; Dean et al., 2021). These works assume that a CBF is pro- +vided, and formulate uncertainty-robust optimization problems for the controller design. They can +be considered complementary to our deterministic but end-to-end approach. Future work should ex- +plore quantifying uncertainty estimates within our framework to robustify the learned policy filters. +Several existing approaches for OOD prevention learn density models of training data that can +be then used to restrict the agent from taking low likelihood actions or moving towards unvis- +ited states (McAllister et al., 2019; Richter and Roy, 2017; Wu et al., 2019; Kumar et al., 2019). +Although some of these methods have been shown to be effective at offline RL settings that are +specially susceptible to distributional shift, the learned density models have no notion of control +invariance and, therefore, do not consider the problem of how to prevent distributional shift over a +long time horizon. A notable exception is the work of Kang et al. (2022) to constrain long-term dis- +tributional shift, in which a min-max Bellman backup operator is constructed so that Lyapunov-like +functions arise as value functions of an offline RL problem. This work however does not consider +the extension to visuomotor control tasks in learned latent spaces. Furthermore, for our approach +we choose not to rely on a min-max backup operator to learn the certificate function and, instead, +use the very suitable theory of CBFs to devise a self-supervised learning framework. +Finally, the work of Wilcox et al. (2022) presents a framework to learn safe sets in a latent state- +space for iterative control tasks. Compared to this work, our framework has the advantage that it is +task-agnostic and does not require any interactions with the environment during training. +3. Background on Control Barrier Functions +We start by introducing some necessary background on Control Barrier Functions, which are tools +from the nonlinear control literature that serve to enforce safety constraints for systems with known +dynamics. As will be clear later, CBFs are particularly well-suited for continuous-time nonlinear +control-affine systems of the form +˙x = f(x) + g(x)u, +(1) +where x ∈ X ⊂ Rn is the state and u ∈ U ⊂ Rm the control input. We assume that f : X → Rn +and g : X → Rn×m are locally Lipschitz continuous. +In the CBF literature, safety is considered as a set invariance problem. In particular, we say that +a control policy π : X → U assures the safety of system (1) with respect to a set Xsafe ⊂ X if the +set Xsafe is forward invariant under the control law π, i.e., for any x0 ∈ Xsafe, the solution x(t) of +system (1) under the control law π remains within Xsafe for all t ≥ 0. +Definition 1 (Control Barrier Function, Ames et al. (2017)) +We say that a continuously differ- +entiable function B : X → R is a Control Barrier Function (CBF) for system (1) with associated +safe-set Xsafe ⊂ X if the following three conditions are satisfied: +3 + +IN-DISTRIBUTION BARRIER FUNCTIONS +B(x) ≥ 0 ∀x ∈ Xsafe, +(2) +B(x) < 0 ∀x ∈ X \ Xsafe, +(3) +∃u ∈ U s.t. +˙B(x, u) + γ(B(x)) ≥ 0 ∀x ∈ X, +(4) +where γ : R → R is an extended class K∞ function. +The existence of a CBF B guarantees that for system (1) any Lipschitz continuous control policy +π satisfying +π(x) ∈ {u ∈ U : ∇B(x)[f(x) + g(x)u] +� +�� +� += ˙B(x,u) ++γ (B(x)) ≥ 0} +(5) +will render the set Xsafe forward invariant (Ames et al., 2017, Corollary 2). +For a given task-specific reference controller πref : X → U that might be safety-agnostic, the +condition of (5) can be used to formulate an optimization problem that, when solved at every time- +step, yields a minimally-invasive policy safety filter (Ames et al., 2014): +πCBF(x) = arg min +u∈U +∥u − πref(x)∥2 +(CBF-QP) +s.t. ∇B(x)[f(x) + g(x)u] + γ(B(x)) ≥ 0. +Assuming that the actuation constraints that define U are linear in u, this problem is a quadratic +program (QP). This is a consequence of the dynamics of the system (1) being control-affine, and it +practically means that the problem can be solved to a high precision very quickly (around 103Hz). +This is critical since the CBF-QP needs to be solved at the real-time control frequency. +4. Problem Statement +The CBF-QP constitutes a very appealing approach for practitioners: it provides a task-agnostic +minimally invasive filter that can wrap safety around any given policy πref, and therefore rewrite +any unsafe control input that πref could output at any time. However, designing a valid CBF is +nontrivial. In fact, it is still an active research topic even when assuming perfect knowledge of the +dynamics of the system (Dai and Permenter, 2022; Choi et al., 2021; Wang et al., 2022). The two +main difficulties in the design of a CBF are the following: first, a control-invariant set Xsafe must be +obtained (which in general is different from the geometric constraint set that could be obtained, for +instance, from a signed-distance field) and, second, a function that satisfies condition (4) must be +found for that set. Furthermore, even after obtaining a CBF, solving the CBF-QP requires perfect +state and dynamics knowledge. +With our framework, we take initial steps towards building a safe policy filter from high- +dimensional observations. Specifically, we take inspiration from CBFs to design an end-to-end +learning framework to constrain deep learning models to remain in-distribution of the training data. +We take as input a dataset of high-dimensional observations of different safe demonstrations, and +build a neural CBF-like function that encourages the system to always stay in-distribution with re- +spect to the observations from the safe demonstrations. This, in turn, significantly improves the +safety of the system during deployment. +More concretely, for a given dataset of N safe trajectories D = +�� +Ii +t, ui +t +�t=Ti +t=0 +�i=N +i=1 we tackle +the problem of designing a policy filter that can be applied to any reference controller πref to detect +and override actions from πref that lead to OOD states. We denote Ii +t and ui +t the high-dimensional +4 + +IN-DISTRIBUTION BARRIER FUNCTIONS +observation and control input, respectively, measured at time t for the ith trajectory. Furthermore, +Ti is the final time-step of trajectory i. The demonstrations in the dataset D might correspond to +different tasks and they do not need to be optimal with respect to any objective. In fact, our only +assumption is that the dataset only contains safe demonstrations (in the sense that these trajectories +should not contain any states from which the system is deemed to fail, even if it has not failed yet), +so that we can encourage long-term constraint satisfaction using CBFs. +5. In-Distribution Barrier Functions +In this section, we introduce a self-supervised approach for synthesizing neural CBF-like functions +whose aim is to constrain the system to remain in-distribution with respect to an offline dataset of +safe demonstrations. We call these functions in-Distribution Barrier Functions (iDBFs). We will +for now assume that we have a parametric continuous-time control-affine model of the dynamics of +the system in a state-space X ⊂ Rn +˙x = fθ(x) + gθ(x)u, +(6) +and present the iDBF learning procedure for this system. Furthermore, for this section we assume +that the dataset D of safe trajectories contains true state measurements, i.e., D = +�� +xi +t, ui +t +�t=Ti +t=0 +�i=N +i=1 , +where xi +t and ui +t are the state and control input, respectively, measured at time t for the ith trajec- +tory. In Section 6, we will provide details on how to learn a dynamics model of this form in a latent +state-space when we have a dataset containing high-dimensional sensory observations. +We parameterize an iDBF Bφ : X → R as a neural network with parameters φ, and construct +an empirical loss function that encourages it to satisfy the three CBF conditions (2), (3) and (4) +with respect to a set Xsafe that is also implicitly learned through self-supervision. To design the +loss function, we take inspiration from previous literature on learning CBFs (Dawson et al., 2022; +Qin et al., 2022; Chang et al., 2019). Nevertheless, instead of assuming that the safe-set Xsafe is +given and that we can sample from it and from its unsafe complement Xunsafe .= X \ Xsafe, we +build our loss function in a self-supervised manner just from the dataset of safe demonstrations. We +accomplish this by leveraging ideas from contrastive learning (Gutmann and Hyv¨arinen, 2010; Oord +et al., 2018; Chopra et al., 2005; Weinberger and Saul, 2009; Schroff et al., 2015). In particular, as +we explain in detail later, we build a contrastive distribution from which to sample candidate unsafe +states, given that we do not have any unsafe demonstrations in our dataset. The loss function we +propose for learning an iDBF takes the following form: +LiDBF = wsafe +Nsafe +� +xsafe +[ϵsafe − Bφ(xsafe)]+ + wunsafe +Nunsafe +� +xunsafe +[ϵunsafe + Bφ(xunsafe)]+ + +wascent +Nsafe +� +(xsafe,usafe) +� +ϵascent − +� +∇Bφ(xsafe)[fθ(xsafe) + gθ(xsafe)usafe] + γ(Bφ(xsafe)) +��+ +, +(7) +where [·]+ := max(0, ·); (xsafe, usafe) are samples from the empirical distribution of the dataset D; +xunsafe are samples from a contrastive distribution that we will define soon; wsafe, wunsafe and wascent +are the weights of the different loss terms; and ϵsafe, ϵunsafe and ϵascent are positive constants that +serve to enforce strictly the inequalities and generalize outside of the training data. +The goal of the first two terms in the loss function is to learn an iDBF that has a positive value +in states that belong to the data distribution of safe demonstrations, and negative everywhere else +(meaning we are encouraging the satisfaction of conditions (2) and (3) of the definition of CBF). +5 + +IN-DISTRIBUTION BARRIER FUNCTIONS +Note that this classification objective is very related to the notion of energy-based models (EBMs) +—neural network density models that assign a low energy value to points close to the training data +distribution and a high value to points that are far from it (Hinton, 2002). In fact, we took inspiration +from the Noise Contrastive Estimation (NCE, Gutmann and Hyv¨arinen (2010)) training procedure +of EBMs, in particular the InfoNCE loss (Oord et al., 2018), to design (7). Intuitively, these methods +use a noise contrastive distribution to generate candidate examples where to increase the value of the +energy of the EBM, while decreasing the energy at the training data points. We precisely want the +opposite result for our problem: a high value of Bφ on the training data distribution, and a low value +everywhere else. However, we have one additional requirement, which is that the iDBF should have +a value of zero at the boundary, as set by conditions (2) and (3). This is the reason why we design +the two first terms of the loss function using the [·]+ operator. +In the third term of the loss (7), note that we do not encourage the satisfaction of condition +(4) over the entire state-space, but only over the dataset of safe demonstrations. However, in the +definition of CBF, if condition (4) is only satisfied ∀x ∈ Xsafe instead of ∀x ∈ X, the CBF still +guarantees the control-invariance of Xsafe. We are therefore using our empirical data distribution of +safe demonstrations as a sampling distribution covering the set Xsafe, which we are also implicitly +learning as the zero-superlevel set of Bφ. Furthermore, compared to prior approaches that encourage +the satisfaction of condition (4) for a single policy (Dawson et al., 2022; Qin et al., 2022), we instead +use all pairs (xsafe, usafe) present in the dataset D to compute this term of the loss. This way, we +force the set of admissible control inputs (5) to be as large as our dataset allows, reducing the +conservatism of the learned iDBF. +In order to generate the contrastive distribution from which to sample xunsafe, as we ultimately +want to learn the iDBF in a latent state-space in which it might not be intuitive how to construct a +noise distribution, we take the following steps. 1) Based on the dataset of safe demonstrations D, +we train a neural behavioral cloning (BC) model that outputs a multi-modal Gaussian distribution +over actions conditioned on the state, with density πBC(u|x). 2) Then, during the training process +of the iDBF, for each xsafe state sampled from D we randomly take Ncandidate control inputs ucandidate +and evaluate their density value based on the BC model πBC(ucandidate|xsafe). 3) If the value of +the density falls below a threshold, then that control input is forward-propagated for one timestep +using the dynamics model (6) to generate a sample xunsafe. This way, we generate a contrastive data +distribution by propagating actions that are unlikely present in the dataset of safe demonstrations. +Furthermore, by only propagating these actions for one timestep, the contrastive distribution is close +to the training data, which is desirable for the learning process (Gutmann and Hirayama, 2011). +6. Learning iDBFs from High-Dimensional Observations +After introducing the training procedure for an iDBF when the state representation and dynamics +model (6) are given, we now relax these assumptions and present an approach to learn a latent +state-space representation and a continuous-time dynamics model of the form (6), suitable to be +integrated in the same end-to-end learning framework. We therefore now consider precisely the +problem setting described in Section 4, in which we only assume having access to a dataset contain- +ing observation-action pairs of safe demonstrations D = +�� +Ii +t, ui +t +�t=Ti +t=0 +�i=N +i=1 . +We use an autoencoder architecture to obtain the latent state-space representation, and employ +the training procedure of Neural Ordinary Differential Equations (Neural ODEs, Chen et al. (2018)) +to learn a dynamics model of the form (6) in the latent state-space. Note that by enforcing the +6 + +IN-DISTRIBUTION BARRIER FUNCTIONS +Figure 1: Our framework’s inference diagram. At each timestep, based on the current observation Ik and the previous +latent state and action, the encoder network Eψ outputs a new latent state xk. Then, the iDBF and dynamics networks give +the values of Bφ(xk), fθ(xk) and gθ(xk) which are passed to the iDBF-QP policy filter. The iDBF-QP takes a reference +control input for the current timestep πref(xk) and returns the closest action that keeps the system in-distribution with +respect to the offline-collected dataset of safe demonstrations. For both of the examples of Section 7, the total inference +time (NN Inference + solving the iDBF-QP) of our framework is less than 5 milliseconds. +continuous-time control-affine structure of the dynamics model, we ensure that the iDBF-QP policy +filter (equivalent to the CBF-QP, see Figure 1) obtained with the learned iDBF and dynamics model +will also be a quadratic program. +The inference procedure of our end-to-end learning framework is depicted in Figure 1. We use a +recursive encoder network Eψ that takes the current measurement Ik, as well as the previous latent +state xk−1 and action uk−1 to generate the new latent state xk at each time-step k. The decoder +network Dξ generates a reconstructed observation ˆIk for each latent state xk. The proposed loss +function for the latent state-space representation and dynamics model penalizes both the prediction +error of the dynamics model and the observation reconstruction error: +Ldyn = +1 +Ndyn(Tpred + 1) +Ndyn +� +j=1 +Tpred +� +k=0 +� +wstate +���˜xtj+k|tj − xtj+k +��� +2 ++wrec1 +���˜Itj+k|tj − Itj+k +��� +2 ++wrec2 +���ˆItj+k − Itj+k +��� +2 � +. +(8) +Here, xtj+k = Eψ(Itj+k, xtj+k−1, utj+k−1) is the latent state at timestep tj + k. ˜xtj+k|tj denotes +the latent state prediction obtained by forward-propagating the dynamics model (6) to timestep +tj + k starting from the state xtj and using zero-order hold on the sequence of control inputs +(utj, utj+1, ..., utj+k−1). Additionally, ˜Itj+k|tj := Dξ(˜xtj+k|tj) is the reconstructed observation +from the dynamics prediction for timestep tj + k. Finally, ˆItj+k := Dξ(xtj+k) is the encoded- +decoded observation at timestep tj + k. +Note that we use a multiple-shooting error for the loss (8), as the prediction horizon Tpred does +not need to coincide with the length of the trajectories in the dataset D. In particular, the loss (8) +is computed by sampling a batch of trajectories from D and then splitting them into Ndyn portions +of length Tpred. The initial timestep of each portion j = 1, ..., Ndyn is denoted as tj. The first two +terms in the loss function are then penalizing the state and reconstruction error of the multistep +predictions of the dynamics model from each initial state xtj. The last term in the loss function +penalizes the reconstruction error of the autoencoder directly, without using the dynamics model. +The recent results of Beintema et al. (2021) show that multiple shooting loss functions lead to more +accurate predictions compared to single-step prediction losses, and to better conditioned learning +problems compared to single-shooting propagation losses. +An iDBF can be learned together with the autoencoder and dynamics model by optimizing +jointly the losses (7) and (8). For the iDBF loss, each xsafe is obtained by encoding the observations +7 + +NN Inference +iDBF-QP +fek), ge(xk) +Neural +ODE +xk-1 +l - T ref(α) 12 +B(xk) +arg min +Et +uEU +Ik +iDBF +Xk +s.t. B(α)[fe(α) +ge(α)u) + (B(α)) ≥ 0 +uk-1 +Plant +ukIN-DISTRIBUTION BARRIER FUNCTIONS +BC Filter +Ensemble Filter +πref +Ours +plow +pmid +phigh +δlow +δmid +δhigh +Collision +Rate (%) +46.72 ± 8.36 0.28 ± 0.27 35.60 ± 7.20 13.86 ± 4.96 2.48 ± 1.57 43.92 ± 7.90 43.82 ± 7.41 42.88 ± 7.41 +Top-Down +Navigation Cumulative +Intervention +0.0 ± 0.0 +109.2 ± 20.1 +85.6 ± 6.8 +146.4 ± 9.6 +189.1 ± 11.6 150.2 ± 19.3 +94.5 ± 16.3 +52.7 ± 11.5 +Collision +Rate (%) +81.00 ± 0.23 1.56 ± 1.20 21.94 ± 1.85 14.44 ± 2.69 8.78 ± 1.86 78.74 ± 0.20 78.60 ± 1.63 81.50 ± 0.27 +Egocentric +Driving +Cumulative +Intervention +0.0 ± 0.0 +278.1 ± 32.6 +713.8 ± 1.4 +726.7 ± 2.6 +750.8 ± 6.9 +28.7 ± 2.1 +42.8 ± 3.6 +208.9 ± 5.7 +Table 1: Evaluation of the collision rate and cumulative filter intervention (a measure of how intrusive the filter is with +respect to the reference controller) for the top-down view robotic navigation example (over 20 simulations of 5-seconds +each with random initial and goal states) and for the egocentric view autonomous driving example (over 20 simulations of +50-seconds each with random initial heading angles). For the BC and ensemble filters, we provide results for 3 different +threshold values: (plow, pmid, phigh) = (0.32, 0.35, 0.38) for the navigation example, and (0.2, 0.5, 0.8) for driving; and +(δlow, δmid, δhigh) = (0.0005, 0.001, 0.002) for both examples. +sampled from the dataset D, and xunsafe is obtained by forward propagating the actions that have a +low probability according to the pretrained BC model, as explained at the end of last section. +Once the iDBF Bφ; dynamics model fθ and gθ; and encoder Eψ networks are trained, we can +construct a policy filter —which we call iDBF-QP in Figure 1— in an equivalent manner to the +CBF-QP that was introduced in Section 3. +Remark 1 It is important to note that our iDBF training procedure encourages the satisfaction of +the CBF conditions (2), (3) and (4) only at a discrete set of training points (which has measure zero). +Because of this, we do not have control invariance guarantees for any particular set, and solving +the iDBF-QP does not theoretically assure that the system will remain in-distribution. Although +obtaining rigorous theoretical guarantees should be a priority for future work, the empirical results +of Section 7 show that our framework takes a promising first step towards building effective policy +filters from raw high-dimensional observations. +7. Examples +In this section, we present the empirical evaluation of our framework on two different simulation +environments: a toy example of a robot navigation task using top-down images of the scene, and an +autonomous driving scenario with egocentric image observations. For both cases, given a safety- +agnostic reference controller πref, we use our iDBF-QP at each timestep with the latest image mea- +surement to find the closest control input to πref among those that prevent the system from entering +OOD states (see Figure 1). For each environment, we train the iDBF, autoencoder and dynamics +model using a dataset containing 64 × 64 RGB images of offline-collected trajectories. +Robot Navigation with Top-Down View Images: In this example, a circular robot with radius +of 1 meter navigates inside of a 10 × 10 meter room that has a square-shaped 4 × 4 meter static +obstacle in the middle, as shown in Figure 2 (left). The underlying dynamics of the robot are those +of a 2D single integrator, with two control inputs corresponding to the x and y velocity commands, +although we do not assume having access to that knowledge. Instead, we only have a dataset of +image-action pairs corresponding to 5000 trajectories of 100 points each (corresponding to 2 sec- +onds since the time-step is 0.02s). These trajectories satisfy two requirements: 1) the robot should +never collide against the obstacle, and 2) the center of the robot should never leave the room limits. +The trajectories are collected applying random actions at each time-step, and we check both condi- +tions before adding a trajectory to the dataset. We use our framework to train an autoencoder with +8 + +IN-DISTRIBUTION BARRIER FUNCTIONS +Figure 2: Example result using our proposed policy filter for a robot top-down visual navigation task. The reference +controller simply tries to bring the robot (blue circle) to a goal state (denoted with ×). Our proposed filter, by keeping the +system in-distribution, prevents the robot from colliding against the obstacle (orange square) and keeps its center-point +inside the limits of the image. A video with several demonstrations of our approach for this task can be found in this link. +latent state-space of dimension 3, a dynamics model, and an iDBF. The reference policy πref simply +applies a velocity in the direction of a goal-point, with magnitude proportional to the distance. In +Figure 2, we show the results of applying our iDBF-QP when the goal state (marked with an ×) is +outside of the room limits and at the other side of the obstacle. Even though the reference controller +is trying to take the shortest path, which would go through the obstacle, the iDBF-QP prevents the +robot from first, colliding with the obstacle, and second, from having its center exit the room limits. +Autonomous Driving with Egocentric View Images: We use the environment provided by +Kahn et al. (2018), which is based on the Bullet physics simulator and the Panda3d graphics engine +(Goslin and Mine, 2004) to obtain egocentric RGB image measurements. The car navigates in a +corridor which has four 90-degree turns to form a square-shaped center-line. One of such turns is +shown in the snapshots of Figure 3. The car has two control inputs: the desired forward velocity +and the steering angle. Given the high-order dynamics of the simulator, we collect data manually to +make sure no trajectories included in the dataset are deemed to collide with any of the walls. We split +the collected data into 450 trajectories of 100 points each (5 seconds since the timestep is 0.05s). +This makes for a much sparser and less diverse (since it is collected by a human) dataset compared +to the previous example. During deployment, we use a reference controller πref that simply drives +the car forward at a constant speed of 3.5m/s. Our iDBF-QP framework of Figure 1, taking the +latest egocentric RGB measurement as input, is very effective at preventing the car from colliding +against the walls, as shown in Table 1. Figure 3 contains snapshots of our iDBF-QP forcing the car +to take a turn as it approaches a corner, even though the reference command is to drive forward. +Using these simulation environments we also aim to compare our proposed approach with other +techniques for avoiding distributional shift. Other works that consider this problem use data density +models to constrain the learned policies (Richter and Roy, 2017; McAllister et al., 2019; Wu et al., +2019), or use uncertainty estimation schemes, such as ensemble models, to avoid taking actions that +lead to highly uncertain states (Chua et al., 2018). We build our baselines upon a conditional BC +density model of the training data and an ensemble of latent state-space dynamics models: +BC Density Filter Baseline: As explained in Section 5, we train a BC multi-modal Gaussian +model that is used to generate the contrastive training distribution for the iDBF. For any state, the +BC model outputs a probability distribution over actions, with density function πBC(u|x). We train +this BC model using privileged true-state information of the system, and use its density values to +build a filter that serves as an apples-to-apples baseline comparison to our approach. Specifically, +the baseline also takes the reference controller πref and, at every timestep, it finds the closest control +9 + +0 +1 +-2 +Filtered +Reference +2 +0 +5 +0 +-1 +-2 +-3 +Filtered +-4 +Reference +2 +3 +time (s)IN-DISTRIBUTION BARRIER FUNCTIONS +Figure 3: Snapshots of egocentric view images of a driving simulation when the car is approaching a corner. The +reference controller just commands the car to drive straight, but our iDBF-QP policy filter forces a left turn as the car +approaches the corner. Therefore, our filter prevents a collision as a result of staying in-distribution with respect to the +safe training data. A video with several demonstrations of our approach for this task can be found in this link. +action to πref(x) that satisfies πBC(u|x) ≥ p, out of 200 randomly sampled actions. If no control +action satisfying that condition is found, the reference control input is applied without filtering. +Given the clear dependence on the threshold value p, we implement this baseline for several values +of p and show the results in Table 1 for three representative cases plow, pmid and phigh. +Ensemble Variance Filter Baseline: We also train an ensemble of independent latent state- +space dynamic models (fθ and gθ), keeping the rest of the framework introduced in Section 6 un- +changed. During deployment, at every timestep we look for the closest control action to πref(x) +that keeps the variance σ2 +ens(x, u) of the predicted dynamics fθ(x) + gθ(x)u under a threshold δ. +As in the previous baseline, we also look over 200 randomly sampled actions at each timestep, and +different threshold levels δlow, δmid and δhigh. Again, if no control action satisfying the threshold +condition is found, the reference control input is applied without filtering. +In Table 1, we provide a summary of the comparison results for both environments. We use the +collision rate as a proxy for distributional shift, since the training data only includes collision-free +trajectories. The collision rate for the robot navigation example is computed as the fraction of time +that the robot spends either in collision with the obstacle or having its center-point outside of the +room limits. For the driving scenario, the collision rate is the fraction of time that the robot is in +collision with any of the walls. For both examples, our method drastically reduces the collision +rate compared to using the reference (unfiltered) controller. Furthermore, we achieve the lowest +collision rates when compared to the baselines. From the baselines, only the BC density filter (with +a very restrictive threshold phigh) manages to achieve small collision rates, at the cost of a very high +cumulative filter intervention rate. The filter intervention rate is computed for both examples as +� +t ∥ut − πref(xt)∥2, where each control input dimension is normalized between −1 and 1. +8. Conclusion +In this work, we take first-steps towards merging control-theoretic CBFs with practical robotic tasks +that involve high-dimensional perception modules. We consider a realistic problem setting in which +no unsafe demonstrations are available, and take a self-supervised learning approach to learn a +function that effectively restricts the system from diverging towards OOD states. By learning this +function in a latent state-space, our framework should be flexible-enough to be applicable to a wide +variety of visuomotor tasks, and should be compatible with the use of large-scale pretrained repre- +sentation learning models. Another important direction for future work would be to use probabilistic +encoding and dynamics models to be able to robustify our proposed filters with respect to prediction +uncertainties. Additionally, exploring the use of loss functions that are not based on reconstruction, +by exploiting the value function nature of the iDBF, could be another promising direction. +10 + +IN-DISTRIBUTION BARRIER FUNCTIONS +Acknowledgments +The authors would like to thank Dr. Jean Mercat, Dr. Hongkai Dai and Dr. Katherine Liu for their +insightful comments and suggestions. +References +Alessandro Abate, Daniele Ahmed, Alec Edwards, Mirco Giacobbe, and Andrea Peruffo. Fossil: a +software tool for the formal synthesis of lyapunov functions and barrier certificates using neural +networks. In Proceedings of the 24th International Conference on Hybrid Systems: Computation +and Control, pages 1–11, 2021. +A. D. Ames, J. W. Grizzle, and P. Tabuada. Control barrier function based quadratic programs +with application to adaptive cruise control. In IEEE Conference on Decision and Control, pages +6271–6278, 2014. +A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada. 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PMLR, 2019. +14 + diff --git a/K9FLT4oBgHgl3EQfLi8r/content/tmp_files/load_file.txt b/K9FLT4oBgHgl3EQfLi8r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9951170c64a9e0bd63cd24c66eb17f0816c87ed --- /dev/null +++ b/K9FLT4oBgHgl3EQfLi8r/content/tmp_files/load_file.txt @@ -0,0 +1,593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf,len=592 +page_content='Proceedings of Machine Learning Research vol XX:1–14, 2023 In-Distribution Barrier Functions: Self-Supervised Policy Filters that Avoid Out-of-Distribution States Fernando Casta˜neda FCASTANEDA@BERKELEY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='EDU University of California, Berkeley Haruki Nishimura HARUKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='NISHIMURA@TRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='GLOBAL Toyota Research Institute, Los Altos, CA Rowan McAllister ROWAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='MCALLISTER@TRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='GLOBAL Toyota Research Institute, Los Altos, CA Koushil Sreenath KOUSHILS@BERKELEY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='EDU University of California, Berkeley Adrien Gaidon ADRIEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='GAIDON@TRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='GLOBAL Toyota Research Institute, Los Altos, CA Abstract Learning-based control approaches have shown great promise in performing complex tasks di- rectly from high-dimensional perception data for real robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Nonetheless, the learned controllers can behave unexpectedly if the trajectories of the system divert from the training data distribution, which can compromise safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In this work, we propose a control filter that wraps any reference policy and effectively encourages the system to stay in-distribution with respect to offline-collected safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Our methodology is inspired by Control Barrier Functions (CBFs), which are model-based tools from the nonlinear control literature that can be used to con- struct minimally invasive safe policy filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' While existing methods based on CBFs require a known low-dimensional state representation, our proposed approach is directly applicable to sys- tems that rely solely on high-dimensional visual observations by learning in a latent state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We demonstrate that our method is effective for two different visuomotor control tasks in simulation environments, including both top-down and egocentric view settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Keywords: Distributional Shift, Control Barrier Functions, State Representation Learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Introduction The modern advances in the representation learning literature have been an enabling factor for the recent surge of a wide variety of methods for robotic control directly from images or high- dimensional sensory observations (Watter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Ebert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Hafner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Lenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Van Hoof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' These approaches for visuomotor planning and control have the potential to solve challenging tasks in which the state of the system might not be directly observable, or even not possible to model analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' While promising, the high- dimensionality of the problem make these methods susceptible to several open challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For example, the exploration requirements of reinforcement learning (RL) algorithms are significantly exacerbated for these tasks, due to the high-dimensionality of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This means that trying to learn safe control policies using RL often requires us to accept that abundant failures will occur during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' On the other hand, supervised learning approaches for control, such as behavioral cloning, would in principle seem less prone to exhibit unsafe behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' However, it is © 2023 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Casta˜neda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Nishimura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' McAllister, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Sreenath & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Gaidon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='12012v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='RO] 27 Jan 2023 IN-DISTRIBUTION BARRIER FUNCTIONS well known that simply because of their data-dependent nature, these methods are still susceptible to a key challenge named distributional shift: if the trajectories of the system divert from the training data distribution, the controller might take unexpected actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' On the other hand, the control theory literature extensively covers the problem of long-horizon constraint satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In particular, Control Barrier Functions (CBFs, Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2014)) are a popular model-based tool used to restrict the trajectories of the system from entering undesirable regions of the state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' One of the properties of CBFs that explain their recent popularity is that they decouple the problem of constraint-satisfaction from any performance objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Specifically, if a CBF is available, then Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2014) showed that we can construct a minimally invasive safety filter that transforms into safety-preserving control actions any unsafe commands that an arbitrary reference policy could output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The main question we want to address in this work naturally emerges from the previous dis- cussion: can we take inspiration from CBFs to avoid out-of-distribution (OOD) states when using data-driven controllers for visuomotor tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Even though CBFs are model-based tools that, as such, require knowledge of the state-space and dynamics of the system, the recent advances on learning latent state-space representations and associated dynamics models clearly set a path for linking data-driven visuomotor policy learning with the use of model-based control-theoretic tools such as CBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Contributions: We present an end-to-end self-supervised approach for learning a task-agnostic policy filter which prevents the system from entering OOD states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We do not assume knowledge of the state-space or system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In addition, our framework only requires an offline-collected dataset of safe demonstrations (where the concept of safety is only linked to the demonstrator’s subjectivity, as it is their responsibility to provide the dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We therefore do not require any un- safe demonstrations to learn a safe policy filter, in contrast to most other works tackling constrained policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, to the best of our knowledge, this is the first work that uses CBFs for constructing policy filters in learned latent state-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This endows our approach with the flexibility of being applicable to systems with high-dimensional sensory observations, in contrast to most prior CBF-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We present simulation experiments on two different visuomotor control tasks, which suggest that our framework, taking only raw RGB images as input, can learn to significantly reduce the distributional shift from safe demonstrations and, consequently, critically improve the safety of both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Related Work There exist some constructive procedures for synthesizing CBFs based on sum-of-squares program- ming (Jarvis-Wloszek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Majumdar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Dai and Permenter, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022) or Hamilton-Jacobi reachability (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' However, these methods require knowl- edge of the dynamics of the system and typically suffer from scalability issues for high-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' More in line with this work, some recent results show that CBFs can be learned from data (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Robey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Lindemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Abate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Jagtap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' None of these works, however, consider systems with high-dimensional observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, the works Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2022) assume a priori knowledge of a control-invariant safe set, and focus on building a CBF for that particular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The line of research of Robey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Lindemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2021) has the most similar problem setup to our work, as they also consider learning from safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 2 IN-DISTRIBUTION BARRIER FUNCTIONS Although, notably, the authors provide formal verification arguments for the learned CBFs, their methods are not applicable to high-dimensional observations, assume a nominal dynamics model is given, and use an algorithmic approach for the detection of the boundary of the dataset that does not scale to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Other recent approaches build signed distance functions from sensory measurements that are obtained from a LiDAR or stereo cameras (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Cosner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' However, these functions are not encouraged to satisfy any set invariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Extensions of the CBF-based control filters to systems with dynamics or measurement-model uncertainty have also been recently proposed (Nguyen and Sreenath, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Casta˜neda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Dhiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' These works assume that a CBF is pro- vided, and formulate uncertainty-robust optimization problems for the controller design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' They can be considered complementary to our deterministic but end-to-end approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Future work should ex- plore quantifying uncertainty estimates within our framework to robustify the learned policy filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Several existing approaches for OOD prevention learn density models of training data that can be then used to restrict the agent from taking low likelihood actions or moving towards unvis- ited states (McAllister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Richter and Roy, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Although some of these methods have been shown to be effective at offline RL settings that are specially susceptible to distributional shift, the learned density models have no notion of control invariance and, therefore, do not consider the problem of how to prevent distributional shift over a long time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' A notable exception is the work of Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2022) to constrain long-term dis- tributional shift, in which a min-max Bellman backup operator is constructed so that Lyapunov-like functions arise as value functions of an offline RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This work however does not consider the extension to visuomotor control tasks in learned latent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, for our approach we choose not to rely on a min-max backup operator to learn the certificate function and, instead, use the very suitable theory of CBFs to devise a self-supervised learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Finally, the work of Wilcox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2022) presents a framework to learn safe sets in a latent state- space for iterative control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Compared to this work, our framework has the advantage that it is task-agnostic and does not require any interactions with the environment during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Background on Control Barrier Functions We start by introducing some necessary background on Control Barrier Functions, which are tools from the nonlinear control literature that serve to enforce safety constraints for systems with known dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' As will be clear later, CBFs are particularly well-suited for continuous-time nonlinear control-affine systems of the form ˙x = f(x) + g(x)u, (1) where x ∈ X ⊂ Rn is the state and u ∈ U ⊂ Rm the control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We assume that f : X → Rn and g : X → Rn×m are locally Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In the CBF literature, safety is considered as a set invariance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In particular, we say that a control policy π : X → U assures the safety of system (1) with respect to a set Xsafe ⊂ X if the set Xsafe is forward invariant under the control law π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', for any x0 ∈ Xsafe, the solution x(t) of system (1) under the control law π remains within Xsafe for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Definition 1 (Control Barrier Function, Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2017)) We say that a continuously differ- entiable function B : X → R is a Control Barrier Function (CBF) for system (1) with associated safe-set Xsafe ⊂ X if the following three conditions are satisfied: 3 IN-DISTRIBUTION BARRIER FUNCTIONS B(x) ≥ 0 ∀x ∈ Xsafe, (2) B(x) < 0 ∀x ∈ X \\ Xsafe, (3) ∃u ∈ U s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' ˙B(x, u) + γ(B(x)) ≥ 0 ∀x ∈ X, (4) where γ : R → R is an extended class K∞ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The existence of a CBF B guarantees that for system (1) any Lipschitz continuous control policy π satisfying π(x) ∈ {u ∈ U : ∇B(x)[f(x) + g(x)u] � �� � = ˙B(x,u) +γ (B(x)) ≥ 0} (5) will render the set Xsafe forward invariant (Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2017, Corollary 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For a given task-specific reference controller πref : X → U that might be safety-agnostic, the condition of (5) can be used to formulate an optimization problem that, when solved at every time- step, yields a minimally-invasive policy safety filter (Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2014): πCBF(x) = arg min u∈U ∥u − πref(x)∥2 (CBF-QP) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' ∇B(x)[f(x) + g(x)u] + γ(B(x)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Assuming that the actuation constraints that define U are linear in u, this problem is a quadratic program (QP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This is a consequence of the dynamics of the system (1) being control-affine, and it practically means that the problem can be solved to a high precision very quickly (around 103Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This is critical since the CBF-QP needs to be solved at the real-time control frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Problem Statement The CBF-QP constitutes a very appealing approach for practitioners: it provides a task-agnostic minimally invasive filter that can wrap safety around any given policy πref, and therefore rewrite any unsafe control input that πref could output at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' However, designing a valid CBF is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In fact, it is still an active research topic even when assuming perfect knowledge of the dynamics of the system (Dai and Permenter, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The two main difficulties in the design of a CBF are the following: first, a control-invariant set Xsafe must be obtained (which in general is different from the geometric constraint set that could be obtained, for instance, from a signed-distance field) and, second, a function that satisfies condition (4) must be found for that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, even after obtaining a CBF, solving the CBF-QP requires perfect state and dynamics knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' With our framework, we take initial steps towards building a safe policy filter from high- dimensional observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Specifically, we take inspiration from CBFs to design an end-to-end learning framework to constrain deep learning models to remain in-distribution of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We take as input a dataset of high-dimensional observations of different safe demonstrations, and build a neural CBF-like function that encourages the system to always stay in-distribution with re- spect to the observations from the safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This, in turn, significantly improves the safety of the system during deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' More concretely, for a given dataset of N safe trajectories D = �� Ii t, ui t �t=Ti t=0 �i=N i=1 we tackle the problem of designing a policy filter that can be applied to any reference controller πref to detect and override actions from πref that lead to OOD states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We denote Ii t and ui t the high-dimensional 4 IN-DISTRIBUTION BARRIER FUNCTIONS observation and control input, respectively, measured at time t for the ith trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, Ti is the final time-step of trajectory i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The demonstrations in the dataset D might correspond to different tasks and they do not need to be optimal with respect to any objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In fact, our only assumption is that the dataset only contains safe demonstrations (in the sense that these trajectories should not contain any states from which the system is deemed to fail, even if it has not failed yet), so that we can encourage long-term constraint satisfaction using CBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In-Distribution Barrier Functions In this section, we introduce a self-supervised approach for synthesizing neural CBF-like functions whose aim is to constrain the system to remain in-distribution with respect to an offline dataset of safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We call these functions in-Distribution Barrier Functions (iDBFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We will for now assume that we have a parametric continuous-time control-affine model of the dynamics of the system in a state-space X ⊂ Rn ˙x = fθ(x) + gθ(x)u, (6) and present the iDBF learning procedure for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, for this section we assume that the dataset D of safe trajectories contains true state measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', D = �� xi t, ui t �t=Ti t=0 �i=N i=1 , where xi t and ui t are the state and control input, respectively, measured at time t for the ith trajec- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In Section 6, we will provide details on how to learn a dynamics model of this form in a latent state-space when we have a dataset containing high-dimensional sensory observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We parameterize an iDBF Bφ : X → R as a neural network with parameters φ, and construct an empirical loss function that encourages it to satisfy the three CBF conditions (2), (3) and (4) with respect to a set Xsafe that is also implicitly learned through self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' To design the loss function, we take inspiration from previous literature on learning CBFs (Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Nevertheless, instead of assuming that the safe-set Xsafe is given and that we can sample from it and from its unsafe complement Xunsafe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='= X \\ Xsafe, we build our loss function in a self-supervised manner just from the dataset of safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We accomplish this by leveraging ideas from contrastive learning (Gutmann and Hyv¨arinen, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Chopra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Weinberger and Saul, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Schroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In particular, as we explain in detail later, we build a contrastive distribution from which to sample candidate unsafe states, given that we do not have any unsafe demonstrations in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The loss function we propose for learning an iDBF takes the following form: LiDBF = wsafe Nsafe � xsafe [ϵsafe − Bφ(xsafe)]+ + wunsafe Nunsafe � xunsafe [ϵunsafe + Bφ(xunsafe)]+ + wascent Nsafe � (xsafe,usafe) � ϵascent − � ∇Bφ(xsafe)[fθ(xsafe) + gθ(xsafe)usafe] + γ(Bφ(xsafe)) ��+ , (7) where [·]+ := max(0, ·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (xsafe, usafe) are samples from the empirical distribution of the dataset D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' xunsafe are samples from a contrastive distribution that we will define soon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' wsafe, wunsafe and wascent are the weights of the different loss terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' and ϵsafe, ϵunsafe and ϵascent are positive constants that serve to enforce strictly the inequalities and generalize outside of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The goal of the first two terms in the loss function is to learn an iDBF that has a positive value in states that belong to the data distribution of safe demonstrations, and negative everywhere else (meaning we are encouraging the satisfaction of conditions (2) and (3) of the definition of CBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 5 IN-DISTRIBUTION BARRIER FUNCTIONS Note that this classification objective is very related to the notion of energy-based models (EBMs) —neural network density models that assign a low energy value to points close to the training data distribution and a high value to points that are far from it (Hinton, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In fact, we took inspiration from the Noise Contrastive Estimation (NCE, Gutmann and Hyv¨arinen (2010)) training procedure of EBMs, in particular the InfoNCE loss (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2018), to design (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Intuitively, these methods use a noise contrastive distribution to generate candidate examples where to increase the value of the energy of the EBM, while decreasing the energy at the training data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We precisely want the opposite result for our problem: a high value of Bφ on the training data distribution, and a low value everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' However, we have one additional requirement, which is that the iDBF should have a value of zero at the boundary, as set by conditions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This is the reason why we design the two first terms of the loss function using the [·]+ operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In the third term of the loss (7), note that we do not encourage the satisfaction of condition (4) over the entire state-space, but only over the dataset of safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' However, in the definition of CBF, if condition (4) is only satisfied ∀x ∈ Xsafe instead of ∀x ∈ X, the CBF still guarantees the control-invariance of Xsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We are therefore using our empirical data distribution of safe demonstrations as a sampling distribution covering the set Xsafe, which we are also implicitly learning as the zero-superlevel set of Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, compared to prior approaches that encourage the satisfaction of condition (4) for a single policy (Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2022), we instead use all pairs (xsafe, usafe) present in the dataset D to compute this term of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This way, we force the set of admissible control inputs (5) to be as large as our dataset allows, reducing the conservatism of the learned iDBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In order to generate the contrastive distribution from which to sample xunsafe, as we ultimately want to learn the iDBF in a latent state-space in which it might not be intuitive how to construct a noise distribution, we take the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 1) Based on the dataset of safe demonstrations D, we train a neural behavioral cloning (BC) model that outputs a multi-modal Gaussian distribution over actions conditioned on the state, with density πBC(u|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 2) Then, during the training process of the iDBF, for each xsafe state sampled from D we randomly take Ncandidate control inputs ucandidate and evaluate their density value based on the BC model πBC(ucandidate|xsafe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 3) If the value of the density falls below a threshold, then that control input is forward-propagated for one timestep using the dynamics model (6) to generate a sample xunsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This way, we generate a contrastive data distribution by propagating actions that are unlikely present in the dataset of safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, by only propagating these actions for one timestep, the contrastive distribution is close to the training data, which is desirable for the learning process (Gutmann and Hirayama, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Learning iDBFs from High-Dimensional Observations After introducing the training procedure for an iDBF when the state representation and dynamics model (6) are given, we now relax these assumptions and present an approach to learn a latent state-space representation and a continuous-time dynamics model of the form (6), suitable to be integrated in the same end-to-end learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We therefore now consider precisely the problem setting described in Section 4, in which we only assume having access to a dataset contain- ing observation-action pairs of safe demonstrations D = �� Ii t, ui t �t=Ti t=0 �i=N i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We use an autoencoder architecture to obtain the latent state-space representation, and employ the training procedure of Neural Ordinary Differential Equations (Neural ODEs, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2018)) to learn a dynamics model of the form (6) in the latent state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Note that by enforcing the 6 IN-DISTRIBUTION BARRIER FUNCTIONS Figure 1: Our framework’s inference diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' At each timestep, based on the current observation Ik and the previous latent state and action, the encoder network Eψ outputs a new latent state xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Then, the iDBF and dynamics networks give the values of Bφ(xk), fθ(xk) and gθ(xk) which are passed to the iDBF-QP policy filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The iDBF-QP takes a reference control input for the current timestep πref(xk) and returns the closest action that keeps the system in-distribution with respect to the offline-collected dataset of safe demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For both of the examples of Section 7, the total inference time (NN Inference + solving the iDBF-QP) of our framework is less than 5 milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' continuous-time control-affine structure of the dynamics model, we ensure that the iDBF-QP policy filter (equivalent to the CBF-QP, see Figure 1) obtained with the learned iDBF and dynamics model will also be a quadratic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The inference procedure of our end-to-end learning framework is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We use a recursive encoder network Eψ that takes the current measurement Ik, as well as the previous latent state xk−1 and action uk−1 to generate the new latent state xk at each time-step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The decoder network Dξ generates a reconstructed observation ˆIk for each latent state xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The proposed loss function for the latent state-space representation and dynamics model penalizes both the prediction error of the dynamics model and the observation reconstruction error: Ldyn = 1 Ndyn(Tpred + 1) Ndyn � j=1 Tpred � k=0 � wstate ���˜xtj+k|tj − xtj+k ��� 2 +wrec1 ���˜Itj+k|tj − Itj+k ��� 2 +wrec2 ���ˆItj+k − Itj+k ��� 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (8) Here, xtj+k = Eψ(Itj+k, xtj+k−1, utj+k−1) is the latent state at timestep tj + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' ˜xtj+k|tj denotes the latent state prediction obtained by forward-propagating the dynamics model (6) to timestep tj + k starting from the state xtj and using zero-order hold on the sequence of control inputs (utj, utj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', utj+k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Additionally, ˜Itj+k|tj := Dξ(˜xtj+k|tj) is the reconstructed observation from the dynamics prediction for timestep tj + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Finally, ˆItj+k := Dξ(xtj+k) is the encoded- decoded observation at timestep tj + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Note that we use a multiple-shooting error for the loss (8), as the prediction horizon Tpred does not need to coincide with the length of the trajectories in the dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In particular, the loss (8) is computed by sampling a batch of trajectories from D and then splitting them into Ndyn portions of length Tpred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The initial timestep of each portion j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', Ndyn is denoted as tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The first two terms in the loss function are then penalizing the state and reconstruction error of the multistep predictions of the dynamics model from each initial state xtj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The last term in the loss function penalizes the reconstruction error of the autoencoder directly, without using the dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The recent results of Beintema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2021) show that multiple shooting loss functions lead to more accurate predictions compared to single-step prediction losses, and to better conditioned learning problems compared to single-shooting propagation losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' An iDBF can be learned together with the autoencoder and dynamics model by optimizing jointly the losses (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For the iDBF loss, each xsafe is obtained by encoding the observations 7 NN Inference iDBF-QP fek), ge(xk) Neural ODE xk-1 l - T ref(α) 12 B(xk) arg min Et uEU Ik iDBF Xk s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' B(α)[fe(α) +ge(α)u) + (B(α)) ≥ 0 uk-1 Plant ukIN-DISTRIBUTION BARRIER FUNCTIONS BC Filter Ensemble Filter πref Ours plow pmid phigh δlow δmid δhigh Collision Rate (%) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='72 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='27 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='60 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='20 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='86 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='48 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='57 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='92 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='90 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='82 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='41 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='88 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='41 Top-Down Navigation Cumulative Intervention 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='0 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='2 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='8 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='4 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='6 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='1 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='6 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='2 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='5 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='7 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='5 Collision Rate (%) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='56 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='20 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='94 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='85 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='44 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='69 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='86 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='20 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='60 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='63 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='27 Egocentric Driving Cumulative Intervention 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='0 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='1 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='6 713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='4 726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='6 750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='8 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='6 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='7 Table 1: Evaluation of the collision rate and cumulative filter intervention (a measure of how intrusive the filter is with respect to the reference controller) for the top-down view robotic navigation example (over 20 simulations of 5-seconds each with random initial and goal states) and for the egocentric view autonomous driving example (over 20 simulations of 50-seconds each with random initial heading angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For the BC and ensemble filters, we provide results for 3 different threshold values: (plow, pmid, phigh) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='32, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='38) for the navigation example, and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='8) for driving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' and (δlow, δmid, δhigh) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='0005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='002) for both examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' sampled from the dataset D, and xunsafe is obtained by forward propagating the actions that have a low probability according to the pretrained BC model, as explained at the end of last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Once the iDBF Bφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' dynamics model fθ and gθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' and encoder Eψ networks are trained, we can construct a policy filter —which we call iDBF-QP in Figure 1— in an equivalent manner to the CBF-QP that was introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Remark 1 It is important to note that our iDBF training procedure encourages the satisfaction of the CBF conditions (2), (3) and (4) only at a discrete set of training points (which has measure zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Because of this, we do not have control invariance guarantees for any particular set, and solving the iDBF-QP does not theoretically assure that the system will remain in-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Although obtaining rigorous theoretical guarantees should be a priority for future work, the empirical results of Section 7 show that our framework takes a promising first step towards building effective policy filters from raw high-dimensional observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Examples In this section, we present the empirical evaluation of our framework on two different simulation environments: a toy example of a robot navigation task using top-down images of the scene, and an autonomous driving scenario with egocentric image observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For both cases, given a safety- agnostic reference controller πref, we use our iDBF-QP at each timestep with the latest image mea- surement to find the closest control input to πref among those that prevent the system from entering OOD states (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For each environment, we train the iDBF, autoencoder and dynamics model using a dataset containing 64 × 64 RGB images of offline-collected trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Robot Navigation with Top-Down View Images: In this example, a circular robot with radius of 1 meter navigates inside of a 10 × 10 meter room that has a square-shaped 4 × 4 meter static obstacle in the middle, as shown in Figure 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The underlying dynamics of the robot are those of a 2D single integrator, with two control inputs corresponding to the x and y velocity commands, although we do not assume having access to that knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Instead, we only have a dataset of image-action pairs corresponding to 5000 trajectories of 100 points each (corresponding to 2 sec- onds since the time-step is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='02s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' These trajectories satisfy two requirements: 1) the robot should never collide against the obstacle, and 2) the center of the robot should never leave the room limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The trajectories are collected applying random actions at each time-step, and we check both condi- tions before adding a trajectory to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We use our framework to train an autoencoder with 8 IN-DISTRIBUTION BARRIER FUNCTIONS Figure 2: Example result using our proposed policy filter for a robot top-down visual navigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The reference controller simply tries to bring the robot (blue circle) to a goal state (denoted with ×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Our proposed filter, by keeping the system in-distribution, prevents the robot from colliding against the obstacle (orange square) and keeps its center-point inside the limits of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' A video with several demonstrations of our approach for this task can be found in this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' latent state-space of dimension 3, a dynamics model, and an iDBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The reference policy πref simply applies a velocity in the direction of a goal-point, with magnitude proportional to the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In Figure 2, we show the results of applying our iDBF-QP when the goal state (marked with an ×) is outside of the room limits and at the other side of the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Even though the reference controller is trying to take the shortest path, which would go through the obstacle, the iDBF-QP prevents the robot from first, colliding with the obstacle, and second, from having its center exit the room limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Autonomous Driving with Egocentric View Images: We use the environment provided by Kahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' (2018), which is based on the Bullet physics simulator and the Panda3d graphics engine (Goslin and Mine, 2004) to obtain egocentric RGB image measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The car navigates in a corridor which has four 90-degree turns to form a square-shaped center-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' One of such turns is shown in the snapshots of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The car has two control inputs: the desired forward velocity and the steering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Given the high-order dynamics of the simulator, we collect data manually to make sure no trajectories included in the dataset are deemed to collide with any of the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We split the collected data into 450 trajectories of 100 points each (5 seconds since the timestep is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='05s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' This makes for a much sparser and less diverse (since it is collected by a human) dataset compared to the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' During deployment, we use a reference controller πref that simply drives the car forward at a constant speed of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content='5m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Our iDBF-QP framework of Figure 1, taking the latest egocentric RGB measurement as input, is very effective at preventing the car from colliding against the walls, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Figure 3 contains snapshots of our iDBF-QP forcing the car to take a turn as it approaches a corner, even though the reference command is to drive forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Using these simulation environments we also aim to compare our proposed approach with other techniques for avoiding distributional shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Other works that consider this problem use data density models to constrain the learned policies (Richter and Roy, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' McAllister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2019), or use uncertainty estimation schemes, such as ensemble models, to avoid taking actions that lead to highly uncertain states (Chua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We build our baselines upon a conditional BC density model of the training data and an ensemble of latent state-space dynamics models: BC Density Filter Baseline: As explained in Section 5, we train a BC multi-modal Gaussian model that is used to generate the contrastive training distribution for the iDBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For any state, the BC model outputs a probability distribution over actions, with density function πBC(u|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We train this BC model using privileged true-state information of the system, and use its density values to build a filter that serves as an apples-to-apples baseline comparison to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Specifically, the baseline also takes the reference controller πref and, at every timestep, it finds the closest control 9 0 1 2 Filtered Reference 2 0 5 0 1 2 3 Filtered 4 Reference 2 3 time (s)IN-DISTRIBUTION BARRIER FUNCTIONS Figure 3: Snapshots of egocentric view images of a driving simulation when the car is approaching a corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The reference controller just commands the car to drive straight, but our iDBF-QP policy filter forces a left turn as the car approaches the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Therefore, our filter prevents a collision as a result of staying in-distribution with respect to the safe training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' A video with several demonstrations of our approach for this task can be found in this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' action to πref(x) that satisfies πBC(u|x) ≥ p, out of 200 randomly sampled actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' If no control action satisfying that condition is found, the reference control input is applied without filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Given the clear dependence on the threshold value p, we implement this baseline for several values of p and show the results in Table 1 for three representative cases plow, pmid and phigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Ensemble Variance Filter Baseline: We also train an ensemble of independent latent state- space dynamic models (fθ and gθ), keeping the rest of the framework introduced in Section 6 un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' During deployment, at every timestep we look for the closest control action to πref(x) that keeps the variance σ2 ens(x, u) of the predicted dynamics fθ(x) + gθ(x)u under a threshold δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' As in the previous baseline, we also look over 200 randomly sampled actions at each timestep, and different threshold levels δlow, δmid and δhigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Again, if no control action satisfying the threshold condition is found, the reference control input is applied without filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' In Table 1, we provide a summary of the comparison results for both environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We use the collision rate as a proxy for distributional shift, since the training data only includes collision-free trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The collision rate for the robot navigation example is computed as the fraction of time that the robot spends either in collision with the obstacle or having its center-point outside of the room limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For the driving scenario, the collision rate is the fraction of time that the robot is in collision with any of the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' For both examples, our method drastically reduces the collision rate compared to using the reference (unfiltered) controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Furthermore, we achieve the lowest collision rates when compared to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' From the baselines, only the BC density filter (with a very restrictive threshold phigh) manages to achieve small collision rates, at the cost of a very high cumulative filter intervention rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' The filter intervention rate is computed for both examples as � t ∥ut − πref(xt)∥2, where each control input dimension is normalized between −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Conclusion In this work, we take first-steps towards merging control-theoretic CBFs with practical robotic tasks that involve high-dimensional perception modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' We consider a realistic problem setting in which no unsafe demonstrations are available, and take a self-supervised learning approach to learn a function that effectively restricts the system from diverging towards OOD states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' By learning this function in a latent state-space, our framework should be flexible-enough to be applicable to a wide variety of visuomotor tasks, and should be compatible with the use of large-scale pretrained repre- sentation learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Another important direction for future work would be to use probabilistic encoding and dynamics models to be able to robustify our proposed filters with respect to prediction uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Additionally, exploring the use of loss functions that are not based on reconstruction, by exploiting the value function nature of the iDBF, could be another promising direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' 10 IN-DISTRIBUTION BARRIER FUNCTIONS Acknowledgments The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Jean Mercat, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Hongkai Dai and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Katherine Liu for their insightful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' References Alessandro Abate, Daniele Ahmed, Alec Edwards, Mirco Giacobbe, and Andrea Peruffo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FLT4oBgHgl3EQfLi8r/content/2301.12012v1.pdf'} +page_content=' Fossil: a software tool for the formal synthesis of lyapunov functions and barrier certificates using neural networks.' 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by +Spin-Polarized Current +Ra´ı M. Menezes,† Jeroen Mulkers,‡ Cl´ecio C. de Souza Silva,¶ Bartel Van +Waeyenberge,‡ and Milorad V. Miloˇsevi´c∗,† +†NANOlab Center of Excellence & Department of Physics, University of Antwerp, +Groenenborgerlaan 171, B-2020 Antwerp, Belgium +‡DyNaMat Lab, Department of Solid State Sciences, Ghent University, Ghent, Belgium +¶Departamento de F´ısica, Universidade Federal de Pernambuco, Cidade Universit´aria, +50670-901, Recife-PE, Brazil +E-mail: milorad.milosevic@uantwerpen.be +Abstract +Spin-waves (magnons) are among the prime candidates for building fast yet energy- +efficient platforms for information transport and computing. We here demonstrate the- +oretically and in state-of-the-art micromagnetic simulation the effects that strategically- +injected spin-polarized current can have on controlling magnonic transport. We reveal +analytically that the Zhang-Li spin-transfer-torque induced by applied current is anal- +ogous to the Dzyaloshinskii-Moriya interaction for scattering the magnons in the linear +regime, to then provide a generalized Snell’s law that describes the spin-wave propaga- +tion across regions with different current densities. We validate the latter in numerical +simulations of realistic systems, and exemplify how these findings may help advance +the design of spin-wave logic and neuromorphic computing devices. +1 +arXiv:2301.04922v1 [cond-mat.mtrl-sci] 12 Jan 2023 + +The development of neuromorphic computing hardware has attracted significant atten- +tion in recent years as such platforms are capable of performing complex information pro- +cessing tasks, such as classification and pattern recognition of various types of data, from +e-commerce to scientific content.1–3 A central challenge of this research is the requirement +of highly interconnected systems, inspired by the biological concepts of the human brain. +Interestingly, wave-based physical systems have been demonstrated to operate as recurrent +neural networks,4 where interference patterns in the propagating substrate can realize an +all-to-all interconnection between points of the substrate that mimic the action of artificial +neurons by scattering and recombining input waves in order to extract their information. +Similar to other wave phenomena in physics, spin-waves (magnons) travel through space +accompanied by a transfer of energy, which if precisely controlled can lead to fast information +transport and computing applications in the nanometric to micrometric scale.5,6 Spin-waves +are readily demonstrated as a promising platform for performing logic operations7,8 and +the recent theoretical advances in wave-based computation can pave the way for spintronic +hardware in the field of artificial intelligence.9 However, even though extensive research has +been carried out in recent years, the precise manipulation of spin waves in nanostructures has +not been entirely mastered and needs to be advanced for the benefit of functional magnonic +devices. +One manner of manipulating the spin textures in magnetic materials is by the application +of spin-polarized (SP) currents, having (part of the) spins of the moving electrons aligned. +The interaction of a SP current with the localized magnetic moments results in a torque on +the magnetization, dubbed a Zhang-Li spin-transfer-torque (STT).10 Being able to affect the +orientation of the magnetization, STT has become nearly unavoidable in the design of spin- +tronic nanodevices.11 The effect and applicability of the STT was demonstrated in racetrack +memory concepts, where the position of local magnetic structures, such as domain walls and +skyrmions, is controlled by in-plane currents.12,13 Moreover, not only do SP currents modify +the equilibrium magnetic state, they can also significantly influence the propagation of spin +2 + +waves (SWs). For instance, in Ref.14 the authors derived the dispersion relation of SWs in +a ferromagnet subjected to a uniform SP current. The applied current introduces a term +proportional to the wavevector k in the spin-wave dispersion relation. The nonreciprocity +in the dispersion relation for waves with a k vector in the same or the opposite direction as +the current flow causes Doppler shift,15 as first validated experimentally by Vlaminck and +Bailleul.16 +A similar nonreciprocal term in the dispersion relation can also be induced by an electric +field17 or by the antisymmetric exchange interaction, also known as Dzyaloshinskii-Moriya +interaction (DMI).18 Analogously to the Doppler shift induced by the currents, a frequency +shift has been measured in ferromagnetic materials possessing DMI.19 For heterochiral ferro- +magnets (i.e., with spatially varying DMI), the reciprocal term causes a nontrivial refraction +of spin waves at interfaces between regions with different DMI, as described by a generalized +Snell’s law in Ref. 20. In this regard, the equivalency in the dispersion relations of SWs in +the presence of DMI and SP current suggests that a spatially varying current density (as +illustrated in Fig. 1) can be used to manipulate the propagation of SWs. +In this Letter, we detail the effect of a SP current on controlling the propagation of spin- +waves. We show that the Zhang-Li STT induced by in-plane current has analogous effect to +DMI for confining and controlling the propagation direction of magnons in the linear regime. +We proceed to derive a Snell’s law to describe the scattering of spin-waves between regions +with different current densities, and validate it by advanced simulations including solving +Poisson’s equation within the micromagnetic framework. Finally we present selected tailored +examples to illustrate how strategically applied current can be employed to advance logic +and neuromorphic computing devices based on spin-waves. +3 + +Figure 1: Schematic illustration of a SW facing a non-uniform distribution of +the SP current. The SW (in red) is induced by the input antenna on the left side and +propagates along the magnetic film. Voltage electrodes induce the distributed SP current +(j) which can be tuned to control the SW propagation. +Results and discussion +Spin-wave dispersion under applied current +Within the micromagnetic framework, we describe the magnetization of a thin extended fer- +romagnetic film by considering the vector field ⃗M(r) = Ms⃗m(r) with constant magnetization +modulus | ⃗M| = Ms and the normalized magnetization direction ⃗m(r) at each point r ∈ R2 +of the film 1. The dynamics of the magnetization is governed by the Landau-Lifshitz-Gilbert +(LLG) equation +˙⃗m = −γ ⃗m × ⃗Heff + α⃗m × ˙⃗m + ⃗τSTT, +(1) +where γ is the gyromagnetic ratio, α the dimensionless damping factor, and ⃗Heff the effective +field, which can be derived from the free energy E[⃗m] by taking the functional derivative +with respect to the magnetization: ⃗Heff = −δE/δ ⃗M. +We extend the LLG equation by adding the torque ⃗τSTT which includes the adiabatic and +1In this paper we use an arrow for 3D vector fields such as the magnetization ⃗m and the effective magnetic +field ⃗Heff and bold variables for 2D dimensional vectors in the plane such as the position r, the k-vector, +and the current u. +4 + +Magnonic antenna +Current Teads +Magnetic film +Xnon-adiabatic STT terms derived by Zhang and Li:10 +⃗τSTT = −⃗m × (⃗m × (u · ∇)⃗m) + β ⃗m × (u · ∇)⃗m, +(2) +where +u = − +µBP +eMs(1 + β2)j. +(3) +Here, ∇ is the two-dimensional differential operator, β is a dimensionless constant that rep- +resents the degree of non-adiabaticity, e the elementary charge and µB the Bohr magneton. +The polarization P is a property of the ferromagnet and does not depend on the magnetiza- +tion. Although u has the units of velocity, we refer to it as current since it is proportional +to the SP current density j. The most prominent effect of the STT is best understood by +assuming small dissipation terms (α ≈ 0 and β ≈ 0). In that case it is easy to prove that +the solution of the LLG equation is given by ⃗m(r − ut; t) if ⃗m(r; t) is the solution in absence +of the STT. Put differently, adding the STT shifts the solution with a velocity u.21 For +instance, due to the STT, relaxed local structures such as domain walls and skyrmions will +move with a velocity v = u when the current is switched on. Accordingly, a SW packet +traveling in the film will gain an additional velocity equal to u. This insight will help to +obtain an intuitive understanding of the results presented in this paper in which we study +the effect of a non-uniform static current u(r) on the propagation of SWs. +The dynamics of the magnetization depends strongly on the characteristics of the mag- +netic film, incorporated in the free energy functional. In this paper, we take into account +the contribution of exchange interaction and Zeeman energy due to applied bias magnetic +field: +E[⃗m] = +�� � +A(∇⃗m)2 − ⃗H · ⃗m +� +dxdy, +(4) +with exchange stiffness A > 0 and bias field ⃗H. +The ground state magnetization is the ferromagnetic (field-polarized) state in which the +magnetization is aligned with the bias field ⃗H. To study small deviations from the ground +5 + +state it is useful to construct the right-handed coordinate system (ˆea, ˆeb, ˆe0) with ˆe0 ∥ ⃗H. +The effective field in this coordinate system becomes +⃗Heff(r, t) = +� +����� +2A +Ms∇2ma(r, t) +2A +Ms∇2mb(r, t) +2A +Ms∇2m0(r, t) + H0 +� +����� +, +(5) +where ⃗m = (ma, mb, m0) and H = (0, 0, H0). Here and in the next section, we focus on +first order deviations from the ground state and consider the SW solution as m0 ≈ 1, +ma = A0ei(k·r−ωt)−µt and mb = iA0ei(k·r−ωt)−µt, where A0 ≪ 1 represents the SW amplitude; +ω is the SW angular frequency; k is the wave vector, and µ represents the damping of the +SW. Substituting that into the LLG equation [Eq. (1)], with STT term given by Eq. (2) and +effective field from Eq. (5), yields the dispersion relation (see supplementary material22) +ω = +γH0 +1 + α2 +� +1 + ξ2k2� ++ 1 + αβ +1 + α2 u · k, +(6) +and damping parameter +µ = αω − βu · k, +(7) +where we define the length scale ξ = +� +2A/H0Ms. Note that the damping µ of the SW +decreases if the wave vector k is parallel to the electron flow u. Consequently, the attenuation +length of a spin wave can be increased by applying a current opposite to the propagation +direction, as predicted earlier by Seo et al.23 +Propagation direction.—The dispersion relation consists out of circular isofrequencies +in k-space, which are shifted away from the origin due to the current. The center of the +circular isofrequency is given by +k0 = − 1 + αβ +2γH0ξ2u, +(8) +6 + +and the radius by +kg = |k − k0| = +� +1 + α2 +γH0ξ2 (ω − ω0), +(9) +with the minimal frequency ω0 = γH0(1 − ξ2k2 +0). +The propagation velocity v of a wave packet is given by the gradient of the frequency in +k-space: +v = ∇kω = 2γH0ξ2 +1 + α2 k + 1 + αβ +1 + α2 u = 2γH0ξ2 +1 + α2 (k − k0), +(10) +where kg = k − k0 defines the propagation direction. The velocity of the wave packet can +be separated as +v∥ = γH0ξ2 +1 + α22k + 1 + αβ +1 + α2 u∥, v⊥ = 1 + αβ +1 + α2 u⊥, +(11) +where ∥ and ⊥ denote the component parallel and perpendicular to the wave vector respec- +tively. Notice that, in general, the propagation direction is not parallel to the wave vector +k, and the SW can be deflected in the presence of applied current. +Generalized Snell’s law +Let us now examine the propagation of SWs when experiencing nonuniform current distri- +butions. For simplicity, we start with the example of a SW propagating between two regions +with different current densities j, i.e. j(x ≤ 0) = 0 and j(x > 0) = j0ˆy. It is well known that +SWs reflect at material boundaries, where the momentum parallel to the interface should +be conserved. In the case of SP current, the change in current density is equivalent to an +interface, and the momentum perpendicular to ∇j, i.e., k · ˆτ ≡ |k − (k · ˆ +∇j) ˆ +∇j|, with ˆτ +the vector tangent to the interface, should be conserved. In our simple example that cor- +responds to k(1) +y += k(2) +y , where the indices 1 and 2 refer to the incident and refracted waves +respectively. If the propagation direction is parallel to the k vector, the well-known Snell’s +law applies: k(1) sin(φ1) = k(2) sin(φ2), with φ1 and φ2 the incident and refracted angles +respectively. However, since in our case the dispersion relation is asymmetric, the Snell’s +7 + +law has to be adjusted as follows +k(1) +g +sin(φ1) + k(1) +0 +· ˆτ = k(2) +g +sin(φ2) + k(2) +0 +· ˆτ, +(12) +where the angles φi are taken with respect to ˆ∇j (i.e., the direction normal to the interface). +Similarly generalized Snell’s laws for the refraction of SWs at domain walls and heterochiral +interfaces were derived in Refs. 24 and 20. +Micromagnetic simulations +To realistically simulate a SW system in the presence of the SP current, we employ the +micromagnetic framework and specifically its mumax3 implementation25,26 to calculate the +SW refraction when propagating between two regions with different current densities. +The SW beams are created by a sinusoidal oscillating field h = h0 sin(ωt)ˆz applied in +a narrow rectangular region (input antenna, see Fig. 2 (a)), where the field amplitude h0 +has a Gaussian profile in the transverse direction and f = ω/2π is the oscillation frequency. +For all simulations we consider h0 = 0.01H0, where H0 = 0.5 T is the bias field applied +along +ˆx direction [see Methods section]. Fig. 2 (a-d) shows snapshots of the simulated +SW propagation across the interface where the current density sharply changes, for different +incident angles φ1. White arrows denote the SW trajectories predicted by our Eq. (12), +which are in excellent agreement with the micromagnetic simulations. Notice that the STT +induced by the SP current can either deflect or confine the SWs. The critical incident angle +φ∗ above which the SW can not propagate for a given applied current, i.e., the SW undergoes +total internal reflection, is obtained from our Snell’s law by imposing that the refracted wave +is parallel to the interface (φ2 = ±π/2), as +φ∗ = ± arcsin +� +±k(2) +g ++ (k(2) +0 +− k(1) +0 ) · ˆτ +k(1) +g +� +. +(13) +Note that due to the asymmetric dispersion relation, the refraction is not symmetric for +8 + +Figure 2: Generalized Snell’s law for SW refraction. (a-d) Snapshots of the simulated +spin-wave propagation across a sharp interface (at x = 0) where the current density changes +from 0 to j = j0ˆy, for different incident angles φ1. White arrows indicate the propagation +direction following the generalized Snell’s law [Eq. (12)]. In these simulations we considered +SW frequency f = 20 GHz, bias field H0 = 0.5 T applied along +ˆx direction, j = 2 × +1012 Am−2ˆy, α = 0.001 and β = 0.002. +positive and negative incident angles. This is seen in the results for φ1 = 60◦ and φ1 = −60◦ +in Fig. 2 (a,d), where only for the second case the total reflection of the SW is achieved. +Poisson solver.—In practice, the current applied into the magnetic film will not exhibit a +step-like distribution as considered in our previous example, but will spread continuously in +the material obeying Poisson’s equation.27 To precisely calculate the interaction of SWs with +such nonuniform current distributions, we implemented a Poisson solver in the micromagnetic +simulation package mumax3.25 Fig. 3 (a) shows a snapshot of the simulated SW propagation +9 + +(a) +(b) +2 +Φ1 =600 +Φ1=450 +[un] +0 +y +Φ1 +Input +-2 +(c) +(d) +2 +Φ1 =-300 +Φ1 =-600 +y[μm] +0 +-2 +-2 +0 +2 -2 +0 +2 +x [μm] +x[μm] +Imzl(×10-2) +0 +1 +2Figure 3: SWs under nonuniform current distribution. +(a) Snapshots of the SW +propagating across a nonuniform current distribution induced between the voltage contacts +(gray regions). White arrows indicate the SW trajectory calculated by iterating Eq. (12) +locally along the current gradient in the propagation direction. Parameters are the same as +in Fig. 2. (b) Contour plot of the SP current density considered in (a). The shown isolines +should be seen as interfaces where the SW is sequentially refracted during propagation. ∆k⊥ +(black arrows) shows the accompanying change in the propagation direction (see text). +across the nonuniform SP current density created between shown finite voltage contacts +at the sample edges. Notice that the SW is pertinently deflected while crossing the region +where current is applied, to finally reach a shifted propagation direction upon leaving the area +pierced by current. Although both the direction and magnitude of the current continuously +10 + +Input +npu +TVchange as the SW propagates, our generalized Snell’s law still applies locally - the SW facing +a local gradient in current density ∇j is effectively analogous to the earlier interface example. +White arrows in Fig. 3 (a) indicate the SW trajectory calculated by iterating Eq. (12) as +the SW propagates in the current distribution, where in each step the angle φi is taken +with respect to the local ˆ∇j. The predicted trajectory is in very good agreement with the +simulation, demonstrating the general applicability of Eq. (12) even for nonuniform current +distributions. Fig. 3 (b) shows the contour plot of the applied current density, where isolines +can be seen as interfaces where the SW is refracted. ∆k⊥ = (k(2) +0 +− k(1) +0 ) · ˆτ quantifies the +change in the propagation direction of the SW, shown as black arrows in Fig. 3 (b). +Multichannel SW selector +Performing logic operations with SWs generally requires combining different input waves +that interfere with each other to generate a desired logic output state.5 The ability to guide +SWs through nanochannels is therefore vital to the development of more complex SW-based +circuitry. In this regard, SP currents can be used to precisely guide the SW in such devices, +for example, to selectively “write” SWs in one of multiple nanotracks or logic gates in a +larger microprocessor. We exemplify here such an application by simulating a multichannel +SW selector, illustrated in Fig. 4 (a). The input SWs are generated in a channel on the +left-hand side of the sample and propagate across a region where SP current is applied. By +tuning the magnitude or direction of the applied current one can precisely deflect the SWs +towards one of the output channels on the right side of the sample, as shown in Figs. 4 (b,c). +Likewise, a frequency selector can be implemented considering the fact that SWs with dif- +ferent frequencies experience different deflections under the same applied current and can +therefore be isolated into separate output channels. +11 + +Figure 4: Guiding SWs for logic and neuromorphic computing. (a) A multichannel +SW selector demonstrated based on one input and two output channels. Voltage leads induce +the SP current in the central region of the sample. (b-c) Snapshots of SW propagation +simulated for the setup shown in (a). By changing magnitude or direction of the applied +current one can guide the SW towards the desired output channel with minimal losses. (d) +Scheme of the envisioned neural network hardware. The (18) input SWs are created on the +left and propagate across the matrix of 80 voltage contacts (grey dots) of 100 nm in diameter +each. (e) SP-current profile induced by the applied voltage that best performs the desired +operation. (f-g) Snapshots of SW simulations after training the neural network. The voltage +(current) pattern was trained to focus the waves of two different frequencies to the desired +outputs. The bar charts show the normalized intensities at the output locations integrated +over 10 ns. In these simulations we considered SW frequency f = 15 GHz (b, c, f) and +f = 14.5 GHz (g), and other parameters same as previously. +12 + +(a) ++ +(b) +(c) ++ +j[(×1012)Am-2] +0.5μm +[mzl(×10-2) +0.5μm +0 +1 +0 +(d) +Input +(f) +15 GHz +4 +01 +3 +口 +n +2 +02 +02 +0 +1 +0 +(e) +(g) +14.5GHz +4 +01 +91 +3 +口 +口 +n +2 +> +02 +% +口 +0 +0 +2 +4 +6 +0 +2 +4 +6 +0.0 +1.0 +x [μm] +j[(×1012)Am-2] +x [μm] +[mzl(×10-2) +Norm. output +0 +0.5 +0 +1 +2Neuromorphic computing +Finally, we demonstrate the use of SP current for the design of neural-network hardware, +based on SW propagation, where weights and interconnections of the network are realized +by a pattern of the SP currents applied to the propagating substrate. Fig. 4 (d) illustrates +the envisioned device. +The input signal is created on the left and propagates across a +region with a matrix of 80 voltage contacts [gray dots in Fig. 4 (d)] of 100 nm in diameter +each. The read-out is taken from the two output antennas on the right side. An arbitrarily +powered voltage matrix induces a distribution of the SP currents in the substrate that +interacts with the input SWs. +Training the neural network is equivalent to finding the +current pattern that realizes the desired input-output mapping, for example, to classify +different input signals by focusing them in different outputs. As suggested in Refs. 4 and 9, +a back-propagation machine learning algorithm can be used for training a similar SW-based +network, which can perform tasks such as vowel recognition and frequency classification. +Here, we demonstrate that a simple Monte Carlo (MC) algorithm can perform the same task +of training the neural network for simple classification problems. In our example, we perform +a frequency-recognition operation, where we consider input SWs with frequencies f = 15 +and 14.5 GHz. The neural network is trained to focus the SWs with 15 GHz to the output +O1 and SWs with 14.5 GHz to the output O2 [see Figs. 4 (f,g)]. The voltage at each contact +is randomly initialized in one of the three values: Ui = −u0, 0 or +u0, with u0 = 0.15 V, +and a Metropolis algorithm is implemented by changing the voltage of one of the contacts +at every MC step [see supplemental material22]. The finally trained configuration [resulting +in the current distribution shown in Fig. 4 (e)] is able to focus SWs of each frequency to the +desired outputs as shown by the bar charts in Figs. 4 (f,g), and can therefore perform the +classification operation. In the supplementary material, we show that the effects of Oersted +and demagnetizing fields are negligible in our example.22 One should note however that the +proposed neuromorphic application can be generalized for arbitrary scenarios, as long as all +the relevant magnetic interactions are considered during the training process of the neural +13 + +Table 1: Material properties of representative low-damping, metallic magnetic materials +that can host spin waves and are therefore candidates for the proposed control of SWs by +SP current. +Ms (MA/m) +A (pJ/m) +α (×10−3) +P (Ωm)−1 +References +Ni80Fe20 +0.7 +10 +7 +0.5 +[ 5,16] +CoFeB +1.3 +15 +4 +0.65 +[ 5,28] +CoFeAlB +1.0 +9 +3 +[ 29] +network. The here proposed fully electronic control enables facilitated reconfiguration of the +neural network to perform different tasks within the same integrated circuit. +To identify the role of different material parameters in the current-induced SW scattering, +and consequently in the proposed applications, let us consider the case of an SW propagating +across two regions with different current densities, as in Fig. 2. In this scenario, for an +incident angle φ1 = 0 and making use of Eqs. (12) and (3), the refraction angle of the SW +is given by φ2 = sin−1 +� +1+αβ +√ +(1+αβ)2+η2 +� +, where η = +4eγ +µB +Ak +Pj . Therefore, the current-induced +SW scattering is maximized when η is minimized. In other words, the SW scattering is +maximized in materials with small exchange stiffness A and for SWs with small wavenumber +k, as well as by increasing the SP current, represented by the polarization P and applied +current density j. Moreover, it is worth mentioning here that applied current j may locally +increase the temperature of the system, and consequently affect the magnetic parameters +such as the saturation magnetization Ms, hence additional scattering may arise from such +non-uniformity of the magnetic landscape in which SWs propagate. In Table 1 we show +material properties of representative low-damping, metallic magnetic materials that can +host spin waves and are therefore convenient candidates for the proposed control of SWs by +SP current. As an additional example, we reproduced the neuromorphic application for the +case of a Ni80Fe20 thin film in the supplementary material.22 +14 + +Conclusions +We have demonstrated the use of non-uniform spin-polarized current for the manipulation of +spin-waves. We showed that the spin-transfer torque induced by the applied current has an +effect analogous to DMI for confining spin-waves and controlling their propagation direction +in the linear regime, and derived a generalized Snell’s law that describes the scattering of +spin-waves between regions with different current densities. Finally, we implemented the +calculation of the current distribution in micromagnetic simulations by solving the Poisson’s +equation within the simulation package mumax3 (to be made available in the upcoming re- +lease of mumax4), in order to (i) validate the derived Snell’s law and (ii) demonstrate how +strategically applied current distributions can be employed in magnonic logic and neuromor- +phic computing devices, thereby advancing the prospects of low-power spintronic hardware +and artificial intelligence. +Methods +Micromagnetic simulations +For the simulations, we employ the micromagnetic framework MUMAX3,25,26 where we con- +sider 10 nm thick magnetic films with saturation magnetization Ms = 0.14 MAm−1, exchange +stiffness A = 3.65 pJm−1 30 and damping constant α = 0.001. The considered free energy +density is given by Eq. (4), and the dynamics of the magnetization is governed by the LLG +equation [Eq. (1)] with STT terms [Eq. (2)] accounting for in-plane applied currents. The +in-plane bias field is set to H = H0ˆx, with H0 = 0.5 T. For all simulations, we consider a +system discretized into cells of size 10×10×10 nm3. The polarization rate of the SP current +is fixed at P = 0.4. The calculation of the current distribution is made by solving the Pois- +son’s equation for the considered geometries, where we assume the electrical conductivity as +σ = 1 × 106 (Ωm)−1. +15 + +Acknowledgement +This work was supported by the Research Foundation - Flanders (FWO-Vlaanderen, un- +der Grant No. K226322N and 12A9223N), EoS ShapeME project, and Brazilian Agencies +FACEPE, CAPES and CNPq. +Supporting Information Available +Calculation of spin-wave dispersion relation under spin-polarized current; Monte Carlo train- +ing of neural network; Effect of Oersted and demagnetizing fields; Example of a spin-wave +neural network in a Ni80Fe20 thin film. +References +(1) Christensen, D. 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Scientific Reports 2017, 7, 1–9. +19 + diff --git a/LNE4T4oBgHgl3EQfJgy4/content/tmp_files/load_file.txt b/LNE4T4oBgHgl3EQfJgy4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15ae6cd883f42376c841d7d8723db9ccefb1344b --- /dev/null +++ b/LNE4T4oBgHgl3EQfJgy4/content/tmp_files/load_file.txt @@ -0,0 +1,579 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf,len=578 +page_content='Towards Magnonic Logic and Neuromorphic Computing: Controlling Spin-Waves by Spin-Polarized Current Ra´ı M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Menezes,† Jeroen Mulkers,‡ Cl´ecio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' de Souza Silva,¶ Bartel Van Waeyenberge,‡ and Milorad V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Miloˇsevi´c∗,† †NANOlab Center of Excellence & Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium ‡DyNaMat Lab, Department of Solid State Sciences, Ghent University, Ghent, Belgium ¶Departamento de F´ısica, Universidade Federal de Pernambuco, Cidade Universit´aria, 50670-901, Recife-PE, Brazil E-mail: milorad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='milosevic@uantwerpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='be Abstract Spin-waves (magnons) are among the prime candidates for building fast yet energy- efficient platforms for information transport and computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We here demonstrate the- oretically and in state-of-the-art micromagnetic simulation the effects that strategically- injected spin-polarized current can have on controlling magnonic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We reveal analytically that the Zhang-Li spin-transfer-torque induced by applied current is anal- ogous to the Dzyaloshinskii-Moriya interaction for scattering the magnons in the linear regime, to then provide a generalized Snell’s law that describes the spin-wave propaga- tion across regions with different current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We validate the latter in numerical simulations of realistic systems, and exemplify how these findings may help advance the design of spin-wave logic and neuromorphic computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='04922v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 The development of neuromorphic computing hardware has attracted significant atten- tion in recent years as such platforms are capable of performing complex information pro- cessing tasks, such as classification and pattern recognition of various types of data, from e-commerce to scientific content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='1–3 A central challenge of this research is the requirement of highly interconnected systems, inspired by the biological concepts of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Interestingly, wave-based physical systems have been demonstrated to operate as recurrent neural networks,4 where interference patterns in the propagating substrate can realize an all-to-all interconnection between points of the substrate that mimic the action of artificial neurons by scattering and recombining input waves in order to extract their information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Similar to other wave phenomena in physics, spin-waves (magnons) travel through space accompanied by a transfer of energy, which if precisely controlled can lead to fast information transport and computing applications in the nanometric to micrometric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5,6 Spin-waves are readily demonstrated as a promising platform for performing logic operations7,8 and the recent theoretical advances in wave-based computation can pave the way for spintronic hardware in the field of artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='9 However, even though extensive research has been carried out in recent years, the precise manipulation of spin waves in nanostructures has not been entirely mastered and needs to be advanced for the benefit of functional magnonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' One manner of manipulating the spin textures in magnetic materials is by the application of spin-polarized (SP) currents, having (part of the) spins of the moving electrons aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The interaction of a SP current with the localized magnetic moments results in a torque on the magnetization, dubbed a Zhang-Li spin-transfer-torque (STT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='10 Being able to affect the orientation of the magnetization, STT has become nearly unavoidable in the design of spin- tronic nanodevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='11 The effect and applicability of the STT was demonstrated in racetrack memory concepts, where the position of local magnetic structures, such as domain walls and skyrmions, is controlled by in-plane currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='12,13 Moreover, not only do SP currents modify the equilibrium magnetic state, they can also significantly influence the propagation of spin 2 waves (SWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' For instance, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='14 the authors derived the dispersion relation of SWs in a ferromagnet subjected to a uniform SP current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The applied current introduces a term proportional to the wavevector k in the spin-wave dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The nonreciprocity in the dispersion relation for waves with a k vector in the same or the opposite direction as the current flow causes Doppler shift,15 as first validated experimentally by Vlaminck and Bailleul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='16 A similar nonreciprocal term in the dispersion relation can also be induced by an electric field17 or by the antisymmetric exchange interaction, also known as Dzyaloshinskii-Moriya interaction (DMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='18 Analogously to the Doppler shift induced by the currents, a frequency shift has been measured in ferromagnetic materials possessing DMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='19 For heterochiral ferro- magnets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=', with spatially varying DMI), the reciprocal term causes a nontrivial refraction of spin waves at interfaces between regions with different DMI, as described by a generalized Snell’s law in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In this regard, the equivalency in the dispersion relations of SWs in the presence of DMI and SP current suggests that a spatially varying current density (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 1) can be used to manipulate the propagation of SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In this Letter, we detail the effect of a SP current on controlling the propagation of spin- waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We show that the Zhang-Li STT induced by in-plane current has analogous effect to DMI for confining and controlling the propagation direction of magnons in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We proceed to derive a Snell’s law to describe the scattering of spin-waves between regions with different current densities, and validate it by advanced simulations including solving Poisson’s equation within the micromagnetic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Finally we present selected tailored examples to illustrate how strategically applied current can be employed to advance logic and neuromorphic computing devices based on spin-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 3 Figure 1: Schematic illustration of a SW facing a non-uniform distribution of the SP current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The SW (in red) is induced by the input antenna on the left side and propagates along the magnetic film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Voltage electrodes induce the distributed SP current (j) which can be tuned to control the SW propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Results and discussion Spin-wave dispersion under applied current Within the micromagnetic framework, we describe the magnetization of a thin extended fer- romagnetic film by considering the vector field ⃗M(r) = Ms⃗m(r) with constant magnetization modulus | ⃗M| = Ms and the normalized magnetization direction ⃗m(r) at each point r ∈ R2 of the film 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The dynamics of the magnetization is governed by the Landau-Lifshitz-Gilbert (LLG) equation ˙⃗m = −γ ⃗m × ⃗Heff + α⃗m × ˙⃗m + ⃗τSTT, (1) where γ is the gyromagnetic ratio, α the dimensionless damping factor, and ⃗Heff the effective field, which can be derived from the free energy E[⃗m] by taking the functional derivative with respect to the magnetization: ⃗Heff = −δE/δ ⃗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We extend the LLG equation by adding the torque ⃗τSTT which includes the adiabatic and 1In this paper we use an arrow for 3D vector fields such as the magnetization ⃗m and the effective magnetic field ⃗Heff and bold variables for 2D dimensional vectors in the plane such as the position r, the k-vector, and the current u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 Magnonic antenna Current Teads Magnetic film Xnon-adiabatic STT terms derived by Zhang and Li:10 ⃗τSTT = −⃗m × (⃗m × (u · ∇)⃗m) + β ⃗m × (u · ∇)⃗m, (2) where u = − µBP eMs(1 + β2)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (3) Here, ∇ is the two-dimensional differential operator, β is a dimensionless constant that rep- resents the degree of non-adiabaticity, e the elementary charge and µB the Bohr magneton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The polarization P is a property of the ferromagnet and does not depend on the magnetiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Although u has the units of velocity, we refer to it as current since it is proportional to the SP current density j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The most prominent effect of the STT is best understood by assuming small dissipation terms (α ≈ 0 and β ≈ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In that case it is easy to prove that the solution of the LLG equation is given by ⃗m(r − ut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' t) if ⃗m(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' t) is the solution in absence of the STT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Put differently, adding the STT shifts the solution with a velocity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='21 For instance, due to the STT, relaxed local structures such as domain walls and skyrmions will move with a velocity v = u when the current is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Accordingly, a SW packet traveling in the film will gain an additional velocity equal to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' This insight will help to obtain an intuitive understanding of the results presented in this paper in which we study the effect of a non-uniform static current u(r) on the propagation of SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The dynamics of the magnetization depends strongly on the characteristics of the mag- netic film, incorporated in the free energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In this paper, we take into account the contribution of exchange interaction and Zeeman energy due to applied bias magnetic field: E[⃗m] = �� � A(∇⃗m)2 − ⃗H · ⃗m � dxdy, (4) with exchange stiffness A > 0 and bias field ⃗H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The ground state magnetization is the ferromagnetic (field-polarized) state in which the magnetization is aligned with the bias field ⃗H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' To study small deviations from the ground 5 state it is useful to construct the right-handed coordinate system (ˆea, ˆeb, ˆe0) with ˆe0 ∥ ⃗H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The effective field in this coordinate system becomes ⃗Heff(r, t) = � ����� 2A Ms∇2ma(r, t) 2A Ms∇2mb(r, t) 2A Ms∇2m0(r, t) + H0 � ����� , (5) where ⃗m = (ma, mb, m0) and H = (0, 0, H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Here and in the next section, we focus on first order deviations from the ground state and consider the SW solution as m0 ≈ 1, ma = A0ei(k·r−ωt)−µt and mb = iA0ei(k·r−ωt)−µt, where A0 ≪ 1 represents the SW amplitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' ω is the SW angular frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' k is the wave vector, and µ represents the damping of the SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Substituting that into the LLG equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (1)], with STT term given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (2) and effective field from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (5), yields the dispersion relation (see supplementary material22) ω = γH0 1 + α2 � 1 + ξ2k2� + 1 + αβ 1 + α2 u · k, (6) and damping parameter µ = αω − βu · k, (7) where we define the length scale ξ = � 2A/H0Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Note that the damping µ of the SW decreases if the wave vector k is parallel to the electron flow u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Consequently, the attenuation length of a spin wave can be increased by applying a current opposite to the propagation direction, as predicted earlier by Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='23 Propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='—The dispersion relation consists out of circular isofrequencies in k-space, which are shifted away from the origin due to the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The center of the circular isofrequency is given by k0 = − 1 + αβ 2γH0ξ2u, (8) 6 and the radius by kg = |k − k0| = � 1 + α2 γH0ξ2 (ω − ω0), (9) with the minimal frequency ω0 = γH0(1 − ξ2k2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The propagation velocity v of a wave packet is given by the gradient of the frequency in k-space: v = ∇kω = 2γH0ξ2 1 + α2 k + 1 + αβ 1 + α2 u = 2γH0ξ2 1 + α2 (k − k0), (10) where kg = k − k0 defines the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The velocity of the wave packet can be separated as v∥ = γH0ξ2 1 + α22k + 1 + αβ 1 + α2 u∥, v⊥ = 1 + αβ 1 + α2 u⊥, (11) where ∥ and ⊥ denote the component parallel and perpendicular to the wave vector respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Notice that, in general, the propagation direction is not parallel to the wave vector k, and the SW can be deflected in the presence of applied current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Generalized Snell’s law Let us now examine the propagation of SWs when experiencing nonuniform current distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' For simplicity, we start with the example of a SW propagating between two regions with different current densities j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' j(x ≤ 0) = 0 and j(x > 0) = j0ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' It is well known that SWs reflect at material boundaries, where the momentum parallel to the interface should be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In the case of SP current, the change in current density is equivalent to an interface, and the momentum perpendicular to ∇j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=', k · ˆτ ≡ |k − (k · ˆ ∇j) ˆ ∇j|, with ˆτ the vector tangent to the interface, should be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In our simple example that cor- responds to k(1) y = k(2) y , where the indices 1 and 2 refer to the incident and refracted waves respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' If the propagation direction is parallel to the k vector, the well-known Snell’s law applies: k(1) sin(φ1) = k(2) sin(φ2), with φ1 and φ2 the incident and refracted angles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' However, since in our case the dispersion relation is asymmetric, the Snell’s 7 law has to be adjusted as follows k(1) g sin(φ1) + k(1) 0 ˆτ = k(2) g sin(φ2) + k(2) 0 ˆτ, (12) where the angles φi are taken with respect to ˆ∇j (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=', the direction normal to the interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Similarly generalized Snell’s laws for the refraction of SWs at domain walls and heterochiral interfaces were derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 24 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Micromagnetic simulations To realistically simulate a SW system in the presence of the SP current, we employ the micromagnetic framework and specifically its mumax3 implementation25,26 to calculate the SW refraction when propagating between two regions with different current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The SW beams are created by a sinusoidal oscillating field h = h0 sin(ωt)ˆz applied in a narrow rectangular region (input antenna, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 2 (a)), where the field amplitude h0 has a Gaussian profile in the transverse direction and f = ω/2π is the oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' For all simulations we consider h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='01H0, where H0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 T is the bias field applied along +ˆx direction [see Methods section].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 2 (a-d) shows snapshots of the simulated SW propagation across the interface where the current density sharply changes, for different incident angles φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' White arrows denote the SW trajectories predicted by our Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (12), which are in excellent agreement with the micromagnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Notice that the STT induced by the SP current can either deflect or confine the SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The critical incident angle φ∗ above which the SW can not propagate for a given applied current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=', the SW undergoes total internal reflection, is obtained from our Snell’s law by imposing that the refracted wave is parallel to the interface (φ2 = ±π/2), as φ∗ = ± arcsin � ±k(2) g + (k(2) 0 − k(1) 0 ) · ˆτ k(1) g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (13) Note that due to the asymmetric dispersion relation, the refraction is not symmetric for 8 Figure 2: Generalized Snell’s law for SW refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (a-d) Snapshots of the simulated spin-wave propagation across a sharp interface (at x = 0) where the current density changes from 0 to j = j0ˆy, for different incident angles φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' White arrows indicate the propagation direction following the generalized Snell’s law [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (12)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In these simulations we considered SW frequency f = 20 GHz, bias field H0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 T applied along +ˆx direction, j = 2 × 1012 Am−2ˆy, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='001 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' positive and negative incident angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' This is seen in the results for φ1 = 60◦ and φ1 = −60◦ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 2 (a,d), where only for the second case the total reflection of the SW is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Poisson solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='—In practice, the current applied into the magnetic film will not exhibit a step-like distribution as considered in our previous example, but will spread continuously in the material obeying Poisson’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='27 To precisely calculate the interaction of SWs with such nonuniform current distributions, we implemented a Poisson solver in the micromagnetic simulation package mumax3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 3 (a) shows a snapshot of the simulated SW propagation 9 (a) (b) 2 Φ1 =600 Φ1=450 [un] 0 y Φ1 Input 2 (c) (d) 2 Φ1 =-300 Φ1 =-600 y[μm] 0 2 2 0 2 -2 0 2 x [μm] x[μm] Imzl(×10-2) 0 1 2Figure 3: SWs under nonuniform current distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (a) Snapshots of the SW propagating across a nonuniform current distribution induced between the voltage contacts (gray regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' White arrows indicate the SW trajectory calculated by iterating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (12) locally along the current gradient in the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (b) Contour plot of the SP current density considered in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The shown isolines should be seen as interfaces where the SW is sequentially refracted during propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' ∆k⊥ (black arrows) shows the accompanying change in the propagation direction (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' across the nonuniform SP current density created between shown finite voltage contacts at the sample edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Notice that the SW is pertinently deflected while crossing the region where current is applied, to finally reach a shifted propagation direction upon leaving the area pierced by current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Although both the direction and magnitude of the current continuously 10 Input npu TVchange as the SW propagates, our generalized Snell’s law still applies locally - the SW facing a local gradient in current density ∇j is effectively analogous to the earlier interface example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' White arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 3 (a) indicate the SW trajectory calculated by iterating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (12) as the SW propagates in the current distribution, where in each step the angle φi is taken with respect to the local ˆ∇j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The predicted trajectory is in very good agreement with the simulation, demonstrating the general applicability of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (12) even for nonuniform current distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 3 (b) shows the contour plot of the applied current density, where isolines can be seen as interfaces where the SW is refracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' ∆k⊥ = (k(2) 0 − k(1) 0 ) · ˆτ quantifies the change in the propagation direction of the SW, shown as black arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Multichannel SW selector Performing logic operations with SWs generally requires combining different input waves that interfere with each other to generate a desired logic output state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 The ability to guide SWs through nanochannels is therefore vital to the development of more complex SW-based circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In this regard, SP currents can be used to precisely guide the SW in such devices, for example, to selectively “write” SWs in one of multiple nanotracks or logic gates in a larger microprocessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We exemplify here such an application by simulating a multichannel SW selector, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The input SWs are generated in a channel on the left-hand side of the sample and propagate across a region where SP current is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' By tuning the magnitude or direction of the applied current one can precisely deflect the SWs towards one of the output channels on the right side of the sample, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Likewise, a frequency selector can be implemented considering the fact that SWs with dif- ferent frequencies experience different deflections under the same applied current and can therefore be isolated into separate output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 11 Figure 4: Guiding SWs for logic and neuromorphic computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (a) A multichannel SW selector demonstrated based on one input and two output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Voltage leads induce the SP current in the central region of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (b-c) Snapshots of SW propagation simulated for the setup shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' By changing magnitude or direction of the applied current one can guide the SW towards the desired output channel with minimal losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (d) Scheme of the envisioned neural network hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The (18) input SWs are created on the left and propagate across the matrix of 80 voltage contacts (grey dots) of 100 nm in diameter each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (e) SP-current profile induced by the applied voltage that best performs the desired operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (f-g) Snapshots of SW simulations after training the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The voltage (current) pattern was trained to focus the waves of two different frequencies to the desired outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The bar charts show the normalized intensities at the output locations integrated over 10 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In these simulations we considered SW frequency f = 15 GHz (b, c, f) and f = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 GHz (g), and other parameters same as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 12 (a) + (b) (c) + j[(×1012)Am-2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5μm [mzl(×10-2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5μm 0 1 0 (d) Input (f) 15 GHz 4 01 3 口 n 2 02 02 0 1 0 (e) (g) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5GHz 4 01 91 3 口 口 n 2 > 02 % 口 0 0 2 4 6 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='0 x [μm] j[(×1012)Am-2] x [μm] [mzl(×10-2) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' output 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 0 1 2Neuromorphic computing Finally, we demonstrate the use of SP current for the design of neural-network hardware, based on SW propagation, where weights and interconnections of the network are realized by a pattern of the SP currents applied to the propagating substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (d) illustrates the envisioned device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The input signal is created on the left and propagates across a region with a matrix of 80 voltage contacts [gray dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (d)] of 100 nm in diameter each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The read-out is taken from the two output antennas on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' An arbitrarily powered voltage matrix induces a distribution of the SP currents in the substrate that interacts with the input SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Training the neural network is equivalent to finding the current pattern that realizes the desired input-output mapping, for example, to classify different input signals by focusing them in different outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' As suggested in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 and 9, a back-propagation machine learning algorithm can be used for training a similar SW-based network, which can perform tasks such as vowel recognition and frequency classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Here, we demonstrate that a simple Monte Carlo (MC) algorithm can perform the same task of training the neural network for simple classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In our example, we perform a frequency-recognition operation, where we consider input SWs with frequencies f = 15 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The neural network is trained to focus the SWs with 15 GHz to the output O1 and SWs with 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 GHz to the output O2 [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (f,g)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The voltage at each contact is randomly initialized in one of the three values: Ui = −u0, 0 or +u0, with u0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='15 V, and a Metropolis algorithm is implemented by changing the voltage of one of the contacts at every MC step [see supplemental material22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The finally trained configuration [resulting in the current distribution shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (e)] is able to focus SWs of each frequency to the desired outputs as shown by the bar charts in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 4 (f,g), and can therefore perform the classification operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In the supplementary material, we show that the effects of Oersted and demagnetizing fields are negligible in our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='22 One should note however that the proposed neuromorphic application can be generalized for arbitrary scenarios, as long as all the relevant magnetic interactions are considered during the training process of the neural 13 Table 1: Material properties of representative low-damping, metallic magnetic materials that can host spin waves and are therefore candidates for the proposed control of SWs by SP current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Ms (MA/m) A (pJ/m) α (×10−3) P (Ωm)−1 References Ni80Fe20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='7 10 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 [ 5,16] CoFeB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='3 15 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='65 [ 5,28] CoFeAlB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='0 9 3 [ 29] network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The here proposed fully electronic control enables facilitated reconfiguration of the neural network to perform different tasks within the same integrated circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' To identify the role of different material parameters in the current-induced SW scattering, and consequently in the proposed applications, let us consider the case of an SW propagating across two regions with different current densities, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In this scenario, for an incident angle φ1 = 0 and making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (12) and (3), the refraction angle of the SW is given by φ2 = sin−1 � 1+αβ √ (1+αβ)2+η2 � , where η = 4eγ µB Ak Pj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Therefore, the current-induced SW scattering is maximized when η is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In other words, the SW scattering is maximized in materials with small exchange stiffness A and for SWs with small wavenumber k, as well as by increasing the SP current, represented by the polarization P and applied current density j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Moreover, it is worth mentioning here that applied current j may locally increase the temperature of the system, and consequently affect the magnetic parameters such as the saturation magnetization Ms, hence additional scattering may arise from such non-uniformity of the magnetic landscape in which SWs propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' In Table 1 we show material properties of representative low-damping, metallic magnetic materials that can host spin waves and are therefore convenient candidates for the proposed control of SWs by SP current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' As an additional example, we reproduced the neuromorphic application for the case of a Ni80Fe20 thin film in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='22 14 Conclusions We have demonstrated the use of non-uniform spin-polarized current for the manipulation of spin-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' We showed that the spin-transfer torque induced by the applied current has an effect analogous to DMI for confining spin-waves and controlling their propagation direction in the linear regime, and derived a generalized Snell’s law that describes the scattering of spin-waves between regions with different current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' we implemented the calculation of the current distribution in micromagnetic simulations by solving the Poisson’s equation within the simulation package mumax3 (to be made available in the upcoming re- lease of mumax4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' in order to (i) validate the derived Snell’s law and (ii) demonstrate how strategically applied current distributions can be employed in magnonic logic and neuromor- phic computing devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' thereby advancing the prospects of low-power spintronic hardware and artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Methods Micromagnetic simulations For the simulations, we employ the micromagnetic framework MUMAX3,25,26 where we con- sider 10 nm thick magnetic films with saturation magnetization Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='14 MAm−1, exchange stiffness A = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='65 pJm−1 30 and damping constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The considered free energy density is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (4), and the dynamics of the magnetization is governed by the LLG equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (1)] with STT terms [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' (2)] accounting for in-plane applied currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The in-plane bias field is set to H = H0ˆx, with H0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' For all simulations, we consider a system discretized into cells of size 10×10×10 nm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The polarization rate of the SP current is fixed at P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' The calculation of the current distribution is made by solving the Pois- son’s equation for the considered geometries, where we assume the electrical conductivity as σ = 1 × 106 (Ωm)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' 15 Acknowledgement This work was supported by the Research Foundation - Flanders (FWO-Vlaanderen, un- der Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' K226322N and 12A9223N), EoS ShapeME project, and Brazilian Agencies FACEPE, CAPES and CNPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Supporting Information Available Calculation of spin-wave dispersion relation under spin-polarized current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Monte Carlo train- ing of neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Effect of Oersted and demagnetizing fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Example of a spin-wave neural network in a Ni80Fe20 thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' References (1) Christensen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=' Dittmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE4T4oBgHgl3EQfJgy4/content/2301.04922v1.pdf'} +page_content=';' metadata={'source': 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a/NtE3T4oBgHgl3EQfBwky/content/tmp_files/2301.04269v1.pdf.txt b/NtE3T4oBgHgl3EQfBwky/content/tmp_files/2301.04269v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3a91a8d5ad3b47b6ca43a84e02c25010e44cb64 --- /dev/null +++ b/NtE3T4oBgHgl3EQfBwky/content/tmp_files/2301.04269v1.pdf.txt @@ -0,0 +1,705 @@ +Photophysics of blue quantum emitters in hexagonal Boron Nitride + +Ivan Zhigulin1,*, Karin Yamamura1,*, Viktor Ivády2,3,4, Angus Gale1, Jake Horder1, Charlene J. +Lobo1, Mehran Kianinia1, Milos Toth1,5, and Igor Aharonovich1,5 + +1School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, +New South Wales 2007, Australia +2Department of Physics of Complex Systems, ELTE Eötvös Loránd University, Egyetem tér 1- +3, 1053 Budapest, Hungary +3MTA–ELTE Lendület "Momentum" NewQubit Research Group, Pázmány Péter sétány 1/A, +1117 Budapest, Hungary +4Department of Physics, Chemistry and Biology, Linköping University, Linköping, Sweden +5ARC Centre of Excellence for Transformative Meta-Optical Systems, University of +Technology Sydney, Ultimo, New South Wales 2007, Australia +* These authors contributed equally. +Corresponding author igor.aharonovich@uts.edu.au + +Abstract + +Colour centres in hexagonal boron nitride (hBN) have emerged as intriguing +contenders for integrated quantum photonics. In this work, we present detailed photophysical +analysis of hBN single emitters emitting at the blue spectral range. The emitters are fabricated +by different electron beam irradiation and annealing conditions and exhibit narrow-band +luminescence centred at 436 nm. Photon statistics as well as rigorous photodynamics analysis +unveils potential level structure of the emitters, which suggests lack of a metastable state, +supported by a theoretical analysis. The potential defect can have an electronic structure with +fully occupied defect state in the lower half of the hBN band gap and empty defect state in the +upper half of the band gap. Overall, our results are important to understand the photophysical +properties of the emerging family of blue quantum emitters in hBN as potential sources for +scalable quantum photonic applications. + +Introduction + +Single photon emitters (SPEs) are widely acknowledged as key enablers to establish +and deploy quantum communication and computing, which involves on-demand generation of +high purity single photon emission1-3. Hexagonal boron nitride (hBN) has gained attention due +to its unique properties of wide layer-dependent bandgap centred around 6 eV, high exciton +binding energies, presence of optically active spin-defects and capability to host room- +temperature (RT) bright SPEs4-11. hBN is also attracting attention for its use as an emerging +optoelectronic material for the deep ultraviolet range12. + +Recently, color centres in hBN emitting at the blue spectral range, termed ‘blue +emitters’, were discovered by cathodoluminescence (CL) measurements 13. This group of +emitters generally displays ultra-bright, spectrally stable and narrowband emission with a zero +phonon line (ZPL) consistently centred around 436 nm 13, 14. It was shown that these defects +are closely related to the presence of a signature UV emission at 4.1 eV 9, 14-16. Pre-irradiation +of hBN, such as high temperature annealing in a nitrogen atmosphere, results in higher yield +of the signature UV emission and thus higher numbers of blue colour centres 15. Additionally, +at cryogenic temperatures, these defects have stable emission with sub-GHz linewidths and +minimal spectral diffusion compared to other quantum emitters in hBN 15. Very recently, two- + +photon interference was demonstrated, opening new opportunities for 2D materials in optical +quantum information 17. +To date, the crystallographic origin of the blue emitters is still under debate, and +likewise, little is known about the photophysical nature of the defects. It is thus important to +investigate the properties and characteristics of this group of defects to gain insight into their +formation mechanisms as well as into their optical properties before they can be fully utilised +for practical applications. +Here, we present detailed photophysical analysis of the blue SPEs in hBN created +using focused electron beam irradiations with varying dosages. We investigate photostability, +saturation behaviour, lifetime and transition rates employing time resolved excitations and time +correlation measurements. We use these results to propose the electronic structure of blue +SPEs and the position of their energy levels relative to the hBN valence and conduction bands. + +We examined in detail over ten blue SPEs found in three different hBN flakes. Although +electron beam irradiation conditions for each flake varied, the obtained results (in terms of the +ZPL emission wavelength) remained consistent, supporting the hypothesis that the blue +quantum emitters have the same defect origin. For clarity, we therefore introduce an emitter +labelling scheme (eg. flake 1, emitter 1 = F1E1), which will be used throughout this manuscript. +We start with analysing the optical properties of emitters, formed by electron irradiation +(schematically shown in figure 1a). First, a spectrum of each emitter was acquired and the key +parameters - namely emission at saturation, excited state fluorescence lifetime, τ1, and +autocorrelation, g(2)(τ), were measured, as shown in Figure 1 for F3E1. Spectral +characteristics of emitters were measured using a continuous-wave (CW) 405 nm laser at +powers ranging from 0.66 mW to 5.00 mW. The photoluminescence (PL) emission of F3E1 at +0.66 mW and 5.00 mW centred at 436 nm is shown in Figure 1b. + + + +a +b +1.0 +5.00 mW +Intensity (arb. unit) +0.8 +0.66 mW +0.6 +0.4 +0.2 +0.0 +420 +440 +460 +480 +Wavelength (nm) +d +c +101 +Data +125 +T1 = 2.15 ± 0.01 ns +PsAT = 3.38 mW +unit) +100 +Fit +INF = 198.30 cps +100 +g(2) = 0.21 +IRF +(arb. +10-1 +Fit +75 +E +10-2 +50 +9 +0 +10-3 +20 +0 +20 +25 +Delay T (ns) +10-4 +0 +5 +10 +15 +1 +2 +3 +4 +5 +Time (ns) +Power (mW)Figure 1. Spectral characteristics of emitter 1 in flake 3 (F3E1). (a) Diagram of hBN lattice and +flake. (b) Photoluminescence spectra of the emitter excited with a 405 nm CW laser at powers +of 0.66 mW (shown in purple) and 5.00 mW (shown in pink) at room temperature. (c) Emission +lifetime measurement of the blue emitter excited with a 402 nm pulsed laser at 0.66 mW and +frequency of 30 MHz. A lifetime τ1 = 2.15 ± 0.01 ns was deduced from an exponential fit (blue +line). Green dots show the instrument response function (IRF) with a corresponding fit that +extracts a value of 0.475 ± 0.005 ns. (d) Power saturation measurement of the same emitter. +Blue line shows the fit of the experimental data (pink dots) using the power dependence model +(shown in Equation (1)). Inset in (d) shows autocorrelation function g(2)(0) of 0.21 proving the +single photon nature of this emitter. + +To study the emission lifetimes, a 402 nm pulsed laser at 0.66 mW and frequency of +30 MHz was employed. All lifetime measurements were conducted at room temperature (RT) +under ambient conditions. Figure 1c shows the emission lifetime τ 1= 2.151 ± 0.009 ns of F3E1, +obtained by fitting the data with a single exponential decay. The value is consistent with +lifetimes measured for other blue emitters found in another flake, 2.019 ± 0.003 ns for F2E1, +1.931 ± 0.003 ns for F2E3, and 2.160 ± 0.017 ns for F2E8. Thus, the lifetime of ~ 2 ns is +consistent with other studies of these emitters 13-15, 18. +To gain the saturation information, PL spectra were acquired at each power for 3 +seconds. The maximum value of each spectrum was used as a point in the saturation plot, +and the saturation power (PSAT) and the maximum obtainable PL intensity at saturation (IINF) +were then extracted from a fit to the data using Equation (1): + + + + +(1) + +Figure 1d shows the fluorescence saturation curve of F3E1 with a calculated value of +3.38 mW for PSAT. Measured PSAT values of other blue emitters are 1.50 mW for F2E1, 2.10 +mW for F2E4, 2.40 mW for F2E5, 6.50 mW for F2E8, and 8.90 mW for F2E9. Thus, examined +emitters have PSAT ranging from 1.50 mW to 8.90 mW, similar to previously reported for blue +emitters in hBN at room temperature, which are slightly above typical saturation values for the +visible (~ 2 eV) emitters in hBN 14, 15. Such relatively high PSAT could be attributed to absorption +cross-section of blue emitters from 405 nm excitation. The inset in Figure 1d shows the +autocorrelation function at zero delay time, g(2)(0) of 0.21 verifying the single photon nature of +this emitter. +Next, we focus on the photostability and photon statistics of the blue emitters, to +understand their energy levels. Saturation power plays a key role in these measurements, +because it is necessary to saturate a system in order to observe an increasing bunching effect +in g(2)(τ) measurements. The relatively high saturation power of blue SPEs makes this +measurement difficult, as it requires prolonged excitation under high power, which causes +bleaching in some of the emitters. We observed repeatedly that exciting a single blue emitter +for > 4 hours at powers greater than 1.00 mW causes the system to go irreversibly into its dark +state. +Figure 2a shows g(2)(τ) measurements of four blue emitters found in Flake 1 acquired +between 0.69 - 2.00 mW for over 4 hours. All emitters have a g(2)(0) value below 0.5 indicating +that they are SPEs. Only slight bunching was observed from the emitters above 1.00 mW, +which becomes marginally more noticeable around 2.00 mW. Note that at longer delay times + +the line is flat, indicating that no multiple shelving states are present 19-23. This photophysical +behaviour is substantially different from the majority of defect based SPEs in diamond, silicon +carbide, or even hBN that often exhibit strong bunching with multiple obvious decay pathways +20, 22, 24. These various decay channels (evident by changes in the slope of the g(2)(τ) at longer +delay times) are often a signature of a metastable state or recurring blinking behaviour25. +These effects are nearly negligible for the blue emitters that behave as a nearly ideal two-level +system, as will also be discussed below. Indeed, g(2)(τ) curves with high PSAT above 5.00 mW +(approximately ~ 25% of emitters) showed no bunching at the measured powers. Note, that +many of the emitters were eventually bleached during the g(2)(τ) acquisition at excitation +powers higher than 2.00 mW after approximately one hour. + + +Figure 2. Photostability and photodynamics analysis of blue SPEs. (a) Power-dependent l +g(2)(τ) measurements at 0.69 mW, 1.00 mW, 1.35 mW, 1.70 mW and 2.00 mW on four different +emitters on Flake 1. (b) Timetraces with 1 second resolution and corresponding histograms of +fluorescence intensity for three emitters. Range of histograms is kept constant at 15 kilo +counts/second. + +The luminescence timetraces (kilo counts/sec) of emitters F1E5, F1E1 and F1E8 at +excitation powers of 0.69 mW, 1.35 mW and 1.70 mW are shown in Figure 2b. Timetraces of +each of these emitters are stable over time, without obvious blinking or fluorescence +intermittency. The ranges of distribution of the photon counts are within 10% of the mean, for +each emitter. This is important, as it suggests that the emitters that remain optically active, +exhibit excellent photostability over long excitation periods 19, 22, 26, 27. + +a +b +110 +F1E8 +(sdo) +1.70 mW +3.0 +105 +F1E7 +2.00 mW +Counts +50 +100 +2.5 +F1E8 +95 +0 +1.70 mW +0 +5 +10 +150 +100 +200 +70 +100 +F1E1 +2.0 +(kcps) +1.35 mW +F1E1 +65 +E +1.35 mW +(2) ( +Counts +50 +g +60 +1.5 +F1E8 +1.00 mW +55 +0 +0 +5 +10 +150 +50 +100 +1.0 +60 +100 +F1E5 +F1E5 +(kcps) +0.69 mW +0.69 mW +55 +Counts +50 +0.5 +50 +45 +0 +10-10 +10-7 +0 +5 +10 +150 +200 +400 +Delay time (s) +Time (min) +Occurence (arb. unit)To investigate the rates quantitatively, we recorded the autocorrelation measurement +as a function of excitation power 19, 20, 22. The g(2)(τ) function taken at various excitation powers +can be fit with a standard bi exponential function using Equation (2): + + + + + + + +(2) + +Where τ1, τ2 correspond to lifetimes of antibunching and bunching respectively, while +a is a parameter that describes bunching amplitude. +Figure 3a shows power-dependent plots of g(2)(τ) measurements of F3E1, with the +value of 𝑔(2)(0) remaining below 0.5 at each excitation. Figure 3b,c show the power- +dependent parameters of τ1, τ2 and the bunching factor a obtained from fitting Equation (2) to +the g(2)(τ) measurements. + + +Figure 3. Power dependent analysis of energy levels of F3E1. (a) Second-order +autocorrelation measurements taken at various excitation powers. 𝑔(2)(𝜏) traces are +normalised and stacked vertically. (b) Lifetime τ1 of transitions from the excited state to the +ground state (green dots) and metastable state lifetime τ2 (purple dots) as a function of laser +power. (c) Pink dots show the bunching factor, a, vs excitation power. + + + + + +From the fitting of τ1, τ2 and a, we calculated the transition rates according to the three- +level rate equation 22, 23. The transition rates between the three states were calculated by + +a +b +50 +10 +1.6 +6.00 mW +40 +5.50 mW +(ns) +1.4 +(ns) +8 +5.00 mW +1.2 +30 ^ +4.50 mW +1.0 +20 +6 +4.00 mW +(1)(z) +0 +2 +4 +6 +g +3.50 mW +c +4 +3.00 mw +0.20 +unit) +2.00 mW +0.15 +(arb. +.00 mw +a +0.10 +0.66 mW +0 +0.05 +-100 +-50 +0 +50 +100 +0 +2 +4 +6 +Delay time (ns) +Power (mw)measuring the power-dependent g2(τ). The excitation rate from the ground state to the excited +state, radiative decay rate from the excited state to the ground state, decay rate from the +excited state to the metastable state, and decay rate from the metastable state to the ground +state are expressed as k12, k21, k23, and k31, respectively. The τ1, τ2 and a can be derived using +Equations (3) and (4). The value of k12 depends on the power, and k21 and k31 are assumed to +be power-dependent states. The transition rates were obtained following a combination of +Equations (3)-(9). + + (3) + + + + + + +(4) + + + + + + + +(5) + + + + + + + +(6) + + + + + + +(7) + + + + + + +(8) + + + + + + + +(9) + +Where, A = k12+k21+k23+k31 and B = k12(k23+k31)+k31(k21+k23), P is the excitation power, +d and C are the coefficients related to the saturation behaviour. The rate per power coefficient, +σ, is determined from fitting τ1 22. The same procedure was repeated for various emitters, as +shown in table 1 below where the calculated transition rates are listed. + +Emitter +k21 (MHz) +k23 (MHz) +k310 (MHz) +σ (MHz/mW) +F3E1 +337.6 +3.5 +10.7 +121.1 +F1E12 +323.9 +41.3 +24.6 +87.4 +F2E1 +344.4 +20.7 +31.7 +137.0 +F2E4 +365.5 +17.4 +22.0 +145.5 +F2E5 +312.0 +23.5 +15.7 +183.0 +Table 1. List of the calculated transition rates of k21, k23, k310 and σ of F3E1, F1E12, F2E1, +F2E4, and F2E5. + + +The calculated transition rates are k21 = 337.6 MHz, k23 = 3.5 MHz, k310 = 10.7 for the +specific emitter F3E1. Similar order of magnitude results were also obtained for the other +emitters. The excited state transition rates are an order of magnitude faster than the transitions +to/from the metastable state, k23/k310. These results are consistent with the fact that the blue +emitters behave as a nearly ideal two-level system. +We can also comment on the lower limit of the quantum efficiency of the blue emitters. +Note that precise measurement of quantum efficiency is challenging as it requires modification +of local dielectric environment28, 29. Most of the investigated emitters yielded a count rate of ~ +500 kHz, at saturation, with ~ 2.1 ns (~ 450 MHz) fluorescence lifetime. The setup at 440 nm +is highly inefficient, mostly driven by the low internal quantum efficiencies of the avalanche +photodetectors (~ 10%). Hence, assuming an upper collection efficiency of our confocal setup +at 4% in the visible (~ 700 nm) range, the efficiency will drop to below 0.5% at the blue spectral +range. This puts a lower bound of the quantum efficiency at ~ 20%, which makes it a bright +solid state quantum system. +Finally, we discuss the potential level structure and excitation cycle for the blue SPEs. +Short wavelength excitations, e.g. the 405 nm laser used in the experiments, may change the +charge state of defects and turn them into a dark state. Such photoionization processes can +induce either blinking or loss of the emitter. The reported robustness of the blue emission +under continuous excitation at moderate powers suggests that the bright charge state of the +underlying defect structure is stable and no photoionization processes taking place. Likewise, +no further fluorescence lines at longer wavelengths were observed, indicating lack of +photochromism 23, 30-32. This observation can be used to narrow down the possible electronic +structures giving rise to the blue emission. +Figure 4. Tentative electronic structure of the blue emitter. (a) Stable and unstable electronic +configuration under 405 nm excitation. Defects with fully occupied defect states in the lower +half of the band gap and empty defect states in the upper half of the band gap are photostable. +(b) Many-body electronic structure of the photostable charge configuration with a closed shell +(CS) singlet ground state (GS) of 1A1’ symmetry. The triplet state decouples from the singlets +and it does not interfere with the optical emission. + +a +b +Notphotostable +Photostable +CB +X +X +Singlet +Triplet +1Y +3y +Excitation +E +405 nm +00 +PO +Negligible ISC rate +PO +405 nm +0 +VB +1Ai +CS singlet GS +PO:partially occupied +O:occupied, +E: emptyTo evaluate the photostability of defect electronic structures, we need to take valence +band-to-defect and defect-to-conduction band transitions into consideration. As illustrated in +Fig. 4a, the energy of the 405 nm photon is approximately half of the band gap of hBN. +Therefore, partially occupied electronic states appearing in the lower half of the band gap are +not photostable, an electron from the valence band can be excited to the defect state and +change the charge state of the defect. Following the same argument, partially occupied states +in the upper half of the band gap are neither photostable. The electron on the defect state can +be excited to the conduction band and change the state again. +From the reported photostability of the blue emitter, we deduce that the most likely +electronic structure of the responsible defect includes fully occupied defect state(s) in the lower +half of the band gap and empty defect state(s) in the upper half of the band gap, see Fig. 4a. +In this case the many-body ground state of the defect is a singlet that transforms according to +the trivial representation of the corresponding point group. For D3h symmetry the ground state +is 1A1’. The optically excited state is also a singlet with an orbital symmetry “Y” that depends +on the symmetry of the partially occupied single particle defect states in the excited states of +the defect. Next to the singlet excited state there must be a triplet excited state, which can be +obtained by flipping one of the electrons on the partially occupied defect states in the 1Y +excited state configuration, as illustrated in Fig. 4b. Since the orbital symmetry does not +change, the triplet state can be labelled as 3Y. Transition from this state to the ground state is +spin forbidden. Furthermore, since the orbital state of the two excited states is the same, spin- +orbit coupling between the singlet and triplet states is also forbidden in first order. Therefore, +the triplet excited state decouples from the optical processes and does not interfere with the +fast radiative decay of the blue emitter, in accord with lack of strong bunching and slow rates +to/from the metastable state. The electronic structure of split interstitial defects fulfils the +criteria deduced from the photostability of the blue emitter 33. However, to identify the +microscopic structure of the blue emitter comprehensive first principles studies and +comparison of theoretical and experimental results are needed. +In summary, we performed detailed photophysical analysis of the blue SPEs in hBN. +Majority of the emitters exhibit excellent photostability and behave like an ideal two-level +system without significant bunching in the autocorrelation function. The ZPLs of the emitters +is consistently 436 nm and the lifetime of the blue SPEs is ~ 2 ns. Based on our excitation +dynamics, the defects’ ground and excited state are located at the middle of the hBN bandgap, +which is advantageous for stability and inhibition of ionisation. 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G., Charge state dynamics of the nitrogen vacancy center in +diamond under 1064-nm laser excitation Physical Review B 2016, 94, (2). +32. +Shotan, Z.; Jayakumar, H.; Considine, C. R.; Mackoit, M.; Fedder, H.; Wrachtrup, J.; +Alkauskas, A.; Doherty, M. W.; Menon, V. M.; Meriles, C. A., Photoinduced Modification of +Single-Photon Emitters in Hexagonal Boron Nitride ACS Photonics 2016, 3, (12), 2490- +2496. +33. +Zhigulin, I. H., J. Ivady, V. White, S. J. U. Gale, A. Li, C. Lobo, C. J. Toth,M. +Aharonovich, I. Kianinia, M., Stark effect of quantum blue emitters in hBN 2022. + +Acknowledgments +This work is supported by the Australian Research Council (CE200100010, FT220100053) +and the Office of Naval Research Global (N62909-22-1-2028). The authors thank the ANFF +node of UTS for access to facilities. V.I acknowledges support from the National Research, +Development, and Innovation Office of Hungary (NKFIH) (Grant No. FK 145395), the Ministry +of Culture and Innovation and the National Research, Development and Innovation Office +within the Quantum Information National Laboratory of Hungary (Grant No. 2022-2.1.1-NL- +2022-00004), and the Knut and Alice Wallenberg Foundation through WBSQD2 project (Grant +No. 2018.0071). + + diff --git a/NtE3T4oBgHgl3EQfBwky/content/tmp_files/load_file.txt b/NtE3T4oBgHgl3EQfBwky/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ef49f9af6f0c9ddba3247bd237b28a71274d849 --- /dev/null +++ b/NtE3T4oBgHgl3EQfBwky/content/tmp_files/load_file.txt @@ -0,0 +1,943 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf,len=942 +page_content='Photophysics of blue quantum emitters in hexagonal Boron Nitride Ivan Zhigulin1,*, Karin Yamamura1,*, Viktor Ivády2,3,4, Angus Gale1, Jake Horder1, Charlene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Lobo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Mehran Kianinia1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Milos Toth1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' and Igor Aharonovich1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 1School of Mathematical and Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' University of Technology Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Ultimo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' New South Wales 2007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Australia 2Department of Physics of Complex Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' ELTE Eötvös Loránd University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Egyetem tér 1- 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 1053 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Hungary 3MTA–ELTE Lendület "Momentum" NewQubit Research Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Pázmány Péter sétány 1/A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 1117 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Hungary 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Chemistry and Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Linköping University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Linköping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Sweden 5ARC Centre of Excellence for Transformative Meta-Optical Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' University of Technology Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Ultimo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' New South Wales 2007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Australia * These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Corresponding author igor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='aharonovich@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='au Abstract Colour centres in hexagonal boron nitride (hBN) have emerged as intriguing contenders for integrated quantum photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' In this work, we present detailed photophysical analysis of hBN single emitters emitting at the blue spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The emitters are fabricated by different electron beam irradiation and annealing conditions and exhibit narrow-band luminescence centred at 436 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Photon statistics as well as rigorous photodynamics analysis unveils potential level structure of the emitters, which suggests lack of a metastable state, supported by a theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The potential defect can have an electronic structure with fully occupied defect state in the lower half of the hBN band gap and empty defect state in the upper half of the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Overall, our results are important to understand the photophysical properties of the emerging family of blue quantum emitters in hBN as potential sources for scalable quantum photonic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Introduction Single photon emitters (SPEs) are widely acknowledged as key enablers to establish and deploy quantum communication and computing, which involves on-demand generation of high purity single photon emission1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Hexagonal boron nitride (hBN) has gained attention due to its unique properties of wide layer-dependent bandgap centred around 6 eV, high exciton binding energies, presence of optically active spin-defects and capability to host room- temperature (RT) bright SPEs4-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' hBN is also attracting attention for its use as an emerging optoelectronic material for the deep ultraviolet range12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Recently, color centres in hBN emitting at the blue spectral range, termed ‘blue emitters’, were discovered by cathodoluminescence (CL) measurements 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' This group of emitters generally displays ultra-bright, spectrally stable and narrowband emission with a zero phonon line (ZPL) consistently centred around 436 nm 13, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' It was shown that these defects are closely related to the presence of a signature UV emission at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='1 eV 9, 14-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Pre-irradiation of hBN, such as high temperature annealing in a nitrogen atmosphere, results in higher yield of the signature UV emission and thus higher numbers of blue colour centres 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Additionally, at cryogenic temperatures, these defects have stable emission with sub-GHz linewidths and minimal spectral diffusion compared to other quantum emitters in hBN 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Very recently, two- photon interference was demonstrated, opening new opportunities for 2D materials in optical quantum information 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' To date, the crystallographic origin of the blue emitters is still under debate, and likewise, little is known about the photophysical nature of the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' It is thus important to investigate the properties and characteristics of this group of defects to gain insight into their formation mechanisms as well as into their optical properties before they can be fully utilised for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Here, we present detailed photophysical analysis of the blue SPEs in hBN created using focused electron beam irradiations with varying dosages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' We investigate photostability, saturation behaviour, lifetime and transition rates employing time resolved excitations and time correlation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' We use these results to propose the electronic structure of blue SPEs and the position of their energy levels relative to the hBN valence and conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' We examined in detail over ten blue SPEs found in three different hBN flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Although electron beam irradiation conditions for each flake varied, the obtained results (in terms of the ZPL emission wavelength) remained consistent, supporting the hypothesis that the blue quantum emitters have the same defect origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' For clarity, we therefore introduce an emitter labelling scheme (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' flake 1, emitter 1 = F1E1), which will be used throughout this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' We start with analysing the optical properties of emitters, formed by electron irradiation (schematically shown in figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' First, a spectrum of each emitter was acquired and the key parameters - namely emission at saturation, excited state fluorescence lifetime, τ1, and autocorrelation, g(2)(τ), were measured, as shown in Figure 1 for F3E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Spectral characteristics of emitters were measured using a continuous-wave (CW) 405 nm laser at powers ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The photoluminescence (PL) emission of F3E1 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW centred at 436 nm is shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' a b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' unit) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 420 440 460 480 Wavelength (nm) d c 101 Data 125 T1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='01 ns PsAT = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='38 mW unit) 100 Fit INF = 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='30 cps 100 g(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='21 IRF (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 10-1 Fit 75 E 10-2 50 9 0 10-3 20 0 20 25 Delay T (ns) 10-4 0 5 10 15 1 2 3 4 5 Time (ns) Power (mW)Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Spectral characteristics of emitter 1 in flake 3 (F3E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (a) Diagram of hBN lattice and flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (b) Photoluminescence spectra of the emitter excited with a 405 nm CW laser at powers of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW (shown in purple) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW (shown in pink) at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (c) Emission lifetime measurement of the blue emitter excited with a 402 nm pulsed laser at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW and frequency of 30 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' A lifetime τ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='01 ns was deduced from an exponential fit (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Green dots show the instrument response function (IRF) with a corresponding fit that extracts a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='475 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='005 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (d) Power saturation measurement of the same emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Blue line shows the fit of the experimental data (pink dots) using the power dependence model (shown in Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Inset in (d) shows autocorrelation function g(2)(0) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='21 proving the single photon nature of this emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' To study the emission lifetimes, a 402 nm pulsed laser at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW and frequency of 30 MHz was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' All lifetime measurements were conducted at room temperature (RT) under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 1c shows the emission lifetime τ 1= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='151 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='009 ns of F3E1, obtained by fitting the data with a single exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The value is consistent with lifetimes measured for other blue emitters found in another flake, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='003 ns for F2E1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='931 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='003 ns for F2E3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='160 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='017 ns for F2E8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Thus, the lifetime of ~ 2 ns is consistent with other studies of these emitters 13-15, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' To gain the saturation information, PL spectra were acquired at each power for 3 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The maximum value of each spectrum was used as a point in the saturation plot, and the saturation power (PSAT) and the maximum obtainable PL intensity at saturation (IINF) were then extracted from a fit to the data using Equation (1): (1) Figure 1d shows the fluorescence saturation curve of F3E1 with a calculated value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='38 mW for PSAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Measured PSAT values of other blue emitters are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='50 mW for F2E1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='10 mW for F2E4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='40 mW for F2E5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='50 mW for F2E8, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='90 mW for F2E9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Thus, examined emitters have PSAT ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='50 mW to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='90 mW, similar to previously reported for blue emitters in hBN at room temperature, which are slightly above typical saturation values for the visible (~ 2 eV) emitters in hBN 14, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Such relatively high PSAT could be attributed to absorption cross-section of blue emitters from 405 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The inset in Figure 1d shows the autocorrelation function at zero delay time, g(2)(0) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='21 verifying the single photon nature of this emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Next, we focus on the photostability and photon statistics of the blue emitters, to understand their energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Saturation power plays a key role in these measurements, because it is necessary to saturate a system in order to observe an increasing bunching effect in g(2)(τ) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The relatively high saturation power of blue SPEs makes this measurement difficult, as it requires prolonged excitation under high power, which causes bleaching in some of the emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' We observed repeatedly that exciting a single blue emitter for > 4 hours at powers greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW causes the system to go irreversibly into its dark state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 2a shows g(2)(τ) measurements of four blue emitters found in Flake 1 acquired between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='69 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW for over 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' All emitters have a g(2)(0) value below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 indicating that they are SPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Only slight bunching was observed from the emitters above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW, which becomes marginally more noticeable around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Note that at longer delay times the line is flat, indicating that no multiple shelving states are present 19-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' This photophysical behaviour is substantially different from the majority of defect based SPEs in diamond, silicon carbide, or even hBN that often exhibit strong bunching with multiple obvious decay pathways 20, 22, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' These various decay channels (evident by changes in the slope of the g(2)(τ) at longer delay times) are often a signature of a metastable state or recurring blinking behaviour25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' These effects are nearly negligible for the blue emitters that behave as a nearly ideal two-level system, as will also be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Indeed, g(2)(τ) curves with high PSAT above 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW (approximately ~ 25% of emitters) showed no bunching at the measured powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Note, that many of the emitters were eventually bleached during the g(2)(τ) acquisition at excitation powers higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW after approximately one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Photostability and photodynamics analysis of blue SPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (a) Power-dependent l g(2)(τ) measurements at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='69 mW, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='35 mW, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='70 mW and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW on four different emitters on Flake 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (b) Timetraces with 1 second resolution and corresponding histograms of fluorescence intensity for three emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Range of histograms is kept constant at 15 kilo counts/second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The luminescence timetraces (kilo counts/sec) of emitters F1E5, F1E1 and F1E8 at excitation powers of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='69 mW, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='35 mW and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='70 mW are shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Timetraces of each of these emitters are stable over time, without obvious blinking or fluorescence intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The ranges of distribution of the photon counts are within 10% of the mean, for each emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' This is important, as it suggests that the emitters that remain optically active, exhibit excellent photostability over long excitation periods 19, 22, 26, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' a b 110 F1E8 (sdo) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='70 mW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 105 F1E7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW Counts 50 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 F1E8 95 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='70 mW 0 5 10 150 100 200 70 100 F1E1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 (kcps) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='35 mW F1E1 65 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='35 mW (2) ( Counts 50 g 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 F1E8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW 55 0 0 5 10 150 50 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 60 100 F1E5 F1E5 (kcps) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='69 mW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='69 mW 55 Counts 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 50 45 0 10-10 10-7 0 5 10 150 200 400 Delay time (s) Time (min) Occurence (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' unit)To investigate the rates quantitatively, we recorded the autocorrelation measurement as a function of excitation power 19, 20, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The g(2)(τ) function taken at various excitation powers can be fit with a standard bi exponential function using Equation (2): (2) Where τ1, τ2 correspond to lifetimes of antibunching and bunching respectively, while a is a parameter that describes bunching amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 3a shows power-dependent plots of g(2)(τ) measurements of F3E1, with the value of 𝑔(2)(0) remaining below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 at each excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 3b,c show the power- dependent parameters of τ1, τ2 and the bunching factor a obtained from fitting Equation (2) to the g(2)(τ) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Power dependent analysis of energy levels of F3E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (a) Second-order autocorrelation measurements taken at various excitation powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 𝑔(2)(𝜏) traces are normalised and stacked vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (b) Lifetime τ1 of transitions from the excited state to the ground state (green dots) and metastable state lifetime τ2 (purple dots) as a function of laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (c) Pink dots show the bunching factor, a, vs excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' From the fitting of τ1, τ2 and a, we calculated the transition rates according to the three- level rate equation 22, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The transition rates between the three states were calculated by a b 50 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='50 mW (ns) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='4 (ns) 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='2 30 ^ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='50 mW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 20 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW (1)(z) 0 2 4 6 g 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='50 mW c 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='20 unit) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='15 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='00 mw a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='66 mW 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='05 -100 -50 0 50 100 0 2 4 6 Delay time (ns) Power (mw)measuring the power-dependent g2(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The excitation rate from the ground state to the excited state, radiative decay rate from the excited state to the ground state, decay rate from the excited state to the metastable state, and decay rate from the metastable state to the ground state are expressed as k12, k21, k23, and k31, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The τ1, τ2 and a can be derived using Equations (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The value of k12 depends on the power, and k21 and k31 are assumed to be power-dependent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The transition rates were obtained following a combination of Equations (3)-(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (3) (4) (5) (6) (7) (8) (9) Where, A = k12+k21+k23+k31 and B = k12(k23+k31)+k31(k21+k23), P is the excitation power, d and C are the coefficients related to the saturation behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The rate per power coefficient, σ, is determined from fitting τ1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The same procedure was repeated for various emitters, as shown in table 1 below where the calculated transition rates are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Emitter k21 (MHz) k23 (MHz) k310 (MHz) σ (MHz/mW) F3E1 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='7 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='1 F1E12 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='3 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='4 F2E1 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='7 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='7 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 F2E4 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 F2E5 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='7 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' List of the calculated transition rates of k21, k23, k310 and σ of F3E1, F1E12, F2E1, F2E4, and F2E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The calculated transition rates are k21 = 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='6 MHz, k23 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5 MHz, k310 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='7 for the specific emitter F3E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Similar order of magnitude results were also obtained for the other emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The excited state transition rates are an order of magnitude faster than the transitions to/from the metastable state, k23/k310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' These results are consistent with the fact that the blue emitters behave as a nearly ideal two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' We can also comment on the lower limit of the quantum efficiency of the blue emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Note that precise measurement of quantum efficiency is challenging as it requires modification of local dielectric environment28, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Most of the investigated emitters yielded a count rate of ~ 500 kHz, at saturation, with ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='1 ns (~ 450 MHz) fluorescence lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The setup at 440 nm is highly inefficient, mostly driven by the low internal quantum efficiencies of the avalanche photodetectors (~ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Hence, assuming an upper collection efficiency of our confocal setup at 4% in the visible (~ 700 nm) range, the efficiency will drop to below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='5% at the blue spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' This puts a lower bound of the quantum efficiency at ~ 20%, which makes it a bright solid state quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Finally, we discuss the potential level structure and excitation cycle for the blue SPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Short wavelength excitations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' the 405 nm laser used in the experiments, may change the charge state of defects and turn them into a dark state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Such photoionization processes can induce either blinking or loss of the emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The reported robustness of the blue emission under continuous excitation at moderate powers suggests that the bright charge state of the underlying defect structure is stable and no photoionization processes taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Likewise, no further fluorescence lines at longer wavelengths were observed, indicating lack of photochromism 23, 30-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' This observation can be used to narrow down the possible electronic structures giving rise to the blue emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Tentative electronic structure of the blue emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (a) Stable and unstable electronic configuration under 405 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Defects with fully occupied defect states in the lower half of the band gap and empty defect states in the upper half of the band gap are photostable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' (b) Many-body electronic structure of the photostable charge configuration with a closed shell (CS) singlet ground state (GS) of 1A1’ symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The triplet state decouples from the singlets and it does not interfere with the optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' a b Notphotostable Photostable CB X X Singlet Triplet 1Y 3y Excitation E 405 nm 00 PO Negligible ISC rate PO 405 nm 0 VB 1Ai CS singlet GS PO:partially occupied O:occupied, E: emptyTo evaluate the photostability of defect electronic structures, we need to take valence band-to-defect and defect-to-conduction band transitions into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 4a, the energy of the 405 nm photon is approximately half of the band gap of hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Therefore, partially occupied electronic states appearing in the lower half of the band gap are not photostable, an electron from the valence band can be excited to the defect state and change the charge state of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Following the same argument, partially occupied states in the upper half of the band gap are neither photostable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The electron on the defect state can be excited to the conduction band and change the state again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' From the reported photostability of the blue emitter, we deduce that the most likely electronic structure of the responsible defect includes fully occupied defect state(s) in the lower half of the band gap and empty defect state(s) in the upper half of the band gap, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' In this case the many-body ground state of the defect is a singlet that transforms according to the trivial representation of the corresponding point group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' For D3h symmetry the ground state is 1A1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The optically excited state is also a singlet with an orbital symmetry “Y” that depends on the symmetry of the partially occupied single particle defect states in the excited states of the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Next to the singlet excited state there must be a triplet excited state, which can be obtained by flipping one of the electrons on the partially occupied defect states in the 1Y excited state configuration, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Since the orbital symmetry does not change, the triplet state can be labelled as 3Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Transition from this state to the ground state is spin forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Furthermore, since the orbital state of the two excited states is the same, spin- orbit coupling between the singlet and triplet states is also forbidden in first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Therefore, the triplet excited state decouples from the optical processes and does not interfere with the fast radiative decay of the blue emitter, in accord with lack of strong bunching and slow rates to/from the metastable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The electronic structure of split interstitial defects fulfils the criteria deduced from the photostability of the blue emitter 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' However, to identify the microscopic structure of the blue emitter comprehensive first principles studies and comparison of theoretical and experimental results are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' In summary, we performed detailed photophysical analysis of the blue SPEs in hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Majority of the emitters exhibit excellent photostability and behave like an ideal two-level system without significant bunching in the autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The ZPLs of the emitters is consistently 436 nm and the lifetime of the blue SPEs is ~ 2 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Based on our excitation dynamics, the defects’ ground and excited state are located at the middle of the hBN bandgap, which is advantageous for stability and inhibition of ionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Our results provide further insights into emerging class of hBN SPEs, that are promising for scalable quantum applications.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Gale, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Lobo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Toth,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Aharonovich, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Kianinia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=', Stark effect of quantum blue emitters in hBN 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' Acknowledgments This work is supported by the Australian Research Council (CE200100010, FT220100053) and the Office of Naval Research Global (N62909-22-1-2028).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' The authors thank the ANFF node of UTS for access to facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='I acknowledges support from the National Research, Development, and Innovation Office of Hungary (NKFIH) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' FK 145395), the Ministry of Culture and Innovation and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 2022-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='1-NL- 2022-00004), and the Knut and Alice Wallenberg Foundation through WBSQD2 project (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} +page_content='0071).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfBwky/content/2301.04269v1.pdf'} diff --git a/O9FKT4oBgHgl3EQfgC6c/content/tmp_files/2301.11832v1.pdf.txt b/O9FKT4oBgHgl3EQfgC6c/content/tmp_files/2301.11832v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd2e1e5567c14e2dfc1a529b07a09b8e9b7297ee --- /dev/null +++ b/O9FKT4oBgHgl3EQfgC6c/content/tmp_files/2301.11832v1.pdf.txt @@ -0,0 +1,2122 @@ +SOBER: Scalable Batch Bayesian Optimization and +Quadrature using Recombination Constraints +Masaki Adachi, Saad Hamid, +Martin Jørgensen, Michael A. Osborne +Machine Learning Reserach Group, University of Oxford, +{masaki, saad, martinj, mosb}@robots.ox.ac.uk +Satoshi Hayakawa, Harald Oberhauser +Mathematical Institute, University of Oxford, +{hayakawa,oberhauser}@maths.ox.ac.uk +Abstract +Batch Bayesian optimisation (BO) has shown to be a sample-efficient method of performing optimisa- +tion where expensive-to-evaluate objective functions can be queried in parallel. However, current +methods do not scale to large batch sizes – a frequent desideratum in practice (e.g. drug discovery +or simulation-based inference). We present a novel algorithm, SOBER, which permits scalable and +diversified batch BO with arbitrary acquisition functions, arbitrary input spaces (e.g. graph), and +arbitrary kernels. The key to our approach is to reformulate batch selection for BO as a Bayesian +quadrature (BQ) problem, which offers computational advantages. This reformulation is beneficial in +solving BQ tasks reciprocally, which introduces the exploitative functionality of BO to BQ. We show +that SOBER offers substantive performance gains in synthetic and real-world tasks, including drug +discovery and simulation-based inference. +1 +Introduction +True function +Vanilla TS +Decoupled TS +SOBER-TS +SOBER-LFI +1st iteration +2nd iteration +3rd iteration +4th iteration +Figure 1: Stuck behaviour in Thompson sampling (TS): Batch +Bayesian optimisation for the 2D Branin function with four al- +gorithms at the 1st, 2nd, 3rd, and 4th batch acquisitions. The +colours represent each acquisition function (𝜋 for SOBER-LFI, +upper confidence bound (UCB) for the others). The white dots, the +black crosses, and the red star indicate the observed query points, +the 30 batch queries, and the true global maximum. +Bayesian optimisation (BO) is a sample efficient surrogate- +model-based black-box optimiser. BO typically uses a +Gaussian process (GP), whose predictive mean and vari- +ance guide the optimiser where to evaluate next. The next +query location is found Bayesian decision theoretically, +by maximising the acquisition function (AF). As such, +BO translates the black-box optimisation problem into a +sequential “inner optimisation” problem. Flexibility and +superb sample efficiency attract a range of expensive-to- +evaluate applications, such as drug discovery [1, 2], ma- +terials discovery [3], hyperparameter optimisation [4, 5], +and graph optimisation [6, 7]. Batch BO parallelises this +by querying multiple locations at the same time (called +batch acquisition), offering much faster convergence. +Motivation. +Despite its many successes, batch BO has +several challenges. Firstly, batch size scalability: The +extensive overhead of many batch BO methods limits the +batch size to be around 10. Scaling to a larger batch size +is preferable for a variety of real-world problems. For +instance, recent high-throughput screening of microwell +plates for a drug discovery contain 384 compounds in +a batch experiment [16]. +Further, in-Silico materials +discovery can query thousands of simulations in parallel +via computer clusters. +As a second challenge, many +batch BO methods are targeted at continuous inputs, yet +the examples mentioned above of drug discovery and +graphs are inherently discrete. Selecting batch acquisition +samples in discrete space leads to combinatorial explosion +arXiv:2301.11832v1 [cs.LG] 27 Jan 2023 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +Table 1: Comparion with batch BO algorithms. The scalability to large batch size is defined by the computational complexity that is +smaller or equivalent complexity than Thompson sampling (TS). The sparsity means the more diversified samples than random +Monte Carlo (MC) samples. 𝑥, 𝑓 , and ℎ spaces refer to the input, function, and hyperparameter spaces, respectively. The black-box +evidence represents the ability to estimate the Bayesian model evidence of the black-box function over input space 𝑥, equivalent to the +task of Bayesian quadrature (BQ), The sampling complexity shows the upper bounds of 𝑁 discrete candidates. Notation remarks: +𝐶reject is the cost of rejection sampling from determinantal point process (DPP), 𝐶acq is the cost of evaluating the acquisition function +(AF) at a point, 𝐶update is the cost of updating the GP, including the Gram matrix inversion and hyperparameter optimisation, 𝐶func is +the cost of evaluating a sampled function at a point, 𝐼MCMC is the number of MCMC iterations until convergence, 𝑀𝐶𝜑 is the cost of +evaluating 𝑀 components of the kernels approximated by Nyström method or random Fourier feature, 𝑃𝑁 +𝑛 := 𝑛!/(𝑁 − 𝑛)! ≫ 𝑛𝑁 is +the total number of all permutations, 𝑁 is the number of candidate points, 𝑛 is the batch size. The computational costs are in order of +𝐶reject ≫ 𝐶update ≫ 𝐶func ≥ 𝐶acq > 𝑀𝐶𝜑, and the number of samples are in order of 𝑁 > 𝑀 > 𝑛, 𝐼MCMC ≫ 𝑛. +Batch BO methods +large +batch +mixed +input +arbitrary +AF +𝑥-space +sparsity +𝑓 -space +sparsity +ℎ-space +sparsity +blackbox +evidence +sampling +complexity +Hallucination [8] + + + + + + + +𝑛(𝑁𝐶acq + 𝐶update + 𝑁 log 𝑁) +Local penalisation [9] + + + + + + + +𝑛(𝑁𝐶acq + 𝑁 log 𝑁) +TS [10] + + + + + + + +𝑛(𝑁𝐶func + 𝑁 log 𝑁) +Decoupled TS [11] + + + + + + + +𝑛(𝑁𝑀𝐶𝜑 + 𝑁 log 𝑁) +DPP [12] + + + + + + + +𝑛3𝑁𝐶reject +DPP-TS [13] + + + + + + + +𝐼MCMC(𝑁𝐶func + 𝑛3) +MC-SAA [14] + + + + + + + +𝑃𝑁 +𝑛 𝐶acq + 𝑃𝑁 +𝑛 log 𝑃𝑁 +𝑛 +GIBBON [15] + + + + + + + +𝑃𝑁 +𝑛 𝐶acq + 𝑃𝑁 +𝑛 log 𝑃𝑁 +𝑛 +SOBER (Ours) + + + + + + + +𝑁𝑀𝐶𝜑 + 𝑁𝐶acq + 𝑛3 log(𝑁/𝑛) +with increasing the batch size. Lastly, batch diversity: scalable methods, such as TS [10, 17], are too exploitative. The +two leftmost columns in Figure 1 exemplifies typical behaviour – getting stuck in a local minimum and wasting batch +samples in the majority of TS methods. This tendency amplifies in noisy and multimodal cases. The larger the batch +size we query, the larger the regret becomes, as the batch samples could have been used for exploring other regions. +Figure 2: 2D UMAP [18] visualisation of 133,055 +molecules on the QM9 dataset coloured by the dipole +moment. Only 5 molecules are over 10 debye, dis- +played with a 100 times large marker size. +Figure 2 shows drug discovery is a case where the input space is noisy +and multimodal. The number of good molecules (over 10 debye) is +very limited (5 out of 133,055). This needle-in-the-haystack situation +is challenging for finding the bias-variance trade-off via optimising GP +hyperparameters. It leads to having two modes: low-noise and high- +noise in the hyperposterior space. Low-noise mode regards every tiny +change as a peak, whereas high-noise sees every peak as noise. While +the low-noise mode tends to get stuck in local minima, the high-noise +mode can not find the best drug. Such problems are usually solved +by adopting fully Bayesian GP (FBGP), which uses the ensemble of +GP models with hyperparameters sampled from hyperposteriors [19]. +However, FBGP results in significant overhead due to Markov chain +Monte Carlo (MCMC) sampling and an expensive ensemble kernel. +This is particularly challenging for expensive batch BO methods, such +as determinantal point process (DPP) [12]. +Batch Bayesian quadrature (BQ), a surrogate-model-based black-box +integration method akin to BO, suffers from ‘over-exploration’. The +batch BQ commonly applied to simulation-based inference, which +estimates the posterior and marginal likelihood (also called evidence) +of black-box functions (non-closed-form simulators). Simulator-based +inference is utilised in many scientific fields, such as exoplanet detec- +tion [20, 21] and battery model selection [22], both of which utilise +computer clusters to run the simulators embarrassingly in parallel for +speed. Generally, posteriors barely overlap prior distribution since +users have weak knowledge of simulator parameters. Thus, the surrogate model in BQ is expected to approximate only +the vicinity of posterior mode (maximum a posteriori (MAP)), whereas existing BO is designed to explore all over the +prior distribution. +Contribution. +We propose the hallucination-free batch acquisition method: solving optimisation as Bayesian +estimation via recombination (SOBER), a scalable batch BO method with supporting light-weight FBGP. The key is +reformulating BO as BQ, which gives a scalable and flexible approach. The two rightmost columns of Figure 1 illustrate +2 + +global +maximum +8 10 12 14 +0 +2 +dipole moment [D]SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +two variants of SOBER. They mitigate the stuck-in-local-minima issue via diversified sampling. Furthermore, SOBER +can solve batch BQ tasks by introducing the exploitive funcitionaly of BO into BQ for faster convergence. Table 1 +summarises a comparison with the other batch BO methods. With the given features, we particularly focus on the +following two real-world applications; drug discovery (large-scale discrete batch BO with graph/string input, batch BO), +and simulation-based inference (simultaneous estimation of both posterior and evidence, batch BQ). Empirically, SOBER +shows better sample-efficiency as well as faster wall-clock time computation than the baseline methods in both batch BO +and BQ. We emphasise that SOBER is an extension of BO and BQ methods and compatible with many existing methods. +2 +Background +2.1 +Batch Bayesian Optimisation +BO is a heuristic for solving the black-box global optimisation problems defined by: +𝑥∗ +true = arg max +𝑥 +𝑓true(𝑥), +(1) +where 𝑥∗ +true is the true global maximum of the true black-box function 𝑓true. The underlying assumptions here are: +1. 𝑓true is a black-box function, and we do not know the function information except for querying the values at +certain locations (oracles). +2. The queries are expensive, and we wish to minimise the number of queries for fast convergence. +3. We can query multiple locations at the same time without additional overhead, and a larger batch size is +desirable for faster convergence in wall-clock time. +4. The total overhead of the batch acquisition algorithm should be negligible over the sequential querying cost. +For sample-efficient global optimisation, BO utilities a surrogate GP model 𝑓 of 𝑓true, specified as: +𝑃(𝑦|D) = GP +� +𝑦; 𝑚(·), 𝐶(·, ·) + 𝜎2 +𝑛I +� +, +(2a) +𝑚(𝑥) = 𝐾(𝑥, Xob)𝐾(Xob, Xob)−1yob, +(2b) +𝐶(𝑥, 𝑥′) = 𝐾(𝑥, 𝑥′) − 𝐾(𝑥, Xob)𝐾(Xob, Xob)−1𝐾(Xob, 𝑥′), +(2c) +where D= (Xob, yob) is the observed dataset, yob := 𝑓true(Xob) are the oracles queried in parallel, 𝑚(·) and 𝐶(·, ·) are the +mean and covariance of the predictive posterior, 𝐾(·, ·|𝜃) is the kernel, 𝜃 is the kernel hyperparameters, and 𝜎2 +𝑛 is the +homoskedastic noise variance of observations. +Queried observations D serially update the GP surrogate model 𝑓 so it can predict the output of 𝑓true more accurately. +𝑚 and 𝐶 guide our beliefs toward the region where the true global maximum 𝑥∗ +true possibly locates. Such a guiding +mechanism is obtained through maximising an AF. +2.2 +Batch Bayesian Quadrature +BQ is a heuristic for evaluating integrals given by: +𝑍true = +∫ +𝑓true(𝑥)d𝜋(𝑥), +(3) +where 𝑓true is the black-box function we wish to integrate against a known probability measure 𝜋. The difference is the +objective being integration, not global optimisation. The integration problem is widely recognised in statistical learning: +expectations, variances, marginalisation, ensembles, Bayesian model selection, and Bayesian model averaging. BQ is, +like BO, solved by GP-surrogate-model-based active learning. The batch acquisition methods are also shared with batch +BO. The methodological differences are: +1. BQ typically assumes a specific kernel to make the integration analytical (e.g. RBF kernel). +2. While BO requires to approximate the black-box function only in the vicinity of the global optimum, BQ needs +to approximate the whole region of interest defined by the probability measure 𝜋. +Thus, BQ is a purely explorative algorithm, and the uncertainty sampling AF is often applied. BQ, being sample-efficient, +is the heuristic of choice if querying 𝑓true is expensive, making other approaches like Monte Carlo troublesome. +2.3 +Bayesian Alternately Subsampled Quadrature (BASQ) +[23] proposed the scalable batch BQ method called Bayesian alternately subsampled quadrature (BASQ). This achieves +fast batch acquisition. It applies to arbitrary kernels and input spaces, e.g. discrete and non-Euclidean spaces. This was +3 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +enabled by a quadrature method called random convex hull quadrature (RCHQ) [24]. RCHQ is a kernel quadrature +(KQ) method that approximates an integral of integrands in reproducing kernel Hilbert space (RKHS) by a weighted +sum of a sample: +∫ +ℎ(𝑥)d𝜋(𝑥) ≈ w⊤ +batchℎ(Xbatch), +(4) +where ℎ is an element of the RKHS H generated by kernel 𝐾. wbatch and Xbatch ∈ R𝑛×𝑑 are weights and batch samples +selected by RCHQ. 𝑛 is batch size, 𝑑 is input dimension. +BASQ reformulated the batch acquisition problem as an “inner integration”. RCHQ is used to choose Xbatch, which +can minimise the worst-case integration error of (4). They are chosen based on the RKHS generated by the kernel +that is the posterior covariance (2c). Such a batch minimisation of integration error is equivalent to posterior variance +minimisation in batch BQ [25]. Hence, RCHQ is utilised as a batch BQ sampler by passing GP predictive covariance 𝐶 +to the kernel 𝐾 in RCHQ alternately. +RCHQ is a subsample-based KQ method, which extracts a weighted 𝑛-point batch sample (wbatch, Xbatch) from 𝑁 (≫ 𝑛) +candidate sample points Xrec with weights wrec. The candidate points are drawn just from the prior 𝜋 or by importance +sampling via another proposal distribution. Such a sample forms an empirical measure 𝜋emp = (wrec, Xrec), which +gives the same integral as the original probability measure 𝜋 in expectation, however, is supported on a tractable number +of points. Based on an 𝑀-point random sample Xnys from 𝜋, we first prepare 𝑛 − 1 test functions 𝜑𝑖(𝑥) := 𝑢⊤ +𝑖 𝐾(Xnys, 𝑥) +following the Nyström method, where 𝑢𝑖 ∈ R𝑀 is the 𝑖-th eigenvector of 𝐾(Xnys, Xnys) and we assume 𝑀 ≥ 𝑛. Next, +recombination [26, 27] extracts a weighted set of 𝑛 points (wbatch, Xbatch) from the candidate points Xrec, so that the +integrals of test functions 𝜑1, . . . , 𝜑𝑛−1 match the integrals against the measure 𝜋emp. This process allows to incorporate +the information of spectral decay of the kernel 𝐾 = 𝐶, accelerating the convergence rate of integration. +With the weights wbatch further updated by GP and quadrature samples Xbatch, we can approximate the mean and variance +of the marginal likelihood of 𝑓 over 𝜋, given by: +E +𝑥∈𝜋[𝑚(𝑥)] ≈ w⊤ +batch𝑚(Xbatch), +(5a) +Var +𝑥∈𝜋[𝑚(𝑥)] ≈ w⊤ +batch𝐶(Xbatch, Xbatch)wbatch +− 2w⊤ +batch𝐶(Xbatch, Xrec)wrec ++ w⊤ +rec𝐶(Xrec, Xrec)wrec. +(5b) +Here, 𝑍 := E𝜋 [𝑚(𝑥)] is an estimate of 𝑍true in (3), and Var𝜋 [𝑚(𝑥)] is the uncertainty of this integration approximation. +The more accurately 𝑓 approximates 𝑓true, the smaller approximation error between 𝑍 and 𝑍true. The empirical +performance of RCHQ is on par with DPP while drastically improving computational cost [24]. +3 +Proposed Method: SOBER +In this section, we present SOBER, which reformulates the batch BO task as the parallelisable batch BQ one, and +introduces the efficient solver for the dual problem. +3.1 +Global Optimisation as Bayesian Quadrature +3.1.1 +Duality in Probability measure +We reframe batch BO as a batch BQ problem. Consider the following dual formulation to Eq. (1) [28]: +𝛿𝑥∗ +true ∈ arg max +𝜋 +∫ +𝑓true(𝑥)d𝜋(𝑥), +(6) +where 𝛿𝑥 is the delta distribution at 𝑥, 𝜋 are probability distributions over the feasible region, and 𝑥∗ +true is the location of the +global maximum of 𝑓true. This black-box integral formulation is identical to BQ, so we can solve it as batch BQ with the +scalable BASQ. The difference to BQ is that 𝜋 is to be updated in each iteration. The more accurate the GP surrogate be- +comes by obtaining more observed points, the “narrower” 𝜋 becomes. Ideally, 𝜋 can eventually reach the global maximum +𝛿𝑥∗ +true in a single global maximum case. As such, solving (6) by updating 𝜋 is equivalent to finding the maximum 𝑥∗ +true. +4 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +3.1.2 +Batch Selection as Kernel Quadrature +The batch selection in SOBER is performed by RCHQ. We propose the ‘objective-RCHQ’, which adds the constraint of +maximising an AF to RCHQ, solving the following problem: +Find an 𝑛-point subset +Xbatch ⊂ Xrec, +(7a) +s.t. +max +w⊤ +batch𝛼(Xbatch) +(7b) +w⊤ +batch𝜑𝑖(Xbatch) = w⊤ +rec𝜑𝑖(Xrec), 1 ≤ 𝑖 < 𝑛, +(7c) +w⊤ +batch1 = w⊤ +rec1 = 1, +wbatch ≥ 0, +(7d) +where 𝛼 is an arbitrary AF, Xrec ∈ R𝑁 ×𝑑 is a sample drawn from 𝜋, and 𝜑𝑖 are test functions determined by another +sample Xnys ∈ R𝑀×𝑑 drawn from 𝜋 (see Section 2.3). Eqs. (7b)-(7d) are the constraints of AF maximisation, batch +uncertainty minimisation (quadrature rules), and weight positivity, respectively. These constraints are vital for batch +diversity [24]. There exist theoretical guarantees for specific choices of AFs depending on the spectral decay of an +integral operator determined by the pair (𝐾, 𝜋) [23, 29]. Empirically, the convergence as 𝑛 increase is fast even without +the AF term. Thus, we can utilize this degree of freedom for incorporating other AFs as in Section 3.4. +3.2 +Approximating the Distribution 𝜋 +To obtain good candidate points Xbatch, it is essential to estimate 𝜋 from the surrogate model 𝑓 . However, as we do not +know the true global maximum 𝑥∗ +true a priori, evaluating the probability of global maximum 𝑃(𝑥∗ +true|D) is unachievable. +Hence, we must approximate 𝜋 with a probability distribution that asymptotically assimilates to 𝑃(𝑥∗ +true|D) when +collecting more observations. +We discuss two variants; TS and likelihood-free inference (LFI), addressing the approximation of 𝜋 differently. +SOBER-TS adopts the current maximum 𝑃(𝑥∗|D) as 𝜋. This definition is well known as TS. We prepare 𝑁 candidates +with parallel decoupled TS [11], which alleviates the sampling overhead and fosters batch diversity via sparsifying +the sampled functions from GP posterior ( 𝑓 -space sparsity). As such, SOBER-TS can be understood as re-selecting +the TS samples that can maximise the sum of the AF and minimise the variance in predictive posterior. However, +TS can not provide the closed-form 𝑃(𝑥∗|D) distribution. SOBER-LFI offers the faster alternative and closed-form +𝜋 based on LFI approach, which is delineated in the next section. +3.3 +Temporary Likelihood by Summary Statistic +Gutmann et al. [30] introduced the “tentative” likelihood 𝑃( 𝑓 (𝑥) > 𝜂|𝑥, D) based on the current global max-value +𝜂 := max 𝑓 (𝑥|D), given by: +𝐿(𝑥|𝜃, 𝜎2 +𝑛, 𝜂) := 𝑃( 𝑓 (𝑥|D) ≥ 𝜂) ∝ Φ +� +𝑚(𝑥|𝜃) − 𝜂 +√︁ +𝐶(𝑥, 𝑥|𝜃) + 𝜎2𝑛 +� +, +(8) +where Φ is cumulative density function (CDF) of normal distribution. Noise-free formulation of this likelihood definition +is the same as the probability of improvement acquisition function [31]. When 𝜂 = 𝑓true(𝑥∗ +true) = 𝑓 (𝑥∗ +true) is satisfied, +𝑃( 𝑓 (𝑥) > 𝜂) becomes the delta function 𝛿𝑥∗ +true. +Now we can estimate posterior belief of the maximum location 𝜋 via the following Bayes’ rule: +𝜋(𝑥) ∝ 𝐿(𝑥)𝜋′(𝑥). +(9) +For each iteration, 𝐿(·|𝜃, 𝜎2 +𝑛, 𝜂) is updated and 𝜋′ is 𝜋 of the previous iteration. In the beginning, 𝜋′ is the prior belief. +In the typical BO setting, we use the bounded uniform prior, but we can incorporate the experts’ knowledge via stronger +prior. We do not examine the difference of prior 𝜋′ in this paper, but the following papers show its strong empirical +performance [32, 33]. +3.4 +Fast Marginalisation by Quadrature Distillation +While BO works well with a single set of optimised hyperparameters (type-II maximum likelihood estimation (MLE)) +on most functions, some noisy cases such as drug discovery shown in Figure 2 requires FBGP modelling. GP +marginalisation is typically done with MC integration via sampling from hyperposterior Π. However, its convergence +rate O(1/√𝑛) is poor, requiring thousands of ensemble GP models for marginalisation. This significantly slows down +batch BO computation as BASQ requires such an ensemble kernel. Hence, we introduce quadrature distillation trick for +faster marginalisation by using the smaller set of weighted hypersamples selected by the quadrature, distilling random +5 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +hypersamples from hyperprior. This method is based on BQ marginalisation in hyperparameter space with the hyper-GP +and hyperlikelihood [34]. We modified this method by sparsifying the BQ weights with a smaller set of positive weights +𝑤QD via RCHQ for faster computation, resulting the following approximation of marginalisation. +𝐿(𝑥) := +∫ +𝐿(𝑥|Θ)dΠ(Θ) ≈ wQD𝐿(𝑥|𝚯QD), +(10) +where 𝑤QD ∈ R𝐻 are the distilled weights. This new algorithm, quadrature distillation, permits fast marginalisation of like- +lihood as well as acquisition function without time-consuming MCMC sampling. This is particularly helpful for expensive +AF, such as information-theoretic AFs [35, 36, 37, 38] and FBGP-based AFs [39, 40, 19]. However, this FBGP modelling +is an additional option of SOBER, of which additional overhead limits the suitable cases. See Appendix A for details. +3.5 +Batch Bayesian Quadrature by Dual GPs +As LFI likelihood is originally introduced to solve simulation-based inference, now our SOBER-LFI is also capable +of solving simulation-based inference. The main difference between BASQ and SOBER is whether or not 𝜋 is updated. +Hence, SOBER-LFI can solve a batch BQ task by inheriting the BASQ modelling. Sampling from 𝜋 efficiently squeezes +the region to be explored only the vicinity of MAP, which is the global maximisation problem as BO. When compared +to the original BO-based LFI [30], SOBER has two benefits; evidence estimation and exact posterior estimation. +While BOLFI is designed to approximate only the posterior distribution using the "approximated" likelihood definition, +SOBER can estimate both the posterior and model evidence in one go, using the exact likelihood definition based on +BASQ. We place dual GPs; one for sampling with BO-LFI, one for BQ modelling with BASQ. Details in Appendix C. +3.6 +Summary of Contribution +In summary, we reformulated the batch BO task as the dual problem defined by Eq. (6). Now, estimating the global +maximum becomes equivalent to updating 𝜋. We introduced two variants: TS and LFI. Both offer approximation +of 𝜋 using the information from the current surrogate model 𝑓 in different ways. Once the empirical measure +𝜋emp = (wrec, Xrec) is constructed by sampling from 𝜋, the ‘objective-RCHQ’ selects the batch samples that minimise +the GP posterior variance and maximise the user-defined AF. Finally, quadrature distillation provides efficient FBGP +modelling, and the BASQ formulation solves batch BQ tasks. +4 +Algorithm +4.1 +Sampling from 𝜋 +Table 2: SOBER algorithm. +Algorithm 1: SOBER +Input: prior 𝜋′, hyperprior Π′(Θ), +observed dataset D = (Xob, yob) +Output: maximum max[Xob], evidence E[𝑚(𝑥)] +1: 𝑓 ← InitialiseGP(D) +2: while convergence: +3: +if FGBP: +4: +wQD, 𝚯QD ← QuadDistil( 𝑓 , Π′(Θ)) +5: +𝜋, 𝛼, 𝐾(·, ·) ← FBGP( 𝑓 , 𝜋′, wQD, 𝚯QD) +6: +else: +7: +𝜋, 𝛼, 𝐾(·, ·) ← Type-II MLE( 𝑓 ) +8: +wrec, Xrec, Xnys ∼ Subsampling(𝜋, 𝜋′) +9: +Xbatch, wbatch ← AutoKQ(wrec, Xrec, Xnys, 𝛼, 𝐾(·, ·)) +10: +ybatch = ParallelQuery( 𝑓true(Xbatch)) +11: +D ← D ∩ Dbatch +12: +𝑓 ← UpdateGP( 𝑓 , D) +13: +𝜋′ ← 𝜋 +14: +E[𝑚(𝑥)], Var[𝑚(𝑥)] ← KQ( 𝑓 , Xbatch, wbatch) +15: return max[Xob], E[𝑚(𝑥)] +SOBER is a sample-based gradient-free approach, and +so can handle discrete, continuous or mixed inputs. +The only difference is the sampler for Xrec. +The simplest scenario is if all discrete candidates +are available a priori and enumerable. As RCHQ +accepts weighted samples 𝜋emp = (wrec, Xrec) for im- +portance sampling, all we have to do is to calculate +the weights wrec. This is simply the normalised pos- +terior 𝜋(Xrec)/ +� +𝜋(Xrec) · 1 +�. If all combinations are +innumerable or unavailable, we sample Xrec from the +discrete prior 𝜋′, which the user can define the arbi- +trarily. Once sampled, the procedure is the same: we +compute wrec, then pass the empirical measure 𝜋emp +to RCHQ. We update the hyperparameters of the prior +𝜋′ via MLE from the weighted sample (wrec, Xrec). +Continuous space can be regarded as innumerable +discrete space, so it can be handled similarly. The +only difference is the prior update. We use weighted +kernel density estimation (KDE) for the update, for +speed and flexibility. +Mixed space is the combination of discrete and con- +tinuous space, which also can be regarded as innumerable discrete space. The prior update is the combination of the +6 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +above two by assuming the discrete and continuous parameters are independent. Importantly, the prior does not need to +precisely approximate 𝜋 as the importance weights wrec will correct the difference. See Appendix B.1 for each detail. +4.2 +Automatic Kernel Quadrature Selection +Batch selection in Section 3.1.2 is performed by RCHQ. However, the sharper 𝜋 becomes (as it inevitably does over +iterations), the slower the RCHQ convergence rate becomes, due to long-tail eigenvalue decay of the kernel matrix. To +avoid this, we propose automatic KQ selection. We automatically switch KQ methods when RCHQ becomes inefficient. +The alternative KQ method is kernel thinning, which is an eigenvalue-decay-independent and subsample-based KQ +method [41]. The choice between these two KQ methods can be made automatically by comparing the worst-case error in +Eq. (5b). The third term in Eq. (5b) is not dependent on the KQ methods, so we can avoid expensive 𝑁 × 𝑁 computations. +RCHQ is selected in the early stage because the smooth kernel makes the eigenvalue decay short-tailed. In the late stage, the +kernel thinning is chosen when the region is narrowed (see Appendix B.2). Table 2 illustrates pseudo-code for SOBER.1 +5 +Related work +Batch Bayesian Optimisation +Batch BO methods are summarised in Table 1. The existing scalable batch BO methods +are all TS-based. Moreover, none of the baseline methods can offer ℎ-space sparsity for fast FBGP and blackbox +evidence estimation (as with the BQ task) when applied to the Bayesian inference task. As for mixed space BOs, existing +works [6, 42] propose ways to avoid the combinatorial explosion, but none of them offer scalable batch acquisition. +Batch Bayesian optimisation for drug discovery +BO for drug discovery can be classified into two categories: +variational autoencoder (VAE) and discrete BO. The former embeds molecules into continuous low dimensional features +using a VAE, performs BO in the latent space, decoding the queried molecules back into the original high dimensional +representation [1, 43]. The latter directly models the discrete spaces with bespoke kernels [44, 45]. Our focus in this +paper is on the batching method and is model agnostic, so we do not compare model variants. +Batch Bayesian Quadrature +BQ has focused on model evidence estimation; few works support posterior inference +simultaneously. The first to study simultaneous estimation of the posterior distribution and evidence was VBMC +[46, 47], based on variational inference. Yet, batch VMBC has not been reported. Batch WSABI [48] was the first +to extend BQ to a batch setting, using local penalisation [9]. Later, BASQ achieved efficient parallelization with +kernel recombination and currently is the only model for scalable batching [23]. However, the original BASQ is +over-explorative, requiring an accurate prior. Log-BASQ [22] mitigates this by introducing the log-warped GP surrogate +model. SOBER balances exploration and exploitation by updating 𝜋. There exists many works on simulation-based +inference, most focus only on posterior inference. BQ and nested sampling (NS) [49] are the only methods for +simultaneous inference of posterior and evidence. NS is a MCMC-based sampler, so it is not sample-efficient and +its state-of-the-art method showed the slower convergence rate than BQ methods [23, 50]. +6 +Experiments +Table 3: Comparison of experimental conditions for batch BO benchmarks. +synthetic functions +real-world datasets +Ackley [42] +Rosenbrock [42] +anti-malaria [51] +polar solvent [52] +total dim 𝑑 +23 +7 +2048 +2048 +continuous dim +3 +1 +- +- +categorical dim +- +6 +- +- +binary dim +20 +- +2,048 +2,048 +countable data +- +- +20,746 +133,055 +batch size 𝑛 +200 +100 +100 +200 +kernel +RBF +RBF +Tanimoto +Tanimoto +prior +mixed +mixed +discrete +discrete +continuous prior +U{−1, 1} +U{0, 10} +- +- +discrete prior +Ber(0.5) +Cat(0.5) +Cat(0.5) +Cat(0.5) +1Code is open sourced at https://github.com/anonymous. +7 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +Ackley +Rosenbrock +Anti-Malarial drug +Polar solvent for batteries +Log10 regret +Log10 regret +Log10 min +−Log10 max +Log10 overhead [s] +random +15 +30 +0 +15 +30 +0 +15 +30 +0 +15 +30 +0 +15 +30 +0 +15 +30 +0 +15 +30 +0 +15 +30 +TS +decoupled TS +DPP-TS +SOBER-TS +SOBER-LFI +Iteration +Iteration +Iteration +Iteration +0 +-2 +3 +-1 +-3 +-0.9 +-1.1 +0 +3 +-3 +0 +3 +-3 +1 +2 +Log10 overhead [s] +Log10 overhead [s] +Log10 overhead [s] +1 +2 +3 +0 +1 +f(x) +f(x) +Figure 3: We evaluate SOBER across 2 synthetic functions and 2 real-world drug discovery datasets. Top: Log regret or log best +observations, Bottom: Log overhead in seconds as function of iterations. Lines and shaded area denote mean ± 1 standard error. The +batch size is 100 or 200 (see Table 3). SOBER-LFI consistently outperforms all four baseline methods. +We now test the sample efficiency and sampling overhead of SOBER against six baseline methods. Our code is built +upon PyTorch-based libraries [53, 54, 14, 2]. All experiments were averaged over 16 repeats, computed in parallel with +multicore CPUs2 for a fair comparison, although GPUs can accelerate the SOBER due to its highly parallelisable nature +and the GPyTorch library. +6.1 +Batch Bayesian Optimisation +For large-scale batch BO methods, we compared with TS, decoupled TS, and DPP-TS based on Table 1. The test +datasets are two synthetic datasets and two real-world datasets shown in Table 3. Synthetic functions are inherited from +the previous work [42], with modifications. Small molecules in the two real-world drug discovery datasets are described +as SMILES [55], which are variable length strings. We transform SMILES to fingerprints [56], which are sparse (2048 +dimensional) bit vectors, and use a Tanimoto kernel [57]. 20,000 candidates (𝑁 = 20, 000) are drawn from the prior +distribution (See details in Appendix D.1). +6.2 +Batch Bayesian Quadrature +Table 4: Comparison of experimental condi- +tions for batch BQ. +2 RC-pair +model +5 RC-pair +model +dim 𝑑 +6 +12 +batch size 𝑛 +100 +100 +kernel +RBF +RBF +prior +Gaussian +Gaussian +We target simultaneous estimation of posterior and model evidence in +simulation-based inference. We compare with batch WSABI, BASQ, and +log-BASQ. We select a Gaussian prior and kernel for a fair comparison +with batch WSABI. Note that SOBER can take arbitrary prior and kernel. +We use two real-world datasets from the lithium-ion battery model selection +[22]. The problem is modified to more challenging conditions in order to +examine the BQ methods. The prior variance is set to be one million times +as large as the true posterior’s (which is known from pre-experiments with +exhaustive MCMC sampling.) See Appendix D.2 for full experimental +details. +6.3 +Results +Figure 3 shows that SOBER-LFI is the top-performing of both synthetic and real-world drug discovery tasks while +maintaining its overhead to be equal to or lower than TS. On Ackley, which has many local maxima, TS-based methods +get stuck in the local maxima, yet SOBER successfully finds the global maxima. Even on the uni-modal Rosenbrock +function, which should be advantageous for hill-climbing algorithms like TS, SOBER outperforms too. This clearly +shows that updating 𝜋 can efficiently squeeze the sampling region around the global maximum. A similar tendency +is found in drug discovery tasks. Particularly, the polar solvent dataset clearly exemplifies the stuck behaviours. TS +2Performed on MacBook Pro 2019, 2.4 GHz 8-Core Intel Core i9, 64 GB 2667 MHz DDR4 +8 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +converges fast in the early stage, but it can not get out of the local maxima, resulting in a final regret equivalent to +random search. Still, SOBER successfully finds better molecules. +2 RC pair +5 RC pair +Log10 |log evidence | +Log10 RMSE of posterior +batchWSABI +10 +20 +0 +10 +20 +0 +10 +20 +0 +10 +20 +BASQ +logBASQ +SOBER-LFI +Iteration +Iteration +3 +1 +5 +3 +1 +0 +2 +3 +1 +2 +Log10 |log evidence | +Log10 RMSE of posterior +5 +3 +1 +Figure 4: +We evaluate SOBER across 2 real-world battery +simulation-based inference tasks. +Top: Log of log evidence. +Bottom: Log RMSE of posterior as function of iterations. Lines +and shaded area denote mean ± 1 std. error. Batch size is 100. +SOBER-LFI consistently outperforms all three baselines. +Figure 4 illustrates that SOBER also outperformed the +BQ baseline methods. When performing inference with a +weakly-informative prior, finding the posterior mode and +reducing the variance only around its vicinity is the key to +fast convergence. While the original BASQ over-explores +the prior distribution and shows plateaus, logBASQ allevi- +ates this behaviour via log-warp modelling. Nonetheless, +SOBER showed significantly faster convergence than all +competitors in both posterior and evidence inference of +all tasks by squeezing 𝜋 toward the posterior mode. +We used type-II MLE optimised LFI AF throughout the +experiments. Appendix D.3 further illustrates the effect of +AF, batch size 𝑛, and hyperparameters (𝑁, 𝑀). Amongst +the AFs, information-theoretic AFs (MES and GIBBON) +can boost the convergence rate than the LFI AF with a +negligible overhead increase. The hyperparameters 𝑁, 𝑀 +are quite intuitive: the discretisation accuracy of the input +𝑥 and function 𝑓 spaces, respectively. Unsurprisingly, the +larger these values become, the faster the convergence +becomes but the larger the overhead is. Our default values +(𝑁 = 20, 000, 𝑀 = 500) are competitive throughout the +experiments. These values can be adjusted to the cost of +queries [23] Moreover, the larger the batch size 𝑛 becomes, +the faster the convergence, even for large batch sizes3. +The ablation study shows that each component (temporary likelihood 𝜋, the iterative 𝜋 update, and the objective-RCHQ) +contributes to faster convergence. Additionally, FBGP with quadrature distillation is shown to be capable of convergence +acceleration, especially in noisy functions, while maintaining the overhead is competitive enough to the baseline +methods. +7 +Discussion +We introduced a hallucination-free approach, SOBER, capable of scalable batch acquisition for both BO and BQ. We +identified three problems of existing batch BO methods: batch diversity, batch size scalability, and combinatorial +explosion in innumerable discrete/mixed inputs. The batch BQ reformulation can make the batch selection more +diversified and parallelisable, the objective RCHQ offers a scalable batch selection solver, and updating 𝜋 can avoid the +combinatorial explosion via squeezing the region to be sampled. Updating 𝜋 also solves the overexploration issue in +batch BQ, resulting in faster convergence on both synthetic and real-world datasets, when compared against existing +batch BO and BQ approaches. +Two limitations of SOBER are that it is not suitable for asynchronous settings and the algorithm cannot be distributed to +each node in computer cluster such as in [10]. Applicability to high-dimensional BO is also an open problem as efficient +sampling from the posterior over the maximiser is difficult. This can be true also for low-dimensional embeddings, such +as those produced by the GPLVM model [58]. However, for some embeddings, such as linear embeddings or VAEs, this +sampling can be done efficiently. +Acknowledgments +We thank Leo Klarner for the insightfuk discussion of Bayesian optimistaion for drug discovery, Samuel Daulton, Binxin +Ru, and Xingchen Wan for the insightful discussion of Bayesian optimisation for graph and mixed space, Ondrej Bajgar +for his helpful comments about improving the paper. Masaki Adachi was supported by the Clarendon Fund, the Oxford +Kobe Scholarship, the Watanabe Foundation, and Toyota Motor Corporation. Harald Oberhauser was supported by +the DataSig Program [EP/S026347/1] and the Hong Kong Innovation and Technology Commission (InnoHK Project +CIMDA). 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In Proceedings of the thirteenth +international conference on artificial intelligence and statistics, pages 541–548. JMLR Workshop and Conference +Proceedings, 2010. +12 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +A +Fast GP Marginalisation +A.1 +Overview +A.1.1 +Sparsifying Hyperposterior Samples by Bayesian Quadrature +First, we apply BQ for sparsifying the hyperposterior samples using the following discrete approximation of the +hyperposterior in the closed form: +𝑃(Θ|D) = 𝑃(D|Θ)𝑃(Θ) +𝑃(D) +, +≈ +𝑚hyper(Θ)Π′(Θ) +∫ +𝑚hyper(Θ)dΠ′(Θ) +, += +𝑚hyper(Θ)Π′(Θ) +�∫ +𝐾hyper(Θ, 𝚯)𝐾hyper(𝚯, 𝚯)−1dΠ′(Θ) +� +𝑃(D|𝚯) +, += +𝑚hyper(Θ)Π′(Θ) +w′⊤ +BQL +, +(11) +where +𝑃(D|Θ) ∼ GP(ℓ; 𝑚hyper(·), 𝐶hyper(·, ·)), +(12a) +w′ +BQ := +∫ +𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1dΠ′(Θ), +(12b) +L := 𝑃(D|𝚯hyper). +(12c) +Here, we place hyper-GP on the marginal likelihood L(·) defined by Eq. (16). We draw hypersamples 𝚯hyper ∈ R𝐵×𝐷, +where 𝐵 is the number of hypersamples and 𝐷 is the number of hyperparameter types, from the hyperprior Π′(·) := 𝑃(Θ). +Then, we evaluate the marginal likelihood L = L(𝚯hyper) in parallel. We select multivariate normal distribution for +hyperprior Π′(Θ) := N (Θ; 𝜇hyper, 𝚺hyper) based on [59], and Gaussian kernel for hyper-GP. Then, weights w′ +BQ in Eq. +(12b) become analytical: +w′ +BQ = +∫ +𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1dΠ′(Θ), += 𝑣 +√︁ +|2𝜋W| +�∫ +N (Θ; 𝚯hyper, W)N (Θ; 𝜇hyper, 𝚺hyper)dΘ +� +𝐾hyper(𝚯hyper, 𝚯hyper)−1, += 𝑣 +√︁ +|2𝜋W|N (𝚯hyper; 𝜇hyper, W + 𝚺hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1, +(13) +(14) +A.2 +Parabolic Transform for Max-value Estimation +Many acquisition functions, including the temporary likelihood 𝐿(·|𝜃, 𝜎2 +𝑛, 𝜂), are dependent on the quadrature hyperpa- +rameters Θ = (𝜃, 𝜎2 +𝑛, 𝜂). Particularly, estimating the current maximum location conditioned on 𝜃 is computationally +challenging. Hence, we inherit the parabolic transform of GP surrogate model from [60]: +𝑓 (𝑥|Θ) = 𝜂 − 1 +2𝑔(𝑥)2, +(15a) +:= GP� 𝑓 (𝑥); 𝑚(𝑥|𝜃, 𝜂), 𝐶(𝑥, 𝑥|𝜃) + 𝜎2 +𝑛I�, +(15b) +𝑔(𝑥|𝜃) := GP�𝑔(·); 𝑚𝑔(·|𝜃), 𝐶𝑔(·, ·|𝜃)�, +(15c) +𝑚(𝑥|𝜃, 𝜂) := 𝜂 − 1 +2 +� +𝑚𝑔(𝑥)2 + 𝐶𝑔(𝑥, 𝑥) +� +, +(15d) +𝐶(𝑥, 𝑥′|𝜃) := 1 +2𝐶𝑔(𝑥, 𝑥′)2 + 𝑚𝑔(𝑥)⊤𝐶𝑔(𝑥, 𝑥′)𝑚𝑔(𝑥′), +(15e) +where 𝑓 (·) is the surrogate model that approximates 𝑓true(·), 𝑔(·) is the square-root warped GP [61] of 𝑓 (·). The +predictive mean 𝑚𝑔(·) and covariance 𝐶𝑔(·, ·) of the warped GP 𝑔(·) are expressed with normal GPs in Eqs. (2b) - (2c). +The predictive mean 𝑚(·) and covariance 𝐶(·, ·) are approximated via moment-matching [50]. D𝑔 = (Xob, y𝑔,ob) is the +observed data for the warped GP, and y𝑔,ob := +√︁ +2(𝜂 − yob). +13 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +Now, 𝜂 becomes a GP hyperparameter via y𝑔,ob. Hence, we can estimate the hyperposterior Π(·) := 𝑃(Θ|D) via the +marginal likelihood of 𝑓 (·), given by: +L(Θ) := 𝑃(D|Θ) = N +� +yob; 𝑚(Xob|Θ), 𝐶(Xob, Xob|𝜃, 𝜎2 +𝑛) +� +. +(16) +Therefore, we can marginalise likelihood via hyperposterior: +𝐿(𝑥) = +∫ +𝐿(𝑥|Θ)dΠ(Θ), +(17) +A.2.1 +Integral Operator Distillation +With the given weighted samples (wBQ, 𝚯hyper), we can approximate the marginalisation. For instance, the marginalised +likelihood 𝐿(·) in Eq. (17) can be approximated as follows: +𝐿(𝑥) ≈ +∫ +𝐿(𝑥|Θ) +𝑚hyper(Θ) +w′⊤ +BQL +dΠ′(Θ), +≈ w⊤ +BQ𝐿(𝑥|𝚯hyper), +(18) +where wBQ := w′ +BQ ◦ L/(w′⊤ +BQL). Both marginalisation for 𝐿(·) and 𝑃(D) share the same signed measure, Eq. (18) +becomes a good approximation [34]. Notably, we can recycle the same weights and samples for other marginalisation, +such as GP predictive posterior. Furthermore, this formulation does not require sampling from hyperposterior. Thus, we +can avoid time-consuming MCMC sampling for marginalisation. +However, even though the random sampling from hyperprior 𝑃(Θ) is fast, it is sample inefficient for integration. Thus, +we wish to approximate the marginalisation with the minimal number of quadrature samples 𝐻 (𝐻 ≪ 𝐵). We adopt +a quadrature distillation trick with RCHQ. The idea is simple; distilling the dataset Dhyper = (𝚯hyper, L(𝚯hyper)) to +small dataset DQD. We set RCHQ arguments as the empirical measure Π(Θ) := 𝚯 and the kernel 𝐶BQ(·, ·), then RCHQ +returns the small sparse samples DQD. Compact wQD and 𝚯QD can be obtained via retraining hyper-GP with DQD. This +permits efficient computation of marginalisation, including expensive AF, such as information-theoretic AFs. +A.3 +Quadrature Distillation Algorithm +Table 5: Quadrature distillation algorithm +Algorithm 2: Quadrature distillation +1: 𝚯hyper ∼ Π′(·) +# random sampling from hyperprior +2: Dhyper = (𝚯hyper, L(𝚯hyper)) +# dataset construction +3: ℓ(Θ) ← TrainGP(Dhyper) +# train hyper-GP +4: Π′ +opt(·), whyper ← HyperpriorOpt(ℓ(·), 𝚯hyper) # optimise hyperprior a posteriori +5: 𝚯QD = RCHQ(ℓ(·), 𝚯hyper, whyper) +# integral operator distillation +6: DQD = (𝚯QD, L(𝚯QD)) +# extract corresponding L from DBQ +7: ℓQD(Θ) ← TrainGP(DQD) +# retrain hyper-GP +8: wQD = BQ(ℓQD(·), Π′ +opt(·)) +# compute BQ weights in Eqs. (33) - (34) +9: return wQD, 𝚯QD +The algorithm flow of the quadrature distillation is shown in Table 5. Each procedure will be explained step by step. +A.3.1 +Train Hyper-GP +First, we place hyper-GP on the hyperlikelihood L(·) with the Gaussian kernel 𝐾hyper(·, ·): +ℓ ∼ GP(ℓ; 𝑚hyper(·), 𝐶hyper(·, ·)) +(19) +𝑚hyper(Θ) = 𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1L +(20) +𝐶hyper(Θ, Θ′) = 𝐾hyper(Θ, Θ′) − 𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1𝐾hyper(𝚯hyper, Θ′) +(21) +𝐾hyper(Θ, 𝚯hyper) := 𝑣 +√︁ +|2𝜋W|N (Θ; 𝚯hyper, W) +(22) +14 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +where 𝑣 is kernel variance and W := 𝑙I is the diagonal covariance matrix whose diagonal elements are the lengthscale +𝑙. Training hyper-GP is done with type-II MLE with L-BFGS-G [62], via maximising the marginal likelihood of the +hyper-GP N +� +L(𝚯hyper); 𝑚(𝚯hyper|𝑡), 𝐶(𝚯hyper, 𝚯hyper|𝑡) +� +, where 𝑡 is the hyper-hyperpameter of the hyper-GP. +A.3.2 +Optimising Hyperprior a Posteriori +Now, hyperevidence 𝑃(D) is the closed-form via Eq. (11). Thus, we can optimise the hyperprior 𝑃(Θ) a posteriori +so as to maximise the hyperevidence 𝑃(D). This is equivalent to minimise the difference between hyperprior Π′(·) +and hyperlikelihood (marginal likleihood) L(·). Hence, this process equals to approximating the mixture of Gaussian +process (the mean of hyper-GP) with a unimodal Gaussian (hyperprior), which can be computed analytically. The mean +predictive posterior 𝑚hyper(·) can be written as the mixture of Gaussians: +𝑚hyper(Θ) = N (Θ; 𝚯hyper, W)L′ +∝ +∑︁ +𝑖 +𝐿′′ +𝑖 N (Θ; Θhyper,𝑖, W) +L′ := 𝑣 +√︁ +|2𝜋W|𝐾hyper(𝚯hyper, 𝚯hyper)−1L +L′′ := L′/(L′1) +(23) +Next, we optimise the hyperprior via maximising the analytical hyperevidence 𝑃(D) = w′⊤ +BQL. The optimised hyperprior +is the approximation of the weighted Gaussian mixture in Eq. (23): +Π′ +opt(Θ) = N (Θ; 𝜇opt +hyper, 𝚺opt +hyper) +(24) +𝜇opt +hyper := L′′𝚯hyper +(25) +𝚺opt +hyper := W + 𝚯⊤ +hyperP𝚯hyper − (𝚯hyperP)𝚯hyperP +(26) +P := diag L′′ +(27) +Now we wish to correct the sample distribution from Π′(·) to Π′ +opt(·). This can be done by the importance sampling, +with the following weights: +whyper := +N (𝚯hyper; 𝜇opt +hyper, 𝚺opt +hyper) +N (𝚯hyper; 𝜇hyper, 𝚺hyper) +(28) += ZN (𝚯hyper; 𝜇IS +hyper, 𝚺IS +hyper) +(29) +where +𝚺IS +hyper := +� +𝚺opt,−1 +hyper − 𝚺−1 +hyper +�−1 +(30) +𝜇IS +hyper := 𝚺IS +hyper +� +𝚺opt,−1 +hyper 𝜇opt +hyper − 𝚺−1 +hyper𝜇hyper +� +(31) +Z := +|𝚺hyper| +|𝚺hyper − 𝚺opt +hyper| +1 +N (𝜇opt +hyper; 𝜇hyper, 𝚺hyper − 𝚺opt +hyper) +(32) +A.3.3 +Integral Operator Distillation +Then, we perform RCHQ with the following arguments: +1. the empirical measure Πemp = (whyper, 𝚯hyper) +2. the kernel 𝐾(Θ, Θ′) = 𝐶hyper(Θ, Θ′) +15 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +Then, the sparse samples 𝚯QD will be returned. Retraining GP yields ℓQD ∼ GP(ℓQD; 𝑚QD(·), 𝐶QD(·, ·)) as the +"distilled" GP. We calculate the BQ weights with this GP. The closed-form BQ weights can be calculated as follows: +w′ +QD = +∫ +𝐾QD(Θ, 𝚯QD)𝐾QD(𝚯QD, 𝚯QD)−1dΠ′ +opt(Θ), += 𝑣 +√︁ +|2𝜋W| +�∫ +N (Θ; 𝚯QD, W)N (Θ; 𝜇opt +hyper, 𝚺opt +hyper)dΘ +� +𝐾QD(𝚯QD, 𝚯QD)−1, += 𝑣 +√︁ +|2𝜋W|N (𝚯QD; 𝜇opt +hyper, W + 𝚺opt +hyper)𝐾QD(𝚯QD, 𝚯QD)−1, +(33) +wQD := w′ +QD ◦ L/(w′⊤ +QDL) +(34) +A.4 +Fast Fully Bayesian Gaussian Process +The resulting measure (wQD, 𝚯QD) can approximate the marginalisation over the hyperposterior Π(·), as shown in Eq. +(17). Other quantities can also be marginalised, such as: +𝑚QD(𝑥) = +∫ +𝑚(𝑥|Θ)dΠ(Θ), +≈ w⊤ +QD𝑚(𝑥|𝚯QD), +(35) +𝑉QD(𝑥) = +∫ +𝐶(𝑥, 𝑥|Θ)dΠ(Θ), +≈ w⊤ +QD +� +𝐶(𝑥, 𝑥|𝚯QD) + 𝑚2(𝑥|𝚯QD) +� +− 𝑚2 +QD(𝑥). +(36) +𝐶QD(𝑥, 𝑥′) = +∫ +𝐶(𝑥, 𝑥′|Θ)dΠ(Θ), +≈ +wQD1 +(wQD1)2 − w2 +QD1 +𝐻 +∑︁ +𝑖 +𝑤i, QD +��𝑚(𝑥|Θi, QD) − 𝑚QD(𝑥)�𝑇 �𝑚(𝑦|Θi, QD) − 𝑚QD(𝑦)�� +, +(37) +This marginalisation approximation is also beneficial to approximate the expensive AFs. Computations of marginal +AFs, information-theoretic AFs and FBGP-based AFs will be explained. +A.4.1 +Marginal Expected Improvement Acquisition Function +The marginal expectation improvement (EI) AF [63] can be calculated with FBGP formulation: +𝛼EI(𝑥) := w⊤ +QD +� +(𝑚(𝑥|𝚯QD) − 𝜼) ◦ Φ(𝑍|𝚯QD) +� ++ w⊤ +QD +�√︃ +𝐶(𝑥, 𝑥|𝚯QD) ◦ 𝜙(𝑍|𝚯QD) +� +(38) +𝑍 := 𝑚(𝑥|𝚯QD) − 𝜼 +√︁ +𝐶(𝑥, 𝑥|𝚯QD) +(39) +where Φ(𝑥), 𝜙(𝑥) are CDF and probability density function (PDF) of the normal distribution, 𝜼 ∈ 𝚯QD is the distilled +max value 𝜂. +A.4.2 +Marginal Upper Confidence Bound Acquisition Function +The marginal upper confidence bound (UCB) AF [64] can be calculated with FBGP formulation: +𝛼UCB(𝑥) := w⊤ +QD𝑚(𝑥|𝚯QD) + +√︁ +𝛽w⊤ +QD +√︃ +𝐶(𝑥, 𝑥|𝚯QD) +(40) +where 𝛽 is the BO hyperparameter, usually 0.2 is selected. +16 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +A.4.3 +Max-value Entropy Search Acquisition Function +The max-value entropy search (MES) AF [37] can be calculated via FITBO formulation [60]: +𝛼FITBO(Xhyper|D) := 𝐻[𝑝(𝑦|D, Xhyper)] − E𝑝(𝜂|D) +� +𝐻[𝑝(𝑦|D, Xhyper, 𝜂)] +� +, +(41) +𝑝(𝑦|D, Xhyper) = +∫ +𝑝(𝑦|D, Xhyper, 𝜂)d𝑝(𝜂|D), +(42) +𝐻[𝑝(𝑦|D, Xhyper)] = +∫ +ln 𝑝(𝑦|D, Xhyper)d𝑝(𝑦|D, Xhyper), +(43) +E𝑝(𝜂|D) +� +𝐻[𝑝(𝑦|D, Xhyper, 𝜂)] +� += +∫ +𝐻[𝑝(𝑦|D, Xhyper, 𝜂)]d𝑝(𝜂|D). +(44) +FITBO AF can be discretised via MC integration: +𝛼FITBO(Xhyper|D) := 𝐻 +� 1 +𝑀 +𝑀 +∑︁ +𝑖 +𝑝(𝑦|D, Xhyper, 𝜃𝑖, 𝜂𝑖) +� +− +1 +2𝑀 +𝑀 +∑︁ +𝑖 +log[2𝜋𝑒(𝐶(Xhyper, Xhyper|D, 𝜃𝑖, 𝜂𝑖) + 𝜎𝑛,𝑖)]. +(45) +Quadrature distillation can approximate the above AF as: +𝛼FITBO(Xhyper|D) ≈ 𝐻 +� +w⊤ +QD𝑚(Xhyper|𝚯QD) +� +− 1 +2w⊤ +QD log[2𝜋𝑒(𝐶(Xhyper, Xhyper|D, 𝚯QD) + 𝝈2 +𝑛,QD)]. +(46) +For faster computation, moment-matching approximation yields the first term as: +𝐻 +� 1 +𝑀 +𝑀 +∑︁ +𝑖 +𝑝(𝑦|D, Xhyper, 𝜃𝑖, 𝜂𝑖) +� +≈ 1 +2 log[2𝜋𝑒(Var[𝑦] + 𝜎2 +𝑛,𝑖)], +(47) +Var[𝑦] = 1 +𝑀 +𝑀 +∑︁ +𝑖 +� +𝐶(Xhyper, Xhyper|𝜃𝑖) + 𝑚2(Xhyper|𝜃𝑖) +� +− E[𝑦Θ𝑖]2, +(48) +E[𝑦] = 1 +𝑀 +𝑀 +∑︁ +𝑖 +𝑚(Xhyper|𝜃𝑖). +(49) +Hence, +𝛼FITBO(Xhyper|D) ≈ 1 +2 log[2𝜋𝑒(Var[𝑦] + w⊤ +QD𝝈2 +𝑛,QD)] +− 1 +2w⊤ +QD log[2𝜋𝑒(𝐶(Xhyper, Xhyper|D, 𝚯QD) + 𝝈2 +𝑛,QD)], +(50) +Var[𝑦] = w⊤ +QD +� +𝐶(Xhyper, Xhyper|𝚯QD) + 𝑚2(Xhyper|𝚯QD) +� +− +� +w⊤ +QD𝑚(Xhyper|𝚯QD) +�2 +. +(51) +A.4.4 +Bayesian Query-by-Committee Acquisition Function +The Bayesian query-by-committee (B-QBC) AF is defined by [19]: +𝛼BQBC(𝑥) := Var𝑝(Θ|D) +� +𝑚(𝑥|Θ) +� +, +(52) += E𝑝(Θ|D) +� +(𝑚(𝑥|Θ) − ˆ𝑚(𝑥))2 +� +. +(53) +The quadrature distillation approximates as follows: +𝛼BQBC(𝑥) ≈ w⊤ +QD +� +(𝑚(𝑥|𝚯QD) − w⊤ +QD𝑚(𝑥|𝚯QD))2 +� +. +(54) +A variant AF, query by mixture of Gaussian process (QB-MGP), can also be approximated by the quadrature distillation: +𝛼QB-MGP(𝑥) := E𝑝(Θ|D) +� +𝐶(𝑥, 𝑥|Θ) +� ++ E𝑝(Θ|D) +� +(𝑚(𝑥|Θ) − ˆ𝑚(𝑥))2 +� +. +(55) +≈ w⊤ +QD𝐶(𝑥, 𝑥|𝚯QD) + w⊤ +QD +� +(𝑚(𝑥|𝚯QD) − w⊤ +QD𝑚(𝑥|𝚯QD))2 +� +. +(56) +17 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +B +Algorithm +B.1 +Subsampling algorithm +The algorithm flow of the subsampling is shown in the Table 6. The details will be explained step by step. +Table 6: Subsampling algorithm +Algorithm 3: Subsampling +1: Xrec ∼ 𝜋′(·) +# sampling from prior +2: wrec = 𝐿(Xrec) +𝜋′(Xrec) · 𝜋′(Xrec)1 +𝐿(Xrec)1 +# compute the weights +3: if len(wrec > 0) < 𝑛 : +4: +𝜋′(·) ← 𝜋′ +initial(·) +# return to the initial prior when overexploitive +5: if continuous: +6: +𝜋(·) = WKDE(wrec, Xrec) +# weighted kernel density estimation +7: +Xrec ∼ 𝜋(·) +# resample from WKDE +8: +wrec = 𝐿(Xrec) +𝜋(Xrec) · 𝜋(Xrec)1 +𝐿(Xrec)1 +# recompute the weights +9: else if discrete and enumerable: +10: +Xrec = 𝜋′(·) +# all discrete candidates +11: +wrec = +𝐿(Xrec) +𝐿(Xrec)⊤1 +# normalised weights +12: else if innumerable discrete: +13: +𝜋(·) ← OptHypersMLE(𝜋′(·), wrec, Xrec) +# MLE hyperparameter optimisation +14: +Xrec ∼ 𝜋(·) +# resample from WKDE +15: +wrec = 𝐿(Xrec) +𝜋(Xrec) · 𝜋(Xrec)1 +𝐿(Xrec)1 +# recompute the weights +16: else mixed: +17: +𝜋(·) ← CombineBothPrior(𝜋′(·), wrec, Xrec) # Combine continuous and discrete prior +18: +wrec = 𝐿(Xrec) +𝜋(Xrec) · 𝜋(Xrec)1 +𝐿(Xrec)1 +# recompute the weights +19: Xnys ∼ Deweighted(wrec, Xrec) +# deweighted random subset extraction +20: return wrec, Xrec, Xnys +B.1.1 +Weighted Kernel Density Estimation +The mean and covariance of the weighted kernel density estimation (WKDE) is estimated with the unbiased data +covariance matrix given by: +𝜇wkde := wrecXrec, +(57) +𝚺wkde := +wrec1 +(wrec1)2 − w2rec1 +𝑁 +∑︁ +𝑖 +𝑤i, rec(𝑋i, rec − 𝜇wkde)𝑇 (𝑋i, rec − 𝜇wkde), +(58) +:= +1 +1 − w2rec1 +𝑁 +∑︁ +𝑖 +𝑤i, rec(𝑋i, rec − 𝜇wkde)𝑇 (𝑋i, rec − 𝜇wkde), +(59) +where 𝑋i, rec ∈ Xrec and 𝑤i, rec ∈ wrec is the 𝑖-th element of Xrec and wrec, respectively. The bandwidth of the kernel is +estimated by the Scott’s method [65]. +B.1.2 +Maximum Likelihood Estimation of Discrete Prior +The optimisation of hyperparamters of the discrete prior distributions was done via MLE from the weighted +samples (wrec, Xrec). +We denote the PDF of Bernoulli distribution (binary) and the categorical distribution as +Ber(𝑥; wBer), Cat(𝑥; wCat), where wBer ∈ R𝑑 and wCat ∈ R𝑑×𝐶 are the weights hyperparaeters, 𝐶 is the number of +categories in the input parameters. The weighted log-PDF can be expressed as follows: +LL := wrec log Ber(Xrec; wBer) +(60) +LL := wrec log Cat(Xrec; wCat) +(61) +We optimise each weight hyperparameters via maximising the log-likelihood (LL) via L-BFGS-B [66]. PyTorch +[53] auto-differentiation gives the gradient for L-BFGS-B. To set a constraint to make weights bounded [0, 1], we +transformed original LL space via the sigmoid function during optimisation. +18 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +B.1.3 +Deweighted sampling +Samples for the Nyström method are better to be spatially sparse to well represent the whole kernel shape. We adopt the +deweighted sampling to constract the small subset of uniformly distributed samples Xnys from the weighted samples +(wrec, Xrec). We resample from the categorical distribution with the inverse weights (1/wrec), then the resampled +samples are uniformly distributed. +B.2 +AutoKQ selection algorithm +Table 7: AutoKQ selection algorithm +Algorithm 4: AutoKQ selection +1: Xrchq, wrchq, Var[𝑚(𝑥)]rchq ← RunRCHQ(wrec, Xrec, 𝝋(·), 𝛼(·), Xnys, 𝑓 (·)) +4: Xkt, wkt, Var[𝑚(𝑥)]kt ← RunKernelThinning(wrec, Xrec, 𝝋(·), 𝛼(·), Xnys, 𝑓 (·) +6: if Var[𝑚(𝑥)]rchq < Var[𝑚(𝑥)]kt: +7: +return Xrchq, wrchq +8: else: +9: +return Xkt, wkt +function RunRCHQ(wrec, Xrec, 𝝋(·), 𝛼(·), Xnys, 𝑓 (·)): +1: 𝝋(·) ← Nyström(Xnys, 𝑓 (·)) +2: Xrchq, wrchq ← RCHQ(wrec, Xrec, 𝝋(·), 𝛼(·)) +3: Var[𝑚(𝑥)]rchq ← KQ( 𝑓 (·), Xbatch, wbatch) +4: return Xrchq, wrchq, Var[𝑚(𝑥)]rchq +function RunKernelThinning(wrec, Xrec, 𝝋(·), 𝛼(·), Xnys, 𝑓 (·)): +4: Xkt, wkt ← KernelThinning(Xrec, 𝑓 (·), 𝛼(·)) +5: Var[𝑚(𝑥)]kt ← KQ( 𝑓 (·), Xbatch, wbatch) +4: return Xkt, wkt, Var[𝑚(𝑥)]kt +Table 7 illustrates the algorithm flow of automatic kernel quadrature selection algorithm. We compare the worst-case +integration error of each algorithm, then pick the batch queries of which integration error is smaller. +C +Simulation-based inference +C.1 +Simulation-based inference +The simulator emulates typically time-evolving signals from the physical device modelled by simultaneous differential +equations. The solution of the differential equation is basically not analytical, requiring numerical approximation such +as the finite element method. Each equation has parameters, such as coefficients of differential terms, which determine +the signal shape. Estimating the parameters that can reproduce the observed signal is a typically tricky task because +simulation is not differentiable with regard to each parameter. Although auto-differentiation can mitigate this problem, +the parameter posterior is typically multimodal, so local optimisation algorithms based on differentiation struggles +to find the global optimum. More importantly, this inverse problem often has no unique solution mathematically. +Hence, rather than estimating one deterministic parameter set, inferring the parameter posterior is more practically +important. Moreover, having dozens of plausible simulators with differing levels of assumption is a common situation +where we need to select the parsimonious model that best describes the given dataset. Bayesian model evidence can +provide a selection criterion. Therefore, estimating both Bayesian model evidence and parameter posterior is a frequent +desideratum in practice. Furthermore, running simulators is expensive to evaluate, so parallelising the computation via +computer clusters is of practical importance. +19 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +Let yobs be the observed signal from the physical device, and we wish to estimate the simulator parameters Θ. This can +be formulated as Bayesian inference, given by: +𝑝(Θ) := 𝜋′(Θ) := N (Θ; 𝜇𝜋, 𝚺𝜋) +(62) +𝑝(D|Θ, 𝑀) := ℓtrue(Θ) := +𝑚 +� +𝑗 +N (err 𝑗 (𝜃); 0, 𝜎2 +noise), +(63) +𝑝(D|𝑀) := N +� +E +𝑥∈𝜋[ℓtrue(Θ)], Var +𝑥∈𝜋[ℓtrue(Θ)] +� +, +(64) +𝑝(Θ|D, 𝑀) = 𝑝(D|Θ, 𝑀)𝑝(Θ) +𝑝(D|𝑀) += +ℓtrue(Θ)𝜋(Θ) +E𝑥∈𝜋 [ℓtrue(Θ)] , +(65) +where +D := {xobs, yobs} ∈ R𝑚×1, +(66) +𝜃 := {𝜃𝑖} ∈ R𝑑−1, +(67) +Θ := {𝜃, 𝜎2 +noise} ∈ R𝑑, +(68) +𝑦sim, 𝑗 (𝜃) := 𝑀(𝜃, xobs), +(69) +err 𝑗 (𝜃) := +� +𝑦obs, 𝑗 − 𝑦sim, 𝑗 (𝜃) +�2 . +(70) +𝑀(𝜃, xobs) is the simulation model, which returns the prediction 𝑦sim, 𝑗 (𝜃) at given simulation parameter 𝜃 at the 𝑗-th +time step. We wish to estimate the model evidence E𝑥∈𝜋 [ℓtrue(Θ)] and the parameter posterior 𝑝(Θ|D, 𝑀). +C.2 +Bayesian Quadrature Formulation +In naive BQ, we place GP on the likelihood as such: +ℓ(Θ) ∼ GP�ℓ(Θ); 𝜇ℓ(Θ), 𝜎ℓ(Θ, Θ′)�. +(71) +The evidence can be estimated via Eq. (5a) in general or Eq. (11) in the Gaussian prior and RBF kernel case. The +posterior can be estimated with the surrogate model and the estimated evidence via Eq. (65). +However, the likelihood is typically transformed into the logarithmic space because its dynamic range is wider than +the numerical over-/underflow limits. Thus, log-warped GP [21, 67, 22] is often applied. Particularly, we consider +moment-matched log-transformed (MMLT) [67] GP, modelled as such: +𝑓 (𝑥) = exp[𝑔(𝑥)] − 1, +(72a) +:= GP� 𝑓 (𝑥); 𝑚(𝑥), 𝐶(𝑥, 𝑥) + 𝜎2 +𝑛I�, +(72b) +𝑔(𝑥) := GP�𝑔(·); 𝑚𝑔(·), 𝐶𝑔(·, ·)�, +(72c) +𝑚(𝑥) := exp +� +𝑚𝑔(𝑥) + 1 +2𝐶𝑔(𝑥, 𝑥) +� +, +(72d) +𝐶(𝑥, 𝑥′) := 𝑚𝑔(𝑥)𝑚𝑔(𝑥′) +� +𝐶𝑔(𝑥, 𝑥) − 1 +� +. +(72e) +The warped GP stores log-transformed values y𝑔 = log(y + 1), so we can avoid the over-/underflows. Adachi et al. [22] +further extended MMLT GP so as to accommodate with BASQ modelling. They adopted the four-layered GP combining +MMLT and parabolic transformation. The reason why they add the parabolic transformation is to copy the exponetiated +function information to not only the surrogate function but also prior update. However, this deep warped structure causes +additional predictive errors due to the cumulative approximation errors from each layer’s moment-matching method. +C.3 +Likelihood-free Inference Formulation +Alternately, BO-based LFI [30] models GP differently. They placed GP on the discrepancy, rather than the likelihood, +defined as : +Δtrue(𝜃) := log ||yobs − ysim(𝜃)|| +(73) +Δ(Θ) ∼ GP�Δ(Θ); 𝜇Δ(𝜃), 𝜎Δ(𝜃, 𝜃′) + 𝜎2 +𝑛I� +(74) +LFI adopts the tentative likelihood defined by Eq. 8 as the likelihood at each iteration. The true likelihood can be +estimated a posteriori. The benefits of this modelling are as follows: +20 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +1. Avoiding extreme dynamic range of likelihood; Δtrue(𝜃) has much more moderate range. +2. We can reformulate BQ as BO. BO is more suitable for solving simulation-based inference as only the vicinity +of the MAP location has meaningful value. As almost everywhere has zero likelihood, so BQ formulation is +over-exploring if the prior is misspecified. +3. We can obtain the “temporary” likelihood 𝐿(𝜃) that approaches the true likelihood ℓtrue(Θ) asymptotically over +iterations. This likelihood can be regarded as "updated prior". This can also mitigate the prior misspecification. +They reformulate the posterior inference as the BO to find the global minimum of the discrepancy Δtrue(𝜃). The resulting +GP surrogate model is used to approximate the posterior. They do not go beyond the posterior inference, so evidence +estimation cannot be done with BOLFI. +C.4 +SOBER-LFI Formulation +We wish to take the best of both world; LFI GP modelling suitable for sampling and exact evidence estimation via BQ +modelling. Thus, we adopt the dual GPs; one for sampling, and the other for BQ modelling. While the sampling GP is +modelled with the inverse discrepancy (−Δtrue(𝜃) so as to be the maximisation objective), the BQ GP is modelled with +log-likelihood with MMLT GP. Importantly, we can query both {Δtrue(𝜃), ℓtrue(Θ)} with negligible overhead as the +time-consuming part is ysim(𝜃). Once we get ysim(𝜃), calculating both {Δtrue(𝜃), ℓtrue(Θ)} are very cheap. +The sampling GP is used for setting up the sampling function 𝜋, in the same manner explained in Section B.1. One +difference is that the 𝜋 becomes extremely sharper than the BO task. WKDE-based sampling can fail to sampling from +𝜋. Hence, we adopted elliptical slice sampling (ESS) [68]. Importance sampling permits using all of the samples from +ESS without the burn-in period. The weights can be calculated via the 𝜋 defined with the sampling GP. Note that ESS +is more expensive than WKDE, so the additional overhead had to be produced instead. As such, the sampling GP +constructs the empirical measure 𝜋emp = (wrec, Xrec). +On the other hand, BQ GP constructs the surrogate model for likelihood. The posterior and evidence inference can be +made in the same manner explained in Section C.2. +Batch acquisition via objective RCHQ becomes a mix of both GPs. The kernel is defined by the BQ GP in Eq. 72e. The +objective with AF is defined by the sampling GP. +D +Experiments +D.1 +Batch Bayesian Optimisation +Anti-Malarial Drug +Polar Solvent +global +minimum +global +maximum +EC50 [μM] +Dipole Moment [D] +number of molecules +Figure 5: The histograms of the target values in the real-world datasets +We examined our method, SOBER, with the following four datasets. All experiments are averaged over 16 iterations +with varied random seeds. The number of candidate samples drawn from the prior distribution are fixed to be 20,000 +(𝑁 = 20, 000) for fair comparison. While SOBER-LFI updates the prior hyperparameters as explained, the others are +fixed with the initial hyperparameters. Implementation of decoupled TS in BoTorch was not compatible with Tanimoto +kernel in GAUCHE [2]. Thus, SOBER-TS had to be sampled from vanilla TS. However, DPP-TS and SOBER-TS are +infeasibly slow. They took about 5 days to run 30 iterations, resulting in 80 days for 16 trials for averaging, even in the +smaller anti-malaria dataset. Hence, we do not compare decoupled TS, DPP-TS, and SOBER-TS for drug discovery. +21 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +Synthetic: Ackley function +Ackley funciton is defined as: +𝑓 (𝑥) := −𝑎 exp +������ +−𝑏 +� +� +� +1 +𝑑 +𝑑 +∑︁ +𝑖=1 +𝑥2 +𝑖 +������ +− exp +� +1 +𝑑 +𝑑 +∑︁ +𝑖=1 +cos(𝑐𝑥𝑖) +� ++ 𝑎 + exp(1) +(75) +where 𝑎 = 20, 𝑐 = 2𝜋, 𝑑 = 23. We take the negative Ackley function as the objective of BO to make this optimisation +problem maximisation. We modified the original Ackley function into a 23-dimensional function with the mixed spaces +of 3 continuous and 20 binary inputs from [0, 1]20, following [42]. The batch size 𝑛 is 200. The continuous prior is the +uniform distribution ranging from [-1, 1]. The binary prior is the Bernoulli distribution with unbiased weights 0.5. We +assume each of continuous and binary priors at each dimension are independent. +Synthetic: Rosenbrock function +Rosenbrock function is defined as: +𝑓 (𝑥) := +�𝑑−1 +∑︁ +𝑖=1 +� +100(𝑥𝑖+1 − 𝑥2 +𝑖 )2 + (𝑥𝑖 − 1)2� +� +(76) +where 𝑑 = 7. We take the negative Rosenbrock function as the objective of BO to make this optimisation problem +maximisation. We modified the original Rosenbrock function into a 7-dimensional function with the mixed spaces of 1 +continuous and 6 discrete variables, following [42]. The first 1 dimension is discretised to be categorical variables, with +4 possible values 𝑥1 ∈ {−5, 0, 5, 10}. The other 6 dimensions are continuous with bounds [−5, 10]6. The batch size 𝑛 +is 100. The continuous prior is the uniform distribution ranging from [-5, 10]. The discrete prior is the categorical +distribution with unbiased weights 0.5. We assume each of continuous and discrete priors at each dimension are +independent. +Real-world: +Anti-Malarial drug discovery +The dataset with 20,746 small molecules represented as 2048- +dimensional features were taken from the P. falciparum whole cell screening derived by the Novatis-GNF Malaria Box +[51]. The target variable is the EC50 value, which is defined as the concentration of the drug which gives half the +maximal response. The lower the concentration, the more effective (better) the drug. We take the nagative EC50 to +make this optimisation problem maximisation. The batch size 𝑛 is 100. We set the categorical prior with unbiased +weights 0.5 for each molecule. +Real-world: Polar solvent for batteries +The dataset with 133,055 small molecules represented as 2048-dimensional +features was optimised and predicted by the quantum-chemical computations using density functional theory, known as +QM9 dataset [52]. The target variable is the dipole moment, which is basically correlated with the solvation capability +in electrolytes in lithium-ion batteries, increasing the ratio of electro-mobile Li-ions. The higher the dipole moment +becomes, the larger (better) the ionic conductivity does. The batch size 𝑛 is 200. We set the categorical prior with +unbiased weights 0.5 for each molecule. +Figure 5 shows the distribution of target values in two real-world drug discovery datasets. The optimal molecules are +outliers from the dataset distribution, so it clearly shows these tasks are needle-in-the-haystack situations. +D.2 +Batch Bayesian Quadrature +We tested our algorithm, SOBER, with the simulation-based inference tasks as the batch BQ method. All experiments +are averaged over 16 iterations with varied random seeds. The number of candidate samples drawn from the prior +distribution is fixed to be 20,000 (𝑁 = 20, 000) for a fair comparison. As the ground truth of posterior and evidence +cannot be obtained for the simulation-based inference, we use the empirical metric to evaluate the quality of each +inference. For posterior evaluation, we drew the 10,000 test samples from the normal distribution centered at the ground +truth parameters and the covariance with the identity matrix of which each element is 5 × 10−6. Then, we computed +the root-mean-squared error (RMSE) between the estimated log-likelihood and true log-likelihood. For evidence, we +simply adopted the negative of estimated evidence. +Real-world: 2 RC Pairs ECM +2 RC Pair equivalent circuit model (ECM) is the simplest lithium-ion battery simulator +with 6-dimensional continuous variables [22]. We generated synthetic signal using the model with 100 frequency steps +equispaced over log-angular frequency regime, then added the Gaussian noise with the amplitude of exp(1) to the +𝑅total = exp(2) signal from the canonical ECM. +Real-world: 5 RC Pairs ECM +5 RC Pair ECM is more complex lithium-ion battery simulator with 12-dimensional +continuous variables [22]. We generated synthetic signal using the model with 100 frequency steps equispaced over +22 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +log-angular frequency regime, then added the Gaussian noise with the amplitude of exp(1) to the 𝑅total = exp(2) signal +from the canonical ECM. +D.3 +Additional Experiments +D.3.1 +Hyperparameter sensitivity +Acquisition Function +batch size n +Nyström samples M +Recombinaiton samples N +Log10 regret +Log10 overhead [s] +0 +-2 +0 +-2 +0 +-2 +3 +2 +1 +0 +15 +0 +15 +0 +15 +0 +15 +0 +0 +15 +0 +15 +0 +15 +0 +15 +3 +2 +1 +0 +3 +2 +1 +0 +3 +2 +1 +0 +Iterations +Iterations +Iterations +Iterations +0 +-2 +Figure 6: Hyperparameter sensitivity analysis using the Ackley function. Lines and shade area denote mean ± 1 standard error. +We tested the hyperparameter sensitivity of SOBER-LFI using the Ackley function. We examined the effect of AFs 𝛼, +batch size 𝑛, the number of Nyström samples 𝑀, and the number of recombination samples 𝑁. We averaged the results +from 16 experiments with varied random seeds, and terminated at the 15th batch acquisition. The baseline conditions +are 𝑛 = 100, 𝛼 = LFI, 𝑀 = 500, and 𝑁 = 20, 000. For AF, the information-theoretic AFs can boost the convergence +rate, whereas the others do not change significantly. For the batch size 𝑛, the convergence rate can be improved in +accordance with the batch size. For quadrature hyperparameters 𝑀 and 𝑁, the larger the number of samples becomes, +the faster the convergence does. However, increasing the number of samples leads to additional overhead increase. Our +default conditions are competitive throughout the experiments. +D.3.2 +Ablation Study +π definitions +ablation study +Log10 regret +Log10 overhead [s] +0 +-2 +0 +-2 +3 +2 +1 +0 +15 +0 +15 +0 +0 +15 +0 +15 +3 +2 +1 +0 +Iterations +Iterations +Figure 7: Ablation study using the Ackley function. Lines and shade area denote mean ± 1 standard error. +23 + +SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints +We performed the ablation study to analyse each algorithm’s effects on convergence rate. Firstly, we compared the +various 𝜋 definitions defined by the AFs. As shown in Figure 7, LFI and PI definitions are the clear performants. This is +because the other AFs are designed to guide the sequential sampling, of which global maxima sensitively changed over +iterations. LFI and PI show the possibility of global maxima, which gradually squeezes the region toward the true global +maxima. Thus, in SOBER-LFI formulation, LFI AF is well-suited as the definition of 𝜋. As another ablation study, we +compared whether or not updating 𝜋 and using AF in the objective RCHQ. As a result, unsurprisingly, updating 𝜋 is the +most influential on the convergence rate. The objective RCHQ does not significantly influence the convergence when we +select LFI as the objective. However, information-theoretic AFs can boost the convergence, as shown in Figure 6. +D.3.3 +Fully Bayesian Gaussian Process +Noise 1E+00 +Log10 regret +Log10 overhead [s] +15 +0 +15 +Iteration +0 +-1 +2 +3 +0 +0 +1 +FBGP Noise +Log10 regret +0 +15 +Iteration +0 +-1 +Log10 overhead [s] +2 +3 +0 +1 +0 +15 +-2 +-3 +MLE-GP Noise +Log10 regret +0 +15 +Iteration +0 +-1 +Log10 overhead [s] +2 +3 +0 +1 +0 +15 +-2 +-3 +Figure 8: Efficacy of Fully Bayesian Gaussian process modelling using the noisy Ackley function. Lines and shade area denote mean +± 1 standard error. +We further tested the effect of FBGP modelling on the convergence rate. To examine the efficacy, we adopted the noisy +Ackley function. We added the Gaussian noise to the queried values from the Ackley function. The amplitude of the noise +is varied from 10−3 to 1 in a logarithmic order. The baseline conditions are 𝑛 = 100, 𝛼 = LFI, 𝑀 = 500, 𝑁 = 20, 000, +and 𝐻 = 50. Figure 8 illustrates that FBGP modelling with quadrature distillation can boost the convergence rate while +maintaining the overhead feasibly small (the overhead of FBGP is smaller than DPP-TS with type-II MLE kernel.) +Hyperweights H +Acquisition Function +Log10 regret +Log10 regret +Log10 overhead [s] +15 +0 +15 +0 +15 +Iteration +Iteration +0 +2 +3 +0 +0 +-1 +1 +Log10 overhead [s] +2 +3 +0 +1 +0 +15 +0 +-1 +-2 +-3 +Figure 9: SOBER-LFI consistently outperforms with small overhead. Lines and shade area denote mean ± 1 standard error. +Furthermore, we examined the effects of the number of hyperweights 𝐻 and the AFs. While 𝐻 are influential on both +convergence rate and overhead, the default value 𝐻 = 50 are reasonably competitive. With regard to the effect of AFs, +QB-MGP AF was the performant. +24 + diff --git a/O9FKT4oBgHgl3EQfgC6c/content/tmp_files/load_file.txt b/O9FKT4oBgHgl3EQfgC6c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3cd7a6721329307eaeadc40b880d3483c1b3424d --- /dev/null +++ b/O9FKT4oBgHgl3EQfgC6c/content/tmp_files/load_file.txt @@ -0,0 +1,1111 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf,len=1110 +page_content='SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Masaki Adachi, Saad Hamid, Martin Jørgensen, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Osborne Machine Learning Reserach Group, University of Oxford, {masaki, saad, martinj, mosb}@robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='uk Satoshi Hayakawa, Harald Oberhauser Mathematical Institute, University of Oxford, {hayakawa,oberhauser}@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='uk Abstract Batch Bayesian optimisation (BO) has shown to be a sample-efficient method of performing optimisa- tion where expensive-to-evaluate objective functions can be queried in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, current methods do not scale to large batch sizes – a frequent desideratum in practice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' drug discovery or simulation-based inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We present a novel algorithm, SOBER, which permits scalable and diversified batch BO with arbitrary acquisition functions, arbitrary input spaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' graph), and arbitrary kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The key to our approach is to reformulate batch selection for BO as a Bayesian quadrature (BQ) problem, which offers computational advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This reformulation is beneficial in solving BQ tasks reciprocally, which introduces the exploitative functionality of BO to BQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We show that SOBER offers substantive performance gains in synthetic and real-world tasks, including drug discovery and simulation-based inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 1 Introduction True function Vanilla TS Decoupled TS SOBER-TS SOBER-LFI 1st iteration 2nd iteration 3rd iteration 4th iteration Figure 1: Stuck behaviour in Thompson sampling (TS): Batch Bayesian optimisation for the 2D Branin function with four al- gorithms at the 1st, 2nd, 3rd, and 4th batch acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The colours represent each acquisition function (𝜋 for SOBER-LFI, upper confidence bound (UCB) for the others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The white dots, the black crosses, and the red star indicate the observed query points, the 30 batch queries, and the true global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Bayesian optimisation (BO) is a sample efficient surrogate- model-based black-box optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BO typically uses a Gaussian process (GP), whose predictive mean and vari- ance guide the optimiser where to evaluate next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The next query location is found Bayesian decision theoretically, by maximising the acquisition function (AF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As such, BO translates the black-box optimisation problem into a sequential “inner optimisation” problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Flexibility and superb sample efficiency attract a range of expensive-to- evaluate applications, such as drug discovery [1, 2], ma- terials discovery [3], hyperparameter optimisation [4, 5], and graph optimisation [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch BO parallelises this by querying multiple locations at the same time (called batch acquisition), offering much faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Despite its many successes, batch BO has several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Firstly, batch size scalability: The extensive overhead of many batch BO methods limits the batch size to be around 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Scaling to a larger batch size is preferable for a variety of real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For instance, recent high-throughput screening of microwell plates for a drug discovery contain 384 compounds in a batch experiment [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Further, in-Silico materials discovery can query thousands of simulations in parallel via computer clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As a second challenge, many batch BO methods are targeted at continuous inputs, yet the examples mentioned above of drug discovery and graphs are inherently discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Selecting batch acquisition samples in discrete space leads to combinatorial explosion arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='11832v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='LG] 27 Jan 2023 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Table 1: Comparion with batch BO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The scalability to large batch size is defined by the computational complexity that is smaller or equivalent complexity than Thompson sampling (TS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The sparsity means the more diversified samples than random Monte Carlo (MC) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑥, 𝑓 , and ℎ spaces refer to the input, function, and hyperparameter spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The black-box evidence represents the ability to estimate the Bayesian model evidence of the black-box function over input space 𝑥, equivalent to the task of Bayesian quadrature (BQ), The sampling complexity shows the upper bounds of 𝑁 discrete candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Notation remarks: 𝐶reject is the cost of rejection sampling from determinantal point process (DPP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐶acq is the cost of evaluating the acquisition function (AF) at a point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐶update is the cost of updating the GP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' including the Gram matrix inversion and hyperparameter optimisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐶func is the cost of evaluating a sampled function at a point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐼MCMC is the number of MCMC iterations until convergence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑀𝐶𝜑 is the cost of evaluating 𝑀 components of the kernels approximated by Nyström method or random Fourier feature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑃𝑁 𝑛 := 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='/(𝑁 − 𝑛)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' ≫ 𝑛𝑁 is the total number of all permutations, 𝑁 is the number of candidate points, 𝑛 is the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The computational costs are in order of 𝐶reject ≫ 𝐶update ≫ 𝐶func ≥ 𝐶acq > 𝑀𝐶𝜑, and the number of samples are in order of 𝑁 > 𝑀 > 𝑛, 𝐼MCMC ≫ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='Batch BO methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='mixed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='arbitrary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='AF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑥-space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='sparsity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑓 -space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='sparsity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='ℎ-space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='sparsity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='blackbox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='evidence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='complexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='Hallucination [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛(𝑁𝐶acq + 𝐶update + 𝑁 log 𝑁) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='Local penalisation [9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛(𝑁𝐶acq + 𝑁 log 𝑁) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='TS [10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛(𝑁𝐶func + 𝑁 log 𝑁) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='Decoupled TS [11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛(𝑁𝑀𝐶𝜑 + 𝑁 log 𝑁) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='DPP [12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛3𝑁𝐶reject ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='DPP-TS [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝐼MCMC(𝑁𝐶func + 𝑛3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='MC-SAA [14] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑃𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛 𝐶acq + 𝑃𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛 log 𝑃𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='GIBBON [15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑃𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛 𝐶acq + 𝑃𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛 log 𝑃𝑁 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='SOBER (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='𝑁𝑀𝐶𝜑 + 𝑁𝐶acq + 𝑛3 log(𝑁/𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='with increasing the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lastly, batch diversity: scalable methods, such as TS [10, 17], are too exploitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The two leftmost columns in Figure 1 exemplifies typical behaviour – getting stuck in a local minimum and wasting batch samples in the majority of TS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This tendency amplifies in noisy and multimodal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The larger the batch size we query, the larger the regret becomes, as the batch samples could have been used for exploring other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Figure 2: 2D UMAP [18] visualisation of 133,055 molecules on the QM9 dataset coloured by the dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Only 5 molecules are over 10 debye, dis- played with a 100 times large marker size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Figure 2 shows drug discovery is a case where the input space is noisy and multimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The number of good molecules (over 10 debye) is very limited (5 out of 133,055).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This needle-in-the-haystack situation is challenging for finding the bias-variance trade-off via optimising GP hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' It leads to having two modes: low-noise and high- noise in the hyperposterior space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Low-noise mode regards every tiny change as a peak, whereas high-noise sees every peak as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While the low-noise mode tends to get stuck in local minima, the high-noise mode can not find the best drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Such problems are usually solved by adopting fully Bayesian GP (FBGP), which uses the ensemble of GP models with hyperparameters sampled from hyperposteriors [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, FBGP results in significant overhead due to Markov chain Monte Carlo (MCMC) sampling and an expensive ensemble kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This is particularly challenging for expensive batch BO methods, such as determinantal point process (DPP) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch Bayesian quadrature (BQ), a surrogate-model-based black-box integration method akin to BO, suffers from ‘over-exploration’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch BQ commonly applied to simulation-based inference, which estimates the posterior and marginal likelihood (also called evidence) of black-box functions (non-closed-form simulators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Simulator-based inference is utilised in many scientific fields, such as exoplanet detec- tion [20, 21] and battery model selection [22], both of which utilise computer clusters to run the simulators embarrassingly in parallel for speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Generally, posteriors barely overlap prior distribution since users have weak knowledge of simulator parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, the surrogate model in BQ is expected to approximate only the vicinity of posterior mode (maximum a posteriori (MAP)), whereas existing BO is designed to explore all over the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We propose the hallucination-free batch acquisition method: solving optimisation as Bayesian estimation via recombination (SOBER), a scalable batch BO method with supporting light-weight FBGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The key is reformulating BO as BQ, which gives a scalable and flexible approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The two rightmost columns of Figure 1 illustrate 2 global maximum 8 10 12 14 0 2 dipole moment [D]SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints two variants of SOBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They mitigate the stuck-in-local-minima issue via diversified sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Furthermore, SOBER can solve batch BQ tasks by introducing the exploitive funcitionaly of BO into BQ for faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Table 1 summarises a comparison with the other batch BO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' With the given features, we particularly focus on the following two real-world applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' drug discovery (large-scale discrete batch BO with graph/string input, batch BO), and simulation-based inference (simultaneous estimation of both posterior and evidence, batch BQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Empirically, SOBER shows better sample-efficiency as well as faster wall-clock time computation than the baseline methods in both batch BO and BQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We emphasise that SOBER is an extension of BO and BQ methods and compatible with many existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Batch Bayesian Optimisation BO is a heuristic for solving the black-box global optimisation problems defined by: 𝑥∗ true = arg max 𝑥 𝑓true(𝑥), (1) where 𝑥∗ true is the true global maximum of the true black-box function 𝑓true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The underlying assumptions here are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓true is a black-box function, and we do not know the function information except for querying the values at certain locations (oracles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The queries are expensive, and we wish to minimise the number of queries for fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We can query multiple locations at the same time without additional overhead, and a larger batch size is desirable for faster convergence in wall-clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The total overhead of the batch acquisition algorithm should be negligible over the sequential querying cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For sample-efficient global optimisation, BO utilities a surrogate GP model 𝑓 of 𝑓true, specified as: 𝑃(𝑦|D) = GP � 𝑦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚(·), 𝐶(·, ·) + 𝜎2 𝑛I � , (2a) 𝑚(𝑥) = 𝐾(𝑥, Xob)𝐾(Xob, Xob)−1yob, (2b) 𝐶(𝑥, 𝑥′) = 𝐾(𝑥, 𝑥′) − 𝐾(𝑥, Xob)𝐾(Xob, Xob)−1𝐾(Xob, 𝑥′), (2c) where D= (Xob, yob) is the observed dataset, yob := 𝑓true(Xob) are the oracles queried in parallel, 𝑚(·) and 𝐶(·, ·) are the mean and covariance of the predictive posterior, 𝐾(·, ·|𝜃) is the kernel, 𝜃 is the kernel hyperparameters, and 𝜎2 𝑛 is the homoskedastic noise variance of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Queried observations D serially update the GP surrogate model 𝑓 so it can predict the output of 𝑓true more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚 and 𝐶 guide our beliefs toward the region where the true global maximum 𝑥∗ true possibly locates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Such a guiding mechanism is obtained through maximising an AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Batch Bayesian Quadrature BQ is a heuristic for evaluating integrals given by: 𝑍true = ∫ 𝑓true(𝑥)d𝜋(𝑥), (3) where 𝑓true is the black-box function we wish to integrate against a known probability measure 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The difference is the objective being integration, not global optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The integration problem is widely recognised in statistical learning: expectations, variances, marginalisation, ensembles, Bayesian model selection, and Bayesian model averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BQ is, like BO, solved by GP-surrogate-model-based active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch acquisition methods are also shared with batch BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The methodological differences are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BQ typically assumes a specific kernel to make the integration analytical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' RBF kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While BO requires to approximate the black-box function only in the vicinity of the global optimum, BQ needs to approximate the whole region of interest defined by the probability measure 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, BQ is a purely explorative algorithm, and the uncertainty sampling AF is often applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BQ, being sample-efficient, is the heuristic of choice if querying 𝑓true is expensive, making other approaches like Monte Carlo troublesome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Bayesian Alternately Subsampled Quadrature (BASQ) [23] proposed the scalable batch BQ method called Bayesian alternately subsampled quadrature (BASQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This achieves fast batch acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' It applies to arbitrary kernels and input spaces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' discrete and non-Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This was 3 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints enabled by a quadrature method called random convex hull quadrature (RCHQ) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' RCHQ is a kernel quadrature (KQ) method that approximates an integral of integrands in reproducing kernel Hilbert space (RKHS) by a weighted sum of a sample: ∫ ℎ(𝑥)d𝜋(𝑥) ≈ w⊤ batchℎ(Xbatch), (4) where ℎ is an element of the RKHS H generated by kernel 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wbatch and Xbatch ∈ R𝑛×𝑑 are weights and batch samples selected by RCHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑛 is batch size, 𝑑 is input dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BASQ reformulated the batch acquisition problem as an “inner integration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' RCHQ is used to choose Xbatch, which can minimise the worst-case integration error of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They are chosen based on the RKHS generated by the kernel that is the posterior covariance (2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Such a batch minimisation of integration error is equivalent to posterior variance minimisation in batch BQ [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, RCHQ is utilised as a batch BQ sampler by passing GP predictive covariance 𝐶 to the kernel 𝐾 in RCHQ alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' RCHQ is a subsample-based KQ method, which extracts a weighted 𝑛-point batch sample (wbatch, Xbatch) from 𝑁 (≫ 𝑛) candidate sample points Xrec with weights wrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The candidate points are drawn just from the prior 𝜋 or by importance sampling via another proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Such a sample forms an empirical measure 𝜋emp = (wrec, Xrec), which gives the same integral as the original probability measure 𝜋 in expectation, however, is supported on a tractable number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Based on an 𝑀-point random sample Xnys from 𝜋, we first prepare 𝑛 − 1 test functions 𝜑𝑖(𝑥) := 𝑢⊤ 𝑖 𝐾(Xnys, 𝑥) following the Nyström method, where 𝑢𝑖 ∈ R𝑀 is the 𝑖-th eigenvector of 𝐾(Xnys, Xnys) and we assume 𝑀 ≥ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Next, recombination [26, 27] extracts a weighted set of 𝑛 points (wbatch, Xbatch) from the candidate points Xrec, so that the integrals of test functions 𝜑1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' , 𝜑𝑛−1 match the integrals against the measure 𝜋emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This process allows to incorporate the information of spectral decay of the kernel 𝐾 = 𝐶, accelerating the convergence rate of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' With the weights wbatch further updated by GP and quadrature samples Xbatch, we can approximate the mean and variance of the marginal likelihood of 𝑓 over 𝜋, given by: E 𝑥∈𝜋[𝑚(𝑥)] ≈ w⊤ batch𝑚(Xbatch), (5a) Var 𝑥∈𝜋[𝑚(𝑥)] ≈ w⊤ batch𝐶(Xbatch, Xbatch)wbatch − 2w⊤ batch𝐶(Xbatch, Xrec)wrec + w⊤ rec𝐶(Xrec, Xrec)wrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (5b) Here, 𝑍 := E𝜋 [𝑚(𝑥)] is an estimate of 𝑍true in (3), and Var𝜋 [𝑚(𝑥)] is the uncertainty of this integration approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The more accurately 𝑓 approximates 𝑓true, the smaller approximation error between 𝑍 and 𝑍true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The empirical performance of RCHQ is on par with DPP while drastically improving computational cost [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3 Proposed Method: SOBER In this section, we present SOBER, which reformulates the batch BO task as the parallelisable batch BQ one, and introduces the efficient solver for the dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Global Optimisation as Bayesian Quadrature 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Duality in Probability measure We reframe batch BO as a batch BQ problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Consider the following dual formulation to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (1) [28]: 𝛿𝑥∗ true ∈ arg max 𝜋 ∫ 𝑓true(𝑥)d𝜋(𝑥), (6) where 𝛿𝑥 is the delta distribution at 𝑥, 𝜋 are probability distributions over the feasible region, and 𝑥∗ true is the location of the global maximum of 𝑓true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This black-box integral formulation is identical to BQ, so we can solve it as batch BQ with the scalable BASQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The difference to BQ is that 𝜋 is to be updated in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The more accurate the GP surrogate be- comes by obtaining more observed points, the “narrower” 𝜋 becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Ideally, 𝜋 can eventually reach the global maximum 𝛿𝑥∗ true in a single global maximum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As such, solving (6) by updating 𝜋 is equivalent to finding the maximum 𝑥∗ true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 4 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Batch Selection as Kernel Quadrature The batch selection in SOBER is performed by RCHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We propose the ‘objective-RCHQ’, which adds the constraint of maximising an AF to RCHQ, solving the following problem: Find an 𝑛-point subset Xbatch ⊂ Xrec, (7a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' max w⊤ batch𝛼(Xbatch) (7b) w⊤ batch𝜑𝑖(Xbatch) = w⊤ rec𝜑𝑖(Xrec), 1 ≤ 𝑖 < 𝑛, (7c) w⊤ batch1 = w⊤ rec1 = 1, wbatch ≥ 0, (7d) where 𝛼 is an arbitrary AF, Xrec ∈ R𝑁 ×𝑑 is a sample drawn from 𝜋, and 𝜑𝑖 are test functions determined by another sample Xnys ∈ R𝑀×𝑑 drawn from 𝜋 (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (7b)-(7d) are the constraints of AF maximisation, batch uncertainty minimisation (quadrature rules), and weight positivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' These constraints are vital for batch diversity [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' There exist theoretical guarantees for specific choices of AFs depending on the spectral decay of an integral operator determined by the pair (𝐾, 𝜋) [23, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Empirically, the convergence as 𝑛 increase is fast even without the AF term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, we can utilize this degree of freedom for incorporating other AFs as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Approximating the Distribution 𝜋 To obtain good candidate points Xbatch, it is essential to estimate 𝜋 from the surrogate model 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, as we do not know the true global maximum 𝑥∗ true a priori, evaluating the probability of global maximum 𝑃(𝑥∗ true|D) is unachievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, we must approximate 𝜋 with a probability distribution that asymptotically assimilates to 𝑃(𝑥∗ true|D) when collecting more observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We discuss two variants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' TS and likelihood-free inference (LFI), addressing the approximation of 𝜋 differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' SOBER-TS adopts the current maximum 𝑃(𝑥∗|D) as 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This definition is well known as TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We prepare 𝑁 candidates with parallel decoupled TS [11], which alleviates the sampling overhead and fosters batch diversity via sparsifying the sampled functions from GP posterior ( 𝑓 -space sparsity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As such, SOBER-TS can be understood as re-selecting the TS samples that can maximise the sum of the AF and minimise the variance in predictive posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, TS can not provide the closed-form 𝑃(𝑥∗|D) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' SOBER-LFI offers the faster alternative and closed-form 𝜋 based on LFI approach, which is delineated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Temporary Likelihood by Summary Statistic Gutmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' [30] introduced the “tentative” likelihood 𝑃( 𝑓 (𝑥) > 𝜂|𝑥, D) based on the current global max-value 𝜂 := max 𝑓 (𝑥|D), given by: 𝐿(𝑥|𝜃, 𝜎2 𝑛, 𝜂) := 𝑃( 𝑓 (𝑥|D) ≥ 𝜂) ∝ Φ � 𝑚(𝑥|𝜃) − 𝜂 √︁ 𝐶(𝑥, 𝑥|𝜃) + 𝜎2𝑛 � , (8) where Φ is cumulative density function (CDF) of normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Noise-free formulation of this likelihood definition is the same as the probability of improvement acquisition function [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' When 𝜂 = 𝑓true(𝑥∗ true) = 𝑓 (𝑥∗ true) is satisfied, 𝑃( 𝑓 (𝑥) > 𝜂) becomes the delta function 𝛿𝑥∗ true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Now we can estimate posterior belief of the maximum location 𝜋 via the following Bayes’ rule: 𝜋(𝑥) ∝ 𝐿(𝑥)𝜋′(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (9) For each iteration, 𝐿(·|𝜃, 𝜎2 𝑛, 𝜂) is updated and 𝜋′ is 𝜋 of the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' In the beginning, 𝜋′ is the prior belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' In the typical BO setting, we use the bounded uniform prior, but we can incorporate the experts’ knowledge via stronger prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We do not examine the difference of prior 𝜋′ in this paper, but the following papers show its strong empirical performance [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4 Fast Marginalisation by Quadrature Distillation While BO works well with a single set of optimised hyperparameters (type-II maximum likelihood estimation (MLE)) on most functions, some noisy cases such as drug discovery shown in Figure 2 requires FBGP modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' GP marginalisation is typically done with MC integration via sampling from hyperposterior Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, its convergence rate O(1/√𝑛) is poor, requiring thousands of ensemble GP models for marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This significantly slows down batch BO computation as BASQ requires such an ensemble kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, we introduce quadrature distillation trick for faster marginalisation by using the smaller set of weighted hypersamples selected by the quadrature, distilling random 5 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints hypersamples from hyperprior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This method is based on BQ marginalisation in hyperparameter space with the hyper-GP and hyperlikelihood [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We modified this method by sparsifying the BQ weights with a smaller set of positive weights 𝑤QD via RCHQ for faster computation, resulting the following approximation of marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐿(𝑥) := ∫ 𝐿(𝑥|Θ)dΠ(Θ) ≈ wQD𝐿(𝑥|𝚯QD), (10) where 𝑤QD ∈ R𝐻 are the distilled weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This new algorithm, quadrature distillation, permits fast marginalisation of like- lihood as well as acquisition function without time-consuming MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This is particularly helpful for expensive AF, such as information-theoretic AFs [35, 36, 37, 38] and FBGP-based AFs [39, 40, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, this FBGP modelling is an additional option of SOBER, of which additional overhead limits the suitable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' See Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5 Batch Bayesian Quadrature by Dual GPs As LFI likelihood is originally introduced to solve simulation-based inference, now our SOBER-LFI is also capable of solving simulation-based inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The main difference between BASQ and SOBER is whether or not 𝜋 is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, SOBER-LFI can solve a batch BQ task by inheriting the BASQ modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Sampling from 𝜋 efficiently squeezes the region to be explored only the vicinity of MAP, which is the global maximisation problem as BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' When compared to the original BO-based LFI [30], SOBER has two benefits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' evidence estimation and exact posterior estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While BOLFI is designed to approximate only the posterior distribution using the "approximated" likelihood definition, SOBER can estimate both the posterior and model evidence in one go, using the exact likelihood definition based on BASQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We place dual GPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' one for sampling with BO-LFI, one for BQ modelling with BASQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='6 Summary of Contribution In summary, we reformulated the batch BO task as the dual problem defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Now, estimating the global maximum becomes equivalent to updating 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We introduced two variants: TS and LFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Both offer approximation of 𝜋 using the information from the current surrogate model 𝑓 in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Once the empirical measure 𝜋emp = (wrec, Xrec) is constructed by sampling from 𝜋, the ‘objective-RCHQ’ selects the batch samples that minimise the GP posterior variance and maximise the user-defined AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Finally, quadrature distillation provides efficient FBGP modelling, and the BASQ formulation solves batch BQ tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 4 Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Sampling from 𝜋 Table 2: SOBER algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Algorithm 1: SOBER Input: prior 𝜋′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' hyperprior Π′(Θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' observed dataset D = (Xob,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' yob) Output: maximum max[Xob],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' evidence E[𝑚(𝑥)] 1: 𝑓 ← InitialiseGP(D) 2: while convergence: 3: if FGBP: 4: wQD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯QD ← QuadDistil( 𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Π′(Θ)) 5: 𝜋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐾(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' ·) ← FBGP( 𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜋′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wQD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯QD) 6: else: 7: 𝜋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐾(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' ·) ← Type-II MLE( 𝑓 ) 8: wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys ∼ Subsampling(𝜋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜋′) 9: Xbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wbatch ← AutoKQ(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝐾(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' ·)) 10: ybatch = ParallelQuery( 𝑓true(Xbatch)) 11: D ← D ∩ Dbatch 12: 𝑓 ← UpdateGP( 𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D) 13: 𝜋′ ← 𝜋 14: E[𝑚(𝑥)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Var[𝑚(𝑥)] ← KQ( 𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wbatch) 15: return max[Xob],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' E[𝑚(𝑥)] SOBER is a sample-based gradient-free approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' and so can handle discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' continuous or mixed inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The only difference is the sampler for Xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The simplest scenario is if all discrete candidates are available a priori and enumerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As RCHQ accepts weighted samples 𝜋emp = (wrec, Xrec) for im- portance sampling, all we have to do is to calculate the weights wrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This is simply the normalised pos- terior 𝜋(Xrec)/ � 𝜋(Xrec) · 1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' If all combinations are innumerable or unavailable, we sample Xrec from the discrete prior 𝜋′, which the user can define the arbi- trarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Once sampled, the procedure is the same: we compute wrec, then pass the empirical measure 𝜋emp to RCHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We update the hyperparameters of the prior 𝜋′ via MLE from the weighted sample (wrec, Xrec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Continuous space can be regarded as innumerable discrete space, so it can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The only difference is the prior update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We use weighted kernel density estimation (KDE) for the update, for speed and flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Mixed space is the combination of discrete and con- tinuous space, which also can be regarded as innumerable discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The prior update is the combination of the 6 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints above two by assuming the discrete and continuous parameters are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Importantly, the prior does not need to precisely approximate 𝜋 as the importance weights wrec will correct the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 for each detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Automatic Kernel Quadrature Selection Batch selection in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 is performed by RCHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, the sharper 𝜋 becomes (as it inevitably does over iterations), the slower the RCHQ convergence rate becomes, due to long-tail eigenvalue decay of the kernel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' To avoid this, we propose automatic KQ selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We automatically switch KQ methods when RCHQ becomes inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The alternative KQ method is kernel thinning, which is an eigenvalue-decay-independent and subsample-based KQ method [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The choice between these two KQ methods can be made automatically by comparing the worst-case error in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The third term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (5b) is not dependent on the KQ methods, so we can avoid expensive 𝑁 × 𝑁 computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' RCHQ is selected in the early stage because the smooth kernel makes the eigenvalue decay short-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' In the late stage, the kernel thinning is chosen when the region is narrowed (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Table 2 illustrates pseudo-code for SOBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 5 Related work Batch Bayesian Optimisation Batch BO methods are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The existing scalable batch BO methods are all TS-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Moreover, none of the baseline methods can offer ℎ-space sparsity for fast FBGP and blackbox evidence estimation (as with the BQ task) when applied to the Bayesian inference task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As for mixed space BOs, existing works [6, 42] propose ways to avoid the combinatorial explosion, but none of them offer scalable batch acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch Bayesian optimisation for drug discovery BO for drug discovery can be classified into two categories: variational autoencoder (VAE) and discrete BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The former embeds molecules into continuous low dimensional features using a VAE, performs BO in the latent space, decoding the queried molecules back into the original high dimensional representation [1, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The latter directly models the discrete spaces with bespoke kernels [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Our focus in this paper is on the batching method and is model agnostic, so we do not compare model variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch Bayesian Quadrature BQ has focused on model evidence estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' few works support posterior inference simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The first to study simultaneous estimation of the posterior distribution and evidence was VBMC [46, 47], based on variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Yet, batch VMBC has not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch WSABI [48] was the first to extend BQ to a batch setting, using local penalisation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Later, BASQ achieved efficient parallelization with kernel recombination and currently is the only model for scalable batching [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, the original BASQ is over-explorative, requiring an accurate prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Log-BASQ [22] mitigates this by introducing the log-warped GP surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' SOBER balances exploration and exploitation by updating 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' There exists many works on simulation-based inference, most focus only on posterior inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BQ and nested sampling (NS) [49] are the only methods for simultaneous inference of posterior and evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' NS is a MCMC-based sampler, so it is not sample-efficient and its state-of-the-art method showed the slower convergence rate than BQ methods [23, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 6 Experiments Table 3: Comparison of experimental conditions for batch BO benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' synthetic functions real-world datasets Ackley [42] Rosenbrock [42] anti-malaria [51] polar solvent [52] total dim 𝑑 23 7 2048 2048 continuous dim 3 1 categorical dim 6 binary dim 20 2,048 2,048 countable data 20,746 133,055 batch size 𝑛 200 100 100 200 kernel RBF RBF Tanimoto Tanimoto prior mixed mixed discrete discrete continuous prior U{−1, 1} U{0, 10} discrete prior Ber(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5) Cat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5) Cat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5) Cat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5) 1Code is open sourced at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='com/anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 7 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Ackley Rosenbrock Anti-Malarial drug Polar solvent for batteries Log10 regret Log10 regret Log10 min −Log10 max Log10 overhead [s] random 15 30 0 15 30 0 15 30 0 15 30 0 15 30 0 15 30 0 15 30 0 15 30 TS decoupled TS DPP-TS SOBER-TS SOBER-LFI Iteration Iteration Iteration Iteration 0 2 3 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 0 3 3 0 3 3 1 2 Log10 overhead [s] Log10 overhead [s] Log10 overhead [s] 1 2 3 0 1 f(x) f(x) Figure 3: We evaluate SOBER across 2 synthetic functions and 2 real-world drug discovery datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Top: Log regret or log best observations, Bottom: Log overhead in seconds as function of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lines and shaded area denote mean ± 1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch size is 100 or 200 (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' SOBER-LFI consistently outperforms all four baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We now test the sample efficiency and sampling overhead of SOBER against six baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Our code is built upon PyTorch-based libraries [53, 54, 14, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' All experiments were averaged over 16 repeats, computed in parallel with multicore CPUs2 for a fair comparison, although GPUs can accelerate the SOBER due to its highly parallelisable nature and the GPyTorch library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Batch Bayesian Optimisation For large-scale batch BO methods, we compared with TS, decoupled TS, and DPP-TS based on Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The test datasets are two synthetic datasets and two real-world datasets shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Synthetic functions are inherited from the previous work [42], with modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Small molecules in the two real-world drug discovery datasets are described as SMILES [55], which are variable length strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We transform SMILES to fingerprints [56], which are sparse (2048 dimensional) bit vectors, and use a Tanimoto kernel [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 20,000 candidates (𝑁 = 20, 000) are drawn from the prior distribution (See details in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Batch Bayesian Quadrature Table 4: Comparison of experimental condi- tions for batch BQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2 RC-pair model 5 RC-pair model dim 𝑑 6 12 batch size 𝑛 100 100 kernel RBF RBF prior Gaussian Gaussian We target simultaneous estimation of posterior and model evidence in simulation-based inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We compare with batch WSABI, BASQ, and log-BASQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We select a Gaussian prior and kernel for a fair comparison with batch WSABI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Note that SOBER can take arbitrary prior and kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We use two real-world datasets from the lithium-ion battery model selection [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The problem is modified to more challenging conditions in order to examine the BQ methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The prior variance is set to be one million times as large as the true posterior’s (which is known from pre-experiments with exhaustive MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=') See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 for full experimental details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Results Figure 3 shows that SOBER-LFI is the top-performing of both synthetic and real-world drug discovery tasks while maintaining its overhead to be equal to or lower than TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' On Ackley, which has many local maxima, TS-based methods get stuck in the local maxima, yet SOBER successfully finds the global maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Even on the uni-modal Rosenbrock function, which should be advantageous for hill-climbing algorithms like TS, SOBER outperforms too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This clearly shows that updating 𝜋 can efficiently squeeze the sampling region around the global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' A similar tendency is found in drug discovery tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Particularly, the polar solvent dataset clearly exemplifies the stuck behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' TS 2Performed on MacBook Pro 2019, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4 GHz 8-Core Intel Core i9, 64 GB 2667 MHz DDR4 8 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints converges fast in the early stage, but it can not get out of the local maxima, resulting in a final regret equivalent to random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Still, SOBER successfully finds better molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2 RC pair 5 RC pair Log10 |log evidence | Log10 RMSE of posterior batchWSABI 10 20 0 10 20 0 10 20 0 10 20 BASQ logBASQ SOBER-LFI Iteration Iteration 3 1 5 3 1 0 2 3 1 2 Log10 |log evidence | Log10 RMSE of posterior 5 3 1 Figure 4: We evaluate SOBER across 2 real-world battery simulation-based inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Top: Log of log evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Bottom: Log RMSE of posterior as function of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lines and shaded area denote mean ± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch size is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' SOBER-LFI consistently outperforms all three baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Figure 4 illustrates that SOBER also outperformed the BQ baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' When performing inference with a weakly-informative prior, finding the posterior mode and reducing the variance only around its vicinity is the key to fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While the original BASQ over-explores the prior distribution and shows plateaus, logBASQ allevi- ates this behaviour via log-warp modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Nonetheless, SOBER showed significantly faster convergence than all competitors in both posterior and evidence inference of all tasks by squeezing 𝜋 toward the posterior mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We used type-II MLE optimised LFI AF throughout the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 further illustrates the effect of AF, batch size 𝑛, and hyperparameters (𝑁, 𝑀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Amongst the AFs, information-theoretic AFs (MES and GIBBON) can boost the convergence rate than the LFI AF with a negligible overhead increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The hyperparameters 𝑁, 𝑀 are quite intuitive: the discretisation accuracy of the input 𝑥 and function 𝑓 spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Unsurprisingly, the larger these values become, the faster the convergence becomes but the larger the overhead is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Our default values (𝑁 = 20, 000, 𝑀 = 500) are competitive throughout the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' These values can be adjusted to the cost of queries [23] Moreover, the larger the batch size 𝑛 becomes, the faster the convergence, even for large batch sizes3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The ablation study shows that each component (temporary likelihood 𝜋, the iterative 𝜋 update, and the objective-RCHQ) contributes to faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Additionally, FBGP with quadrature distillation is shown to be capable of convergence acceleration, especially in noisy functions, while maintaining the overhead is competitive enough to the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 7 Discussion We introduced a hallucination-free approach, SOBER, capable of scalable batch acquisition for both BO and BQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We identified three problems of existing batch BO methods: batch diversity, batch size scalability, and combinatorial explosion in innumerable discrete/mixed inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch BQ reformulation can make the batch selection more diversified and parallelisable, the objective RCHQ offers a scalable batch selection solver, and updating 𝜋 can avoid the combinatorial explosion via squeezing the region to be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Updating 𝜋 also solves the overexploration issue in batch BQ, resulting in faster convergence on both synthetic and real-world datasets, when compared against existing batch BO and BQ approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Two limitations of SOBER are that it is not suitable for asynchronous settings and the algorithm cannot be distributed to each node in computer cluster such as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Applicability to high-dimensional BO is also an open problem as efficient sampling from the posterior over the maximiser is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This can be true also for low-dimensional embeddings, such as those produced by the GPLVM model [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, for some embeddings, such as linear embeddings or VAEs, this sampling can be done efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Acknowledgments We thank Leo Klarner for the insightfuk discussion of Bayesian optimistaion for drug discovery, Samuel Daulton, Binxin Ru, and Xingchen Wan for the insightful discussion of Bayesian optimisation for graph and mixed space, Ondrej Bajgar for his helpful comments about improving the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Masaki Adachi was supported by the Clarendon Fund, the Oxford Kobe Scholarship, the Watanabe Foundation, and Toyota Motor Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Harald Oberhauser was supported by the DataSig Program [EP/S026347/1] and the Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Martin Jørgensen was supported by the Carlsberg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3We emphasise the convergence acceleration in the large batch is not typically achievable with other baseline methods, such as GIBBON (See 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Marginalisation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Overview A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Sparsifying Hyperposterior Samples by Bayesian Quadrature First, we apply BQ for sparsifying the hyperposterior samples using the following discrete approximation of the hyperposterior in the closed form: 𝑃(Θ|D) = 𝑃(D|Θ)𝑃(Θ) 𝑃(D) , ≈ 𝑚hyper(Θ)Π′(Θ) ∫ 𝑚hyper(Θ)dΠ′(Θ) , = 𝑚hyper(Θ)Π′(Θ) �∫ 𝐾hyper(Θ, 𝚯)𝐾hyper(𝚯, 𝚯)−1dΠ′(Θ) � 𝑃(D|𝚯) , = 𝑚hyper(Θ)Π′(Θ) w′⊤ BQL , (11) where 𝑃(D|Θ) ∼ GP(ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚hyper(·), 𝐶hyper(·, ·)), (12a) w′ BQ := ∫ 𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1dΠ′(Θ), (12b) L := 𝑃(D|𝚯hyper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (12c) Here, we place hyper-GP on the marginal likelihood L(·) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We draw hypersamples 𝚯hyper ∈ R𝐵×𝐷, where 𝐵 is the number of hypersamples and 𝐷 is the number of hyperparameter types, from the hyperprior Π′(·) := 𝑃(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Then, we evaluate the marginal likelihood L = L(𝚯hyper) in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We select multivariate normal distribution for hyperprior Π′(Θ) := N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇hyper, 𝚺hyper) based on [59], and Gaussian kernel for hyper-GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Then, weights w′ BQ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (12b) become analytical: w′ BQ = ∫ 𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1dΠ′(Θ), = 𝑣 √︁ |2𝜋W| �∫ N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯hyper, W)N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇hyper, 𝚺hyper)dΘ � 𝐾hyper(𝚯hyper, 𝚯hyper)−1, = 𝑣 √︁ |2𝜋W|N (𝚯hyper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇hyper, W + 𝚺hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1, (13) (14) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Parabolic Transform for Max-value Estimation Many acquisition functions, including the temporary likelihood 𝐿(·|𝜃, 𝜎2 𝑛, 𝜂), are dependent on the quadrature hyperpa- rameters Θ = (𝜃, 𝜎2 𝑛, 𝜂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Particularly, estimating the current maximum location conditioned on 𝜃 is computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, we inherit the parabolic transform of GP surrogate model from [60]: 𝑓 (𝑥|Θ) = 𝜂 − 1 2𝑔(𝑥)2, (15a) := GP� 𝑓 (𝑥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚(𝑥|𝜃, 𝜂), 𝐶(𝑥, 𝑥|𝜃) + 𝜎2 𝑛I�, (15b) 𝑔(𝑥|𝜃) := GP�𝑔(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚𝑔(·|𝜃), 𝐶𝑔(·, ·|𝜃)�, (15c) 𝑚(𝑥|𝜃, 𝜂) := 𝜂 − 1 2 � 𝑚𝑔(𝑥)2 + 𝐶𝑔(𝑥, 𝑥) � , (15d) 𝐶(𝑥, 𝑥′|𝜃) := 1 2𝐶𝑔(𝑥, 𝑥′)2 + 𝑚𝑔(𝑥)⊤𝐶𝑔(𝑥, 𝑥′)𝑚𝑔(𝑥′), (15e) where 𝑓 (·) is the surrogate model that approximates 𝑓true(·), 𝑔(·) is the square-root warped GP [61] of 𝑓 (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The predictive mean 𝑚𝑔(·) and covariance 𝐶𝑔(·, ·) of the warped GP 𝑔(·) are expressed with normal GPs in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (2b) - (2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The predictive mean 𝑚(·) and covariance 𝐶(·, ·) are approximated via moment-matching [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D𝑔 = (Xob, y𝑔,ob) is the observed data for the warped GP, and y𝑔,ob := √︁ 2(𝜂 − yob).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 13 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Now, 𝜂 becomes a GP hyperparameter via y𝑔,ob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, we can estimate the hyperposterior Π(·) := 𝑃(Θ|D) via the marginal likelihood of 𝑓 (·), given by: L(Θ) := 𝑃(D|Θ) = N � yob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚(Xob|Θ), 𝐶(Xob, Xob|𝜃, 𝜎2 𝑛) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (16) Therefore, we can marginalise likelihood via hyperposterior: 𝐿(𝑥) = ∫ 𝐿(𝑥|Θ)dΠ(Θ), (17) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Integral Operator Distillation With the given weighted samples (wBQ, 𝚯hyper), we can approximate the marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For instance, the marginalised likelihood 𝐿(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (17) can be approximated as follows: 𝐿(𝑥) ≈ ∫ 𝐿(𝑥|Θ) 𝑚hyper(Θ) w′⊤ BQL dΠ′(Θ), ≈ w⊤ BQ𝐿(𝑥|𝚯hyper), (18) where wBQ := w′ BQ ◦ L/(w′⊤ BQL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Both marginalisation for 𝐿(·) and 𝑃(D) share the same signed measure, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (18) becomes a good approximation [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Notably, we can recycle the same weights and samples for other marginalisation, such as GP predictive posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Furthermore, this formulation does not require sampling from hyperposterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, we can avoid time-consuming MCMC sampling for marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, even though the random sampling from hyperprior 𝑃(Θ) is fast, it is sample inefficient for integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, we wish to approximate the marginalisation with the minimal number of quadrature samples 𝐻 (𝐻 ≪ 𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We adopt a quadrature distillation trick with RCHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The idea is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' distilling the dataset Dhyper = (𝚯hyper, L(𝚯hyper)) to small dataset DQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We set RCHQ arguments as the empirical measure Π(Θ) := 𝚯 and the kernel 𝐶BQ(·, ·), then RCHQ returns the small sparse samples DQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Compact wQD and 𝚯QD can be obtained via retraining hyper-GP with DQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This permits efficient computation of marginalisation, including expensive AF, such as information-theoretic AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Quadrature Distillation Algorithm Table 5: Quadrature distillation algorithm Algorithm 2: Quadrature distillation 1: 𝚯hyper ∼ Π′(·) # random sampling from hyperprior 2: Dhyper = (𝚯hyper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' L(𝚯hyper)) # dataset construction 3: ℓ(Θ) ← TrainGP(Dhyper) # train hyper-GP 4: Π′ opt(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' whyper ← HyperpriorOpt(ℓ(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯hyper) # optimise hyperprior a posteriori 5: 𝚯QD = RCHQ(ℓ(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯hyper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' whyper) # integral operator distillation 6: DQD = (𝚯QD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' L(𝚯QD)) # extract corresponding L from DBQ 7: ℓQD(Θ) ← TrainGP(DQD) # retrain hyper-GP 8: wQD = BQ(ℓQD(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Π′ opt(·)) # compute BQ weights in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (33) - (34) 9: return wQD, 𝚯QD The algorithm flow of the quadrature distillation is shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Each procedure will be explained step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Train Hyper-GP First, we place hyper-GP on the hyperlikelihood L(·) with the Gaussian kernel 𝐾hyper(·, ·): ℓ ∼ GP(ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚hyper(·), 𝐶hyper(·, ·)) (19) 𝑚hyper(Θ) = 𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1L (20) 𝐶hyper(Θ, Θ′) = 𝐾hyper(Θ, Θ′) − 𝐾hyper(Θ, 𝚯hyper)𝐾hyper(𝚯hyper, 𝚯hyper)−1𝐾hyper(𝚯hyper, Θ′) (21) 𝐾hyper(Θ, 𝚯hyper) := 𝑣 √︁ |2𝜋W|N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯hyper, W) (22) 14 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints where 𝑣 is kernel variance and W := 𝑙I is the diagonal covariance matrix whose diagonal elements are the lengthscale 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Training hyper-GP is done with type-II MLE with L-BFGS-G [62], via maximising the marginal likelihood of the hyper-GP N � L(𝚯hyper);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚(𝚯hyper|𝑡), 𝐶(𝚯hyper, 𝚯hyper|𝑡) � , where 𝑡 is the hyper-hyperpameter of the hyper-GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Optimising Hyperprior a Posteriori Now, hyperevidence 𝑃(D) is the closed-form via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, we can optimise the hyperprior 𝑃(Θ) a posteriori so as to maximise the hyperevidence 𝑃(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This is equivalent to minimise the difference between hyperprior Π′(·) and hyperlikelihood (marginal likleihood) L(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, this process equals to approximating the mixture of Gaussian process (the mean of hyper-GP) with a unimodal Gaussian (hyperprior), which can be computed analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The mean predictive posterior 𝑚hyper(·) can be written as the mixture of Gaussians: 𝑚hyper(Θ) = N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯hyper, W)L′ ∝ ∑︁ 𝑖 𝐿′′ 𝑖 N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Θhyper,𝑖, W) L′ := 𝑣 √︁ |2𝜋W|𝐾hyper(𝚯hyper, 𝚯hyper)−1L L′′ := L′/(L′1) (23) Next, we optimise the hyperprior via maximising the analytical hyperevidence 𝑃(D) = w′⊤ BQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The optimised hyperprior is the approximation of the weighted Gaussian mixture in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (23): Π′ opt(Θ) = N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇opt hyper, 𝚺opt hyper) (24) 𝜇opt hyper := L′′𝚯hyper (25) 𝚺opt hyper := W + 𝚯⊤ hyperP𝚯hyper − (𝚯hyperP)𝚯hyperP (26) P := diag L′′ (27) Now we wish to correct the sample distribution from Π′(·) to Π′ opt(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This can be done by the importance sampling, with the following weights: whyper := N (𝚯hyper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇opt hyper, 𝚺opt hyper) N (𝚯hyper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇hyper, 𝚺hyper) (28) = ZN (𝚯hyper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇IS hyper, 𝚺IS hyper) (29) where 𝚺IS hyper := � 𝚺opt,−1 hyper − 𝚺−1 hyper �−1 (30) 𝜇IS hyper := 𝚺IS hyper � 𝚺opt,−1 hyper 𝜇opt hyper − 𝚺−1 hyper𝜇hyper � (31) Z := |𝚺hyper| |𝚺hyper − 𝚺opt hyper| 1 N (𝜇opt hyper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇hyper, 𝚺hyper − 𝚺opt hyper) (32) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Integral Operator Distillation Then, we perform RCHQ with the following arguments: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' the empirical measure Πemp = (whyper, 𝚯hyper) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' the kernel 𝐾(Θ, Θ′) = 𝐶hyper(Θ, Θ′) 15 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Then, the sparse samples 𝚯QD will be returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Retraining GP yields ℓQD ∼ GP(ℓQD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚QD(·), 𝐶QD(·, ·)) as the "distilled" GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We calculate the BQ weights with this GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The closed-form BQ weights can be calculated as follows: w′ QD = ∫ 𝐾QD(Θ, 𝚯QD)𝐾QD(𝚯QD, 𝚯QD)−1dΠ′ opt(Θ), = 𝑣 √︁ |2𝜋W| �∫ N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝚯QD, W)N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇opt hyper, 𝚺opt hyper)dΘ � 𝐾QD(𝚯QD, 𝚯QD)−1, = 𝑣 √︁ |2𝜋W|N (𝚯QD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇opt hyper, W + 𝚺opt hyper)𝐾QD(𝚯QD, 𝚯QD)−1, (33) wQD := w′ QD ◦ L/(w′⊤ QDL) (34) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4 Fast Fully Bayesian Gaussian Process The resulting measure (wQD, 𝚯QD) can approximate the marginalisation over the hyperposterior Π(·), as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Other quantities can also be marginalised, such as: 𝑚QD(𝑥) = ∫ 𝑚(𝑥|Θ)dΠ(Θ), ≈ w⊤ QD𝑚(𝑥|𝚯QD), (35) 𝑉QD(𝑥) = ∫ 𝐶(𝑥, 𝑥|Θ)dΠ(Θ), ≈ w⊤ QD � 𝐶(𝑥, 𝑥|𝚯QD) + 𝑚2(𝑥|𝚯QD) � − 𝑚2 QD(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (36) 𝐶QD(𝑥, 𝑥′) = ∫ 𝐶(𝑥, 𝑥′|Θ)dΠ(Θ), ≈ wQD1 (wQD1)2 − w2 QD1 𝐻 ∑︁ 𝑖 𝑤i, QD ��𝑚(𝑥|Θi, QD) − 𝑚QD(𝑥)�𝑇 �𝑚(𝑦|Θi, QD) − 𝑚QD(𝑦)�� , (37) This marginalisation approximation is also beneficial to approximate the expensive AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Computations of marginal AFs, information-theoretic AFs and FBGP-based AFs will be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Marginal Expected Improvement Acquisition Function The marginal expectation improvement (EI) AF [63] can be calculated with FBGP formulation: 𝛼EI(𝑥) := w⊤ QD � (𝑚(𝑥|𝚯QD) − 𝜼) ◦ Φ(𝑍|𝚯QD) � + w⊤ QD �√︃ 𝐶(𝑥, 𝑥|𝚯QD) ◦ 𝜙(𝑍|𝚯QD) � (38) 𝑍 := 𝑚(𝑥|𝚯QD) − 𝜼 √︁ 𝐶(𝑥, 𝑥|𝚯QD) (39) where Φ(𝑥), 𝜙(𝑥) are CDF and probability density function (PDF) of the normal distribution, 𝜼 ∈ 𝚯QD is the distilled max value 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Marginal Upper Confidence Bound Acquisition Function The marginal upper confidence bound (UCB) AF [64] can be calculated with FBGP formulation: 𝛼UCB(𝑥) := w⊤ QD𝑚(𝑥|𝚯QD) + √︁ 𝛽w⊤ QD √︃ 𝐶(𝑥, 𝑥|𝚯QD) (40) where 𝛽 is the BO hyperparameter, usually 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 16 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Max-value Entropy Search Acquisition Function The max-value entropy search (MES) AF [37] can be calculated via FITBO formulation [60]: 𝛼FITBO(Xhyper|D) := 𝐻[𝑝(𝑦|D, Xhyper)] − E𝑝(𝜂|D) � 𝐻[𝑝(𝑦|D, Xhyper, 𝜂)] � , (41) 𝑝(𝑦|D, Xhyper) = ∫ 𝑝(𝑦|D, Xhyper, 𝜂)d𝑝(𝜂|D), (42) 𝐻[𝑝(𝑦|D, Xhyper)] = ∫ ln 𝑝(𝑦|D, Xhyper)d𝑝(𝑦|D, Xhyper), (43) E𝑝(𝜂|D) � 𝐻[𝑝(𝑦|D, Xhyper, 𝜂)] � = ∫ 𝐻[𝑝(𝑦|D, Xhyper, 𝜂)]d𝑝(𝜂|D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (44) FITBO AF can be discretised via MC integration: 𝛼FITBO(Xhyper|D) := 𝐻 � 1 𝑀 𝑀 ∑︁ 𝑖 𝑝(𝑦|D, Xhyper, 𝜃𝑖, 𝜂𝑖) � − 1 2𝑀 𝑀 ∑︁ 𝑖 log[2𝜋𝑒(𝐶(Xhyper, Xhyper|D, 𝜃𝑖, 𝜂𝑖) + 𝜎𝑛,𝑖)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (45) Quadrature distillation can approximate the above AF as: 𝛼FITBO(Xhyper|D) ≈ 𝐻 � w⊤ QD𝑚(Xhyper|𝚯QD) � − 1 2w⊤ QD log[2𝜋𝑒(𝐶(Xhyper, Xhyper|D, 𝚯QD) + 𝝈2 𝑛,QD)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (46) For faster computation, moment-matching approximation yields the first term as: 𝐻 � 1 𝑀 𝑀 ∑︁ 𝑖 𝑝(𝑦|D, Xhyper, 𝜃𝑖, 𝜂𝑖) � ≈ 1 2 log[2𝜋𝑒(Var[𝑦] + 𝜎2 𝑛,𝑖)], (47) Var[𝑦] = 1 𝑀 𝑀 ∑︁ 𝑖 � 𝐶(Xhyper, Xhyper|𝜃𝑖) + 𝑚2(Xhyper|𝜃𝑖) � − E[𝑦Θ𝑖]2, (48) E[𝑦] = 1 𝑀 𝑀 ∑︁ 𝑖 𝑚(Xhyper|𝜃𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (49) Hence, 𝛼FITBO(Xhyper|D) ≈ 1 2 log[2𝜋𝑒(Var[𝑦] + w⊤ QD𝝈2 𝑛,QD)] − 1 2w⊤ QD log[2𝜋𝑒(𝐶(Xhyper, Xhyper|D, 𝚯QD) + 𝝈2 𝑛,QD)], (50) Var[𝑦] = w⊤ QD � 𝐶(Xhyper, Xhyper|𝚯QD) + 𝑚2(Xhyper|𝚯QD) � − � w⊤ QD𝑚(Xhyper|𝚯QD) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (51) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4 Bayesian Query-by-Committee Acquisition Function The Bayesian query-by-committee (B-QBC) AF is defined by [19]: 𝛼BQBC(𝑥) := Var𝑝(Θ|D) � 𝑚(𝑥|Θ) � , (52) = E𝑝(Θ|D) � (𝑚(𝑥|Θ) − ˆ𝑚(𝑥))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (53) The quadrature distillation approximates as follows: 𝛼BQBC(𝑥) ≈ w⊤ QD � (𝑚(𝑥|𝚯QD) − w⊤ QD𝑚(𝑥|𝚯QD))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (54) A variant AF, query by mixture of Gaussian process (QB-MGP), can also be approximated by the quadrature distillation: 𝛼QB-MGP(𝑥) := E𝑝(Θ|D) � 𝐶(𝑥, 𝑥|Θ) � + E𝑝(Θ|D) � (𝑚(𝑥|Θ) − ˆ𝑚(𝑥))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (55) ≈ w⊤ QD𝐶(𝑥, 𝑥|𝚯QD) + w⊤ QD � (𝑚(𝑥|𝚯QD) − w⊤ QD𝑚(𝑥|𝚯QD))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (56) 17 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints B Algorithm B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Subsampling algorithm The algorithm flow of the subsampling is shown in the Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The details will be explained step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Table 6: Subsampling algorithm Algorithm 3: Subsampling 1: Xrec ∼ 𝜋′(·) # sampling from prior 2: wrec = 𝐿(Xrec) 𝜋′(Xrec) · 𝜋′(Xrec)1 𝐿(Xrec)1 # compute the weights 3: if len(wrec > 0) < 𝑛 : 4: 𝜋′(·) ← 𝜋′ initial(·) # return to the initial prior when overexploitive 5: if continuous: 6: 𝜋(·) = WKDE(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec) # weighted kernel density estimation 7: Xrec ∼ 𝜋(·) # resample from WKDE 8: wrec = 𝐿(Xrec) 𝜋(Xrec) · 𝜋(Xrec)1 𝐿(Xrec)1 # recompute the weights 9: else if discrete and enumerable: 10: Xrec = 𝜋′(·) # all discrete candidates 11: wrec = 𝐿(Xrec) 𝐿(Xrec)⊤1 # normalised weights 12: else if innumerable discrete: 13: 𝜋(·) ← OptHypersMLE(𝜋′(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec) # MLE hyperparameter optimisation 14: Xrec ∼ 𝜋(·) # resample from WKDE 15: wrec = 𝐿(Xrec) 𝜋(Xrec) · 𝜋(Xrec)1 𝐿(Xrec)1 # recompute the weights 16: else mixed: 17: 𝜋(·) ← CombineBothPrior(𝜋′(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec) # Combine continuous and discrete prior 18: wrec = 𝐿(Xrec) 𝜋(Xrec) · 𝜋(Xrec)1 𝐿(Xrec)1 # recompute the weights 19: Xnys ∼ Deweighted(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec) # deweighted random subset extraction 20: return wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Weighted Kernel Density Estimation The mean and covariance of the weighted kernel density estimation (WKDE) is estimated with the unbiased data covariance matrix given by: 𝜇wkde := wrecXrec, (57) 𝚺wkde := wrec1 (wrec1)2 − w2rec1 𝑁 ∑︁ 𝑖 𝑤i, rec(𝑋i, rec − 𝜇wkde)𝑇 (𝑋i, rec − 𝜇wkde), (58) := 1 1 − w2rec1 𝑁 ∑︁ 𝑖 𝑤i, rec(𝑋i, rec − 𝜇wkde)𝑇 (𝑋i, rec − 𝜇wkde), (59) where 𝑋i, rec ∈ Xrec and 𝑤i, rec ∈ wrec is the 𝑖-th element of Xrec and wrec, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The bandwidth of the kernel is estimated by the Scott’s method [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Maximum Likelihood Estimation of Discrete Prior The optimisation of hyperparamters of the discrete prior distributions was done via MLE from the weighted samples (wrec, Xrec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We denote the PDF of Bernoulli distribution (binary) and the categorical distribution as Ber(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wBer), Cat(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wCat), where wBer ∈ R𝑑 and wCat ∈ R𝑑×𝐶 are the weights hyperparaeters, 𝐶 is the number of categories in the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The weighted log-PDF can be expressed as follows: LL := wrec log Ber(Xrec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wBer) (60) LL := wrec log Cat(Xrec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wCat) (61) We optimise each weight hyperparameters via maximising the log-likelihood (LL) via L-BFGS-B [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' PyTorch [53] auto-differentiation gives the gradient for L-BFGS-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' To set a constraint to make weights bounded [0, 1], we transformed original LL space via the sigmoid function during optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 18 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Deweighted sampling Samples for the Nyström method are better to be spatially sparse to well represent the whole kernel shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We adopt the deweighted sampling to constract the small subset of uniformly distributed samples Xnys from the weighted samples (wrec, Xrec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We resample from the categorical distribution with the inverse weights (1/wrec), then the resampled samples are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 AutoKQ selection algorithm Table 7: AutoKQ selection algorithm Algorithm 4: AutoKQ selection 1: Xrchq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wrchq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Var[𝑚(𝑥)]rchq ← RunRCHQ(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝝋(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓 (·)) 4: Xkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Var[𝑚(𝑥)]kt ← RunKernelThinning(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝝋(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓 (·) 6: if Var[𝑚(𝑥)]rchq < Var[𝑚(𝑥)]kt: 7: return Xrchq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wrchq 8: else: 9: return Xkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wkt function RunRCHQ(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝝋(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓 (·)): 1: 𝝋(·) ← Nyström(Xnys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓 (·)) 2: Xrchq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wrchq ← RCHQ(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝝋(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼(·)) 3: Var[𝑚(𝑥)]rchq ← KQ( 𝑓 (·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wbatch) 4: return Xrchq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wrchq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Var[𝑚(𝑥)]rchq function RunKernelThinning(wrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝝋(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xnys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓 (·)): 4: Xkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wkt ← KernelThinning(Xrec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑓 (·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝛼(·)) 5: Var[𝑚(𝑥)]kt ← KQ( 𝑓 (·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Xbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wbatch) 4: return Xkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' wkt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Var[𝑚(𝑥)]kt Table 7 illustrates the algorithm flow of automatic kernel quadrature selection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We compare the worst-case integration error of each algorithm, then pick the batch queries of which integration error is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' C Simulation-based inference C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Simulation-based inference The simulator emulates typically time-evolving signals from the physical device modelled by simultaneous differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The solution of the differential equation is basically not analytical, requiring numerical approximation such as the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Each equation has parameters, such as coefficients of differential terms, which determine the signal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Estimating the parameters that can reproduce the observed signal is a typically tricky task because simulation is not differentiable with regard to each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Although auto-differentiation can mitigate this problem, the parameter posterior is typically multimodal, so local optimisation algorithms based on differentiation struggles to find the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' More importantly, this inverse problem often has no unique solution mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, rather than estimating one deterministic parameter set, inferring the parameter posterior is more practically important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Moreover, having dozens of plausible simulators with differing levels of assumption is a common situation where we need to select the parsimonious model that best describes the given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Bayesian model evidence can provide a selection criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Therefore, estimating both Bayesian model evidence and parameter posterior is a frequent desideratum in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Furthermore, running simulators is expensive to evaluate, so parallelising the computation via computer clusters is of practical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 19 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Let yobs be the observed signal from the physical device, and we wish to estimate the simulator parameters Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This can be formulated as Bayesian inference, given by: 𝑝(Θ) := 𝜋′(Θ) := N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇𝜋, 𝚺𝜋) (62) 𝑝(D|Θ, 𝑀) := ℓtrue(Θ) := 𝑚 � 𝑗 N (err 𝑗 (𝜃);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 0, 𝜎2 noise), (63) 𝑝(D|𝑀) := N � E 𝑥∈𝜋[ℓtrue(Θ)], Var 𝑥∈𝜋[ℓtrue(Θ)] � , (64) 𝑝(Θ|D, 𝑀) = 𝑝(D|Θ, 𝑀)𝑝(Θ) 𝑝(D|𝑀) = ℓtrue(Θ)𝜋(Θ) E𝑥∈𝜋 [ℓtrue(Θ)] , (65) where D := {xobs, yobs} ∈ R𝑚×1, (66) 𝜃 := {𝜃𝑖} ∈ R𝑑−1, (67) Θ := {𝜃, 𝜎2 noise} ∈ R𝑑, (68) 𝑦sim, 𝑗 (𝜃) := 𝑀(𝜃, xobs), (69) err 𝑗 (𝜃) := � 𝑦obs, 𝑗 − 𝑦sim, 𝑗 (𝜃) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (70) 𝑀(𝜃, xobs) is the simulation model, which returns the prediction 𝑦sim, 𝑗 (𝜃) at given simulation parameter 𝜃 at the 𝑗-th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We wish to estimate the model evidence E𝑥∈𝜋 [ℓtrue(Θ)] and the parameter posterior 𝑝(Θ|D, 𝑀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Bayesian Quadrature Formulation In naive BQ, we place GP on the likelihood as such: ℓ(Θ) ∼ GP�ℓ(Θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇ℓ(Θ), 𝜎ℓ(Θ, Θ′)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (71) The evidence can be estimated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (5a) in general or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (11) in the Gaussian prior and RBF kernel case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The posterior can be estimated with the surrogate model and the estimated evidence via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, the likelihood is typically transformed into the logarithmic space because its dynamic range is wider than the numerical over-/underflow limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, log-warped GP [21, 67, 22] is often applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Particularly, we consider moment-matched log-transformed (MMLT) [67] GP, modelled as such: 𝑓 (𝑥) = exp[𝑔(𝑥)] − 1, (72a) := GP� 𝑓 (𝑥);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚(𝑥), 𝐶(𝑥, 𝑥) + 𝜎2 𝑛I�, (72b) 𝑔(𝑥) := GP�𝑔(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝑚𝑔(·), 𝐶𝑔(·, ·)�, (72c) 𝑚(𝑥) := exp � 𝑚𝑔(𝑥) + 1 2𝐶𝑔(𝑥, 𝑥) � , (72d) 𝐶(𝑥, 𝑥′) := 𝑚𝑔(𝑥)𝑚𝑔(𝑥′) � 𝐶𝑔(𝑥, 𝑥) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' (72e) The warped GP stores log-transformed values y𝑔 = log(y + 1), so we can avoid the over-/underflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Adachi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' [22] further extended MMLT GP so as to accommodate with BASQ modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They adopted the four-layered GP combining MMLT and parabolic transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The reason why they add the parabolic transformation is to copy the exponetiated function information to not only the surrogate function but also prior update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, this deep warped structure causes additional predictive errors due to the cumulative approximation errors from each layer’s moment-matching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Likelihood-free Inference Formulation Alternately, BO-based LFI [30] models GP differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They placed GP on the discrepancy, rather than the likelihood, defined as : Δtrue(𝜃) := log ||yobs − ysim(𝜃)|| (73) Δ(Θ) ∼ GP�Δ(Θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 𝜇Δ(𝜃), 𝜎Δ(𝜃, 𝜃′) + 𝜎2 𝑛I� (74) LFI adopts the tentative likelihood defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 8 as the likelihood at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The true likelihood can be estimated a posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The benefits of this modelling are as follows: 20 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Avoiding extreme dynamic range of likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Δtrue(𝜃) has much more moderate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We can reformulate BQ as BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' BO is more suitable for solving simulation-based inference as only the vicinity of the MAP location has meaningful value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As almost everywhere has zero likelihood, so BQ formulation is over-exploring if the prior is misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We can obtain the “temporary” likelihood 𝐿(𝜃) that approaches the true likelihood ℓtrue(Θ) asymptotically over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This likelihood can be regarded as "updated prior".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This can also mitigate the prior misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They reformulate the posterior inference as the BO to find the global minimum of the discrepancy Δtrue(𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The resulting GP surrogate model is used to approximate the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They do not go beyond the posterior inference, so evidence estimation cannot be done with BOLFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='4 SOBER-LFI Formulation We wish to take the best of both world;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' LFI GP modelling suitable for sampling and exact evidence estimation via BQ modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, we adopt the dual GPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' one for sampling, and the other for BQ modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While the sampling GP is modelled with the inverse discrepancy (−Δtrue(𝜃) so as to be the maximisation objective), the BQ GP is modelled with log-likelihood with MMLT GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Importantly, we can query both {Δtrue(𝜃), ℓtrue(Θ)} with negligible overhead as the time-consuming part is ysim(𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Once we get ysim(𝜃), calculating both {Δtrue(𝜃), ℓtrue(Θ)} are very cheap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The sampling GP is used for setting up the sampling function 𝜋, in the same manner explained in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' One difference is that the 𝜋 becomes extremely sharper than the BO task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' WKDE-based sampling can fail to sampling from 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, we adopted elliptical slice sampling (ESS) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Importance sampling permits using all of the samples from ESS without the burn-in period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The weights can be calculated via the 𝜋 defined with the sampling GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Note that ESS is more expensive than WKDE, so the additional overhead had to be produced instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As such, the sampling GP constructs the empirical measure 𝜋emp = (wrec, Xrec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' On the other hand, BQ GP constructs the surrogate model for likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The posterior and evidence inference can be made in the same manner explained in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Batch acquisition via objective RCHQ becomes a mix of both GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The kernel is defined by the BQ GP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 72e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The objective with AF is defined by the sampling GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D Experiments D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Batch Bayesian Optimisation Anti-Malarial Drug Polar Solvent global minimum global maximum EC50 [μM] Dipole Moment [D] number of molecules Figure 5: The histograms of the target values in the real-world datasets We examined our method, SOBER, with the following four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' All experiments are averaged over 16 iterations with varied random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The number of candidate samples drawn from the prior distribution are fixed to be 20,000 (𝑁 = 20, 000) for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While SOBER-LFI updates the prior hyperparameters as explained, the others are fixed with the initial hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Implementation of decoupled TS in BoTorch was not compatible with Tanimoto kernel in GAUCHE [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, SOBER-TS had to be sampled from vanilla TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, DPP-TS and SOBER-TS are infeasibly slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' They took about 5 days to run 30 iterations, resulting in 80 days for 16 trials for averaging, even in the smaller anti-malaria dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Hence, we do not compare decoupled TS, DPP-TS, and SOBER-TS for drug discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 21 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints Synthetic: Ackley function Ackley funciton is defined as: 𝑓 (𝑥) := −𝑎 exp ������ −𝑏 � � � 1 𝑑 𝑑 ∑︁ 𝑖=1 𝑥2 𝑖 ������ − exp � 1 𝑑 𝑑 ∑︁ 𝑖=1 cos(𝑐𝑥𝑖) � + 𝑎 + exp(1) (75) where 𝑎 = 20, 𝑐 = 2𝜋, 𝑑 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We take the negative Ackley function as the objective of BO to make this optimisation problem maximisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We modified the original Ackley function into a 23-dimensional function with the mixed spaces of 3 continuous and 20 binary inputs from [0, 1]20, following [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch size 𝑛 is 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The continuous prior is the uniform distribution ranging from [-1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The binary prior is the Bernoulli distribution with unbiased weights 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We assume each of continuous and binary priors at each dimension are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Synthetic: Rosenbrock function Rosenbrock function is defined as: 𝑓 (𝑥) := �𝑑−1 ∑︁ 𝑖=1 � 100(𝑥𝑖+1 − 𝑥2 𝑖 )2 + (𝑥𝑖 − 1)2� � (76) where 𝑑 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We take the negative Rosenbrock function as the objective of BO to make this optimisation problem maximisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We modified the original Rosenbrock function into a 7-dimensional function with the mixed spaces of 1 continuous and 6 discrete variables, following [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The first 1 dimension is discretised to be categorical variables, with 4 possible values 𝑥1 ∈ {−5, 0, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The other 6 dimensions are continuous with bounds [−5, 10]6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch size 𝑛 is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The continuous prior is the uniform distribution ranging from [-5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The discrete prior is the categorical distribution with unbiased weights 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We assume each of continuous and discrete priors at each dimension are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Real-world: Anti-Malarial drug discovery The dataset with 20,746 small molecules represented as 2048- dimensional features were taken from the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' falciparum whole cell screening derived by the Novatis-GNF Malaria Box [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The target variable is the EC50 value, which is defined as the concentration of the drug which gives half the maximal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The lower the concentration, the more effective (better) the drug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We take the nagative EC50 to make this optimisation problem maximisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch size 𝑛 is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We set the categorical prior with unbiased weights 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5 for each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Real-world: Polar solvent for batteries The dataset with 133,055 small molecules represented as 2048-dimensional features was optimised and predicted by the quantum-chemical computations using density functional theory, known as QM9 dataset [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The target variable is the dipole moment, which is basically correlated with the solvation capability in electrolytes in lithium-ion batteries, increasing the ratio of electro-mobile Li-ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The higher the dipole moment becomes, the larger (better) the ionic conductivity does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The batch size 𝑛 is 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We set the categorical prior with unbiased weights 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='5 for each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Figure 5 shows the distribution of target values in two real-world drug discovery datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The optimal molecules are outliers from the dataset distribution, so it clearly shows these tasks are needle-in-the-haystack situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Batch Bayesian Quadrature We tested our algorithm, SOBER, with the simulation-based inference tasks as the batch BQ method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' All experiments are averaged over 16 iterations with varied random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The number of candidate samples drawn from the prior distribution is fixed to be 20,000 (𝑁 = 20, 000) for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As the ground truth of posterior and evidence cannot be obtained for the simulation-based inference, we use the empirical metric to evaluate the quality of each inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For posterior evaluation, we drew the 10,000 test samples from the normal distribution centered at the ground truth parameters and the covariance with the identity matrix of which each element is 5 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Then, we computed the root-mean-squared error (RMSE) between the estimated log-likelihood and true log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For evidence, we simply adopted the negative of estimated evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Real-world: 2 RC Pairs ECM 2 RC Pair equivalent circuit model (ECM) is the simplest lithium-ion battery simulator with 6-dimensional continuous variables [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We generated synthetic signal using the model with 100 frequency steps equispaced over log-angular frequency regime, then added the Gaussian noise with the amplitude of exp(1) to the 𝑅total = exp(2) signal from the canonical ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Real-world: 5 RC Pairs ECM 5 RC Pair ECM is more complex lithium-ion battery simulator with 12-dimensional continuous variables [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We generated synthetic signal using the model with 100 frequency steps equispaced over 22 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints log-angular frequency regime, then added the Gaussian noise with the amplitude of exp(1) to the 𝑅total = exp(2) signal from the canonical ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Additional Experiments D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='1 Hyperparameter sensitivity Acquisition Function batch size n Nyström samples M Recombinaiton samples N Log10 regret Log10 overhead [s] 0 2 0 2 0 2 3 2 1 0 15 0 15 0 15 0 15 0 0 15 0 15 0 15 0 15 3 2 1 0 3 2 1 0 3 2 1 0 Iterations Iterations Iterations Iterations 0 2 Figure 6: Hyperparameter sensitivity analysis using the Ackley function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lines and shade area denote mean ± 1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We tested the hyperparameter sensitivity of SOBER-LFI using the Ackley function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We examined the effect of AFs 𝛼, batch size 𝑛, the number of Nyström samples 𝑀, and the number of recombination samples 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We averaged the results from 16 experiments with varied random seeds, and terminated at the 15th batch acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The baseline conditions are 𝑛 = 100, 𝛼 = LFI, 𝑀 = 500, and 𝑁 = 20, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For AF, the information-theoretic AFs can boost the convergence rate, whereas the others do not change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For the batch size 𝑛, the convergence rate can be improved in accordance with the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' For quadrature hyperparameters 𝑀 and 𝑁, the larger the number of samples becomes, the faster the convergence does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, increasing the number of samples leads to additional overhead increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Our default conditions are competitive throughout the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='2 Ablation Study π definitions ablation study Log10 regret Log10 overhead [s] 0 2 0 2 3 2 1 0 15 0 15 0 0 15 0 15 3 2 1 0 Iterations Iterations Figure 7: Ablation study using the Ackley function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lines and shade area denote mean ± 1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 23 SOBER: Scalable Batch Bayesian Optimization and Quadrature using Recombination Constraints We performed the ablation study to analyse each algorithm’s effects on convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Firstly, we compared the various 𝜋 definitions defined by the AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As shown in Figure 7, LFI and PI definitions are the clear performants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' This is because the other AFs are designed to guide the sequential sampling, of which global maxima sensitively changed over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' LFI and PI show the possibility of global maxima, which gradually squeezes the region toward the true global maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Thus, in SOBER-LFI formulation, LFI AF is well-suited as the definition of 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As another ablation study, we compared whether or not updating 𝜋 and using AF in the objective RCHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' As a result, unsurprisingly, updating 𝜋 is the most influential on the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The objective RCHQ does not significantly influence the convergence when we select LFI as the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' However, information-theoretic AFs can boost the convergence, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content='3 Fully Bayesian Gaussian Process Noise 1E+00 Log10 regret Log10 overhead [s] 15 0 15 Iteration 0 1 2 3 0 0 1 FBGP Noise Log10 regret 0 15 Iteration 0 1 Log10 overhead [s] 2 3 0 1 0 15 2 3 MLE-GP Noise Log10 regret 0 15 Iteration 0 1 Log10 overhead [s] 2 3 0 1 0 15 2 3 Figure 8: Efficacy of Fully Bayesian Gaussian process modelling using the noisy Ackley function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lines and shade area denote mean ± 1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We further tested the effect of FBGP modelling on the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' To examine the efficacy, we adopted the noisy Ackley function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' We added the Gaussian noise to the queried values from the Ackley function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The amplitude of the noise is varied from 10−3 to 1 in a logarithmic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' The baseline conditions are 𝑛 = 100, 𝛼 = LFI, 𝑀 = 500, 𝑁 = 20, 000, and 𝐻 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Figure 8 illustrates that FBGP modelling with quadrature distillation can boost the convergence rate while maintaining the overhead feasibly small (the overhead of FBGP is smaller than DPP-TS with type-II MLE kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=') Hyperweights H Acquisition Function Log10 regret Log10 regret Log10 overhead [s] 15 0 15 0 15 Iteration Iteration 0 2 3 0 0 1 1 Log10 overhead [s] 2 3 0 1 0 15 0 1 2 3 Figure 9: SOBER-LFI consistently outperforms with small overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Lines and shade area denote mean ± 1 standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' Furthermore, we examined the effects of the number of hyperweights 𝐻 and the AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' While 𝐻 are influential on both convergence rate and overhead, the default value 𝐻 = 50 are reasonably competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' With regard to the effect of AFs, QB-MGP AF was the performant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FKT4oBgHgl3EQfgC6c/content/2301.11832v1.pdf'} +page_content=' 24' metadata={'source': 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Mephisto provides abstractions that cover a broad set of +task designs and data collection workflows, and provides a +simple user experience to make best-practices easy defaults. +In this whitepaper we discuss the current state of data collec- +tion and annotation in ML research, establish the motivation +for building a shared framework to enable researchers to cre- +ate and open-source data collection and annotation tools as +part of their publication, and outline a set of suggested re- +quirements for a system to facilitate these goals. We then step +through our resolution in Mephisto, explaining the abstrac- +tions we use, our design decisions around the user experi- +ence, and share implementation details and where they align +with the original motivations. We also discuss current limi- +tations, as well as future work towards continuing to deliver +on the framework’s initial goals. Mephisto is available as an +open source project1, and its documentation can be found at +www.mephisto.ai. +1 +Motivation +In the current era of Machine Learning (ML) research, we +find a recurring gap between the discovery and widespread +use of novel and best-practice methodology for high-quality +crowdsourcing. This disconnect drives bad experiences both +for researchers trying to collect high-quality data and work- +ers who end up stuck with low-quality tasks and work prac- +tices. The thinking is, while some projects certainly aim to +incorporate state of the art methodologies for crowdsourc- +ing, those who do are left reproducing implementation work, +and those who don’t are left with worse results. It’s impos- +sible to directly confirm this claim, however studies show +that even commonly used datasets have clear quality is- +sues (Paullada et al. 2021; Northcutt, Athalye, and Mueller +2021). +We believe much of this is caused by the common practice +of publishing research papers without technical implemen- +tations of accompanying data collection methodologies. In a +quick audit in November 2022 of PapersWithCode, a pub- +lic repository of ML publications and datasets, we examine +the top 35 cited datasets to see which ones provided code +implementations. Of these, 31 describe their crowdsourcing +1https://www.github.org/facebookresearch/mephisto +methodology in varying levels of depth as part of the re- +search, 18 papers were accompanied with usage code at the +time of writing, yet only 3 provide code for the collection +and quality assurance portion of the paper. If a researcher +wanted to extend one of these existing common datasets, +for instance for debiasing reasons, they would have to re- +implement the collection or annotation flow from scratch. +This is in contrast to the modeling side of ML research, +where code implementations have become much more stan- +dard. According to PapersWithCode’s trends2, 8.5% of pa- +pers in 2015 had code implementations compared to 27.75% +of papers from 2022. This increase closely follows the re- +lease and adoption of the TensorFlow and later PyTorch +frameworks, and holds despite a huge increase in the raw +count of ML and AI papers published every year. As 66- +68% of the repositories from 2022 are now built on PyTorch, +it would be hard to conclude these frameworks had no role +in driving increased code sharing. +Given the rise of data-driven ML solutions and adoption +of “big data” more broadly (Sarker 2021) there’s even more +pressure to improve the standards for data collection and an- +notation. Standard deep model architectures may be able to +trade off between label accuracy and training time (Rolnick +et al. 2017), but as we begin to run into scaling laws on large +datasets (Kaplan et al. 2020) improving the quality of the +dataset becomes a route for improving models again. While +this is often done with pruning (Sorscher et al. 2022) it could +also be handled at the start of the data funnel. Training time +and cost are also becoming more central issues as large mod- +els can take months to train on energy-hungry hardware. +If data quality is a known issue, we have to investi- +gate why it isn’t resolved. While annotation providers of- +ten try to build out tooling to improve labelling for common +cases, these don’t cover cases on the boundaries of research. +Works that examine and analyze the issue, like Paullada +et al. (2021); Vaughan (2018); and Sheng and Zhang (2019), +stop understandably short of providing researchers with im- +proved tooling. Further, change is difficult. It’s not simple +for most research labs to adopt better methodologies for run- +ning data collection, especially if they already have an es- +tablished system or they cannot afford to invest the time and +capital into building alternatives. +2https://paperswithcode.com/trends +arXiv:2301.05154v1 [cs.AI] 12 Jan 2023 + +To resolve this need in the longer term, we aim to provide +a route for efficient community collaboration for collecting +high-quality data by using easily-to-publish shared tooling. +This tooling is Mephisto. Making it easy to share, refine, and +use best practices at the start of the data funnel can have im- +pact on collection costs, training costs, fairness, and model +accuracy. +The project is named Mephisto, alluding to the succes- +sor3 of the original “man-in-the-machine” chess playing +‘automaton’ called the Mechanical Turk. As Mephisto im- +proved upon the original “man-in-the-machine” operator +with a remotely operated one, our framework aims to im- +prove upon the complexities of traditional crowdsourcing, +by abstracting its complexities further away and hopefully +improving the experience for both workers and researchers +in the process. +2 +Related Works +Data annotation is certainly not an overlooked area, how- +ever publications in the space of collection methodologies +tend to fall roughly into a few categories. Specific area pa- +pers will devise a collection scheme for a specific task, and +may explain the process. Surveys attempt to synthesize the +learnings from these collections into broader themes. Tools +and platforms attempt to solve a slice of the problem with +clear code implementations. +2.1 +Novel Methodologies +Across the field, there is no shortage of specific techniques +that researchers have used to collect datasets. The following +is entirely non-exhaustive, and a much more complete listing +of these works can be found in Vaughan (2018). +Task-specific interfaces have been developed for complex +in-domain tasks (Torralba, Russell, and Yuen 2010; Chang +et al. 2015). +Model-in-the-loop setups have been used for evaluation +(Xu et al. 2020), sample-efficient data collection (Settles +2012), and live service (Lasecki et al. 2017; Kamar 2016). +Offline model evaluation is used as well (Ribeiro, Singh, and +Guestrin 2016; Li, Weston, and Roller 2019). +Different techniques have been surfaced for ensuring +data quality, expanding beyond inter-annotator agreement +to more advanced approaches (Ghosh, Kale, and McAfee +2011). Some studies also specifically analysis the trade-off +between data quality and budget (Khetan and Oh 2016). +As seen in the top datasets though, publishing of these +methodologies doesn’t necessarily come with accompany- +ing code (though some in the above do on Mephisto). +2.2 +Crowdsourcing Surveys +To attempt to bring shared signal out of the spread of sug- +gestions and methods, surveys of the field attempt to collect, +group, and evaluate various techniques. These often provide +suggestions for how others may include their own collec- +tions (Vaughan 2018; Sheng and Zhang 2019). They under- +standably stop short of providing researchers with compre- +hensive tooling. +3https://en.wikipedia.org/wiki/Mephisto (automaton) +In many cases, surveys refer to crowdsourcing as a key +part of the data lifecycle, and try to shed light on the com- +plexities and next steps within a specific domain (Ashmore, +Calinescu, and Paterson 2021; Sambasivan et al. 2021). +Some works have outlined entire workflows for developing +higher-quality datasets (Hutchinson et al. 2021). While the +insights are certainly valuable, few papers directly refer to +these works when building out their tooling, and few of those +end up releasing actionable code assets. +One survey (Neves and Seva 2020) attempts to find and +document all of the available annotation tools, though this +list is certainly non-exhaustive. +2.3 +Full-code Solutions +Often, works that release crowdsourcing code do so as part +of a paper solving a contained problem space, like how Li +et al. (2021) sets up compositional dialogue tasks, or making +the experiment flow for research somewhat easier for a spe- +cific purpose, such as how Gureckis et al. (2016) attempts to +abstract the complexities of interacting with the Mechanical +Turk platform. +Mephisto falls into this last category of work as well. We +branch out of ParlAI-MTurk (Miller et al. 2017), a project +that was designed to make dialogue-based research easier. +Acknowledging the risk of becoming another bygone stan- +dard4 we attempt to make the platform general enough to +support integrating any of the above works, and intend to +help ground the conversation in usable tools for researchers. +3 +Project Goals +With Mephisto, we seek to address the core problems that +prevent current crowdsourcing work from being easy to +write, use, and distribute. For this we outline core values and +elements we believe a research annotation platform should +follow, such that we can evaluate our progress. +3.1 +Distribution, Reproduction, and Extension +In a research setting, each step of the process of distribu- +tion, reproduction, and extension are relevant for a work to +contribute to forwarding the field. Distribution puts the tech- +niques into the hands of other researchers, and can include +sharing just methodology through the entire code setup. Re- +production allows new individuals to try out work, and the +barrier for reproduction is often inversely proportional to +how much of the process was able to be shared. Extension +is the next step on reproduction, and pushes the initial work +forward into new research. +To help facilitate distribution, reproduction, and extension +of work, an annotation platform should make it easy to dis- +tribute all of the code related to a research project, and that +code should be easy for new readers to set up, run, and mod- +ify on their own. +3.2 +Flexibility +As research is a moving target, any platform that aims to +support the varying needs of research should be casting a +4For those familiar with the trend of https://xkcd.com/927/ + +wide net for functionality. This complexity however runs +counter to ease-of-use for a platform, which can raise the +barrier of entry get researcher buy-in. An ideal platform +should find a way to balance these two needs. +Abstracting Crowdsourcing and Implementations +In +order to adapt to new scenarios and situations, it’s valuable +to examine the core elements of a crowdsourcing task, and +isolate these into a coherent data model. From these build- +ing blocks multiple interoperable implementations can be +built up, thus allowing for a high degree of control over spe- +cialized collection systems. For the average user, basic im- +plementations with simpler controls can be provided to get +them up-and-running with as little context as possible. +This type of approach allows work created on the plat- +form to avoid the pitfalls of work such as in (Gureckis et al. +2016), which helps users get started but ties them to a spe- +cific platform for crowdsourcing. +Hooks with Default Best Practices +Within individual +components of the platform, it should be possible for re- +searchers to exact a high degree of control to run spe- +cific jobs. This includes over portions such as worker- +task pairing, collection pipelines and workflows, automated +review tooling, and any other considerations. To prevent +overwhelming new users, each of these should have best- +practices provided by default, allowing new users to benefit +from the shared knowledge of current best approaches. +This avoids the issues of both Hutchinson et al. (2021) and +Miller et al. (2017), where in the former there’s too much +flexibility at the onset (considering you would have to im- +plement it all yourself), and in the latter all tasks are forced +into Dialogue data collection best practices and techniques. +3.3 +Data Quality considerations +Any project aiming to facilitate crowdsourcing must con- +sider data quality to be a priority, as no matter how easy it +is to use and share, it isn’t particularly useful if the output +data is low-quality. We raise a few important considerations +in this space, many under the lens that research work is of- +ten time-limited and it can take a number of revisions and +iterations to have something worthwhile. +UI/UX and worker quality of life +The designed user- +interface (UI) of a task is a significant contributor to task +result quality (Finnerty et al. 2013). Tasks that are well de- +signed may contain clear criteria, include examples, give for- +mat specifications, reduce cognitive demand, etc. (Wu and +Quinn 2017). An ideal framework should help encode some +of these best practices for the busy researcher. +Workers are also more likely to return to tasks that are +designed with a good user experience (UX) in mind. Tasks +should minimize user frustration, both in terms of design and +usability. Tasks should also be architected so that they are +resilient to errors. Hitting error cases should be clear to users +at the least, and at most triggering some form of alerting so +that the researcher can respond swiftly and appropriately. +Incorporating feedback channels is also a great way to +identify and improve upon design blindspots that may oc- +cur. We consider these blindspots to be the norm, not the +exception. An ideal crowdsourcing framework should pro- +vide researchers with feedback mechanisms that serve as a +catch-all for any oversights on their parts. In implementa- +tion, this would allow for easy communication from work- +ers back to the researchers through some feedback channel. +Giving workers an opportunity to share feedback with re- +searchers can create for a better worker experience. (Bragg, +Mausam, and Weld 2018). +Quality Assurance Practices +Standard quality assurance +practices, such as worker qualification, gold-labelling, inter- +annotator agreement, etc. should be easy to discover and +enable without getting into the literature. Encouraging re- +searchers to design their tasks with these elements in the +forefront will result in better data quality than tasks with +these elements added as afterthoughts. Further, common +workflows like pilots and worker communication can be crit- +ical, and thus should be easy to enact. +Worker Diversity and Representation +When collecting +a dataset, one element of quality comes from ensuring the +data is worked on from as large and representative a col- +lection of contributors as achievable. Often, this is limited +by the tools of the company providing the crowd, and at +times considered private information. Still, a strong crowd- +sourcing platform should provide tools to encourage re- +searchers to extend their crowd with best-practices for sim- +plified onboarding, task maximums per worker, and the abil- +ity to use multiple crowdsourcing platforms. At the very +least, it should be able to report some metrics for the source +crowd, possibly integrating with something like data cards +(Pushkarna, Zaldivar, and Kjartansson 2022). +4 +Current implementation +In following alongside the values and principles from the +previous section, we designed Mephisto with an underlying +set of abstractions, a few initial implementations, and then +some best-practice elements both for task quality as well as +researcher experience. This section aims to give a technical +overview of how Mephisto operates today. +4.1 +Abstractions +We’ll start off by describing Mephisto’s underlying abstrac- +tions, which aim to break the complexity of crowdsourcing +into components to build architecture around. After describ- +ing the data model, getting the rest of Mephisto is almost as +easy as Architect, Blueprint, CrowdProvider, Database5. +What is in a task? The Data Model +In order to reason +about crowdsourcing, we break out a number of definitions +that represent underlying elements of the data model. +The first is a Task, which can be considered as a group of +directly related work that needs to be done, such as “Label +50,000 images with varying segmentation masks”. +Beneath this level is a TaskRun, which can be consid- +ered an individual job you may have run. Of the 50,000 im- +ages above, you may want to label the first 1,000 with 3 pos- +sible mask labels for a pilot. This would be an appropriate +5Initialization sequence not by design, we promise + +Figure 1: Mephisto Crowdsourcing Data Model Overview +TaskRun. (A Task may have just one TaskRun, but will +often have many). +Within a TaskRun, you may have many Assignments, +which can be considered a discrete element you need done. +This starts to be at the level of what you’ll show a worker, +such as “Label these 5 images with segmentation masks”. +An Assignment may be broken up into many Units, +which represent the contribution that one individual may +have on a task. For some Assignments, there may be just +one Unit, however for example you may have two Units +on an Assignment that you want to have labelled twice +to check inter-annotator agreement, or on a dialogue where +you need two workers to communicate with one another at +the same time. +For those actually doing the work, we have Workers +which keep track of everything an individual has ever done +for you for all tasks. +In order to distinguish the full Worker history, we also +have Agents, which can be considered as a pairing between +a Unit and a Worker representing the work that worker +did for that particular unit. +These are the underlying data model components that +back Mephisto, and we can begin to reason about the rest +of the flow for an annotation Task with this terminology. +Hosting the Job: Architects +Architects comprise the +scripts to set up a server that runs a task in Mephisto. They +allow researchers to use Mephisto with different cloud con- +figurations. For this, they cover the server lifecycle during +a task, and thus should implement methods for preparing, +deploying, and shutting down servers. They also define the +interface for which external workers are able to connect to +the Mephisto back-end. +The Tasks: Blueprints +Blueprints are the center of +Mephisto’s different tasks, and aim to capture both task +flows common to a task and configuration settings that can +allow someone to customize and extend that task. They de- +fine the inputs and outputs for a specific task as well as the +overall task interface. The specific abstraction requires a few +important components, listed below: +• The AgentState defines the format of the data that +will be saved during collection of a Unit. +• The TaskRunner defines any back-end logic that is re- +quired to execute a task. +• The TaskBuilder defines any resources that need to +be built before a job. Usually this includes the front-end +to be hosted as part of a task. +• The SharedTaskState can be used to hold live +state information shared between all of the Units in a +TaskRun, often referred to when assigning work. +The Workers: Crowd Providers +CrowdProviders are +what enable Mephisto to connect differing crowds to your +task. These interact closely with the abstract Workers, +Agents, and Units in the following way: +• Workers comprise the long-term identity for +a worker, and are an interface where Mephisto can in- +clude worker-specific functionality that interfaces with a +provider’s API. This may include blocking, giving quali- +fications, and direct communication. +• Units are an interface to the remote hook of +a job posting, or similar. They need to keep track of ex- +ternal status, and should also provide the interface for +registering and expiring a work request with a provider. +• Agents cover the link between a worker and a +single Unit, and must implement methods for checking +their remote status, as well as marking work as completed +or rejected. +Results Storage: Databases +Databases are what en- +able Mephisto to store your results, regardless of the server +setup you are using. For this, we provide the MephistoDB +abstraction, which lists all of the required database calls one +would need to implement to run Mephisto. +4.2 +Mephisto Architecture +In practice, Mephisto is able to handle any arbitrary config- +uration of Blueprint, Architect, CrowdProvider, +and Database and coordinate the initialization, deploy- +ment, monitoring, and shutdown of each over the TaskRun +they comprise. Over the course of such a LiveTaskRun, it +also reports metrics and saves partial results. One key goal is +that any “business logic” that people would like to customize +has a clear hook for doing so in the abstractions, such that +most users don’t need to deal with the complexity of how +these interfaces are coordinated. After data collection has +concluded, Mephisto provides tools that allow one to inter- +act with and explore the data stored in the MephistoDB. + +TASK +contains many +TASK-RUN +WORKER +viewed as +given +is comprised of +丰 +ASSIGNMENT +AGENT +GRANTED-QUALIFICATION +split into +works on +instanceof +UNIT +QUALIFICATION4.3 +Blueprints +It’s our goal that in the majority of cases, most of Mephisto +users should be able to rely on existing Blueprints, rather +than needing to write new ones from scratch. To this end, we +provide a few useful implementations that cover a wide set +of use cases. +• The ReactStaticBlueprint is a setup where one +can provide any simple data collection front-end applica- +tion written in React that can be considered a single turn. +In short: The worker is provided some data, they work on +it remotely, and then return the result. +• The RemoteProcedureBlueprint allows a more +complex setup, where the front-end application is able +to make direct queries to some back-end specified dur- +ing task setup. This allows for doing processing that +wouldn’t be possible on the worker’s side, such as run- +ning a model in the loop. +• The StaticHTMLBlueprint stands as the easiest +onboarding ramp to Mephisto, in that it accepts stan- +dard .html files that researchers may be more familiar +with than React. It isn’t as feature rich as other offerings +though. +Beyond these, the ParlAIChatBlueprint stands as +a good example of a live task with a specified and highly +configurable flow, catered towards dialogue-focused jobs. +Quality Assurance Mixins +We provide a handful of mix- +ins for quality assurance which are available to be used on +anything run from the ReactStaticBlueprint or the +RemoteProcedureBlueprint. Including these mixins +into a Blueprint means that blueprint has the specified +functionality enabled and knows how to handle it. +• The OnboardingRequired mixin allows researchers +to set up a separate flow for workers who haven’t done +the task before, allowing them to learn what the require- +ments are. +• The UseGoldUnit mixin provides a familiar flow for +providing known-good examples for which workers will +be evaluated against periodically as a quality check. +• The ScreenTaskRequired mixin allows researchers +to have the first actual Unit that a Worker works on for +a job be a specified (usually easy-to-verify) unit, which +allows researchers to do automated analysis and valida- +tion of before giving more work. +Researchers can use the primitives for these methods to in- +corporate strong quality assurance flows into their tasks, +without needing to build any complex machinery on top of +the underlying validation measures for their task. Of course, +it still requires some initial rounds of piloting and tweaking +to ensure the validators are well calibrated. +4.4 +Crowd Providers +The main crowd providers we have implemented at the mo- +ment are the MTurkProvider and the MockProvider. +The former allows for direct interfacing with the Amazon +Mechanical Turk platform, while the latter allows for testing +tasks locally, or allowing people to access while “mocking” +a specified worker. Adding more Crowd Providers is ongo- +ing work. +4.5 +Architects +The currently available architects at the time of writing +are the HerokuArchitect and the LocalArchitect. +The former allows launching using Heroku cloud services as +a provider. The latter allows hosting on the machine running +Mephisto, which is useful for testing locally or collecting +from research participants on the same local network. We +also have an EC2-based architect which requires registering +a domain name with AWS for use. +4.6 +Front-end Packages +Another component of launching a crowdsourcing task is de- +signing the task’s UI. Mephisto allows researchers to launch +tasks with UI implemented with either plain browser HTML, +or for more advanced cases, with the React JavaScript li- +brary. +For +the +React +implementation, +Mephisto +provides +an +npm +(Node +Package +Manager) +package +named +mephisto-task. +The +package +enables +researchers +to +interface +seamlessly +with +the +Mephisto +back-end +Blueprints from their front-end code. It surfaces the task +data provided from the back-end, callbacks to handle task +submission for the CrowdProvider, as well as boolean +flags that can be used to conditionally display different +views (e.g. task preview, onboarding, errors, submissions, +etc.). +The +package +also +exposes +three +React +Hooks: +useMephistoTask, +useMephistoLiveTask, +and +useMephistoRemoteProcedureTask. +The +latter two can be used for more advanced tasks, such as +chat-bots or model-in-the-loop tasks, respectively. We +particularly see model-in-the-loop as an opportunity to +increase task result quality by augmenting worker per- +formance in real-time, minimizing tedious and rote work, +improving perceived UX, and providing real-time validation +and feedback, though more research is needed in this +area. Model-in-the-loop approaches can also be used to +dynamically generate subsequent tasks based on prior tasks, +providing customized control over what tasks get launched +next while a task run is already underway. +The examples/ folder in the Mephisto GitHub reposi- +tory provides sample task templates using the simple setup, +as well as advanced setups including chat-bots and model- +in-the-loop functionality. +4.7 +Worker Feedback +With Mephisto’s extensible architecture, creating plugins is +easy as well. We provide two first-party plugins through the +npm package mephisto-worker-addons to help im- +prove the worker experience for tasks. Specifically, we pro- +vide the Feedback and Tips React components. +The Feedback component allows workers to provide sug- +gestions back to the reseacher as they’re working through +tasks. This could include questions, bugs they’ve found, + +or positive acknowledgement. This communication channel +back to the researcher can be a way to improve worker senti- +ment and improve task quality. Researchers can also choose +to tip or give bonuses to submitters who provide valuable +feedback. Communicating this reward scheme can also cre- +ate a helpful incentive mechanism for gathering tips. +The Tips components allows workers to create a shared +FAQ-style wiki that other workers can benefit from. This +comes with built-in moderation as submitted Tips need to +be approved by the researcher before they’re visible to other +workers. Aside from being helpful, these examples indicate +a few ways of how Mephisto can be made extensible to suit +custom research needs. +4.8 +Review Tooling +Mephisto also includes a Python based command-line in- +terface (CLI) tool to allow users to review a task’s results, +or more generally any arbitrary data. The command accepts +an input data source as well as a “review template”, and +launches a local webserver to allow for browsing the data. +For any arbitrary data, one can just pipe in an input file: +cat input.jsonl | mephisto review +--json my-review-interface --stdout +Or for using specifically with a Mephisto task run, one +can use the --db flag: +mephisto review --db task-name +my-review-interface --stdout +To facilitate review, we provide a React template based +on create-react-app that implements a modular ren- +dering architecture. This architecture allows researchers to +easily define how a “data item” should be rendered by imple- +menting their own custom renderer as a single React com- +ponent. Out of the box, we ship a few default renderers; for +example, a JSON renderer and a Word Cloud renderer for +text-heavy tasks. +Once a task run is complete and a dataset has been +accumulated, researchers can share results along with the +Mephisto-based review and visualization tool as part of their +publication. Mephisto’s base review tooling was used by the +Ego4D project (Grauman et al. 2021) to share their collected +3,000 hours of egocentric video6. +4.9 +Worker Qualifications +Mephisto provides a simple setup for tagging workers for +any reason, wherein you can create and assign arbitrary +Qualifications to any Worker. We find this is use- +ful in setting up allow and block lists, querying or selecting +workers based on skills you’ve noted them for, and creating +complex task flows (such as those where participating in one +role disqualifies another). +5 +Future work +Mephisto is an evolving system, and we continue to iterate +and develop it alongside the values listed in this document. +6https://ego4d-data.org/docs/viz/ +As we discover new powerful methods for crowdsourcing +we aim to include them in Mephisto as top-level functional- +ity. We also aim to provide easy ways for anyone on the plat- +form to build new hooks and functionality, and share them +with others who are developing tasks. Further, we hope to +extend the base set of existing tooling that Mephisto sup- +ports out-of-box. Lastly, we aim to extend the portability of +the platform, such that it can be used with as many providers, +on as many hosting solutions, and with as many tasks as pos- +sible. +We also hope to continue to build along the dimensions of +task-design - making it easier to share and use community- +sourced design and task templates, opt into UI and UX best- +practices as they emerge, and experiment with new primi- +tives to improve worker experience, such as gamification. +Even with these steps though, we’re only scratching the +surface of implementing the best practices of today, let alone +accommodating those of tomorrow. We hope this work can +stand as a foundation that future work will build upon. Our +roadmap is available on the Github project page, and we’re +open to feedback on where we should take the project. +5.1 +Contributing +Mephisto is an open source project, and we value contribu- +tions from our users. We welcome anyone to join in and help +with the vision of easy, reproducible crowdsourcing with +best-practices built in on our GitHub7. Feel like we’re do- +ing something wrong, or are missing a technique that people +should be using immediately? Great! File an issue, or better +yet open a PR. +Ethical statement +Mephisto is provided as a crowdsourcing software with a +permissive license on use. While the Mephisto platform +aims to improve annotation methodologies and facilitate co- +operation towards resolving data collection issues, it cer- +tainly is still a work in progress towards those goals. It +doesn’t directly impose them as constraints on its users, so +while we try to make currently agreed upon best practices +the defaults, they can be overridden. +As such, issues such as underpayment or mistreatment of +workers, collection of biased datasets, and data licensing is- +sues may still arise. 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CoRR, +abs/2010.07079. + diff --git a/QdE4T4oBgHgl3EQfkw2x/content/tmp_files/load_file.txt b/QdE4T4oBgHgl3EQfkw2x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c75fe3ffe0b47a8def92d4896d6e3e4a1a7a2f3d --- /dev/null +++ b/QdE4T4oBgHgl3EQfkw2x/content/tmp_files/load_file.txt @@ -0,0 +1,651 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf,len=650 +page_content='Mephisto: A Framework for Portable, Reproducible, and Iterative Crowdsourcing Jack Urbanek and Pratik Ringshia Meta AI jju@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='com, tikir@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='com Abstract We introduce Mephisto, a framework to make crowdsourcing for research more reproducible, transparent, and collabora- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Mephisto provides abstractions that cover a broad set of task designs and data collection workflows, and provides a simple user experience to make best-practices easy defaults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' In this whitepaper we discuss the current state of data collec- tion and annotation in ML research, establish the motivation for building a shared framework to enable researchers to cre- ate and open-source data collection and annotation tools as part of their publication, and outline a set of suggested re- quirements for a system to facilitate these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We then step through our resolution in Mephisto, explaining the abstrac- tions we use, our design decisions around the user experi- ence, and share implementation details and where they align with the original motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We also discuss current limi- tations, as well as future work towards continuing to deliver on the framework’s initial goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Mephisto is available as an open source project1, and its documentation can be found at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='mephisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 1 Motivation In the current era of Machine Learning (ML) research, we find a recurring gap between the discovery and widespread use of novel and best-practice methodology for high-quality crowdsourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This disconnect drives bad experiences both for researchers trying to collect high-quality data and work- ers who end up stuck with low-quality tasks and work prac- tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The thinking is, while some projects certainly aim to incorporate state of the art methodologies for crowdsourc- ing, those who do are left reproducing implementation work, and those who don’t are left with worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' It’s impos- sible to directly confirm this claim, however studies show that even commonly used datasets have clear quality is- sues (Paullada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Northcutt, Athalye, and Mueller 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We believe much of this is caused by the common practice of publishing research papers without technical implemen- tations of accompanying data collection methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' In a quick audit in November 2022 of PapersWithCode, a pub- lic repository of ML publications and datasets, we examine the top 35 cited datasets to see which ones provided code implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Of these, 31 describe their crowdsourcing 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='org/facebookresearch/mephisto methodology in varying levels of depth as part of the re- search, 18 papers were accompanied with usage code at the time of writing, yet only 3 provide code for the collection and quality assurance portion of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' If a researcher wanted to extend one of these existing common datasets, for instance for debiasing reasons, they would have to re- implement the collection or annotation flow from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This is in contrast to the modeling side of ML research, where code implementations have become much more stan- dard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' According to PapersWithCode’s trends2, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='5% of pa- pers in 2015 had code implementations compared to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='75% of papers from 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This increase closely follows the re- lease and adoption of the TensorFlow and later PyTorch frameworks, and holds despite a huge increase in the raw count of ML and AI papers published every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' As 66- 68% of the repositories from 2022 are now built on PyTorch, it would be hard to conclude these frameworks had no role in driving increased code sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Given the rise of data-driven ML solutions and adoption of “big data” more broadly (Sarker 2021) there’s even more pressure to improve the standards for data collection and an- notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Standard deep model architectures may be able to trade off between label accuracy and training time (Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2017), but as we begin to run into scaling laws on large datasets (Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2020) improving the quality of the dataset becomes a route for improving models again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' While this is often done with pruning (Sorscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2022) it could also be handled at the start of the data funnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Training time and cost are also becoming more central issues as large mod- els can take months to train on energy-hungry hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' If data quality is a known issue, we have to investi- gate why it isn’t resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' While annotation providers of- ten try to build out tooling to improve labelling for common cases, these don’t cover cases on the boundaries of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Works that examine and analyze the issue, like Paullada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Vaughan (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' and Sheng and Zhang (2019), stop understandably short of providing researchers with im- proved tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Further, change is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' It’s not simple for most research labs to adopt better methodologies for run- ning data collection, especially if they already have an es- tablished system or they cannot afford to invest the time and capital into building alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2https://paperswithcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='com/trends arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='05154v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='AI] 12 Jan 2023 To resolve this need in the longer term, we aim to provide a route for efficient community collaboration for collecting high-quality data by using easily-to-publish shared tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This tooling is Mephisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Making it easy to share, refine, and use best practices at the start of the data funnel can have im- pact on collection costs, training costs, fairness, and model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The project is named Mephisto, alluding to the succes- sor3 of the original “man-in-the-machine” chess playing ‘automaton’ called the Mechanical Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' As Mephisto im- proved upon the original “man-in-the-machine” operator with a remotely operated one, our framework aims to im- prove upon the complexities of traditional crowdsourcing, by abstracting its complexities further away and hopefully improving the experience for both workers and researchers in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2 Related Works Data annotation is certainly not an overlooked area, how- ever publications in the space of collection methodologies tend to fall roughly into a few categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Specific area pa- pers will devise a collection scheme for a specific task, and may explain the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Surveys attempt to synthesize the learnings from these collections into broader themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Tools and platforms attempt to solve a slice of the problem with clear code implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='1 Novel Methodologies Across the field, there is no shortage of specific techniques that researchers have used to collect datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The following is entirely non-exhaustive, and a much more complete listing of these works can be found in Vaughan (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Task-specific interfaces have been developed for complex in-domain tasks (Torralba, Russell, and Yuen 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Model-in-the-loop setups have been used for evaluation (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2020), sample-efficient data collection (Settles 2012), and live service (Lasecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Kamar 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Offline model evaluation is used as well (Ribeiro, Singh, and Guestrin 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Li, Weston, and Roller 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Different techniques have been surfaced for ensuring data quality, expanding beyond inter-annotator agreement to more advanced approaches (Ghosh, Kale, and McAfee 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Some studies also specifically analysis the trade-off between data quality and budget (Khetan and Oh 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' As seen in the top datasets though, publishing of these methodologies doesn’t necessarily come with accompany- ing code (though some in the above do on Mephisto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='2 Crowdsourcing Surveys To attempt to bring shared signal out of the spread of sug- gestions and methods, surveys of the field attempt to collect, group, and evaluate various techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' These often provide suggestions for how others may include their own collec- tions (Vaughan 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Sheng and Zhang 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' They under- standably stop short of providing researchers with compre- hensive tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 3https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='org/wiki/Mephisto (automaton) In many cases, surveys refer to crowdsourcing as a key part of the data lifecycle, and try to shed light on the com- plexities and next steps within a specific domain (Ashmore, Calinescu, and Paterson 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Sambasivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Some works have outlined entire workflows for developing higher-quality datasets (Hutchinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' While the insights are certainly valuable, few papers directly refer to these works when building out their tooling, and few of those end up releasing actionable code assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' One survey (Neves and Seva 2020) attempts to find and document all of the available annotation tools, though this list is certainly non-exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='3 Full-code Solutions Often, works that release crowdsourcing code do so as part of a paper solving a contained problem space, like how Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (2021) sets up compositional dialogue tasks, or making the experiment flow for research somewhat easier for a spe- cific purpose, such as how Gureckis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (2016) attempts to abstract the complexities of interacting with the Mechanical Turk platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Mephisto falls into this last category of work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We branch out of ParlAI-MTurk (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2017), a project that was designed to make dialogue-based research easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Acknowledging the risk of becoming another bygone stan- dard4 we attempt to make the platform general enough to support integrating any of the above works, and intend to help ground the conversation in usable tools for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 3 Project Goals With Mephisto, we seek to address the core problems that prevent current crowdsourcing work from being easy to write, use, and distribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For this we outline core values and elements we believe a research annotation platform should follow, such that we can evaluate our progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='1 Distribution, Reproduction, and Extension In a research setting, each step of the process of distribu- tion, reproduction, and extension are relevant for a work to contribute to forwarding the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Distribution puts the tech- niques into the hands of other researchers, and can include sharing just methodology through the entire code setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Re- production allows new individuals to try out work, and the barrier for reproduction is often inversely proportional to how much of the process was able to be shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Extension is the next step on reproduction, and pushes the initial work forward into new research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' To help facilitate distribution, reproduction, and extension of work, an annotation platform should make it easy to dis- tribute all of the code related to a research project, and that code should be easy for new readers to set up, run, and mod- ify on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='2 Flexibility As research is a moving target, any platform that aims to support the varying needs of research should be casting a 4For those familiar with the trend of https://xkcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='com/927/ wide net for functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This complexity however runs counter to ease-of-use for a platform, which can raise the barrier of entry get researcher buy-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' An ideal platform should find a way to balance these two needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Abstracting Crowdsourcing and Implementations In order to adapt to new scenarios and situations, it’s valuable to examine the core elements of a crowdsourcing task, and isolate these into a coherent data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' From these build- ing blocks multiple interoperable implementations can be built up, thus allowing for a high degree of control over spe- cialized collection systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For the average user, basic im- plementations with simpler controls can be provided to get them up-and-running with as little context as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This type of approach allows work created on the plat- form to avoid the pitfalls of work such as in (Gureckis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2016), which helps users get started but ties them to a spe- cific platform for crowdsourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Hooks with Default Best Practices Within individual components of the platform, it should be possible for re- searchers to exact a high degree of control to run spe- cific jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This includes over portions such as worker- task pairing, collection pipelines and workflows, automated review tooling, and any other considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' To prevent overwhelming new users, each of these should have best- practices provided by default, allowing new users to benefit from the shared knowledge of current best approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This avoids the issues of both Hutchinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (2021) and Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (2017), where in the former there’s too much flexibility at the onset (considering you would have to im- plement it all yourself), and in the latter all tasks are forced into Dialogue data collection best practices and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='3 Data Quality considerations Any project aiming to facilitate crowdsourcing must con- sider data quality to be a priority, as no matter how easy it is to use and share, it isn’t particularly useful if the output data is low-quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We raise a few important considerations in this space, many under the lens that research work is of- ten time-limited and it can take a number of revisions and iterations to have something worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' UI/UX and worker quality of life The designed user- interface (UI) of a task is a significant contributor to task result quality (Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Tasks that are well de- signed may contain clear criteria, include examples, give for- mat specifications, reduce cognitive demand, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (Wu and Quinn 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' An ideal framework should help encode some of these best practices for the busy researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Workers are also more likely to return to tasks that are designed with a good user experience (UX) in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Tasks should minimize user frustration, both in terms of design and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Tasks should also be architected so that they are resilient to errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Hitting error cases should be clear to users at the least, and at most triggering some form of alerting so that the researcher can respond swiftly and appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Incorporating feedback channels is also a great way to identify and improve upon design blindspots that may oc- cur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We consider these blindspots to be the norm, not the exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' An ideal crowdsourcing framework should pro- vide researchers with feedback mechanisms that serve as a catch-all for any oversights on their parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' In implementa- tion, this would allow for easy communication from work- ers back to the researchers through some feedback channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Giving workers an opportunity to share feedback with re- searchers can create for a better worker experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (Bragg, Mausam, and Weld 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Quality Assurance Practices Standard quality assurance practices, such as worker qualification, gold-labelling, inter- annotator agreement, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' should be easy to discover and enable without getting into the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Encouraging re- searchers to design their tasks with these elements in the forefront will result in better data quality than tasks with these elements added as afterthoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Further, common workflows like pilots and worker communication can be crit- ical, and thus should be easy to enact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Worker Diversity and Representation When collecting a dataset, one element of quality comes from ensuring the data is worked on from as large and representative a col- lection of contributors as achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Often, this is limited by the tools of the company providing the crowd, and at times considered private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Still, a strong crowd- sourcing platform should provide tools to encourage re- searchers to extend their crowd with best-practices for sim- plified onboarding, task maximums per worker, and the abil- ity to use multiple crowdsourcing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' At the very least, it should be able to report some metrics for the source crowd, possibly integrating with something like data cards (Pushkarna, Zaldivar, and Kjartansson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4 Current implementation In following alongside the values and principles from the previous section, we designed Mephisto with an underlying set of abstractions, a few initial implementations, and then some best-practice elements both for task quality as well as researcher experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This section aims to give a technical overview of how Mephisto operates today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='1 Abstractions We’ll start off by describing Mephisto’s underlying abstrac- tions, which aim to break the complexity of crowdsourcing into components to build architecture around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' After describ- ing the data model, getting the rest of Mephisto is almost as easy as Architect, Blueprint, CrowdProvider, Database5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' What is in a task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The Data Model In order to reason about crowdsourcing, we break out a number of definitions that represent underlying elements of the data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The first is a Task, which can be considered as a group of directly related work that needs to be done, such as “Label 50,000 images with varying segmentation masks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Beneath this level is a TaskRun, which can be consid- ered an individual job you may have run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Of the 50,000 im- ages above, you may want to label the first 1,000 with 3 pos- sible mask labels for a pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This would be an appropriate 5Initialization sequence not by design, we promise Figure 1: Mephisto Crowdsourcing Data Model Overview TaskRun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' (A Task may have just one TaskRun, but will often have many).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Within a TaskRun, you may have many Assignments, which can be considered a discrete element you need done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This starts to be at the level of what you’ll show a worker, such as “Label these 5 images with segmentation masks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' An Assignment may be broken up into many Units, which represent the contribution that one individual may have on a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For some Assignments, there may be just one Unit, however for example you may have two Units on an Assignment that you want to have labelled twice to check inter-annotator agreement, or on a dialogue where you need two workers to communicate with one another at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For those actually doing the work, we have Workers which keep track of everything an individual has ever done for you for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' In order to distinguish the full Worker history, we also have Agents, which can be considered as a pairing between a Unit and a Worker representing the work that worker did for that particular unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' These are the underlying data model components that back Mephisto, and we can begin to reason about the rest of the flow for an annotation Task with this terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Hosting the Job: Architects Architects comprise the scripts to set up a server that runs a task in Mephisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' They allow researchers to use Mephisto with different cloud con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For this, they cover the server lifecycle during a task, and thus should implement methods for preparing, deploying, and shutting down servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' They also define the interface for which external workers are able to connect to the Mephisto back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The Tasks: Blueprints Blueprints are the center of Mephisto’s different tasks, and aim to capture both task flows common to a task and configuration settings that can allow someone to customize and extend that task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' They de- fine the inputs and outputs for a specific task as well as the overall task interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The specific abstraction requires a few important components, listed below: The AgentState defines the format of the data that will be saved during collection of a Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The TaskRunner defines any back-end logic that is re- quired to execute a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The TaskBuilder defines any resources that need to be built before a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Usually this includes the front-end to be hosted as part of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The SharedTaskState can be used to hold live state information shared between all of the Units in a TaskRun, often referred to when assigning work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The Workers: Crowd Providers CrowdProviders are what enable Mephisto to connect differing crowds to your task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' These interact closely with the abstract Workers, Agents, and Units in the following way: Workers comprise the long-term identity for a worker, and are an interface where Mephisto can in- clude worker-specific functionality that interfaces with a provider’s API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This may include blocking, giving quali- fications, and direct communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Units are an interface to the remote hook of a job posting, or similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' They need to keep track of ex- ternal status, and should also provide the interface for registering and expiring a work request with a provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Agents cover the link between a worker and a single Unit, and must implement methods for checking their remote status, as well as marking work as completed or rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Results Storage: Databases Databases are what en- able Mephisto to store your results, regardless of the server setup you are using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For this, we provide the MephistoDB abstraction, which lists all of the required database calls one would need to implement to run Mephisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='2 Mephisto Architecture In practice, Mephisto is able to handle any arbitrary config- uration of Blueprint, Architect, CrowdProvider, and Database and coordinate the initialization, deploy- ment, monitoring, and shutdown of each over the TaskRun they comprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Over the course of such a LiveTaskRun, it also reports metrics and saves partial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' One key goal is that any “business logic” that people would like to customize has a clear hook for doing so in the abstractions, such that most users don’t need to deal with the complexity of how these interfaces are coordinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' After data collection has concluded, Mephisto provides tools that allow one to inter- act with and explore the data stored in the MephistoDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' TASK contains many TASK-RUN WORKER viewed as given is comprised of 丰 ASSIGNMENT AGENT GRANTED-QUALIFICATION split into works on instanceof UNIT QUALIFICATION4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='3 Blueprints It’s our goal that in the majority of cases, most of Mephisto users should be able to rely on existing Blueprints, rather than needing to write new ones from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' To this end, we provide a few useful implementations that cover a wide set of use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The ReactStaticBlueprint is a setup where one can provide any simple data collection front-end applica- tion written in React that can be considered a single turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' In short: The worker is provided some data, they work on it remotely, and then return the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The RemoteProcedureBlueprint allows a more complex setup, where the front-end application is able to make direct queries to some back-end specified dur- ing task setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This allows for doing processing that wouldn’t be possible on the worker’s side, such as run- ning a model in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The StaticHTMLBlueprint stands as the easiest onboarding ramp to Mephisto, in that it accepts stan- dard .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='html files that researchers may be more familiar with than React.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' It isn’t as feature rich as other offerings though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Beyond these, the ParlAIChatBlueprint stands as a good example of a live task with a specified and highly configurable flow, catered towards dialogue-focused jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Quality Assurance Mixins We provide a handful of mix- ins for quality assurance which are available to be used on anything run from the ReactStaticBlueprint or the RemoteProcedureBlueprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Including these mixins into a Blueprint means that blueprint has the specified functionality enabled and knows how to handle it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The OnboardingRequired mixin allows researchers to set up a separate flow for workers who haven’t done the task before, allowing them to learn what the require- ments are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The UseGoldUnit mixin provides a familiar flow for providing known-good examples for which workers will be evaluated against periodically as a quality check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The ScreenTaskRequired mixin allows researchers to have the first actual Unit that a Worker works on for a job be a specified (usually easy-to-verify) unit, which allows researchers to do automated analysis and valida- tion of before giving more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Researchers can use the primitives for these methods to in- corporate strong quality assurance flows into their tasks, without needing to build any complex machinery on top of the underlying validation measures for their task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Of course, it still requires some initial rounds of piloting and tweaking to ensure the validators are well calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='4 Crowd Providers The main crowd providers we have implemented at the mo- ment are the MTurkProvider and the MockProvider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The former allows for direct interfacing with the Amazon Mechanical Turk platform, while the latter allows for testing tasks locally, or allowing people to access while “mocking” a specified worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Adding more Crowd Providers is ongo- ing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='5 Architects The currently available architects at the time of writing are the HerokuArchitect and the LocalArchitect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The former allows launching using Heroku cloud services as a provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The latter allows hosting on the machine running Mephisto, which is useful for testing locally or collecting from research participants on the same local network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We also have an EC2-based architect which requires registering a domain name with AWS for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='6 Front-end Packages Another component of launching a crowdsourcing task is de- signing the task’s UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Mephisto allows researchers to launch tasks with UI implemented with either plain browser HTML, or for more advanced cases, with the React JavaScript li- brary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For the React implementation, Mephisto provides an npm (Node Package Manager) package named mephisto-task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The package enables researchers to interface seamlessly with the Mephisto back-end Blueprints from their front-end code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' It surfaces the task data provided from the back-end, callbacks to handle task submission for the CrowdProvider, as well as boolean flags that can be used to conditionally display different views (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' task preview, onboarding, errors, submissions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The package also exposes three React Hooks: useMephistoTask, useMephistoLiveTask, and useMephistoRemoteProcedureTask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The latter two can be used for more advanced tasks, such as chat-bots or model-in-the-loop tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We particularly see model-in-the-loop as an opportunity to increase task result quality by augmenting worker per- formance in real-time, minimizing tedious and rote work, improving perceived UX, and providing real-time validation and feedback, though more research is needed in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Model-in-the-loop approaches can also be used to dynamically generate subsequent tasks based on prior tasks, providing customized control over what tasks get launched next while a task run is already underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The examples/ folder in the Mephisto GitHub reposi- tory provides sample task templates using the simple setup, as well as advanced setups including chat-bots and model- in-the-loop functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='7 Worker Feedback With Mephisto’s extensible architecture, creating plugins is easy as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We provide two first-party plugins through the npm package mephisto-worker-addons to help im- prove the worker experience for tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Specifically, we pro- vide the Feedback and Tips React components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The Feedback component allows workers to provide sug- gestions back to the reseacher as they’re working through tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This could include questions, bugs they’ve found, or positive acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This communication channel back to the researcher can be a way to improve worker senti- ment and improve task quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Researchers can also choose to tip or give bonuses to submitters who provide valuable feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Communicating this reward scheme can also cre- ate a helpful incentive mechanism for gathering tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The Tips components allows workers to create a shared FAQ-style wiki that other workers can benefit from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This comes with built-in moderation as submitted Tips need to be approved by the researcher before they’re visible to other workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Aside from being helpful, these examples indicate a few ways of how Mephisto can be made extensible to suit custom research needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='8 Review Tooling Mephisto also includes a Python based command-line in- terface (CLI) tool to allow users to review a task’s results, or more generally any arbitrary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' The command accepts an input data source as well as a “review template”, and launches a local webserver to allow for browsing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' For any arbitrary data, one can just pipe in an input file: cat input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='jsonl | mephisto review --json my-review-interface --stdout Or for using specifically with a Mephisto task run, one can use the --db flag: mephisto review --db task-name my-review-interface --stdout To facilitate review, we provide a React template based on create-react-app that implements a modular ren- dering architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' This architecture allows researchers to easily define how a “data item” should be rendered by imple- menting their own custom renderer as a single React com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Out of the box, we ship a few default renderers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' for example, a JSON renderer and a Word Cloud renderer for text-heavy tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Once a task run is complete and a dataset has been accumulated, researchers can share results along with the Mephisto-based review and visualization tool as part of their publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Mephisto’s base review tooling was used by the Ego4D project (Grauman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 2021) to share their collected 3,000 hours of egocentric video6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='9 Worker Qualifications Mephisto provides a simple setup for tagging workers for any reason, wherein you can create and assign arbitrary Qualifications to any Worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We find this is use- ful in setting up allow and block lists, querying or selecting workers based on skills you’ve noted them for, and creating complex task flows (such as those where participating in one role disqualifies another).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 5 Future work Mephisto is an evolving system, and we continue to iterate and develop it alongside the values listed in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 6https://ego4d-data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='org/docs/viz/ As we discover new powerful methods for crowdsourcing we aim to include them in Mephisto as top-level functional- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We also aim to provide easy ways for anyone on the plat- form to build new hooks and functionality, and share them with others who are developing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Further, we hope to extend the base set of existing tooling that Mephisto sup- ports out-of-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Lastly, we aim to extend the portability of the platform, such that it can be used with as many providers, on as many hosting solutions, and with as many tasks as pos- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We also hope to continue to build along the dimensions of task-design - making it easier to share and use community- sourced design and task templates, opt into UI and UX best- practices as they emerge, and experiment with new primi- tives to improve worker experience, such as gamification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Even with these steps though, we’re only scratching the surface of implementing the best practices of today, let alone accommodating those of tomorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We hope this work can stand as a foundation that future work will build upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Our roadmap is available on the Github project page, and we’re open to feedback on where we should take the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content='1 Contributing Mephisto is an open source project, and we value contribu- tions from our users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' We welcome anyone to join in and help with the vision of easy, reproducible crowdsourcing with best-practices built in on our GitHub7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Feel like we’re do- ing something wrong, or are missing a technique that people should be using immediately?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Great!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' File an issue, or better yet open a PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Ethical statement Mephisto is provided as a crowdsourcing software with a permissive license on use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' While the Mephisto platform aims to improve annotation methodologies and facilitate co- operation towards resolving data collection issues, it cer- tainly is still a work in progress towards those goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' It doesn’t directly impose them as constraints on its users, so while we try to make currently agreed upon best practices the defaults, they can be overridden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' As such, issues such as underpayment or mistreatment of workers, collection of biased datasets, and data licensing is- sues may still arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' A researcher using Mephisto still must to do their due diligence to ensure they are up-to-date on the best methodologies for their collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Acknowledgements We’d like to thank all of Mephisto’s public contributors as well as the ParlAI team and other early pilot users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' References Ashmore, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=';' metadata={'source': 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Girdhar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Hamburger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} +page_content=';' 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+page_content='07079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQfkw2x/content/2301.05154v1.pdf'} diff --git a/QtE4T4oBgHgl3EQf_A4a/content/tmp_files/2301.05367v1.pdf.txt b/QtE4T4oBgHgl3EQf_A4a/content/tmp_files/2301.05367v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f06f469fc0e0b614b60c9c1d66aaca4aafe02345 --- /dev/null +++ b/QtE4T4oBgHgl3EQf_A4a/content/tmp_files/2301.05367v1.pdf.txt @@ -0,0 +1,1956 @@ +Received August 25, 2022; +Revised January 12, 2023; +Accepted January 12, 2023 +DOI: xxx/xxxx +ARTICLE TYPE +Surface magnetic field of the A-type metallic-line star +omicron Pegasi revisited +Yoichi Takeda +11-2 Enomachi, Naka-ku, Hiroshima-shi, +730-0851, Japan +Correspondence +Email: ytakeda@js2.so-net.ne.jp +The bright A-type metallic-line star 표 Peg was reported in the early 1990s to have a +surface magnetic field of ∼ 2 kG by analyzing the widths and strengths of spectral +lines. In respect that those old studies were of rather empirical or approximate nature +and the quality of observational data was not sufficient, this problem has been newly +reinvestigated based on physically more rigorous simulations of line flux profiles, +along with the observed equivalent widths (푊 ) and full-widths at half-maximum (ℎ) +of 198 Fe I and 182 Fe II lines measured from the high-quality spectra. Given the +Fe abundance derived from the conventional analysis, theoretical 푊 and ℎ values +calculated for various sets of parameters were compared with the observed ones, +which lead to the following conclusion regarding ⟨퐻⟩ (mean field strength). (1) An +analysis of 푊 yielded ⟨퐻⟩ ∼1–1.5 kG from Fe II lines with the microturbulence of +푣t ∼ 1.5 km s−1. (2) A comparison of ℎ resulted in ⟨퐻⟩ ∼1.5–2 kG as well as the +projected rotational velocity of 푣e sin 푖 ≃ 5 km s−1. (3) Accordingly, the existence of +mean magnetic field on the order of ⟨퐻⟩ ∼ 1–2 kG in 표 Peg was confirmed, which +is almost consistent with the consequence of the previous work. +KEYWORDS: +stars: atmospheres — stars: chemically peculiar — stars: early-type stars: individual (표 Peg) — stars: +magnetic fields +1 +INTRODUCTION +The star 표 Peg (= HD 214994 = HR 8641 = HIP 112051; +spectral type is A1 IV) is one of the most frequently studied +A-type stars in stellar spectroscopy, because of its apparent +brightness (푉 = 4.79) and sharp-line nature with low pro- +jected rotational velocity (푣e sin 푖 ≲ 10 km s−1). As is often the +case with slowly-rotating stars in the upper main sequence, it +is a chemically peculiar (CP) star; more specifically, because +of the conspicuous overabundances in the elements heavier +than the Fe group, it is regarded as a hot metallic-lined A- +type (Am) star (e.g., Adelman 1988, Hill & Landstreet 1993). +0Abbreviations: LTE, local thermodynamic equilibrium; FWHM, full width +at half maximum +Definite variability has not been confirmed, despite of the des- +ignation “Si-variable” and “variable radial velocity” in Renson +& Manfroid’s (2009) catalogue of CP stars. +A remarkable and rare feature of this star is that a sur- +face magnetic field was spectroscopically detected in the past, +despite that Am stars are generally not magnetic (unlike other +groups of CP stars such as the SrCrEu type). 1 +— It was Mathys & Lanz (1990; hereinafter referred to as +ML90) who first reported the existence of a magnetic field of +퐻 ∼ 2 kG in 표 Peg, where they employed two independent +1The meaning of this statement (non-magnetic nature of Am stars in general) +is that appreciable magnetism (e.g., field strength on the order of ∼kG) found in +magnetic CP stars is generally absent in Am stars. Recent very high-precision spec- +tropolarimetric observations have revealed that an extremely weak magnetic field +(on the order of ∼G) is detectable in several hot Am stars; Sirius (Petit et al. 2009), +훽 UMa and 휃 Leo (Blazère et al. 2016a), Alhena (훾 Gem) (Blazère et al. 2016b, +2020). +arXiv:2301.05367v1 [astro-ph.SR] 13 Jan 2023 + +2 +Y. TAKEDA +techniques: (i) statistical line-width analysis (Stenflo & Lin- +degren 1977) and (ii) empirical relation for the 퐻-dependent +difference of equivalent width (푊 ) between Fe II 6147.7 and +6149.2 lines (which have almost the same strengths in the non- +magnetic case; cf. Mathys 1990). +— Successively, numerically solving the transfer equation of +polarized radiation in the presence of a magnetic field accord- +ing to Takeda (1991a), Takeda (1991b; hereinafter referred to +as T91b) explained the 퐻-dependence of 푊6147.7∕푊6149.2 dif- +ference as due to the desaturation effect caused by magnetic +broadening (while proposing the simultaneous use of simi- +lar line pair Fe II 4416.8/4385.4), and derived 퐻 ∼ 2–3 kG +(assuming a microturbulence of 푣t ∼ 0 km s−1) for 표 Peg, +which is favorably compared with ML90. +— Further, Takeda (1993; hereinafter T93) devised a method +for determining the magnetic field based on the 푊 values +of many lines, which is regarded as a refined version of the +classical Hensberge & De Loore’s (1974) technique and can +establish the (퐻, 푣t) solution by requiring the consistency of +abundances derived from lines of various strengths. Regarding +표 Peg, T93 concluded 퐻 ∼ 2 kG (and 푣t ∼ 1.5 km s−1), which +is again consistent with ML90. +Since then, however, little progress seems to have been made +as to the corroboration of these findings. As already noted +by ML90, the spectropolarimetric technique (most commonly +used for investigating magnetic fields of CP stars) has been +unsuccessful to accomplish a meaningful detection in 표 Peg +(though extremely weak magnetic field might as well be still +detectable as remarked in footnote 1). Actually, besides the +pioneering work of Babcock (1958), Shorlin et al. (2002) could +not detect any ⟨퐻푧⟩ (disk-averaged line-of-sight component of +the field) of significance in 표 Peg by their spectropolarimet- +ric observation coupled with Least-Squares-Decomposition +(LSD) technique.2 +This suggests that ⟨퐻푧⟩ happens to be too small (presum- +ably because components of opposite signs are cancelled out +by disk integration) to produce a significant signal of circular +polarization. +Accordingly, in order to study the magnetic nature of 표 Peg, +the best way would be to analyze the equivalent widths or line +widths of many spectral lines (which contain information of +“absolute” magnetic field strengths) as previously done. How- +ever, it may be premature to regard the results of those old +studies as sufficiently reliable because several problems are +involved from a methodological point of view. +2According to the compilation of Bychkov, Bychkova, & Madej (2009), 표 Peg’s +⟨퐻푧⟩ resulting from Shorlin et al.’s (2002) observation was 32(±20) G, which would +have been regarded as under the level of significance. +• The analysis of ML90 was not based on a rigorous mod- +eling but a rather tentatively postulated analytical rela- +tion between the line width, line strength, and Zeeman- +broadening parameters. Especially, since the original +Stenflo & Lindegren’s (1977) work (on which their study +is based) was intended to estimate the solar magnetic field +at the disk center, the effect of rotational broadening on +the line width was not taken into account; thus, how the +stellar projected rotational velocity (푣e sin 푖) affects the +functional form of the relation is unclear. +• Although T91b simulated the emergent profiles and the +strengths of spectral line pairs by correctly treating the +transport of polarized radiation in a magnetic field, only +the specific intensity profiles at the disk center (휇 = +cos 휃 = 1) were calculated for several single-valued field +strengths (퐻) and different angles between the magnetic +field vector and the line of sight (휓) , which must be unre- +alistic for comparing with the flux profiles of magnetic +stars. Another problem is the choice of microturbulence: +푣t ∼ 0 km s−1 adopted in T91b is not consistent with that +obtained later by T93 (∼ 1.5 km s−1); if the latter were +chosen, a considerably higher magnetic field (∼ 4–5 kG) +would have resulted (cf. Sect. 4.1 in T91b). +• The results of T93 were actually not robust but rather del- +icate, because 퐻 could be firmly established only from +Fe II lines among the three species (finding a definite +solution was difficult for Fe I and Ti II lines). Above all, +the validity of the approximate method for evaluating the +flux equivalent width under a magnetic field proposed by +T93 (denoted as 푊푏, which is a simple mean between the +minimum and maximum intensification cases) should be +quantitatively checked in the first place. +• Attention should also be paid to the adopted observa- +tional data of 표 Peg. ML90’s measurements of line widths +for their application of Stenflo–Lindegren technique were +done on spectra with a signal-to-noise ratio (S/N) of ∼ +100 covering 3700–4650 Å resulting from the co-added +photographic spectrogram with a reciprocal dispersion of +2.4 Å mm−1 obtained at Dominion Astrophysical Obser- +vatory (Adelman, Cowley, & Hill 1988). Similarly, the +equivalent width data of Fe I, Fe II, and Ti II lines for +표 Peg adopted in the analysis of T93 were taken from +Adelman’s (1988) Table 5, for which he used the same +co-added Dominion spectra. Since the quality of these +photographic spectra is not sufficient from the viewpoint +of present-day standard and using only the blue-region +data is not advantageous for the purpose of magnetic- +field detection, it is desirable to employ modern spectra +of much higher quality covering wider (from blue through +red) wavelength range. + +Y. TAKEDA +3 +As such, it is worth reinvestigating the magnetic field of +표 Peg by analyzing the widths and strengths for a number of +spectral lines as done by ML90 and T93 but based on physi- +cally rigorous theoretical line-profile modeling along with new +observational data. Given this situation and now that CCD +spectra of very high S/N for this star are now available, I +decided to freshly revisit this problem, while making use of the +profile simulation program based on disk integration coupled +with the solution code of polarized radiation transfer in a mag- +netic field. The purpose of this paper is to report the results of +this new analysis. +2 +OBSERVATIONAL DATA +2.1 +Observed spectra of 표 Peg +The high-dispersion spectra of 표 Peg adopted in this investiga- +tion are the same as used in Takeda et al.’s (2012) abundance +study of alkali elements for A-type stars. The observations +were done on UT 2008 October 4 (8 frames of 1200 s exposure +within a time span of ∼ 6 hr), October 7 (1 frame of 1800 s +exposure), October 8 (8 frames of 1200 s exposure within a +time span of ∼ 3 hr), and October 9 (6 frames of 1200 s expo- +sure within a time span of ∼ 2 hr) by using the HIDES (HIgh +Dispersion Echelle Spectrograph) placed at the coudé focus +of the 1.88 m reflector at Okayama Astrophysical Observa- +tory. Equipped with three mosaicked 4K×2K CCD detectors +at the camera focus, echelle spectra covering 4100–7800 Å +(in the mode of red cross-disperser) with a resolving power of +푅 ∼ 100000 (corresponding to the slit width of 100 휇m) were +obtained. +The reduction of the spectra (bias subtraction, flat-fielding, +scattered-light subtraction, spectrum extraction, wavelength +calibration, and continuum normalization) was performed by +using the “echelle” package of the software IRAF3 in a stan- +dard manner. In order to improve the signal-to-ratio, all avail- +able frames were co-added, by which very high S/N (typically +around ∼ 1000 on the average) could be accomplished in the +final spectra as shown in Fig. 1 (S/N ∼ +√ +푐표푢푛푡). +2.2 +Selected lines and measurements +In this study, we concentrate to using only spectral lines of +Fe I and Fe II, as they are available in larger number over +a wide range of strengths than other species. The candidate +lines to be analyzed were carefully sorted out by inspecting the +observed spectral feature while comparing it with the calcu- +lated strengths of neighboring lines as well as the synthesized +3IRAF is distributed by the National Optical Astronomy Observatories, which +is operated by the Association of Universities for Research in Astronomy, Inc. under +cooperative agreement with the National Science Foundation. +4000 +4500 +5000 +0 +1000000 +2000000 +3000000 +(a) +5500 +6000 +6500 +0 +1000000 +2000000 +3000000 +(b) +count +7000 +7500 +0 +1000000 +2000000 +3000000 +(c) +Wavelength (Å) +FIGURE 1 Panels (a), (b), and (c) show the distributions of +accumulated photoelectron counts of CCD for the final spec- +tra of 표 Peg, each corresponding to three wavelength regions +(4100–5300/5300–6600/6600–7900 Å) comprising 32/20/13 +orders, respectively. Note that the signal-to-noise ratio can +be estimated as S/N ∼ +√ +푐표푢푛푡 in the present photon-noise- +limited case. The spectra in each of the echelle orders show +characteristic distributions of the blaze function. +theoretical spectrum, as done by Takeda (2020). Because of +the necessity of calculating the Zeeman components, those +lines lacking the information of quantum numbers (퐿, 푆, 퐽) +for the lower and upper levels were excluded. As a result, +198 Fe I and 182 Fe II lines were selected. The equivalent +widths (푊휆) of these lines were measured by fitting the line- +depth profile (푅휆 ≡ 1 − 푓휆∕푓cont) with the Gaussian function +(∝ exp[−(휆 − 휆0)2∕푎2]), while the line widths (ℎ휆; defined as +the FWHM of 푅휆) were directly evaluated from the profiles. +The 푊휆 values of these Fe lines range from ∼ 1 mÅ to +∼ 200 mÅ (Fig. 2a) and their effective Landé factors are +between 0 ≲ 푔L +eff ≲ 3 (Fig. 2b). As seen from the empirical +curves of growth depicted in Fig. 2c, the linear part and the +shoulder/flat part are roughly separated around 106푊휆∕휆 ∼ +10 (typically several tens mÅ in 푊휆). A comparison of the +equivalent widths with those published by Adelman (1988) +is displayed in Fig. 2d, where a reasonable consistency is +observed. Fig. 2e shows that the directly measured FWHM +values (ℎ휆) are mostly in agreement with the corresponding +Gaussian-fit ones (derived from the 푒-folding half-width as +ℎG.F. +휆 +≡ 2 +√ +ln 2 푎). The dependence of ℎ푣 (in the velocity unit + +4 +Y. TAKEDA +derived as ℎ휆푐∕휆: 푐 is the speed of light) upon the equivalent +width is illustrated in Fig. 2f, which shows that ℎ푣 begins to +exhibit a systematic 푊휆-dependence at 106푊휆∕휆 ≳ 10. +The measured 푊 and ℎ values of these 380 Fe lines along +with their atomic data (wavelength, excitation potential, oscil- +lator strengths, damping constants, term information, effective +Landé factor, etc.) mostly taken from the VALD database +(Ryabchikova et al. 2015) are summarized in “felines.dat” of +the online material, where the original profile data of all lines +are also given in “obsprofiles.dat”. +3 +CONVENTIONAL ANALYSIS OF +EQUIVALENT WIDTHS FOR +MICROTURBULENCE +Before dealing with the main issue of magnetic field estimation +to be described in Sect. 4 and 5, we first conduct a prepara- +tory analysis of determining the microturbulence based on +the equivalent widths by applying the conventional procedure +(while assuming as if no magnetic field exists). +3.1 +Atmospheric model and parameters +Regarding the standard model atmosphere of 표 Peg, we +adopted Kurucz’s (1993) ATLAS9 solar abundance model +with 푇eff = 9500 K, log 푔 = 3.60 (cgs unit). These atmo- +spheric parameters were chosen by inspecting the various +literature data summarized in Table 1. As recognized from this +table, these values (though rather rounded) are consistent with +those derived in many of the past studies (especially the recent +ones published after 2000). Besides, this choice is in accord +with 푀 = 2.8푀⊙ (mass) and 푅 = 4.4푅⊙ (radius) evaluated +from the position on the log 퐿 (luminosity) vs. 푇eff diagram +in comparison with theoretical evolutionary tracks (see, e.g., +Fig. 1 in Takeda et al. 2012). Abundance determination from +an equivalent width for a given microturbulence was done +by using Kurucz’s (1993) WIDTH9 program while assuming +LTE. The adopted line parameters are given in “felines.dat” +(see Sect. 2.2). +Two approaches are tried in this 푣t determination test; (1) +usual method of finding the minimum abundance dispersion, +and (2) alternative method requiring the overall consistency +between the observed and theoretical 푊 . +3.2 +Method 1: minimum abundance +dispersion +The effect of 푣t on abundance determination appreciably +depends on line strengths: abundances determined from weak +lines in the linear part of the curve of growth are essentially free +from 푣t, while those from stronger lines on the shoulder-to- +flat part are considerably 푣t-dependent. Therefore, 푣t is usually +established by requiring that abundances derived from lines of +various equivalent widths (푊 ) be consistent with each other. +Among the several practical procedures for accomplishing +this requirement, Blackwell et al.’s (1976) method is used here. +–(1) For each line 푛, a set of abundances (퐴푘 +푛; 푘 = 1, 2, … , 퐾) +are derived from 푊푛 while incrementally changing the micro- +turbulence (푣푘 +t ; 푘 = 1, 2, … , 퐾). +–(2) Then, the mean abundance (⟨퐴⟩푘) averaged over 푁 lines +and the standard deviation 휎푘 +퐴 are calculated for each of the 퐾 +microturbulences (푣푘 +t ). +–(3) By inspecting the resulting standard deviation (휎푘 +퐴; 푘 = +1, 2, … , 퐾), the location of minimum 휎퐴 corresponds to +desired solution of 푣t. +This procedure was applied to our data set of Fe lines. The +resulting 퐴 vs. 푣t relations for each of the lines along with the +corresponding 휎퐴 vs. 푣t curve are shown in Figs. 3a and 3b +for Fe I (푁1 = 198) and Fe II lines (푁2 = 182), respectively. +As seen from these figures, we obtain (1.71 km s−1, 7.75)4 and +(1.76 km s−1, 7.82) as the results of (푣t, ⟨퐴⟩) for Fe I and Fe II; +and the corresponding 퐴푛 for each line is plotted against 푊푛 in +Figs. 3c and 3d, respectively . +3.3 +Method 2: minimum equivalent width +dispersion +Next, another method for 푣t determination is tried. Since 퐴 = +7.8 may be regarded as the fiducial Fe abundance of 표 Peg from +the results of 7.75 (Fe I) and 7.82 (Fe II) derived in the previous +subsection, we can calculate the theoretical equivalent widths +(푊 푘,7.8 +cal,푛 ; 푛 = 1, 2, … , 푁) for each of the lines with this fixed +Fe abundance but incrementally changing 푣푘 +t , which are to be +compared with the observed ones (푊obs,푛; 푛 = 1, 2, … , 푁). +Then, by examining the standard deviations 휎푘 +푊 evaluated for +various 푣푘 +t values, +휎푘 +푊 ≡ +√ +√ +√ +√ +푁 +∑ +푛=1 +(푊 푘,7.8 +cal,푛 − 푊obs,푛)2∕푁, +(1) +we can find the solution of 푣t as that accomplishing the mini- +mum 휎푘 +푊 . Although the final 푊cal calculated with such deter- +mined 푣t is not exactly equal to the observed 푊obs (because of +the fixed Fe abundance), the primary aim of line-independent +consistency (i.e., without any global 푊 -dependent trend) can +be accomplished. +The differences 푊 푘,7.8 +cal,푛 − 푊obs,푛 for each lines are plotted +against 푣t in Fig. 4a (Fe I) and 4b (Fe II), where the correspond- +ing 휎푊 versus 푣t curves are also depicted. From these figures, +4퐴 is the logarithmic number abundance of Fe relative to H, which is expressed +in the usual normalization of log[푁(Fe)∕푁(H)] + 12. + +Y. TAKEDA +5 +4000 5000 6000 7000 8000 +0.5 +1 +5 +10 +50 +100 +λ (Å) +Wλ (mÅ) +(a) +0 +10 +20 +30 +0 +1 +2 +3 +(Wλ/λ)106 +geff +L +(b) +-8 +-6 +-4 +-2 +0 +-1 +0 +1 +2 +log gf - χlow(5040/Teff) +log(106Wλ/λ) +(c) +0.1 +0.2 +0.3 +0.4 +0.1 +0.2 +0.3 +0.4 +hλ (Å) +hλ +G.F. (Å) +(e) +0 +10 +20 +30 +8 +10 +12 +14 +(Wλ/λ)106 +hv (km s-1) +(f) +1 +5 10 +50 100 +1 +5 +10 +50 +100 +Wλ +Takeda (mÅ) +Wλ +Adelman (mÅ) +(d) +FIGURE 2 (a) Equivalent width vs. wavelength. (b) Effective Landé 푔 factor vs. reduced equivalent width. (c) Empirical curves +of growth for the Fe I and Fe II lines, where log 푔푓 −휒low(5040∕푇eff) is taken in the abscissa (푔: statistical weight of the lower level, +푓: oscillator strength, 휒low: lower excitation potential in eV, 푇eff: effective temperature in K). (d) Comparison of the equivalent +widths measured in this study with those published by Adelman (1988) for 97 lines (65 Fe I lines and 32 Fe II lines) in common. (e) +Correlation between the directly measured full-widths at half-maximum (ℎ휆) with the corresponding Gaussian-fit values (ℎG.F. +휆 +). +(f) Full-widths at half-maximum in the velocity unit (ℎ푣 ≡ 푐ℎ휆∕휆; 푐 is the speed of light) plotted against the reduced equivalent +widths. In each panel, the data for Fe I and Fe II lines are discriminated in blue filled symbols and red open symbols, respectively. +we obtain 1.38 km s−1 and 1.77 km s−1 for Fe I and Fe II, +respectively. The differences 푊 푘,7.8 +cal,푛 −푊obs,푛 for each lines cor- +responding to these 푣t solutions are plotted against 푊obs,푛 in +Fig. 4c (Fe I) and 4d (Fe II), + +6 +Y. TAKEDA +TABLE 1 Atmospheric parameters of 표 Peg published so far. +Authors +푇eff +log 푔 +푣t +[Fe/H] +푣e sin 푖 +Wolff (1967) +9330 +3.2 +2.0 +0.30 +Conti & Strom (1968) +9500 +4.0 +3.0 +0.2푎 +Adelman (1973) +10100 +4.0 +3.0 +0.1 +Allen (1977) +9600 +3.8 +3.2 +−0.09 +Mitton (1977) +9500 +3.5 +1.6 +0.08푏 +Adelman et al. (1984) +9625 +3.45 +1.6 +0.20 +Adelman & Fuhr (1985) +1.9 +0.26 +Adelman (1988) +9600 +3.60 +1.3 +0.10 +6 +Castelli & Hack (1988) +9590 +3.55 +1.9 +0.14 +6 +Kocer et al. (1988) +9500 +3.50 +1.5 +0.04 +Sadakane (1988) +9500 +3.50 +2.0 +0.02 +Van’t Veer et al. (1988) +9350 +3.50 +1.5 +0.02 +Burkhart & Coupry (1991) +9650 +3.6 +0.1 +Hill & Landstreet (1993) +9680 +3.71 +1.5푐 +0.03 +Abt & Morrell (1995) +10 +Hill (1995) +1.7 +0.19 +6.3 +Sokolov (1995) +10050 +Blackwell & Lynas-Gray (1998) +9443 +Di Benedetto (1998) +9720 +Hui-Bon-Hoa (2000) +9650 +3.6 +1.5 +0.42 +≤ 10 +Adelman et al. (2002) +9591 +3.64 +Adelman et al. (2002) +9525 +3.70 +Royer et al. (2002) +14 +Royer et al. (2007) +14 +Landstreet et al. (2009) +9500 +3.62 +2.0 +0.14푑 +7 +Zorec et al. (2009) +9930 +Prugniel et al. (2011) +9373 +3.73 +−0.14 +Takeda et al. (2012) +9453 +3.64 +3.1 +0.13푒 +6.0 +Zorec & Royer (2012) +9506 +3.73푓 +14 +Gray (2014) +9600 +3.7 +0.0푔 +6.00ℎ +Takeda et al. (2018) +9453 +3.64 +2.7 +0.18푖 +6.6 +Summarized here are the effective temperature (in K), logarithmic surface gravity in c.g.s unit (in dex), microturbulence (in km s−1), Fe abun- +dance relative to the Sun, and projected rotational velocity (in km s−1) of 표 Peg taken from previous publications. Since these parameters were +determined in variously different methods, the original references should be consulted for the details. Regarding [Fe/H], if only those derived +from Fe I and Fe II lines are available, a simply averaged value of these two is listed here. Besides, in case that the reference solar Fe abundance +is not explicitly given, an appropriate value widely used at the time of publication was tentatively adopted. +푎Relative to the mean of 4 standard stars. +푏Relative to Procyon. +푐Assumed. +푑Solar Fe abundance of log(Fe∕H)⊙ = −4.49 was assumed. +푒Solar Fe abundance of 퐴⊙(Fe) = 7.50 was assumed. +푓Derived from 퐿 (bolometric luminosity), 푇eff, and 푀 (mass). +푔Assumed. +ℎRadial-tangential macroturbulence of 휁RT = 5.7 km s−1 was adopted. +푖Relative to Procyon. +A comparison of 푣t(Method 1) derived in Sect. 3.2 with +this 푣t(Method 2) suggests that, while we can confirm a good +agreement for the case of Fe II lines (1.76/1.77 km s−1), a dis- +crepancy is seen for 푣t based on Fe I lines (1.71/1.38 km s−1). +As a matter of fact, the results from Fe I lines appear to be +somewhat problematic. As can be seen in Fig. 3c, the distribu- +tion of 푊 푘,7.8 +cal,푛 − 푊obs,푛 differences for Fe I lines of medium- +to-large strengths (푊obs ≳ 10 mÅ) shows some asymmetric +feature (i.e., positive for lines of 20–50 mÅ while negative +for those of ≳ 50 mÅ). This trend has made the 휎푘 +푊 curve +shallower with a less clear minimum (Fig. 4a), which eventu- +ally leads to larger uncertainties in 푣t determination. Generally +speaking, since only a very tiny fraction of Fe atoms remain +neutral (those in Fe II and Fe III stages are dominant) in the +atmosphere of A-type stars, the formation of Fe I lines is con- +siderably 푇 -dependent and vulnerable to model atmosphere +structure, while Fe II lines are more robust in this respect (see +Appendix A2 of Takeda 2020). Accordingly, Fe II lines may +yield more reliable results than Fe I lines. + +Y. TAKEDA +7 +0 +1 +2 +3 +7 +7.5 +8 +8.5 +9 +0.1 +0.15 +0.2 +0.25 +0.3 +vt (km s-1) +A1 +σA1 +1.71 km s-1 +Fe I lines +(a) +0 +1 +2 +3 +7 +7.5 +8 +8.5 +9 +0.1 +0.15 +0.2 +0.25 +0.3 +vt (km s-1) +A2 +σA2 +1.76 km s-1 +Fe II lines +(b) +1 +5 10 +50100 +7 +7.5 +8 +8.5 +9 +Wλ (mÅ) +A1 +vt = 1.71 km s-1 + = 7.75 +(c) +1 +5 10 +50100 +7 +7.5 +8 +8.5 +9 +Wλ (mÅ) +A2 +vt = 1.76 km s-1 + = 7.82 +(d) +FIGURE 3 Upper panels (a), (b): solid lines show how +the Fe abundance (퐴) of each line varies by changing 푣t, +where weaker lines (106푊휆∕휆 < 10) and stronger lines +(106푊휆∕휆 > 10), are distinguished by gray and black lines, +respectively. In addition, 휎퐴 (standard deviation of 퐴) is +plotted against 푣t by the thick green solid line (its scale is +marked in the right axis), and the 푣t solution corresponding +to the minimum 휎퐴) is also indicated. Lower panels (c), (d): +Fe abundances corresponding to the 푣t solution are plotted +against the observed equivalent widths. The mean abundance +(⟨퐴⟩) is also indicated by the horizontal dashed line. The +left-hand and right-hand panels are for Fe I and Fe II lines, +respectively. +4 +LINE PROFILE SIMULATION OF A +MAGNETIC STAR +4.1 +Magnetic field model +As already mentioned in Sect. 1, the main aim of this inves- +tigation is to check the previously reported results (possible +existence of a magnetic field on the order of ∼ 2 kG in +표 Peg) based on physically legitimate simulations of Zeeman- +broadened line profiles. What matters here is the choice of +rotating magnetic star models (field configuration,inclination +of magnetic/rotational axes viewed by an observer, etc.) among +diversified possibilities. In the context of lacking information, +we tentatively assume a model which is as simple as possible +but does not yield results contradicting the known observa- +tional facts. In any case, given that we are primarily interested +in the value of ⟨퐻⟩ (mean field strength averaged over the +disk; cf. Eq. (3)), we do not need to be too much particular +0 +1 +2 +3 +-40 +-20 +0 +20 +40 +0 +5 +10 +15 +vt (km s-1) +Wcal +7.8 - Wobs (mÅ) +σW +1.38 km s-1 +Fe I lines +(a) +0 +1 +2 +3 +-40 +-20 +0 +20 +40 +0 +5 +10 +15 +vt (km s-1) +Wcal +7.8 - Wobs (mÅ) +σW +1.77 km s-1 +Fe II lines +(b) +1 +5 10 +50100 +-40 +-20 +0 +20 +40 +Wobs (mÅ) +Wcal +7.8 - Wobs (mÅ) +vt = 1.38 km s-1 +(c) +1 +5 10 +50100 +-40 +-20 +0 +20 +40 +Wobs (mÅ) +Wcal +7.8 - Wobs (mÅ) +vt = 1.77 km s-1 +(d) +FIGURE 4 Upper panels (a), (b): solid lines show how the +difference between the theoretical equivalent width calcu- +lated for 퐴 = 7.8 (푊 7.8 +cal ) and the observed equivalent width +(푊obs) varies by changing 푣t, where weaker and stronger lines +are distinguished by gray and black lines, respectively (as +in Fig. 3). The standard deviation (휎푊 ) of the differences +defined by Eq. (1) is depicted against 푣t in thick green line +(scale is in the right axis). Lower panels (c), (d): Equivalent +width differences (푊 7.8 +cal −푊obs) corresponding to the 푣t value +of minimum 휎푊 are plotted against 푊obs. The left-hand and +right-hand panels are for Fe I and Fe II lines, respectively. +about this issue, because the functionality of ℎ(⟨퐻⟩, 푣e sin 푖) +or 푊 (⟨퐻⟩, 푣t) would not be very sensitive to any choice of +models in the first approximation, +Following this policy, a simple dipole model is adopted in +this study, which is represented in the spherical coordinate +system as follows: +퐻푟 = 퐻pol(푅∕푟)3 cos 휃 +퐻휃 = (퐻pol∕2)(푅∕푟)3 sin 휃 +퐻휙 = 0, +(2) +where 퐻pol is the magnetic field strength at the magnetic +pole on the stellar surface (푟 = 푅, 휃 = 0◦), which yields +(퐻푟, 퐻휃, 퐻휙) = (퐻pol cos 휃, 퐻pol∕2 sin 휃, 0) at the stellar sur- +face (푟 = 푅). Accordingly, the absolute strength of the surface +field is largest at the magnetic pole (휃 = 0◦, |퐇| = 퐻pol) and +smallest at the equator (휃 = 90◦, |퐇| = 퐻pol∕2) +As to the axis orientation and view angle of this model star, +we assume that the magnetic and rotational axes are in line with +each other and perpendicular to the observer’s line of sight; + +8 +Y. TAKEDA +i.e., 푖 = 훼 = 90◦ (as usual, 푖 and 훼 are the angles of rotational +and magnetic axes in reference to the line of sight), as shown +in the upper illustration of Fig. 5. This simple assumption is +reasonable in context of the observational characteristics of +표 Peg, because (i) the magnetic field configuration viewed by +the observer does not depend upon the rotational phase (i.e., +no appreciable variability) and (ii) the line-of-sight component +of the field is cancelled out by averaging over the disk to result +in ⟨퐻푧⟩ = 0 (meaningful circular polarization signal is unde- +tected). The observed aspect of surface magnetic field in this +model is schematically depicted in the lower-left panel (merid- +ional cross section) and lower-middle panel (observer’s view) +of Fig. 5. Besides, the magnetic field strengths |퐇| (in unit of +퐻pol) at various points on the visible disk are plotted against +cos 휓 (휓 is the angle between the field vector and the line of +sight) in the lower-right panel of Fig. 5, from which we can see +that |퐇| is between 퐻pol∕2 and 퐻pol. +4.2 +Calculation of local 퐼 profiles +As to the calculation of specific intensity profile (퐼휆) of a line +under the presence of a magnetic field, Unno’s (1956) radiative +transfer equation in terms of the Stokes parameters (퐼, 푄, 푉 ) +was solved with the help of Takeda’s (1991a) numerical pro- +cedure (cf. Sect. 2 therein) as done by T91b. The line opacity +profiles of Zeeman-split 휋, 휎+, and 휎− components (derived +from 푆, 퐿, and 퐽 of upper and lower levels by assuming LS +coupling) for a given magnetic field were evaluated by making +use of the line opacity data of non-magnetic case calculated +by the WIDTH9 program (with the same model atmosphere +as adopted in Sect. 3). The necessary parameters for comput- +ing the 퐼휆 profile emergent from a disk point are |퐇|, 휓, 휇 +(direction cosine of the angle between the surface normal and +the line of sight), along with 퐴 (Fe abundance) and 푣t (micro- +turbulence). Accordingly, a grid of emergent 퐼grid +휆 +profiles (up +to 1Å from the line center with a step of 0.005Å) were com- +puted in advance for each of the 380 lines for combinations of +31 |퐇| values (0, 200, 400, … 5800, 6000 G), 10 휓 values (0, +10, 20, …, 80, 90◦), 10 휇 values (0.1, 0.2, 0.3, ⋯, 0.9, 1.0), +and 7 푣t values (0.0, 0.5, 1.0, …, 2.5, 3.0 km −1), while the Fe +abundance was fixed at 퐴 = 7.80 (cf. Sect. 3). +4.3 +Line flux profile by disk integration +The flux profile 퐹휆 of a spectral line can then be simulated by +integrating the 퐼휆 at each point of the visible disk (evaluated +by interpolating the grid of 퐼grid +휆 +corresponding to the local +physical condition), while adequately taking into account the +Doppler shift due to the line-of-sight velocity. For this purpose, +we modified the program CALSPEC (Takeda, Kawanomoto, +& Ohishi 2008) which simulates the spectral line profile of a +rotating star by dividing its surface into 180×360 segments. +Since only the case of slow rotation is considered, the effects of +gravity darkening and gravitational distortion were neglected; +therefore, the star is spherical and homogeneously covered +with the solar abundance atmosphere of 푇eff = 9500 K and +log 푔 = 3.60. The parameters of 퐻pol and 푣e sin 푖(= 푣e) have +to be assigned (along with 푣t and 퐴) in this modeling of line +flux profile. +The calculations of 퐹휆 for each line were done for 13 퐻pol +values (0, 500, 1000, … 5500, 6000 G), 7 푣e sin 푖 values (0.0, +2.5, 5.0, …, 12.5, 15.0 km s−1), and 7 푣t values (0.0, 0.5, 1.0, +…, 2.5, 3.0 km −1), again at the fixed 퐴 = 7.80. Further, the +equivalent widths (푊cal) and FWHMs (ℎcal) were also evalu- +ated from these line profiles. As an example of simulation, the +퐹휆 results derived for representative three lines (Fe I 4383.544, +Fe II 6147.734, and Fe II 6149.246) are displayed in Fig. 6, +where the corresponding observed profiles are also shown for +comparison. +For the sake of future discussion, the mean absolute field +strength averaged over the visible stellar disk (⟨퐻⟩) is defined +as follows: +⟨퐻⟩ ≡ ∫ +disk +∫ |퐇|(푥, 푦)퐼cont(푥, 푦)d푥d푦 +/ +∫ +disk +∫ 퐼cont(푥, 푦)d푥d푦 , +(3) +where |퐇|(푥, 푦) and 퐼cont(푥, 푦) are the absolute field strength +and the continuum specific intensity (to the observer) at the +disk point (푥, 푦), respectively. Naturally, ⟨퐻⟩ is in propor- +tion to 퐻pol with the proportionality constant of ⟨퐻⟩∕퐻pol = +0.642 in the postulated magnetic field configuration (훼 = 90◦). +Likewise, the disk-averaged line-of-sight component (in the +푧-direction) of the magnetic field (⟨퐻푧⟩) is definable in the +similar manner and ⟨퐻푧⟩ = 0 holds in the present case. +5 +MAGNETIC FIELD DETERMINATION +5.1 +Equivalent widths analysis +Let us first try to establish (퐻pol, 푣t) from equivalent widths +(푊 ). Here, Method 2 described in Sect. 3.3 is applied, in which +푊 7.8 +cal (theoretical equivalent width calculated with 퐴 = 7.8)5 is +compared with 푊obs. Since theoretical 푊 7.8 +cal data are prepared +for combinations of 퐻pol and 푣t (while results for 푣e sin 푖 = +0 were adopted because of its irrelevance in this case), 휎푊 +defined by Eq. (1) is also regarded as a function of these two +parameters. +5The integrated strengths (equivalent widths) of unsaturated weak lines in the +linear part of the curve of growth, which essentially determine the abundance, +are practically free from any Zeeman broadening effect (like the effect of micro- +turbulence). Accordingly, the Fe abundance of 퐴 = 7.8 derived in Sect. 3.2 by +the conventional analysis is invariably valid irrespective of the existence of any +magnetic field. + +Y. TAKEDA +9 +P + To +observer +-1 -0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +cos ψ +|H| / Hpol +P +FIGURE 5 The upper figure schematically describes the adopted rotating star model with a dipole magnetic field (parameterized +by 퐻pol; field strength at the pole P), where the observer’s line of sight is perpendicular to both of the rotational axis (푖 = 90◦) and +the magnetic axis (훼 = 90◦). The lower three figures represent the observed characteristics of the magnetic field in this model: +Left — surface field vectors (indicated by arrows) in the meridional plane. Center — schematic illustration of surface magnetic +field lines viewed by an observer. Right — Correlation between |퐇|∕퐻pol (absolute field strength in unit of 퐻pol and cos 휓 (휓 is +the angle between the magnetic field vector and the line of sight) at each point of the visible disk. +The resulting 휎푊 (퐻pol, 푣t) values derived from Fe I and Fe II +lines are given in Table 2, and the contours of 휎푊 on the 퐻pol– +푣t plane are depicted in Fig. 7. As seen from the locations +of minimum 휎푊 , 퐻pol solutions for Fe I (∼ 0 G) and Fe II +(∼ 2000 G) are rather conflicting, though 푣t ∼ 1.5 km s−1 is +consistently obtained irrespective of the species. +5.2 +Line widths analysis +Next, we extract information of magnetic field from the line +width (ℎ), where the contribution of 푣e sin 푖 plays an important +role. For this purpose, the observed width (ℎobs) is compared +with the theoretical width (ℎ7.8 +cal) calculated for various combi- +nations of 퐻pol and 푣e sin 푖 but for fixed 퐴 = 7.8 and 푣t = +1.5 km s−1 (according to the result of Sect. 5.1). Similarly to +Eq. (1), we define 휎ℎ (function of 퐻pol and 푣e sin 푖) as +휎ℎ ≡ +√ +√ +√ +√ +푁 +∑ +푛=1 +(ℎ7.8 +푣,cal,푛 − ℎ0 +푣,obs,푛)2∕푁. +(4) +Here, ℎ0 +푣,obs ≡ +√ +ℎ2 +푣,obs − 32 is the observed line width (in +km s−1) corrected for the instrumental effect (FWHM of +3 km s−1), where the fact that line profiles are well approx- +imated by Gaussian function (cf. Fig. 2e) was taken into +account. +The resulting 휎ℎ(퐻pol, 푣e sin 푖) values derived from Fe I and +Fe II lines are given in Table 3, and the contours of 휎ℎ on +the 퐻pol–푣e sin 푖 plane are depicted in Fig. 8. Inspecting the +locations of minimum 휎ℎ, we obtain 퐻pol ∼ 3000 G and +푣e sin 푖 ∼ 5 km s−1 for both Fe I and Fe II lines. +6 +DISCUSSION +6.1 +Results and their characteristics +In Sect. 5, we derived the magnetic field of 표 Peg (퐻pol or ⟨퐻⟩) +and the related line-broadening parameters (푣t and 푣e sin 푖) by +comparing the observed and simulated equivalent widths (푊 ) +and line widths (ℎ). The results are summarized in Table 4. + +rotation axis += magnetic axis +observer +/=α=90°10 +Y. TAKEDA +-0.4 -0.2 0 +0.2 0.4 +0.2 +Fe I 4383.544 +Hpol = 0 kG +2 kG +4 kG +6 kG +∆λ (Å) +-0.4 -0.2 0 +0.2 0.4 +0.2 +Fe II 6147.734 +Hpol = 0 kG +2 kG +4 kG +6 kG +vesini = 0 +5 +10 +15 +∆λ (Å) +-0.4 -0.2 0 +0.2 0.4 +0.2 +Fe II 6149.246 +Hpol = 0 kG +2 kG +4 kG +6 kG +∆λ (Å) +FIGURE 6 Demonstrative examples of theoretical flux profiles simulated by disk integration of unpolarized specific intensities +(Stokes 퐼) for three representative lines: Fe I 4383.544 (left), Fe II 6147.734 (center), and Fe II 6149.246 (right). Shown here are +results calculated with 푣t = 1.5 km s−1 for four 퐻pol values (0, 2, 4, and 6 kG) and four 푣e sin 푖 values (0, 5, 10, and 15 km s−1). +In addition, the actually observed profiles of 표 Peg are also displayed at the bottom for comparison. These simulated profiles of +Fe II 6147.734 and 6149.246 may be compared with Fig. 2a and Fig. 2b of T91b, where the Zeeman-split structures are more +manifest (because they are specific intensity profiles for single-valued magnetic field along with the assumption of 푣t = 0 km s−1 +without rotational broadening). +Since the velocity parameter solutions (휂푘∗ at the grid node +푘 = 푘∗, where 휂 denotes either 푣t or 푣e sin 푖) have rounded +values because the grids are rather coarse, 휎 was analyti- +cally expressed by a second-order polynomial of 휂 by using +휎(푘∗ −1), 휎(푘∗), and 휎(푘∗ +1), from which the new 휂 solution +(휂est) was estimated as corresponding to the minimum of this +parabolic 휎(휂). Such derived 푣est +t +and 푣e sin 푖est are also given +in Table 4. + +Y. TAKEDA +11 +TABLE 2 Calculated 휎푊 values as functions of 퐻pol and 푣t. +푣t +퐻pol = 0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +⟨퐻⟩ = 0 +321 +642 +962 +1283 +1604 +1925 +2246 +2566 +(Fe I lines) +3.0 +9.004 +9.037 +9.128 +9.274 +9.471 +9.711 +9.985 +10.285 +10.607 +2.5 +6.607 +6.647 +6.757 +6.932 +7.165 +7.444 +7.760 +8.101 +8.462 +2.0 +4.445 +4.491 +4.614 +4.810 +5.068 +5.376 +5.721 +6.091 +6.476 +1.5 +3.164 +3.192 +3.273 +3.414 +3.615 +3.873 +4.176 +4.511 +4.866 +1.0 +3.552 +3.530 +3.483 +3.438 +3.428 +3.473 +3.581 +3.748 +3.961 +0.5 +4.613 +4.557 +4.420 +4.239 +4.057 +3.909 +3.815 +3.786 +3.823 +0.0 +5.105 +5.037 +4.870 +4.647 +4.411 +4.199 +4.036 +3.935 +3.901 +(Fe II lines) +3.0 +13.143 +13.239 +13.500 +13.914 +14.463 +15.129 +15.892 +16.734 +17.642 +2.5 +8.508 +8.610 +8.890 +9.337 +9.933 +10.658 +11.486 +12.400 +13.379 +2.0 +5.097 +5.163 +5.357 +5.707 +6.226 +6.903 +7.714 +8.631 +9.629 +1.5 +5.323 +5.247 +5.085 +4.934 +4.903 +5.078 +5.491 +6.122 +6.925 +1.0 +8.094 +7.933 +7.543 +7.021 +6.477 +6.017 +5.740 +5.717 +5.973 +0.5 +10.537 +10.334 +9.838 +9.158 +8.403 +7.669 +7.041 +6.595 +6.398 +0.0 +11.481 +11.261 +10.730 +10.001 +9.186 +8.376 +7.655 +7.095 +6.759 +Given in this table are the values of 휎푊 [standard deviation between the observed and calculated equivalent widths in unit of mÅ; defined by +Eq. (1)] calculated for each combination of 퐻pol (field strength at the magnetic pole in unit of G; see the top row) and 푣t (microturbulence in +unit of km s−1; see the leftmost column). At the second row, the mean field strengths (in G) averaged over the stellar disk [⟨퐻⟩; cf. Eq.(3)] +corresponding to each 퐻pol are given. The minimum 휎푊 among each group is indicated by an underline. +TABLE 3 Calculated 휎ℎ values as functions of 퐻pol and 푣e sin 푖. +푣e sin 푖 +퐻pol = 0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +⟨퐻⟩ = 0 +321 +642 +962 +1283 +1604 +1925 +2246 +2566 +(Fe I lines) +15.0 +14.936 +14.946 +14.977 +15.029 +15.100 +15.191 +15.303 +15.438 +15.597 +12.5 +10.608 +10.621 +10.661 +10.727 +10.819 +10.940 +11.089 +11.268 +11.480 +10.0 +6.343 +6.361 +6.414 +6.503 +6.628 +6.792 +6.997 +7.243 +7.532 +7.5 +2.265 +2.288 +2.359 +2.480 +2.654 +2.886 +3.178 +3.530 +3.942 +5.0 +2.116 +2.082 +1.988 +1.841 +1.661 +1.491 +1.405 +1.485 +1.762 +2.5 +5.133 +5.079 +4.928 +4.677 +4.337 +3.929 +3.490 +3.066 +2.724 +0.0 +6.127 +6.073 +5.916 +5.645 +5.263 +4.793 +4.274 +3.751 +3.280 +(Fe II lines) +15.0 +13.736 +13.749 +13.789 +13.853 +13.940 +14.050 +14.184 +14.344 +14.532 +12.5 +9.675 +9.691 +9.739 +9.818 +9.928 +10.069 +10.244 +10.454 +10.702 +10.0 +5.684 +5.705 +5.767 +5.871 +6.017 +6.207 +6.445 +6.732 +7.071 +7.5 +1.937 +1.963 +2.040 +2.175 +2.372 +2.638 +2.977 +3.389 +3.873 +5.0 +2.149 +2.109 +1.999 +1.829 +1.631 +1.466 +1.442 +1.645 +2.063 +2.5 +4.736 +4.675 +4.503 +4.221 +3.847 +3.421 +3.005 +2.684 +2.548 +0.0 +5.588 +5.525 +5.344 +5.034 +4.611 +4.120 +3.629 +3.207 +2.923 +Given in this table are the values of 휎ℎ [standard deviation between the observed and calculated full-width at half-maximum in unit of km s−1; +defined by Eq. (4)] calculated for each combination of 퐻pol (field strength at the magnetic pole in unit of G; see the top row) and 푣e sin 푖 (projected +rotational velocity in unit of km s−1; see the leftmost column). The minimum 휎ℎ among each section is indicated by an underline. Otherwise, +the same as in Table 2. +By inspecting these tables, we can read the following con- +sequences regarding the magnetic field strength ⟨퐻⟩ (as well +as 푣t and 푣e sin 푖) of 표 Peg. +• Regarding the equivalent width analysis, contradicting +results are obtained for ⟨퐻⟩ (∼ 0 kG from Fe I lines +and ∼ 1.3 kG from Fe II lines). However, since the for- +mer is likely to be less reliable for the reason described +in Sect. 3.3, we preferentially adopt the latter solution of +⟨퐻⟩ ∼ 1.3 kG. Meanwhile, 푣t is consistently settled at +∼ 1.5 km s−1. +• As to the line width analysis, mean field strengths of +⟨퐻⟩ ∼ 1.9 kG are derived for both Fe I and Fe II lines. The +projected rotational velocity is concluded to be 푣e sin 푖 ∼ +5 km s−1. +• Based on these results, although the ⟨퐻⟩ value from 푊 +tends to be somewhat lower than that from ℎ, the mean + +12 +Y. TAKEDA +FIGURE 7 Graphical display of the contours of 휎푊 on the +퐻pol–푣t plane, where the results for Fe I and Fe II lines are +separately displayed in the upper and lower panels, respec- +tively. In each panel, the position of (퐻∗ +pol, 푣∗ +t ) corresponding +to the minimum 휎푊 is indicated by an asterisk (*). +magnetic field on the order of ⟨퐻⟩ ∼ 1.5–2 kG in 표 Peg is +anyhow confirmed. Accordingly, the consequence of our +new analysis is almost consistent with the conclusion of +previous studies (ML90, T91b, T93), which reported the +existence of 퐻 ∼ 2 kG in this star. +• We may state that the impact of magnetic field is not very +significant on the spectroscopic determination of 푣t and +푣e sin 푖, because the resulting values (∼ 1.5 km s−1 and +∼ 5 km s−1) are not much different from those derived +by neglecting the magnetic effect (푣t ∼ 1.7–1.8 km s−1 +derived in Sect. 3.2, and the typical recent literature val- +ues of 푣e sin 푖 are ∼ 6–7 km s−1 as seen in Table 1). +Regarding 푣t, this is a reconfirmation of the argument in +T93 (but not that in T91b). +6.2 +Precision check of T93 approximation +In the analysis of equivalent widths (Sect. 5.1), we could estab- +lish the magnetic field of 표 Peg from Fe II lines (퐻pol ∼ 1.5– +2 kG), but not from Fe I lines (i.e., a well-defined minimum +is not found in 휎푊 which continues to decline with a decrease +in 퐻pol until 퐻pol → 0). This situation is rather similar to the +FIGURE 8 Graphical display of the contours of 휎ℎ on the +퐻pol–푣e sin 푖 plane, where the results for Fe I and Fe II +lines are separately displayed in the upper and lower panels, +respectively. In each panel, the position of (퐻∗ +pol, 푣e sin 푖∗) +corresponding to the minimum 휎ℎ is indicated by an asterisk +(*). +case of T93, where a successful result was obtained from Fe II +lines but not from Fe I lines (see the run of 휎b depicted in the +middle-row panels of Fig. 2 in T93). +In T93, a practical (but approximate) method was used for +evaluating the line flux equivalent width under the existence of +magnetic field, in which the conventional spectrum-synthesis +code is applicable without any necessity of solving the trans- +fer equation of polarized radiation (cf. Sect. 2 in T93 for a +detailed explanation). Briefly speaking, in this method, two +equivalent widths are calculated for a given 퐻 correspond- +ing to the minimum intensification (푊a; using only 휎− and +휎+ components but independently from each other) and max- +imum intensification (푊c; use of 휎−, 휎+, and 휋 components +altogether while assuming as if no polarization effect exists). +Then, it was assumed in T93 that the theoretical equivalent +width to be adopted (which should be between 푊a and 푊c) is +given by the “simple mean” of these minimum and maximum +as 푊b(퐻) ≡ [푊a(퐻) + 푊c(퐻)]∕2. +In order to examine the precision of this approximation, 푊b +values were calculated at various field strengths (퐻) for all +of the 380 Fe lines (with 퐴 = 7.8 and 푣t = 1.5 km s−1), + +Y. TAKEDA +13 +TABLE 4 Summary of solutions based on line-strength or +line-width analysis. +(Line strengths analysis) +Species +퐻∗ +pol +⟨퐻⟩∗ +푣∗ +t +푣est +t +Fe I +0 +0 +1.5 +1.37 +Fe II +2000 +1283 +1.5 +1.52 +(Line widths analysis) +Species +퐻∗ +pol +⟨퐻⟩∗ +푣e sin 푖∗ +푣e sin 푖est +Fe I +3000 +1925 +5.0 +5.10 +Fe II +3000 +1925 +5.0 +5.01 +Quantities with asterisks (*) in column 2–4 are the solu- +tions corresponding to the minimum of 휎푊 or 휎ℎ, while that +in column 5 is the estimated solution derived by quadratic +interpolation (see Sect. 6.1). +which were then compared with the corresponding 푊cal values +simulated in Sect. 4.3. +The resulting 푊 vs. 퐻 relations (based on different meth- +ods of T93 and this study) for three representative lines (Fe I +4383.544, Fe II 6147.734, and Fe II 6149.246) are compared +in Figs. 9a, 9b, and 9c, respectively. We can see from these +figures that 푊b(퐻) (solid line) is a reasonable approximation +of 푊cal(⟨퐻⟩) (symbols), though some systematic departure +is observed at larger 퐻 depending on lines.6 The logarith- +mic differences between 푊b and 푊cal for all lines are plotted +against 푊cal in Figs. 9d, 9e, and 9f for different field strengths +of 0, 2, and 4 kG, respectively. These figures indicate that +| log(푊b∕푊cal)| is typically a few hundredths dex at most (i.e., +several or ≲ 10 percent in 푊 ) even at the magnetic field of +4 kG. Accordingly, the practical approach proposed by T93 +for calculating equivalent widths of a magnetic star may be +regarded as a reasonable approximation of moderate precision +(especially when the lines to be used are carefully chosen). +6.3 +Implication from the line-pair method +Finally, as an application of the simulations done in Sect. 4, we +estimate the magnetic field of 표 Peg based on the very simple +approach using the strengths of specific line pairs belonging +to the same multiplet, which was first tried by ML90 and then +6We may state that a line with single (or practically single) 휎− or 휎+ component +(such like the cases of Fe I 4383.544 or Fe II 6149.246) tends to suffer an appreciable +deviation, since the difference between 푊a and 푊c is comparatively large because +푊a is 퐻-independent and constant (cf. Figs. 9a and 9c). In contrast, if 휎− (or 휎+) +components of a line are sufficiently apart (like Fe II 6147.734), 푊b makes a fairly +good approximation for 푊cal, because 푊a and 푊c increase with 퐻 in somewhat +similar manner and the difference tends to be small (cf. Fig. 9b). +extended by T91b. This method makes use of the relative dif- +ference of equivalent widths for the two lines (1 and 2) defined +as 훿 ≡ 2(푊1 − 푊2)∕(푊1 + 푊2). While 훿 ≃ 0 in the non- +magnetic case, 훿 begins to depart from zero with an increase +in 퐻 (because of the different 퐻-sensitivity between lines 1 +and 2). Accordingly, 퐻 may be estimated by comparing the +observed 훿obs with the known 훿 vs. 퐻 relation. +Here, two line pairs are relevant, which are called after T91b +as “red pair (R)” (Fe II 6147.7 and 6149.2) and “blue pair (B)” +(Fe II 4416.8 and 4385.4). See Table 1 of T91b for more details +on these line pairs. Since these 4 Fe II lines are included in +our 380 target lines, 훿R and 훿B at various field strengths can be +evaluated from their 푊cal results7 calculated at 퐻pol = 0, 500, +1000, …, 5500, and 6000 G (along with 푣t = 1.5 km s−1 and +퐴 = 7.8). +The resulting theoretical 훿R and 훿B are plotted against ⟨퐻⟩ +in Fig. 10, where the positions of 훿R = +0.025 and 훿B = +−0.020 derived from the observed equivalent widths in mÅ +(푊1,R∕푊2,R = 49.8∕48.6, 푊1,B∕푊2,B = 91.5∕93.4) are also +indicated. The following trends can be read from this figure. +— A comparison of theoretical and observed 훿R yields a mean +magnetic field strength of ⟨퐻⟩ ∼ 2.4 kG. +— Regarding 훿B, a unique solution can not be found. What can +be said from Fig. 10 is that ⟨퐻⟩ is either ∼ 1.3 kG or ∼ 2.9 kG. +— In any case, these results do not contradict the consequence +of the main analysis (detection of ⟨퐻⟩ on the order of ∼ 2 kG; +cf. Sect. 6.1). +6.4 +Magnetic nature of 표 Peg +Our analysis on the line strengths and widths has thus corrob- +orated that an appreciable magnetic field of ⟨퐻⟩ ∼1–2 kG +(mean field strength averaged over the disk) exists in the +Am star 표 Peg. Here, we should recall that previous spec- +tropolarimetric observations conducted so far failed to detect +any meaningful signal of circular polarization in this star (cf. +Sect. 1), which means that ⟨퐻푧⟩ (mean line-of-sight compo- +nent of the field averaged over the disk) is very weak. Although +detection of ultra-weak ⟨퐻푧⟩ on the order of several G might +as well be possible by using higher-precision observations (see +footnote 1), we can at least state that ⟨퐻푧⟩ is negligibly weak +compared to ⟨퐻⟩. +One possibility to explain this marked disagreement is that +the magnetic field is not globally organized but has a com- +plex structure (such as suggested by ML90). If several or +more strong magnetic regions of smaller scale with different +polarities exist on the stellar disk (such as sunspots), the net +7Since the log 푔푓 values of the lines consisting the pair given in the VALD +database (which we adopted in this study) are rather discrepant from each other, +the requirement of 훿 ≃ 0 in the non-magnetic case is not fulfilled if log 푔푓(VALD) +data were used. Therefore, the log 푔푓 values presented in Table 1 of T91b were +exceptionally employed here for both of the red-pair lines and blue-pair lines. + +14 +Y. TAKEDA +0 +1000 2000 3000 4000 +100 +120 +140 +160 +H or (G) +Wabc, Wcal (mÅ) +Fe I 4383.544 +Wc +Wb +Wa +geff +L = 1.300 +(a) +0 +1000 2000 3000 4000 +30 +40 +50 +60 +70 +H or (G) +Wabc, Wcal (mÅ) +Fe II 6147.734 +geff +L = 0.833 +(b) +0 +1000 2000 3000 4000 +40 +50 +60 +70 +80 +H or (G) +Wabc, Wcal (mÅ) +Fe II 6149.246 +geff +L = 1.333 +(c) +0.5 1 +5 10 +50100 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +Wcal ( = 0 kG) +log (Wb/Wcal) +0 kG +(d) +0.5 1 +5 10 +50100 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +Wcal ( = 2 kG) +log (Wb/Wcal) +2 kG +(e) +0.5 1 +5 10 +50100 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +Wcal ( = 4 kG) +log (Wb/Wcal) +4 kG +(f) +FIGURE 9 Left panels (a–c) show how the flux equivalent width varies by changing the magnetic field strength for three rep- +resentative lines: (a) Fe I 4383.544, (b) Fe II 6147.734, and (c) Fe II 6149.246 (their Zeeman patterns are shown in the inset +of each panel). Those evaluated by the WIDTH9 program with three kinds of approximations proposed in T93, 푊a (minimum +intensification involving only 휎− and 휎+ components), 푊c (maximum intensification for the case of neglecting the polarization +effect), and 푊b (simple mean of 푊a and 푊c; finally adopted in T93), are depicted in dashed line, dash-dotted line, and solid line, +respectively. Meanwhile, those calculated based on our dipole magnetic field model by disk integration of local 퐼 profiles (푊cal) +are plotted by filled symbols. Note that these 푊cal values are plotted against ⟨퐻⟩ (not 퐻pol). All these 푊 calculations were done +with 퐴 = 7.80 and 푣t = 1.5 km s−1. The logarithmic differences evaluated for all 380 Fe lines [log(푊b∕푊cal)] are plotted against +푊cal in the right panels (d–f) for different mean field strengths (⟨퐻⟩): (d) 0 kG, (e) 2 kG, and (f) 4 kG. +line-of-sight component of the field (⟨퐻푧⟩) would almost van- +ish while the mean magnetic field strength (⟨퐻⟩) still remains +detectable. However, it seems that very strong magnetic spots +or patches (with strengths considerably exceeding ∼ 2 kG) are + +Y. TAKEDA +15 +0 +2000 +4000 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 + (G) +δR, δB +R +B +FIGURE 10 Relative differences of equivalent +widths [훿 ≡ 2(푊1 − 푊2)∕(푊1 + 푊2)] for the red +(R) and blue (B) pair lines (cf. Table 1 in T91b) are +plotted against the mean field strength (⟨퐻⟩) by +solid lines, which were calculated with 퐴 = 7.80 +and 푣t = 1.5 km s−1. The observed values (훿R = ++0.025, 훿B = −0.020) are indicated by horizontal +dotted lines. +rather unlikely in the present case, because they should give +rise to some kind of appreciable peculiarities in the profiles +of magnetically-sensitive lines. For example, if assumed that +1/3 of the stellar disk is covered by a strong magnetic patch +of 퐻 ∼ 6 kG while the remaining 2/3 is non-magnetic, Fe II +6149.246 line would show a complex profile as expected from +the simulation; but such an anomalous feature is absent in the +actual profile which is nearly Gaussian (cf. the right panel of +Fig. 6). +Accordingly, whichever configuration of the magnetic field, +the field contrast over the stellar disk would not be distinctly +large (i.e., not so much like spots/patches as rather gradual). In +this context, the simple rotating dipole model of aligned rota- +tional/magnetic axis viewed almost equator-on (푖 ≃ 훼 ≃ 90◦), +which was assumed in the simulation of this study, may be +regarded as the likely solution for 표 Peg, because it naturally +explains the observational fact of ⟨퐻푧⟩ ∼ 0. Although it is +not easy to check this hypothesis observationally, some weak +rotational modulation of circular polarization might as well be +detected if 훼 is not exactly (but slightly deviates from) 90◦. +In this case, the rotation period is estimated as 푃 ≃ 42 d by +combining 푣e(≃ 푣e sin 푖) ≃ 5 km s−1 and 푅 ≃ 4.4푅⊙. It may +thus be worthwhile to examine whether a modulation period +of ∼ 40 d is observed for this star by ultra high-precision +spectropolarimetry. +7 +SUMMARY AND CONCLUSION +The star 표 Peg is a representative A-type star (classified as a hot +Am star from its abundance characteristics), which has been +frequently studied by a number of investigators because of its +brightness and sharp-line nature. +In the early 1990s, several authors (ML90, T91b, T93) +reported the existence of surface magnetic field on the order of +∼ 2 kG in this star based on the analysis of widths or strengths +of many spectral lines, which was a significant finding because +the conventional spectropolarimetry has been unsuccessful in +detecting any meaningful signal of ⟨퐻푧⟩. +However, the techniques employed by these old studies were +not necessarily founded on a physically legitimate basis but +rather empirical or approximate in character. Besides, the qual- +ity of the adopted spectra of 표 Peg, on which the observational +data of line widths and strengths were measured, was not +satisfactory as viewed from the present-day standard. +Given that this detection does not seem to have been corrob- +orated since then, I decided to revisit this issue based on (i) an +improved modeling of theoretical line flux profile of a rotat- +ing magnetic star (by disk integration of local intensity profiles +obtained by correctly solving the transfer equation of polarized +radiation) and (ii) using the high-resolution (푅 ∼ 100000) and +very high S/N (∼ 1000) spectra of 표 Peg. +The magnetic and rotational axes of this model star (with a +dipole field) were assumed to be in line with each other and +perpendicular to the observer’s line of sight (푖 = 훼 = 90◦), by +which ⟨퐻푧⟩ = 0 is attained in accordance with observations. +As for the spectral lines whose full-widths at half-maximum +(ℎ) and equivalent widths (푊 ) are to be analyzed, 380 Fe lines +(198 Fe I and 182 Fe II lines) were carefully selected, which +are free from any serious blending effect. +The conventional analysis (without taking into account the +effect of magnetic intensification) of equivalent widths was +first carried out, which resulted in 퐴 ≃ 7.8 (Fe abundance) and +푣t ≃ 1.7–1.8 km s−1 (microturbulence). This result of 퐴(Fe) = +7.8 was used as the fiducial abundance to be fixed throughout +the subsequent magnetic field analyses. +By requiring the minimum dispersion between the theo- +retical 푊 values simulated with the magnetic field model +(depending on the field strength and microturbulence) and the +observed ones, we found that ⟨퐻⟩ ∼ 1.3 kG (from Fe II lines) +and 푣t ∼ 1.5 km s−1. +Similarly, by comparing the simulated ℎ values (function +of projected rotational velocity and magnetic field strength) +with the measured ones, the best solutions accomplishing the +least dispersion were derived as ⟨퐻⟩ ∼ 1.9 kG and 푣e sin 푖 ∼ +5 km s−1. +Based on these results, although the ⟨퐻⟩ value from the +analysis of 푊 tends to be somewhat lower than that from ℎ, + +16 +Y. TAKEDA +the mean magnetic field on the order of ⟨퐻⟩ ∼ 1–2 kG has +been confirmed in 표 Peg. +In addition, supplementary applications of the simulated 푊 +results were also conducted for checking purposes: (i) The pre- +cision of the practical method proposed by T93 for evaluating +푊 in the presence of a magnetic field was examined and con- +firmed to be a reasonable and useful approximation. (ii) The +line-pair method used by ML90 and T91b was applied based on +the newly simulated 푊 values of specific line pairs and found +that 퐻 is in the range of ∼ 1–3 kG. +In summary, the consequence resulting from our analysis on +the 푊 and ℎ data of Fe lines is almost consistent with the con- +clusion of previous studies (ML90, T91b, T93) which reported +퐻 ∼ 2 kG for this star. +Regarding the reason for the marked discrepancy between +⟨퐻⟩ ∼ (1–2 kG) and ⟨퐻푧⟩(∼ 0), an accidental accomplish- +ment of 푖 ≃ 훼 ≃ 90◦ in the poloidal configuration might as +well be ponderable, rather than invoking a complex structure +with small-scale magnetic regions of different polarities. +ACKNOWLEDGMENTS +This research has made use of the SIMBAD database, oper- +ated by CDS, Strasbourg, France. This work has also made +use of the VALD database, operated at Uppsala University, the +Institute of Astronomy RAS in Moscow, and the University of +Vienna. +DATA AVAILABILITY +The basic data and results underlying this article are pre- +sented as the online supplementary material. The line profile +data used for measurements are given in “obsprofiles.dat”, +while the original spectra of 표 Peg are in the public domain +and available at https://smoka.nao.ac.jp/index.jsp (SMOKA +Science Archive site). +SUPPORTING INFORMATION +This article accompanies the following online materials. +• ReadMe.txt +• felines.dat +• obsprofiles.dat +REFERENCES +Abt, H. A., & Morrell, N. I., 1995, ApJS, 99, 135. +Adelman, S. J., 1973, ApJ, 183, 95. +Adelman, S. J., 1988, MNRAS, 230, 671. +Adelman, S. J., & Fuhr, J. R., 1985, A&A, 152, 434. +Adelman, S. J., Cowley, C. R., & Hill, G., 1988, in Elemental Abun- +dance Analyses, ed. S. J., Adelman & T. 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F., & Martayan, C., 2009, A&A, 501, 297. +Zorec, J., & Royer, F., 2012, A&A, 537, A120. + diff --git a/QtE4T4oBgHgl3EQf_A4a/content/tmp_files/load_file.txt b/QtE4T4oBgHgl3EQf_A4a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..253731a64139c8e69241f1855d4fa0b20579acf2 --- /dev/null +++ b/QtE4T4oBgHgl3EQf_A4a/content/tmp_files/load_file.txt @@ -0,0 +1,1362 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf,len=1361 +page_content='Received August 25, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Revised January 12, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accepted January 12, 2023 DOI: xxx/xxxx ARTICLE TYPE Surface magnetic field of the A-type metallic-line star omicron Pegasi revisited Yoichi Takeda 11-2 Enomachi, Naka-ku, Hiroshima-shi, 730-0851, Japan Correspondence Email: ytakeda@js2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='so-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='jp The bright A-type metallic-line star 표 Peg was reported in the early 1990s to have a surface magnetic field of ∼ 2 kG by analyzing the widths and strengths of spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In respect that those old studies were of rather empirical or approximate nature and the quality of observational data was not sufficient, this problem has been newly reinvestigated based on physically more rigorous simulations of line flux profiles, along with the observed equivalent widths (푊 ) and full-widths at half-maximum (ℎ) of 198 Fe I and 182 Fe II lines measured from the high-quality spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Given the Fe abundance derived from the conventional analysis, theoretical 푊 and ℎ values calculated for various sets of parameters were compared with the observed ones, which lead to the following conclusion regarding ⟨퐻⟩ (mean field strength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1) An analysis of 푊 yielded ⟨퐻⟩ ∼1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 kG from Fe II lines with the microturbulence of 푣t ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2) A comparison of ℎ resulted in ⟨퐻⟩ ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5–2 kG as well as the projected rotational velocity of 푣e sin 푖 ≃ 5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (3) Accordingly, the existence of mean magnetic field on the order of ⟨퐻⟩ ∼ 1–2 kG in 표 Peg was confirmed, which is almost consistent with the consequence of the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' KEYWORDS: stars: atmospheres — stars: chemically peculiar — stars: early-type stars: individual (표 Peg) — stars: magnetic fields 1 INTRODUCTION The star 표 Peg (= HD 214994 = HR 8641 = HIP 112051;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' spectral type is A1 IV) is one of the most frequently studied A-type stars in stellar spectroscopy, because of its apparent brightness (푉 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='79) and sharp-line nature with low pro- jected rotational velocity (푣e sin 푖 ≲ 10 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As is often the case with slowly-rotating stars in the upper main sequence, it is a chemically peculiar (CP) star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' more specifically, because of the conspicuous overabundances in the elements heavier than the Fe group, it is regarded as a hot metallic-lined A- type (Am) star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', Adelman 1988, Hill & Landstreet 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 0Abbreviations: LTE, local thermodynamic equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' FWHM, full width at half maximum Definite variability has not been confirmed, despite of the des- ignation “Si-variable” and “variable radial velocity” in Renson & Manfroid’s (2009) catalogue of CP stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' A remarkable and rare feature of this star is that a sur- face magnetic field was spectroscopically detected in the past, despite that Am stars are generally not magnetic (unlike other groups of CP stars such as the SrCrEu type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 1 — It was Mathys & Lanz (1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' hereinafter referred to as ML90) who first reported the existence of a magnetic field of 퐻 ∼ 2 kG in 표 Peg, where they employed two independent 1The meaning of this statement (non-magnetic nature of Am stars in general) is that appreciable magnetism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', field strength on the order of ∼kG) found in magnetic CP stars is generally absent in Am stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Recent very high-precision spec- tropolarimetric observations have revealed that an extremely weak magnetic field (on the order of ∼G) is detectable in several hot Am stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sirius (Petit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2009), 훽 UMa and 휃 Leo (Blazère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2016a), Alhena (훾 Gem) (Blazère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2016b, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='05367v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='SR] 13 Jan 2023 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA techniques: (i) statistical line-width analysis (Stenflo & Lin- degren 1977) and (ii) empirical relation for the 퐻-dependent difference of equivalent width (푊 ) between Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7 and 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 lines (which have almost the same strengths in the non- magnetic case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Mathys 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' — Successively, numerically solving the transfer equation of polarized radiation in the presence of a magnetic field accord- ing to Takeda (1991a), Takeda (1991b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' hereinafter referred to as T91b) explained the 퐻-dependence of 푊6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7∕푊6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 dif- ference as due to the desaturation effect caused by magnetic broadening (while proposing the simultaneous use of simi- lar line pair Fe II 4416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8/4385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4), and derived 퐻 ∼ 2–3 kG (assuming a microturbulence of 푣t ∼ 0 km s−1) for 표 Peg, which is favorably compared with ML90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' — Further, Takeda (1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' hereinafter T93) devised a method for determining the magnetic field based on the 푊 values of many lines, which is regarded as a refined version of the classical Hensberge & De Loore’s (1974) technique and can establish the (퐻, 푣t) solution by requiring the consistency of abundances derived from lines of various strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Regarding 표 Peg, T93 concluded 퐻 ∼ 2 kG (and 푣t ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1), which is again consistent with ML90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since then, however, little progress seems to have been made as to the corroboration of these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As already noted by ML90, the spectropolarimetric technique (most commonly used for investigating magnetic fields of CP stars) has been unsuccessful to accomplish a meaningful detection in 표 Peg (though extremely weak magnetic field might as well be still detectable as remarked in footnote 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Actually, besides the pioneering work of Babcock (1958), Shorlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2002) could not detect any ⟨퐻푧⟩ (disk-averaged line-of-sight component of the field) of significance in 표 Peg by their spectropolarimet- ric observation coupled with Least-Squares-Decomposition (LSD) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 This suggests that ⟨퐻푧⟩ happens to be too small (presum- ably because components of opposite signs are cancelled out by disk integration) to produce a significant signal of circular polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, in order to study the magnetic nature of 표 Peg, the best way would be to analyze the equivalent widths or line widths of many spectral lines (which contain information of “absolute” magnetic field strengths) as previously done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' How- ever, it may be premature to regard the results of those old studies as sufficiently reliable because several problems are involved from a methodological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2According to the compilation of Bychkov, Bychkova, & Madej (2009), 표 Peg’s ⟨퐻푧⟩ resulting from Shorlin et al.’s (2002) observation was 32(±20) G, which would have been regarded as under the level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The analysis of ML90 was not based on a rigorous mod- eling but a rather tentatively postulated analytical rela- tion between the line width, line strength, and Zeeman- broadening parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Especially, since the original Stenflo & Lindegren’s (1977) work (on which their study is based) was intended to estimate the solar magnetic field at the disk center, the effect of rotational broadening on the line width was not taken into account;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' thus, how the stellar projected rotational velocity (푣e sin 푖) affects the functional form of the relation is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Although T91b simulated the emergent profiles and the strengths of spectral line pairs by correctly treating the transport of polarized radiation in a magnetic field, only the specific intensity profiles at the disk center (휇 = cos 휃 = 1) were calculated for several single-valued field strengths (퐻) and different angles between the magnetic field vector and the line of sight (휓) , which must be unre- alistic for comparing with the flux profiles of magnetic stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Another problem is the choice of microturbulence: 푣t ∼ 0 km s−1 adopted in T91b is not consistent with that obtained later by T93 (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' if the latter were chosen, a considerably higher magnetic field (∼ 4–5 kG) would have resulted (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 in T91b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The results of T93 were actually not robust but rather del- icate, because 퐻 could be firmly established only from Fe II lines among the three species (finding a definite solution was difficult for Fe I and Ti II lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Above all, the validity of the approximate method for evaluating the flux equivalent width under a magnetic field proposed by T93 (denoted as 푊푏, which is a simple mean between the minimum and maximum intensification cases) should be quantitatively checked in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Attention should also be paid to the adopted observa- tional data of 표 Peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' ML90’s measurements of line widths for their application of Stenflo–Lindegren technique were done on spectra with a signal-to-noise ratio (S/N) of ∼ 100 covering 3700–4650 Å resulting from the co-added photographic spectrogram with a reciprocal dispersion of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 Å mm−1 obtained at Dominion Astrophysical Obser- vatory (Adelman, Cowley, & Hill 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Similarly, the equivalent width data of Fe I, Fe II, and Ti II lines for 표 Peg adopted in the analysis of T93 were taken from Adelman’s (1988) Table 5, for which he used the same co-added Dominion spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since the quality of these photographic spectra is not sufficient from the viewpoint of present-day standard and using only the blue-region data is not advantageous for the purpose of magnetic- field detection, it is desirable to employ modern spectra of much higher quality covering wider (from blue through red) wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 3 As such, it is worth reinvestigating the magnetic field of 표 Peg by analyzing the widths and strengths for a number of spectral lines as done by ML90 and T93 but based on physi- cally rigorous theoretical line-profile modeling along with new observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Given this situation and now that CCD spectra of very high S/N for this star are now available, I decided to freshly revisit this problem, while making use of the profile simulation program based on disk integration coupled with the solution code of polarized radiation transfer in a mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The purpose of this paper is to report the results of this new analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2 OBSERVATIONAL DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Observed spectra of 표 Peg The high-dispersion spectra of 표 Peg adopted in this investiga- tion are the same as used in Takeda et al.’s (2012) abundance study of alkali elements for A-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The observations were done on UT 2008 October 4 (8 frames of 1200 s exposure within a time span of ∼ 6 hr), October 7 (1 frame of 1800 s exposure), October 8 (8 frames of 1200 s exposure within a time span of ∼ 3 hr), and October 9 (6 frames of 1200 s expo- sure within a time span of ∼ 2 hr) by using the HIDES (HIgh Dispersion Echelle Spectrograph) placed at the coudé focus of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='88 m reflector at Okayama Astrophysical Observa- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Equipped with three mosaicked 4K×2K CCD detectors at the camera focus, echelle spectra covering 4100–7800 Å (in the mode of red cross-disperser) with a resolving power of 푅 ∼ 100000 (corresponding to the slit width of 100 휇m) were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The reduction of the spectra (bias subtraction, flat-fielding, scattered-light subtraction, spectrum extraction, wavelength calibration, and continuum normalization) was performed by using the “echelle” package of the software IRAF3 in a stan- dard manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In order to improve the signal-to-ratio, all avail- able frames were co-added, by which very high S/N (typically around ∼ 1000 on the average) could be accomplished in the final spectra as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 1 (S/N ∼ √ 푐표푢푛푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Selected lines and measurements In this study, we concentrate to using only spectral lines of Fe I and Fe II, as they are available in larger number over a wide range of strengths than other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The candidate lines to be analyzed were carefully sorted out by inspecting the observed spectral feature while comparing it with the calcu- lated strengths of neighboring lines as well as the synthesized 3IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' under cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4000 4500 5000 0 1000000 2000000 3000000 (a) 5500 6000 6500 0 1000000 2000000 3000000 (b) count 7000 7500 0 1000000 2000000 3000000 (c) Wavelength (Å) FIGURE 1 Panels (a), (b), and (c) show the distributions of accumulated photoelectron counts of CCD for the final spec- tra of 표 Peg, each corresponding to three wavelength regions (4100–5300/5300–6600/6600–7900 Å) comprising 32/20/13 orders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Note that the signal-to-noise ratio can be estimated as S/N ∼ √ 푐표푢푛푡 in the present photon-noise- limited case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The spectra in each of the echelle orders show characteristic distributions of the blaze function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' theoretical spectrum, as done by Takeda (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Because of the necessity of calculating the Zeeman components, those lines lacking the information of quantum numbers (퐿, 푆, 퐽) for the lower and upper levels were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As a result, 198 Fe I and 182 Fe II lines were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The equivalent widths (푊휆) of these lines were measured by fitting the line- depth profile (푅휆 ≡ 1 − 푓휆∕푓cont) with the Gaussian function (∝ exp[−(휆 − 휆0)2∕푎2]), while the line widths (ℎ휆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' defined as the FWHM of 푅휆) were directly evaluated from the profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The 푊휆 values of these Fe lines range from ∼ 1 mÅ to ∼ 200 mÅ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2a) and their effective Landé factors are between 0 ≲ 푔L eff ≲ 3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As seen from the empirical curves of growth depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2c, the linear part and the shoulder/flat part are roughly separated around 106푊휆∕휆 ∼ 10 (typically several tens mÅ in 푊휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' A comparison of the equivalent widths with those published by Adelman (1988) is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2d, where a reasonable consistency is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2e shows that the directly measured FWHM values (ℎ휆) are mostly in agreement with the corresponding Gaussian-fit ones (derived from the 푒-folding half-width as ℎG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 휆 ≡ 2 √ ln 2 푎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The dependence of ℎ푣 (in the velocity unit 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA derived as ℎ휆푐∕휆: 푐 is the speed of light) upon the equivalent width is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2f, which shows that ℎ푣 begins to exhibit a systematic 푊휆-dependence at 106푊휆∕휆 ≳ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The measured 푊 and ℎ values of these 380 Fe lines along with their atomic data (wavelength, excitation potential, oscil- lator strengths, damping constants, term information, effective Landé factor, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=') mostly taken from the VALD database (Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2015) are summarized in “felines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='dat” of the online material, where the original profile data of all lines are also given in “obsprofiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='dat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3 CONVENTIONAL ANALYSIS OF EQUIVALENT WIDTHS FOR MICROTURBULENCE Before dealing with the main issue of magnetic field estimation to be described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4 and 5, we first conduct a prepara- tory analysis of determining the microturbulence based on the equivalent widths by applying the conventional procedure (while assuming as if no magnetic field exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Atmospheric model and parameters Regarding the standard model atmosphere of 표 Peg, we adopted Kurucz’s (1993) ATLAS9 solar abundance model with 푇eff = 9500 K, log 푔 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='60 (cgs unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' These atmo- spheric parameters were chosen by inspecting the various literature data summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As recognized from this table, these values (though rather rounded) are consistent with those derived in many of the past studies (especially the recent ones published after 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Besides, this choice is in accord with 푀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8푀⊙ (mass) and 푅 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4푅⊙ (radius) evaluated from the position on the log 퐿 (luminosity) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푇eff diagram in comparison with theoretical evolutionary tracks (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 1 in Takeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Abundance determination from an equivalent width for a given microturbulence was done by using Kurucz’s (1993) WIDTH9 program while assuming LTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The adopted line parameters are given in “felines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='dat” (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Two approaches are tried in this 푣t determination test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1) usual method of finding the minimum abundance dispersion, and (2) alternative method requiring the overall consistency between the observed and theoretical 푊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Method 1: minimum abundance dispersion The effect of 푣t on abundance determination appreciably depends on line strengths: abundances determined from weak lines in the linear part of the curve of growth are essentially free from 푣t, while those from stronger lines on the shoulder-to- flat part are considerably 푣t-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Therefore, 푣t is usually established by requiring that abundances derived from lines of various equivalent widths (푊 ) be consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Among the several practical procedures for accomplishing this requirement, Blackwell et al.’s (1976) method is used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' –(1) For each line 푛, a set of abundances (퐴푘 푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푘 = 1, 2, … , 퐾) are derived from 푊푛 while incrementally changing the micro- turbulence (푣푘 t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푘 = 1, 2, … , 퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' –(2) Then, the mean abundance (⟨퐴⟩푘) averaged over 푁 lines and the standard deviation 휎푘 퐴 are calculated for each of the 퐾 microturbulences (푣푘 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' –(3) By inspecting the resulting standard deviation (휎푘 퐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푘 = 1, 2, … , 퐾), the location of minimum 휎퐴 corresponds to desired solution of 푣t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This procedure was applied to our data set of Fe lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The resulting 퐴 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푣t relations for each of the lines along with the corresponding 휎퐴 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푣t curve are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3a and 3b for Fe I (푁1 = 198) and Fe II lines (푁2 = 182), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As seen from these figures, we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='71 km s−1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='75)4 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='76 km s−1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='82) as the results of (푣t, ⟨퐴⟩) for Fe I and Fe II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' and the corresponding 퐴푛 for each line is plotted against 푊푛 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3c and 3d, respectively .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 Method 2: minimum equivalent width dispersion Next, another method for 푣t determination is tried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 may be regarded as the fiducial Fe abundance of 표 Peg from the results of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='75 (Fe I) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='82 (Fe II) derived in the previous subsection, we can calculate the theoretical equivalent widths (푊 푘,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal,푛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푛 = 1, 2, … , 푁) for each of the lines with this fixed Fe abundance but incrementally changing 푣푘 t , which are to be compared with the observed ones (푊obs,푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푛 = 1, 2, … , 푁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Then, by examining the standard deviations 휎푘 푊 evaluated for various 푣푘 t values, 휎푘 푊 ≡ √ √ √ √ 푁 ∑ 푛=1 (푊 푘,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal,푛 − 푊obs,푛)2∕푁, (1) we can find the solution of 푣t as that accomplishing the mini- mum 휎푘 푊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Although the final 푊cal calculated with such deter- mined 푣t is not exactly equal to the observed 푊obs (because of the fixed Fe abundance), the primary aim of line-independent consistency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', without any global 푊 -dependent trend) can be accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The differences 푊 푘,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal,푛 − 푊obs,푛 for each lines are plotted against 푣t in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4a (Fe I) and 4b (Fe II), where the correspond- ing 휎푊 versus 푣t curves are also depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' From these figures, 4퐴 is the logarithmic number abundance of Fe relative to H, which is expressed in the usual normalization of log[푁(Fe)∕푁(H)] + 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 5 4000 5000 6000 7000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1 5 10 50 100 λ (Å) Wλ (mÅ) (a) 0 10 20 30 0 1 2 3 (Wλ/λ)106 geff L (b) 8 6 4 2 0 1 0 1 2 log gf - χlow(5040/Teff) log(106Wλ/λ) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 hλ (Å) hλ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (Å) (e) 0 10 20 30 8 10 12 14 (Wλ/λ)106 hv (km s-1) (f) 1 5 10 50 100 1 5 10 50 100 Wλ Takeda (mÅ) Wλ Adelman (mÅ) (d) FIGURE 2 (a) Equivalent width vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (b) Effective Landé 푔 factor vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' reduced equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (c) Empirical curves of growth for the Fe I and Fe II lines, where log 푔푓 −휒low(5040∕푇eff) is taken in the abscissa (푔: statistical weight of the lower level, 푓: oscillator strength, 휒low: lower excitation potential in eV, 푇eff: effective temperature in K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (d) Comparison of the equivalent widths measured in this study with those published by Adelman (1988) for 97 lines (65 Fe I lines and 32 Fe II lines) in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (e) Correlation between the directly measured full-widths at half-maximum (ℎ휆) with the corresponding Gaussian-fit values (ℎG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 휆 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (f) Full-widths at half-maximum in the velocity unit (ℎ푣 ≡ 푐ℎ휆∕휆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푐 is the speed of light) plotted against the reduced equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In each panel, the data for Fe I and Fe II lines are discriminated in blue filled symbols and red open symbols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' we obtain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='38 km s−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='77 km s−1 for Fe I and Fe II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The differences 푊 푘,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal,푛 −푊obs,푛 for each lines cor- responding to these 푣t solutions are plotted against 푊obs,푛 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4c (Fe I) and 4d (Fe II), 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA TABLE 1 Atmospheric parameters of 표 Peg published so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Authors 푇eff log 푔 푣t [Fe/H] 푣e sin 푖 Wolff (1967) 9330 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='30 Conti & Strom (1968) 9500 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2푎 Adelman (1973) 10100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Allen (1977) 9600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='09 Mitton (1977) 9500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='08푏 Adelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1984) 9625 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='20 Adelman & Fuhr (1985) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='26 Adelman (1988) 9600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='10 6 Castelli & Hack (1988) 9590 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='14 6 Kocer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1988) 9500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 Sadakane (1988) 9500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 Van’t Veer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1988) 9350 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 Burkhart & Coupry (1991) 9650 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Hill & Landstreet (1993) 9680 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5푐 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='03 Abt & Morrell (1995) 10 Hill (1995) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 Sokolov (1995) 10050 Blackwell & Lynas-Gray (1998) 9443 Di Benedetto (1998) 9720 Hui-Bon-Hoa (2000) 9650 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='42 ≤ 10 Adelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2002) 9591 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='64 Adelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2002) 9525 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='70 Royer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2002) 14 Royer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2007) 14 Landstreet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2009) 9500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='14푑 7 Zorec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2009) 9930 Prugniel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2011) 9373 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='14 Takeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2012) 9453 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='13푒 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 Zorec & Royer (2012) 9506 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='73푓 14 Gray (2014) 9600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0푔 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='00ℎ Takeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2018) 9453 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='18푖 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 Summarized here are the effective temperature (in K), logarithmic surface gravity in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='s unit (in dex), microturbulence (in km s−1), Fe abun- dance relative to the Sun, and projected rotational velocity (in km s−1) of 표 Peg taken from previous publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since these parameters were determined in variously different methods, the original references should be consulted for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Regarding [Fe/H], if only those derived from Fe I and Fe II lines are available, a simply averaged value of these two is listed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Besides, in case that the reference solar Fe abundance is not explicitly given, an appropriate value widely used at the time of publication was tentatively adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푎Relative to the mean of 4 standard stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푏Relative to Procyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푐Assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푑Solar Fe abundance of log(Fe∕H)⊙ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='49 was assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푒Solar Fe abundance of 퐴⊙(Fe) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='50 was assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푓Derived from 퐿 (bolometric luminosity), 푇eff, and 푀 (mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푔Assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' ℎRadial-tangential macroturbulence of 휁RT = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7 km s−1 was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푖Relative to Procyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' A comparison of 푣t(Method 1) derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 with this 푣t(Method 2) suggests that, while we can confirm a good agreement for the case of Fe II lines (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='76/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='77 km s−1), a dis- crepancy is seen for 푣t based on Fe I lines (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='71/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='38 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As a matter of fact, the results from Fe I lines appear to be somewhat problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3c, the distribu- tion of 푊 푘,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal,푛 − 푊obs,푛 differences for Fe I lines of medium- to-large strengths (푊obs ≳ 10 mÅ) shows some asymmetric feature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', positive for lines of 20–50 mÅ while negative for those of ≳ 50 mÅ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This trend has made the 휎푘 푊 curve shallower with a less clear minimum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4a), which eventu- ally leads to larger uncertainties in 푣t determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Generally speaking, since only a very tiny fraction of Fe atoms remain neutral (those in Fe II and Fe III stages are dominant) in the atmosphere of A-type stars, the formation of Fe I lines is con- siderably 푇 -dependent and vulnerable to model atmosphere structure, while Fe II lines are more robust in this respect (see Appendix A2 of Takeda 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, Fe II lines may yield more reliable results than Fe I lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 7 0 1 2 3 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 vt (km s-1) A1 σA1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='71 km s-1 Fe I lines (a) 0 1 2 3 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 vt (km s-1) A2 σA2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='76 km s-1 Fe II lines (b) 1 5 10 50100 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 9 Wλ (mÅ) A1 vt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='71 km s-1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='75 (c) 1 5 10 50100 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 9 Wλ (mÅ) A2 vt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='76 km s-1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='82 (d) FIGURE 3 Upper panels (a), (b): solid lines show how the Fe abundance (퐴) of each line varies by changing 푣t, where weaker lines (106푊휆∕휆 < 10) and stronger lines (106푊휆∕휆 > 10), are distinguished by gray and black lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In addition, 휎퐴 (standard deviation of 퐴) is plotted against 푣t by the thick green solid line (its scale is marked in the right axis), and the 푣t solution corresponding to the minimum 휎퐴) is also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Lower panels (c), (d): Fe abundances corresponding to the 푣t solution are plotted against the observed equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The mean abundance (⟨퐴⟩) is also indicated by the horizontal dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The left-hand and right-hand panels are for Fe I and Fe II lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4 LINE PROFILE SIMULATION OF A MAGNETIC STAR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Magnetic field model As already mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 1, the main aim of this inves- tigation is to check the previously reported results (possible existence of a magnetic field on the order of ∼ 2 kG in 표 Peg) based on physically legitimate simulations of Zeeman- broadened line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' What matters here is the choice of rotating magnetic star models (field configuration,inclination of magnetic/rotational axes viewed by an observer, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=') among diversified possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In the context of lacking information, we tentatively assume a model which is as simple as possible but does not yield results contradicting the known observa- tional facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In any case, given that we are primarily interested in the value of ⟨퐻⟩ (mean field strength averaged over the disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (3)), we do not need to be too much particular 0 1 2 3 40 20 0 20 40 0 5 10 15 vt (km s-1) Wcal 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 - Wobs (mÅ) σW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='38 km s-1 Fe I lines (a) 0 1 2 3 40 20 0 20 40 0 5 10 15 vt (km s-1) Wcal 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 - Wobs (mÅ) σW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='77 km s-1 Fe II lines (b) 1 5 10 50100 40 20 0 20 40 Wobs (mÅ) Wcal 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 - Wobs (mÅ) vt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='38 km s-1 (c) 1 5 10 50100 40 20 0 20 40 Wobs (mÅ) Wcal 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 - Wobs (mÅ) vt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='77 km s-1 (d) FIGURE 4 Upper panels (a), (b): solid lines show how the difference between the theoretical equivalent width calcu- lated for 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 (푊 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal ) and the observed equivalent width (푊obs) varies by changing 푣t, where weaker and stronger lines are distinguished by gray and black lines, respectively (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The standard deviation (휎푊 ) of the differences defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1) is depicted against 푣t in thick green line (scale is in the right axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Lower panels (c), (d): Equivalent width differences (푊 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal −푊obs) corresponding to the 푣t value of minimum 휎푊 are plotted against 푊obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The left-hand and right-hand panels are for Fe I and Fe II lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' about this issue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' because the functionality of ℎ(⟨퐻⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푣e sin 푖) or 푊 (⟨퐻⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푣t) would not be very sensitive to any choice of models in the first approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Following this policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' a simple dipole model is adopted in this study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' which is represented in the spherical coordinate system as follows: 퐻푟 = 퐻pol(푅∕푟)3 cos 휃 퐻휃 = (퐻pol∕2)(푅∕푟)3 sin 휃 퐻휙 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (2) where 퐻pol is the magnetic field strength at the magnetic pole on the stellar surface (푟 = 푅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 휃 = 0◦),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' which yields (퐻푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 퐻휃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 퐻휙) = (퐻pol cos 휃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 퐻pol∕2 sin 휃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 0) at the stellar sur- face (푟 = 푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, the absolute strength of the surface field is largest at the magnetic pole (휃 = 0◦, |퐇| = 퐻pol) and smallest at the equator (휃 = 90◦, |퐇| = 퐻pol∕2) As to the axis orientation and view angle of this model star, we assume that the magnetic and rotational axes are in line with each other and perpendicular to the observer’s line of sight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', 푖 = 훼 = 90◦ (as usual, 푖 and 훼 are the angles of rotational and magnetic axes in reference to the line of sight), as shown in the upper illustration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This simple assumption is reasonable in context of the observational characteristics of 표 Peg, because (i) the magnetic field configuration viewed by the observer does not depend upon the rotational phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', no appreciable variability) and (ii) the line-of-sight component of the field is cancelled out by averaging over the disk to result in ⟨퐻푧⟩ = 0 (meaningful circular polarization signal is unde- tected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The observed aspect of surface magnetic field in this model is schematically depicted in the lower-left panel (merid- ional cross section) and lower-middle panel (observer’s view) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Besides, the magnetic field strengths |퐇| (in unit of 퐻pol) at various points on the visible disk are plotted against cos 휓 (휓 is the angle between the field vector and the line of sight) in the lower-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5, from which we can see that |퐇| is between 퐻pol∕2 and 퐻pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Calculation of local 퐼 profiles As to the calculation of specific intensity profile (퐼휆) of a line under the presence of a magnetic field, Unno’s (1956) radiative transfer equation in terms of the Stokes parameters (퐼, 푄, 푉 ) was solved with the help of Takeda’s (1991a) numerical pro- cedure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2 therein) as done by T91b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The line opacity profiles of Zeeman-split 휋, 휎+, and 휎− components (derived from 푆, 퐿, and 퐽 of upper and lower levels by assuming LS coupling) for a given magnetic field were evaluated by making use of the line opacity data of non-magnetic case calculated by the WIDTH9 program (with the same model atmosphere as adopted in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The necessary parameters for comput- ing the 퐼휆 profile emergent from a disk point are |퐇|, 휓, 휇 (direction cosine of the angle between the surface normal and the line of sight), along with 퐴 (Fe abundance) and 푣t (micro- turbulence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, a grid of emergent 퐼grid 휆 profiles (up to 1Å from the line center with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='005Å) were com- puted in advance for each of the 380 lines for combinations of 31 |퐇| values (0, 200, 400, … 5800, 6000 G), 10 휓 values (0, 10, 20, …, 80, 90◦), 10 휇 values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3, ⋯, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0), and 7 푣t values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0, …, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 km −1), while the Fe abundance was fixed at 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='80 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 Line flux profile by disk integration The flux profile 퐹휆 of a spectral line can then be simulated by integrating the 퐼휆 at each point of the visible disk (evaluated by interpolating the grid of 퐼grid 휆 corresponding to the local physical condition), while adequately taking into account the Doppler shift due to the line-of-sight velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' For this purpose, we modified the program CALSPEC (Takeda, Kawanomoto, & Ohishi 2008) which simulates the spectral line profile of a rotating star by dividing its surface into 180×360 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since only the case of slow rotation is considered, the effects of gravity darkening and gravitational distortion were neglected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' therefore, the star is spherical and homogeneously covered with the solar abundance atmosphere of 푇eff = 9500 K and log 푔 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The parameters of 퐻pol and 푣e sin 푖(= 푣e) have to be assigned (along with 푣t and 퐴) in this modeling of line flux profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The calculations of 퐹휆 for each line were done for 13 퐻pol values (0, 500, 1000, … 5500, 6000 G), 7 푣e sin 푖 values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0, …, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 km s−1), and 7 푣t values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0, …, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 km −1), again at the fixed 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Further, the equivalent widths (푊cal) and FWHMs (ℎcal) were also evalu- ated from these line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As an example of simulation, the 퐹휆 results derived for representative three lines (Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544, Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734, and Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246) are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6, where the corresponding observed profiles are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' For the sake of future discussion, the mean absolute field strength averaged over the visible stellar disk (⟨퐻⟩) is defined as follows: ⟨퐻⟩ ≡ ∫ disk ∫ |퐇|(푥, 푦)퐼cont(푥, 푦)d푥d푦 / ∫ disk ∫ 퐼cont(푥, 푦)d푥d푦 , (3) where |퐇|(푥, 푦) and 퐼cont(푥, 푦) are the absolute field strength and the continuum specific intensity (to the observer) at the disk point (푥, 푦), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Naturally, ⟨퐻⟩ is in propor- tion to 퐻pol with the proportionality constant of ⟨퐻⟩∕퐻pol = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='642 in the postulated magnetic field configuration (훼 = 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Likewise, the disk-averaged line-of-sight component (in the 푧-direction) of the magnetic field (⟨퐻푧⟩) is definable in the similar manner and ⟨퐻푧⟩ = 0 holds in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5 MAGNETIC FIELD DETERMINATION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Equivalent widths analysis Let us first try to establish (퐻pol, 푣t) from equivalent widths (푊 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Here, Method 2 described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 is applied, in which 푊 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal (theoretical equivalent width calculated with 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8)5 is compared with 푊obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since theoretical 푊 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal data are prepared for combinations of 퐻pol and 푣t (while results for 푣e sin 푖 = 0 were adopted because of its irrelevance in this case), 휎푊 defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1) is also regarded as a function of these two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5The integrated strengths (equivalent widths) of unsaturated weak lines in the linear part of the curve of growth, which essentially determine the abundance, are practically free from any Zeeman broadening effect (like the effect of micro- turbulence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, the Fe abundance of 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 by the conventional analysis is invariably valid irrespective of the existence of any magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 9 P To observer 1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 1 cos ψ |H| / Hpol P FIGURE 5 The upper figure schematically describes the adopted rotating star model with a dipole magnetic field (parameterized by 퐻pol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' field strength at the pole P), where the observer’s line of sight is perpendicular to both of the rotational axis (푖 = 90◦) and the magnetic axis (훼 = 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The lower three figures represent the observed characteristics of the magnetic field in this model: Left — surface field vectors (indicated by arrows) in the meridional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Center — schematic illustration of surface magnetic field lines viewed by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Right — Correlation between |퐇|∕퐻pol (absolute field strength in unit of 퐻pol and cos 휓 (휓 is the angle between the magnetic field vector and the line of sight) at each point of the visible disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The resulting 휎푊 (퐻pol, 푣t) values derived from Fe I and Fe II lines are given in Table 2, and the contours of 휎푊 on the 퐻pol– 푣t plane are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As seen from the locations of minimum 휎푊 , 퐻pol solutions for Fe I (∼ 0 G) and Fe II (∼ 2000 G) are rather conflicting, though 푣t ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1 is consistently obtained irrespective of the species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Line widths analysis Next, we extract information of magnetic field from the line width (ℎ), where the contribution of 푣e sin 푖 plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' For this purpose, the observed width (ℎobs) is compared with the theoretical width (ℎ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 cal) calculated for various combi- nations of 퐻pol and 푣e sin 푖 but for fixed 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 and 푣t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1 (according to the result of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1), we define 휎ℎ (function of 퐻pol and 푣e sin 푖) as 휎ℎ ≡ √ √ √ √ 푁 ∑ 푛=1 (ℎ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 푣,cal,푛 − ℎ0 푣,obs,푛)2∕푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (4) Here, ℎ0 푣,obs ≡ √ ℎ2 푣,obs − 32 is the observed line width (in km s−1) corrected for the instrumental effect (FWHM of 3 km s−1), where the fact that line profiles are well approx- imated by Gaussian function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2e) was taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The resulting 휎ℎ(퐻pol, 푣e sin 푖) values derived from Fe I and Fe II lines are given in Table 3, and the contours of 휎ℎ on the 퐻pol–푣e sin 푖 plane are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Inspecting the locations of minimum 휎ℎ, we obtain 퐻pol ∼ 3000 G and 푣e sin 푖 ∼ 5 km s−1 for both Fe I and Fe II lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6 DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 Results and their characteristics In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5, we derived the magnetic field of 표 Peg (퐻pol or ⟨퐻⟩) and the related line-broadening parameters (푣t and 푣e sin 푖) by comparing the observed and simulated equivalent widths (푊 ) and line widths (ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The results are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' rotation axis = magnetic axis observer /=α=90°10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544 Hpol = 0 kG 2 kG 4 kG 6 kG ∆λ (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734 Hpol = 0 kG 2 kG 4 kG 6 kG vesini = 0 5 10 15 ∆λ (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246 Hpol = 0 kG 2 kG 4 kG 6 kG ∆λ (Å) FIGURE 6 Demonstrative examples of theoretical flux profiles simulated by disk integration of unpolarized specific intensities (Stokes 퐼) for three representative lines: Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544 (left), Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734 (center), and Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Shown here are results calculated with 푣t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1 for four 퐻pol values (0, 2, 4, and 6 kG) and four 푣e sin 푖 values (0, 5, 10, and 15 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In addition, the actually observed profiles of 표 Peg are also displayed at the bottom for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' These simulated profiles of Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734 and 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246 may be compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2b of T91b, where the Zeeman-split structures are more manifest (because they are specific intensity profiles for single-valued magnetic field along with the assumption of 푣t = 0 km s−1 without rotational broadening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since the velocity parameter solutions (휂푘∗ at the grid node 푘 = 푘∗, where 휂 denotes either 푣t or 푣e sin 푖) have rounded values because the grids are rather coarse, 휎 was analyti- cally expressed by a second-order polynomial of 휂 by using 휎(푘∗ −1), 휎(푘∗), and 휎(푘∗ +1), from which the new 휂 solution (휂est) was estimated as corresponding to the minimum of this parabolic 휎(휂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Such derived 푣est t and 푣e sin 푖est are also given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 11 TABLE 2 Calculated 휎푊 values as functions of 퐻pol and 푣t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푣t 퐻pol = 0 500 1000 1500 2000 2500 3000 3500 4000 ⟨퐻⟩ = 0 321 642 962 1283 1604 1925 2246 2566 (Fe I lines) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='004 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='037 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='128 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='274 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='471 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='711 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='985 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='285 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='607 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='607 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='647 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='757 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='932 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='165 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='595 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='481 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='261 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='730 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='001 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='186 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='376 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='655 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='095 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='759 Given in this table are the values of 휎푊 [standard deviation between the observed and calculated equivalent widths in unit of mÅ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (1)] calculated for each combination of 퐻pol (field strength at the magnetic pole in unit of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' see the top row) and 푣t (microturbulence in unit of km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' see the leftmost column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' At the second row, the mean field strengths (in G) averaged over the stellar disk [⟨퐻⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (3)] corresponding to each 퐻pol are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The minimum 휎푊 among each group is indicated by an underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TABLE 3 Calculated 휎ℎ values as functions of 퐻pol and 푣e sin 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 푣e sin 푖 퐻pol = 0 500 1000 1500 2000 2500 3000 3500 4000 ⟨퐻⟩ = 0 321 642 962 1283 1604 1925 2246 2566 (Fe I lines) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='936 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='946 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='977 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='029 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='100 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='191 15.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='421 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='005 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='684 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='588 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='525 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='344 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='034 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='611 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='120 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='629 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='207 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='923 Given in this table are the values of 휎ℎ [standard deviation between the observed and calculated full-width at half-maximum in unit of km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (4)] calculated for each combination of 퐻pol (field strength at the magnetic pole in unit of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' see the top row) and 푣e sin 푖 (projected rotational velocity in unit of km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' see the leftmost column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The minimum 휎ℎ among each section is indicated by an underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Otherwise, the same as in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' By inspecting these tables, we can read the following con- sequences regarding the magnetic field strength ⟨퐻⟩ (as well as 푣t and 푣e sin 푖) of 표 Peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Regarding the equivalent width analysis, contradicting results are obtained for ⟨퐻⟩ (∼ 0 kG from Fe I lines and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 kG from Fe II lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' However, since the for- mer is likely to be less reliable for the reason described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3, we preferentially adopt the latter solution of ⟨퐻⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Meanwhile, 푣t is consistently settled at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As to the line width analysis, mean field strengths of ⟨퐻⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='9 kG are derived for both Fe I and Fe II lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The projected rotational velocity is concluded to be 푣e sin 푖 ∼ 5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Based on these results, although the ⟨퐻⟩ value from 푊 tends to be somewhat lower than that from ℎ, the mean 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA FIGURE 7 Graphical display of the contours of 휎푊 on the 퐻pol–푣t plane, where the results for Fe I and Fe II lines are separately displayed in the upper and lower panels, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In each panel, the position of (퐻∗ pol, 푣∗ t ) corresponding to the minimum 휎푊 is indicated by an asterisk (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' magnetic field on the order of ⟨퐻⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5–2 kG in 표 Peg is anyhow confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, the consequence of our new analysis is almost consistent with the conclusion of previous studies (ML90, T91b, T93), which reported the existence of 퐻 ∼ 2 kG in this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' We may state that the impact of magnetic field is not very significant on the spectroscopic determination of 푣t and 푣e sin 푖, because the resulting values (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1 and ∼ 5 km s−1) are not much different from those derived by neglecting the magnetic effect (푣t ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 km s−1 derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2, and the typical recent literature val- ues of 푣e sin 푖 are ∼ 6–7 km s−1 as seen in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Regarding 푣t, this is a reconfirmation of the argument in T93 (but not that in T91b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 Precision check of T93 approximation In the analysis of equivalent widths (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1), we could estab- lish the magnetic field of 표 Peg from Fe II lines (퐻pol ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5– 2 kG), but not from Fe I lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', a well-defined minimum is not found in 휎푊 which continues to decline with a decrease in 퐻pol until 퐻pol → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This situation is rather similar to the FIGURE 8 Graphical display of the contours of 휎ℎ on the 퐻pol–푣e sin 푖 plane, where the results for Fe I and Fe II lines are separately displayed in the upper and lower panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In each panel, the position of (퐻∗ pol, 푣e sin 푖∗) corresponding to the minimum 휎ℎ is indicated by an asterisk (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' case of T93, where a successful result was obtained from Fe II lines but not from Fe I lines (see the run of 휎b depicted in the middle-row panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2 in T93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In T93, a practical (but approximate) method was used for evaluating the line flux equivalent width under the existence of magnetic field, in which the conventional spectrum-synthesis code is applicable without any necessity of solving the trans- fer equation of polarized radiation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 2 in T93 for a detailed explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Briefly speaking, in this method, two equivalent widths are calculated for a given 퐻 correspond- ing to the minimum intensification (푊a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' using only 휎− and 휎+ components but independently from each other) and max- imum intensification (푊c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' use of 휎−, 휎+, and 휋 components altogether while assuming as if no polarization effect exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Then, it was assumed in T93 that the theoretical equivalent width to be adopted (which should be between 푊a and 푊c) is given by the “simple mean” of these minimum and maximum as 푊b(퐻) ≡ [푊a(퐻) + 푊c(퐻)]∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In order to examine the precision of this approximation, 푊b values were calculated at various field strengths (퐻) for all of the 380 Fe lines (with 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 and 푣t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1), Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 13 TABLE 4 Summary of solutions based on line-strength or line-width analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (Line strengths analysis) Species 퐻∗ pol ⟨퐻⟩∗ 푣∗ t 푣est t Fe I 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='37 Fe II 2000 1283 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='52 (Line widths analysis) Species 퐻∗ pol ⟨퐻⟩∗ 푣e sin 푖∗ 푣e sin 푖est Fe I 3000 1925 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='10 Fe II 3000 1925 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='01 Quantities with asterisks (*) in column 2–4 are the solu- tions corresponding to the minimum of 휎푊 or 휎ℎ, while that in column 5 is the estimated solution derived by quadratic interpolation (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' which were then compared with the corresponding 푊cal values simulated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The resulting 푊 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 퐻 relations (based on different meth- ods of T93 and this study) for three representative lines (Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544, Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734, and Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246) are compared in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 9a, 9b, and 9c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' We can see from these figures that 푊b(퐻) (solid line) is a reasonable approximation of 푊cal(⟨퐻⟩) (symbols), though some systematic departure is observed at larger 퐻 depending on lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6 The logarith- mic differences between 푊b and 푊cal for all lines are plotted against 푊cal in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 9d, 9e, and 9f for different field strengths of 0, 2, and 4 kG, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' These figures indicate that | log(푊b∕푊cal)| is typically a few hundredths dex at most (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', several or ≲ 10 percent in 푊 ) even at the magnetic field of 4 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, the practical approach proposed by T93 for calculating equivalent widths of a magnetic star may be regarded as a reasonable approximation of moderate precision (especially when the lines to be used are carefully chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 Implication from the line-pair method Finally, as an application of the simulations done in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 4, we estimate the magnetic field of 표 Peg based on the very simple approach using the strengths of specific line pairs belonging to the same multiplet, which was first tried by ML90 and then 6We may state that a line with single (or practically single) 휎− or 휎+ component (such like the cases of Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544 or Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246) tends to suffer an appreciable deviation, since the difference between 푊a and 푊c is comparatively large because 푊a is 퐻-independent and constant (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 9a and 9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In contrast, if 휎− (or 휎+) components of a line are sufficiently apart (like Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734), 푊b makes a fairly good approximation for 푊cal, because 푊a and 푊c increase with 퐻 in somewhat similar manner and the difference tends to be small (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' extended by T91b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This method makes use of the relative dif- ference of equivalent widths for the two lines (1 and 2) defined as 훿 ≡ 2(푊1 − 푊2)∕(푊1 + 푊2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' While 훿 ≃ 0 in the non- magnetic case, 훿 begins to depart from zero with an increase in 퐻 (because of the different 퐻-sensitivity between lines 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, 퐻 may be estimated by comparing the observed 훿obs with the known 훿 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 퐻 relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Here, two line pairs are relevant, which are called after T91b as “red pair (R)” (Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7 and 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2) and “blue pair (B)” (Fe II 4416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 and 4385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' See Table 1 of T91b for more details on these line pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Since these 4 Fe II lines are included in our 380 target lines, 훿R and 훿B at various field strengths can be evaluated from their 푊cal results7 calculated at 퐻pol = 0, 500, 1000, …, 5500, and 6000 G (along with 푣t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1 and 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The resulting theoretical 훿R and 훿B are plotted against ⟨퐻⟩ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 10, where the positions of 훿R = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='025 and 훿B = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='020 derived from the observed equivalent widths in mÅ (푊1,R∕푊2,R = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8∕48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='6, 푊1,B∕푊2,B = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5∕93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4) are also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The following trends can be read from this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' — A comparison of theoretical and observed 훿R yields a mean magnetic field strength of ⟨퐻⟩ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' — Regarding 훿B, a unique solution can not be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' What can be said from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 10 is that ⟨퐻⟩ is either ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 kG or ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='9 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' — In any case, these results do not contradict the consequence of the main analysis (detection of ⟨퐻⟩ on the order of ∼ 2 kG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4 Magnetic nature of 표 Peg Our analysis on the line strengths and widths has thus corrob- orated that an appreciable magnetic field of ⟨퐻⟩ ∼1–2 kG (mean field strength averaged over the disk) exists in the Am star 표 Peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Here, we should recall that previous spec- tropolarimetric observations conducted so far failed to detect any meaningful signal of circular polarization in this star (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 1), which means that ⟨퐻푧⟩ (mean line-of-sight compo- nent of the field averaged over the disk) is very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Although detection of ultra-weak ⟨퐻푧⟩ on the order of several G might as well be possible by using higher-precision observations (see footnote 1), we can at least state that ⟨퐻푧⟩ is negligibly weak compared to ⟨퐻⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' One possibility to explain this marked disagreement is that the magnetic field is not globally organized but has a com- plex structure (such as suggested by ML90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' If several or more strong magnetic regions of smaller scale with different polarities exist on the stellar disk (such as sunspots), the net 7Since the log 푔푓 values of the lines consisting the pair given in the VALD database (which we adopted in this study) are rather discrepant from each other, the requirement of 훿 ≃ 0 in the non-magnetic case is not fulfilled if log 푔푓(VALD) data were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Therefore, the log 푔푓 values presented in Table 1 of T91b were exceptionally employed here for both of the red-pair lines and blue-pair lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 0 1000 2000 3000 4000 100 120 140 160 H or (G) Wabc, Wcal (mÅ) Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544 Wc Wb Wa geff L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='300 (a) 0 1000 2000 3000 4000 30 40 50 60 70 H or (G) Wabc, Wcal (mÅ) Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734 geff L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='833 (b) 0 1000 2000 3000 4000 40 50 60 70 80 H or (G) Wabc, Wcal (mÅ) Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246 geff L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='333 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1 5 10 50100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='06 Wcal ( = 0 kG) log (Wb/Wcal) 0 kG (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1 5 10 50100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='06 Wcal ( = 2 kG) log (Wb/Wcal) 2 kG (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 1 5 10 50100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='06 Wcal ( = 4 kG) log (Wb/Wcal) 4 kG (f) FIGURE 9 Left panels (a–c) show how the flux equivalent width varies by changing the magnetic field strength for three rep- resentative lines: (a) Fe I 4383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='544, (b) Fe II 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='734, and (c) Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246 (their Zeeman patterns are shown in the inset of each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Those evaluated by the WIDTH9 program with three kinds of approximations proposed in T93, 푊a (minimum intensification involving only 휎− and 휎+ components), 푊c (maximum intensification for the case of neglecting the polarization effect), and 푊b (simple mean of 푊a and 푊c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' finally adopted in T93), are depicted in dashed line, dash-dotted line, and solid line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Meanwhile, those calculated based on our dipole magnetic field model by disk integration of local 퐼 profiles (푊cal) are plotted by filled symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Note that these 푊cal values are plotted against ⟨퐻⟩ (not 퐻pol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' All these 푊 calculations were done with 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='80 and 푣t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The logarithmic differences evaluated for all 380 Fe lines [log(푊b∕푊cal)] are plotted against 푊cal in the right panels (d–f) for different mean field strengths (⟨퐻⟩): (d) 0 kG, (e) 2 kG, and (f) 4 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' line-of-sight component of the field (⟨퐻푧⟩) would almost van- ish while the mean magnetic field strength (⟨퐻⟩) still remains detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' However, it seems that very strong magnetic spots or patches (with strengths considerably exceeding ∼ 2 kG) are Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA 15 0 2000 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='2 (G) δR, δB R B FIGURE 10 Relative differences of equivalent widths [훿 ≡ 2(푊1 − 푊2)∕(푊1 + 푊2)] for the red (R) and blue (B) pair lines (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Table 1 in T91b) are plotted against the mean field strength (⟨퐻⟩) by solid lines, which were calculated with 퐴 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='80 and 푣t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The observed values (훿R = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='025, 훿B = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='020) are indicated by horizontal dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' rather unlikely in the present case, because they should give rise to some kind of appreciable peculiarities in the profiles of magnetically-sensitive lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' For example, if assumed that 1/3 of the stellar disk is covered by a strong magnetic patch of 퐻 ∼ 6 kG while the remaining 2/3 is non-magnetic, Fe II 6149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='246 line would show a complex profile as expected from the simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' but such an anomalous feature is absent in the actual profile which is nearly Gaussian (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Accordingly, whichever configuration of the magnetic field, the field contrast over the stellar disk would not be distinctly large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', not so much like spots/patches as rather gradual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In this context, the simple rotating dipole model of aligned rota- tional/magnetic axis viewed almost equator-on (푖 ≃ 훼 ≃ 90◦), which was assumed in the simulation of this study, may be regarded as the likely solution for 표 Peg, because it naturally explains the observational fact of ⟨퐻푧⟩ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Although it is not easy to check this hypothesis observationally, some weak rotational modulation of circular polarization might as well be detected if 훼 is not exactly (but slightly deviates from) 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In this case, the rotation period is estimated as 푃 ≃ 42 d by combining 푣e(≃ 푣e sin 푖) ≃ 5 km s−1 and 푅 ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='4푅⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' It may thus be worthwhile to examine whether a modulation period of ∼ 40 d is observed for this star by ultra high-precision spectropolarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' 7 SUMMARY AND CONCLUSION The star 표 Peg is a representative A-type star (classified as a hot Am star from its abundance characteristics), which has been frequently studied by a number of investigators because of its brightness and sharp-line nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In the early 1990s, several authors (ML90, T91b, T93) reported the existence of surface magnetic field on the order of ∼ 2 kG in this star based on the analysis of widths or strengths of many spectral lines, which was a significant finding because the conventional spectropolarimetry has been unsuccessful in detecting any meaningful signal of ⟨퐻푧⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' However, the techniques employed by these old studies were not necessarily founded on a physically legitimate basis but rather empirical or approximate in character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Besides, the qual- ity of the adopted spectra of 표 Peg, on which the observational data of line widths and strengths were measured, was not satisfactory as viewed from the present-day standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Given that this detection does not seem to have been corrob- orated since then, I decided to revisit this issue based on (i) an improved modeling of theoretical line flux profile of a rotat- ing magnetic star (by disk integration of local intensity profiles obtained by correctly solving the transfer equation of polarized radiation) and (ii) using the high-resolution (푅 ∼ 100000) and very high S/N (∼ 1000) spectra of 표 Peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The magnetic and rotational axes of this model star (with a dipole field) were assumed to be in line with each other and perpendicular to the observer’s line of sight (푖 = 훼 = 90◦), by which ⟨퐻푧⟩ = 0 is attained in accordance with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' As for the spectral lines whose full-widths at half-maximum (ℎ) and equivalent widths (푊 ) are to be analyzed, 380 Fe lines (198 Fe I and 182 Fe II lines) were carefully selected, which are free from any serious blending effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The conventional analysis (without taking into account the effect of magnetic intensification) of equivalent widths was first carried out, which resulted in 퐴 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 (Fe abundance) and 푣t ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 km s−1 (microturbulence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This result of 퐴(Fe) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='8 was used as the fiducial abundance to be fixed throughout the subsequent magnetic field analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' By requiring the minimum dispersion between the theo- retical 푊 values simulated with the magnetic field model (depending on the field strength and microturbulence) and the observed ones, we found that ⟨퐻⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='3 kG (from Fe II lines) and 푣t ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Similarly, by comparing the simulated ℎ values (function of projected rotational velocity and magnetic field strength) with the measured ones, the best solutions accomplishing the least dispersion were derived as ⟨퐻⟩ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='9 kG and 푣e sin 푖 ∼ 5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Based on these results, although the ⟨퐻⟩ value from the analysis of 푊 tends to be somewhat lower than that from ℎ, 16 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' TAKEDA the mean magnetic field on the order of ⟨퐻⟩ ∼ 1–2 kG has been confirmed in 표 Peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In addition, supplementary applications of the simulated 푊 results were also conducted for checking purposes: (i) The pre- cision of the practical method proposed by T93 for evaluating 푊 in the presence of a magnetic field was examined and con- firmed to be a reasonable and useful approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' (ii) The line-pair method used by ML90 and T91b was applied based on the newly simulated 푊 values of specific line pairs and found that 퐻 is in the range of ∼ 1–3 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' In summary, the consequence resulting from our analysis on the 푊 and ℎ data of Fe lines is almost consistent with the con- clusion of previous studies (ML90, T91b, T93) which reported 퐻 ∼ 2 kG for this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Regarding the reason for the marked discrepancy between ⟨퐻⟩ ∼ (1–2 kG) and ⟨퐻푧⟩(∼ 0), an accidental accomplish- ment of 푖 ≃ 훼 ≃ 90◦ in the poloidal configuration might as well be ponderable, rather than invoking a complex structure with small-scale magnetic regions of different polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research has made use of the SIMBAD database, oper- ated by CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' This work has also made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' DATA AVAILABILITY The basic data and results underlying this article are pre- sented as the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' The line profile data used for measurements are given in “obsprofiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='dat”, while the original spectra of 표 Peg are in the public domain and available at https://smoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='jp/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='jsp (SMOKA Science Archive site).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' SUPPORTING INFORMATION This article accompanies the following online materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' ReadMe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='txt felines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='dat obsprofiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content='dat REFERENCES Abt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', & Morrell, N.' metadata={'source': 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2009, A&A, 501, 297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=' Zorec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', & Royer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} +page_content=', 2012, A&A, 537, A120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQf_A4a/content/2301.05367v1.pdf'} diff --git a/R9AzT4oBgHgl3EQf0f4J/vector_store/index.faiss b/R9AzT4oBgHgl3EQf0f4J/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b55fb015605907c86bc9618ca56f359ace147ab9 --- /dev/null +++ b/R9AzT4oBgHgl3EQf0f4J/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:76c51554194643514ffe19c686a2eb80ef89d646a4fa60a99affc73d3fca9a6a +size 5308461 diff --git a/SdAzT4oBgHgl3EQfJPus/content/tmp_files/2301.01077v1.pdf.txt b/SdAzT4oBgHgl3EQfJPus/content/tmp_files/2301.01077v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..17190ead92bebd652201ea96850220b52a846f0c --- /dev/null +++ b/SdAzT4oBgHgl3EQfJPus/content/tmp_files/2301.01077v1.pdf.txt @@ -0,0 +1,812 @@ +arXiv:2301.01077v1 [math.AG] 3 Jan 2023 +Holomorphic tensors on Vaisman manifolds +Liviu Ornea1, Misha Verbitsky2 +Dedicated to Professor Valentin Poenaru at his ninetieth birthday +Abstract +An LCK (locally conformally K¨ahler) manifold is a com- +plex manifold admitting a Hermitian form ω which sat- +isfies dω = ω ∧ θ, where θ is a closed 1-form, called the +Lee form. An LCK manifold is called Vaisman if the Lee +form is parallel with respect to the Levi-Civita connection. +The dual vector field, called the Lee field, is holomorphic +and Killing. We prove that any holomorphic tensor on a +Vaisman manifold is invariant with respect to the Lee field. +This is used to compute the Kodaira dimension of Vais- +man manifolds. We prove that the Kodaira dimension of a +Vaisman manifold obtained as a Z-quotient of an algebraic +cone over a projective manifold X is equal to the Kodaira +dimension of X. This can be applied to prove the defor- +mational stability of the Kodaira dimension of Vaisman +manifolds. +Contents +1 +Introduction +2 +2 +Preliminaries +3 +2.1 +Locally conformally K¨ahler manifolds +. . . . . . . . . . . . . +3 +2.2 +LCK manifolds with potential . . . . . . . . . . . . . . . . . . +4 +2.2.1 +Algebraic cones . . . . . . . . . . . . . . . . . . . . . . +5 +2.2.2 +Vaisman manifolds . . . . . . . . . . . . . . . . . . . . +6 +3 +Holomorphic tensors on LCK manifolds with potential +7 +3.1 +Extension of reflexive coherent sheaves . . . . . . . . . . . . . +7 +3.2 +Invariance of holomorphic tensor fields . . . . . . . . . . . . . +8 +1Liviu Ornea is partially supported by Romanian Ministry of Education and Research, +Program PN-III, Project number PN-III-P4-ID-PCE-2020-0025, Contract 30/04.02.2021 +2Misha Verbitsky is partially supported by the HSE University Basic Research Pro- +gram, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2. +Keywords: Locally conformally K¨ahler, LCK potential, algebraic group, Zariski closure +2010 Mathematics Subject Classification: 53C55, 32G05. +1 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +4 +Zariski closures and the Chevalley theorem +9 +5 +Holomorphic tensors on Vaisman manifolds +11 +6 +An application: the Kodaira dimension of Vaisman mani- +folds +12 +1 +Introduction +Locally conformally K¨ahler geometry is probably the simplest complex ge- +ometry beyond the K¨ahler which admits many examples, explicit and non- +explicit. A manifold is called LCK (locally conformally K¨ahler) if it +admits a K¨ahler covering, with the deck group acting by holomorphic homo- +theties (Definition 2.1). Every Hopf manifold is LCK, because its universal +cover admits a K¨ahler potential, with the deck transform multiplying the +potential by a number. A manifold with this property is called an LCK +manifold with potential; an LCK manifold admits a potential if and only +if it can be realized as a complex submanifold in a Hopf manifold ([OV3]). +Recall that a diagonal Hopf manifold is a manifold of form Cn\0 +⟨A⟩ , +where A is a linear contraction which can be diagonalized in appropriate +basis. +A more special class of LCK manifolds with potential are Vais- +man manifolds, Subsection 2.2.2, which can be characterized as complex +manifolds admitting a holomorphic embedding to a diagonal Hopf manifold +([OV9, Theorem 4.9]). +Every Vaisman manifold M is equipped with an action of the Lee and +anti-Lee fields, which are Killing and holomorphic (Proposition 2.17). In +[Ve2], it was shown that any stable holomorphic bundle on M is equivariant +with respect to this action. From Theorem 2.18, it follows that any complex +subvariety is invariant under this action as well. +Let G be the group of holomorphic isometries of a Vaisman manifold M +obtained as the closure of the group generated by the Lee and the anti-Lee +flow. In this paper, we prove that any holomorphic tensor field on M is +G-invariant. +For holomorphic vector fields and differential forms this result was ob- +tained by K. Tsukada in [T1, Theorem 3.3, Theorem 4.2]. +We prove a similar result for LCK manifolds with potential. Let M be an +LCK manifold with potential, and ˜ +M its K¨ahler Z-cover. We assume that ˜ +M +is an open algebraic cone (Definition 2.10); this is true automatically when +dimC M ⩾ 3 ([OV8]). When dimC M = 2, this would follow if we assume +the GSS conjecture ([OV7, Section 25.6]). Since ˜ +M is algebraic, it makes +– 2 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +sense to speak of the Zariski closure of the Z-action on +˜ +M. We consider +the Zariski closure G of the Z-action as the smallest algebraic subgroup +of Aut( ˜ +M) containing this Z-action; this group is clearly commutative and +positive-dimensional. Since the action of G commutes with the action of the +deck transform group Z ⊂ G on ˜ +M, the quotient G/Z naturally acts on M. +We prove that any holomorphic tensor field on an LCK manifold with +potential is G-invariant (Proposition 3.3). +Let H = Cn\0 +⟨A⟩ +be a diagonal Hopf manfold, and G the Zariski closure +of ⟨A⟩. Let αi be the eigenvalues of A, and A1 an operator which has the +eigenvalues |αi| in the same basis. In Theorem 4.3, we show that the linear +field logA1 belongs to the Zariski closure of ⟨A⟩, and in Theorem 5.1, step 3, +we prove that log A1 can be obtained as the Lee field of a Vaisman structure +on H. +This implies that the Lee and anti-Lee field of a Vaisman manifold belong +to the Lie algebra of G (the Zariski closure of the Z-action). Therefore, any +holomorphic tensor field is Lee and anti-Lee invariant. +2 +Preliminaries +For the material in this section, we refer the reader to [OV7]. +2.1 +Locally conformally K¨ahler manifolds +Definition 2.1: A complex manifold (M, I) is called locally conformally +K¨ahler (LCK, for short) if it admits a covering ( ˜ +M, I) equipped with a +K¨ahler metric ˜ω such that the deck group of the cover acts on ( ˜ +M, ˜ω) by +holomorphic homotheties. +An LCK metric on an LCK manifold is an +Hermitian metric on (M, I) such that its pullback to ˜ +M is conformal with +˜ω. +Proposition 2.2: A Hermitian manifold (M, I, g) is LCK if and only if there +exists a closed 1-form θ (called the Lee form) such that the fundamental +form ω(x, y) := g(Ix, y) satisfies dω = θ ∧ ω. +Definition 2.3: An LCK structure (ω, θ) on a complex manifold (M, I) is +called strict if the Lee form is not exact. When the Lee form is exact, the +structure (ω, θ) is called globally conformally K¨ahler (GCK). +Example 2.4: Let H := (Cn \ 0)/⟨A⟩, A = λ Id, |λ| > 1, be a classical +– 3 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Hopf manifold. It is LCK because the flat K¨ahler metric ˜g0 = � dzi ⊗dzi +is multiplied by 2 by the deck group Z. The Lee form on the covering Cn \0 +is θ = −d log |z|2. Since the Hopf manifold is diffeomorphic with S1 ×S2n−1, +the LCK structure is strict. +Remark 2.5: All complex submanifolds of an LCK manifold are still LCK +(but not necessarily strict). +2.2 +LCK manifolds with potential +Definition 2.6: An LCK manifold has LCK potential if it admits a K¨ahler +covering on which the K¨ahler metric has a global and positive potential +function ψ such that the deck group multiplies ψ by a constant.1 In this +case, M is called LCK manifold with potential. +Example 2.7: The diagonal Hopf manifold in Example 2.4 is LCK with +potential because ˜g0 has the global automorphic potential ψ := � |zi|2. +By [OV3], compact LCK manifolds with potential are stable to small +deformations of the complex structure. This implies that all linear Hopf +manifolds (Cn \ 0)/⟨A⟩, A ∈ GL(n, C), with eigenvalues of absolute value +> 1, are LCK with potential. +Remark 2.8: One can easily prove that all complex submanifolds of an +LCK manifold with potential are still LCK with potential. +One of the most important properties of LCK manifolds with potential +is the following Kodaira-type embedding result: +Theorem 2.9: ([OV3]) Let M be a compact LCK manifold with potential +of complex dimension at least 3. Then M admits a holomorphic embedding +into a linear Hopf manifold. +The idea of the proof is that the K¨ahler cover of an LCK manifold with +potential is not a cone in the Riemannian sense (it lacks the homothety), +but is very close to the idea of cone, because it can be completed with just +one (singular) point. This completion can be realized using a theorem by +Rossi and Andreotti-Siu which only works in dimension greater than 3. In +dimension 2, the result is true, if one assumes the GSS conjecture ([OV6, +1In the sequel, a differential form which is multiplied by a constant factor by the action +of the deck group is called automorphic. +– 4 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Theorem 5.6]), which is a long-standing conjecture, often assumed to be +true. +Below, we give a precise description of the K¨ahler cover of an LCK +manifold with potential. +2.2.1 +Algebraic cones +Definition 2.10: ([OV4]) A closed algebraic cone is an affine variety C +admitting a C∗-action ρ with a unique fixed point x0, called the origin or +the apex, and satisfying the following: +(i) C is smooth outside of x0, and C∗ acts freely on C\x0. +(ii) ρ acts on the Zariski tangent space Tx0C with all eigenvalues |αi| < 1. +An open algebraic cone is a complex manifold which is biholomorphic to +a closed algebraic cone without the origin. +Theorem 2.11: ([OV8]) Let M be an LCK manifold with proper potential, +and ˜ +M its K¨ahler Z-cover. Then ˜ +M is an open algebraic cone. +Remark 2.12: The restriction to proper potentials is not significant, since +any LCK metric with potential can be approximated in the C∞-topology +(while keeping the complex structure fixed) by an LCK metric with proper +potential ([OV2, OV5]). +As a topological space, the closed algebraic cone +˜ +Mc is obtained as a +one-point completion of an open algebraic cone ˜ +M. However, the complex +structure on a closed algebraic cone is not uniquely determined by ˜ +M. We +refer the reader to [OV8] for this intricate and fascinating subject. +The manifold ˜ +M does not define the LCK manifold M uniquely: M = +˜ +M +Z +is determined by the choice of the Z-action which may vary. +In [OV8], we proved that an algebraic cone of an LCK manifold with +potential is isomorphic to an affine cone over a projective orbifold. This +nearly transforms the geometry of LCK manifolds with potential into a +chapter of algebraic geometry. +We end these preliminaries by citing a result which gives a method of +constructing LCK manifolds with potential once the algebraic cone which +will be the covering is known. We first need the following notion. +– 5 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Definition 2.13: Let ˜ +M be an open algebraic cone, ˜ +Mc the corresponding +closed cone, and ⃗r ∈ T ˜ +Mc a holomorphic vector field such that for all t > 0 +the diffeomorphism et⃗r is a holomorphic contraction of ˜ +Mc to the origin. In +this situation, the 1-parameter family et⃗r is called the flow of contractions +(or contraction flow). A strictly pseudoconvex hypersurface S ⊂ +˜ +M is +called a pseudoconvex shell if S intersects each orbit of et⃗r, t ∈ R, exactly +once. +Pseudoconvex shells are used to obtain automorphic plurisubharmonic +functions, as follows. +Theorem 2.14: ([OV4]) Let ˜ +M be an algebraic cone, et⃗r a contraction flow, +and S ⊂ ˜ +M a pseudoconvex shell. Then for each λ ∈ R there exists a unique +function ϕλ such that Lie⃗r ϕ = λϕ and ϕλ +���S = 1. Moreover, such ϕλ is +strictly plurisubharmonic when λ ≫ 0. +2.2.2 +Vaisman manifolds +Among the LCK manifolds with potential, the best understood subclass is +formed by the Vaisman manifolds. These are LCK manifolds with the +Lee form parallel with respect to the Levi-Civita connection of the LCK +metric. +Let (M, I, ω, θ) be a Vaisman manifold and π : +˜ +M → M a K¨ahler cover +on which π∗θ is exact. Then it can be seen that the ˜ω-squared norm of π∗θ +is a positive, automorphic potential for the K¨ahler metric ˜ω ([OV3]). +Example 2.15: Some examples of Vaisman manifolds: +(i) Diagonal Hopf manifolds (Cn\0)/⟨A⟩ where A is semi-simple and with +eigenvalues αi of absolute value > 1, [GO, OV4]. +(ii) Elliptic complex surfaces (see [B] for the complete classification of Vais- +man compact surfaces; see also [VVO]). +(iii) All compact submanifolds of a Vaisman manifold ([Ve1]). +Remark 2.16: The class of Vaisman manifolds is strict: neither the LCK +Inoue surfaces, nor the non-diagonal Hopf manifolds can bear Vaisman met- +rics ([B], [OV4]). +– 6 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +The metric dual θ♯ of the Lee form is called the Lee field. +Proposition 2.17: ([Vai1, Vai2]) Let (M, g, I, θ) be a Vaisman manifold. +Then the Lee field θ♯ is Killing and holomorphic; moreover, it commutes +with Iθ♯. +Denote by Σ the holomorphic 1-dimensional foliation generated by θ♯ +and Iθ♯. It is called the canonical foliation (the motivation is given in +the next theorem). +Theorem 2.18: Let M be a compact Vaisman manifold, and Σ ⊂ TM its +canonical foliation. Then: +(i) Σ is independent from the choice of the Vaisman metric ([T3]). +(ii) dcθ = ω − θ ∧ Iθ ([Vai2]) and the exact (1,1)-form ω0 := dcθ is semi- +positive ([Ve1]). Therefore, Σ = ker ω0, and ω0 is transversally K¨ahler +with respect to Σ. +The following result is almost obvious. We give its proof for consistency. +Lemma 2.19: Let M be a compact Vaisman manifold, and G the closure +of the Lie group generated by the action of the Lee and the anti-Lee fields +on M. Then G is a compact torus. +Proof: +Since the Lee and anti-Lee fields are acting on M by isometries +(Proposition 2.17), the group G is a closed subgroup of the isometry group +of M, and the latter is compact. Then G is compact; since it is the closure +of an abelian group, it is abelian. +3 +Holomorphic tensors on LCK manifolds with +potential +The purpose of this section is to prove Proposition 3.3. For its proof, we +shall recall some facts related to coherent sheaves. +3.1 +Extension of reflexive coherent sheaves +Given a coherent sheaf F over X, let F∗ := Hom(F, OX) be the dual +sheaf, that is, the sheaf of module homomorphisms to the ring of regular +– 7 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +functions. The natural morphism of sheaves F −→ F∗∗ does not need to +be an isomorphism: for example, its kernel contains the torsion of F. A +coherent sheaf is called reflexive if the natural map F −→ F∗∗ is an iso- +morphism. For an introduction to the reflexive sheaves, see [OSS]. Here we +state some results which are relevant to our work. +First of all, notice that the sheaf F∗ is already reflexive ([OSS], Ch. II, +Lemma 1.1.8). Moreover, the singular set of a reflexive sheaf over a normal +variety has codimension ⩾ 3; in particular, a reflexive sheaf over a complex +surface is locally free ([OSS], Ch. II, 1.1.10). Also, a reflexive sheaf of rank +one over a smooth manifold is locally free ([OSS], Ch. II, Lemma 1.1.15). +The most important property of reflexive sheaves is normality. Let Z ⊂ +X be a subvariety of a normal complex variety, codimX Z ⩾ 2. Consider the +open embedding map j : X\Z ֒→ X, and let j∗ and j∗ be the sheaf pullback +and pushforward, For any sheaf F, there exists a natural sheaf morphism +F −→ j∗j∗F taking a section of F to its restriction to X\Z. A coherent +sheaf F is normal if the natural map F −→ j∗j∗F is an isomorphism, for +any subvariety Z ⊂ X of codimension ⩾ 2. +The following theorem can be understood as a sheaf version of the Har- +togs extension theorem. +Theorem 3.1: A coherent sheaf F over a normal complex variety is normal +if and only if it is reflexive. +Proof: +[Se, Proposition 7]. +Theorem 3.2: ([OV7, Theorem 27.6]) Let M be a normal complex vari- +ety with isolated singularities, dimC M ⩾ 3, x ∈ M a point, and M0 := +M\{x} +j֒→ M. Consider a reflexive sheaf F on M0. Then j∗F is a reflexive +coherent sheaf. Moreover, j∗F is the unique coherent reflexive sheaf on M +which is isomorphic to F on M0. +Proof: +Uniqueness of a reflexive extension follows from the fact that a +coherent sheaf F over a normal variety is normal if and only if it is reflexive +(Theorem 3.1). The existence of coherent extension of F follows from [AS, +Proposition 6.1]. +3.2 +Invariance of holomorphic tensor fields +We are now ready to prove the main result of this section. +– 8 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Proposition 3.3: Let M be a compact LCK manifold with potential, ˜ +M its +K¨ahler Z-cover, considered as an open algebraic cone, and Φ ∈ H0(M, B) +be a holomorphic tensor field on M, where B = (Ω1M)⊗k ⊗ TM⊗l. Denote +by ˜Φ its lift to ˜ +M, and let G be the Zariski closure of the Z-action on ˜ +M. +Then ˜Φ is G-invariant. +Proof. +Step 1: +Consider a Z-action on a finite-dimensional space +W. +Then any Z-invariant vector w ∈ W is invariant under the Zariski +closure of Z. This is why the statement of Proposition 3.3 is not surprising. +However, the space of tensor fields on +˜ +M is not finite-dimensional, which +makes Proposition 3.3 non-trivial. Denote by m the maximal ideal of the +origin in the closed algebraic cone ˜ +Mc. We think of the quotients O ˜ +Mc/mk +as of the spaces of jets of holomorphic (or algebraic) functions. As in [OV8], +we choose ˜ +Mc normal, so that the Z-action is extended to ˜ +Mc. Then any +Z-invariant jet u ∈ O ˜ +Mc/mk is also G-invariant. +Step 2: From now on, we use the same letter Φ to denote the lift of +Φ to +˜ +M. +Consider Φ as a Z-invariant section of the appropriate tensor +bundle B = (Ω1M)⊗k ⊗ TM⊗l, which is by construction Z-equivariant. +Using Theorem 3.2, we extend B to a reflexive coherent sheaf Bc on ˜ +Mc. +Since coherent reflexive sheaves are normal ([OSS, Ch. II, Lemma 1.1.12], +[Se, Proposition 7]), the section Φ admits a holomorphic extension to ˜ +Mc, +denoted as Φc ∈ H0( ˜ +Mc, Bc). Consider the space Jk +B := +Bc +mk+1Bc of k-jets of +Bc. The Z-action on H0( ˜ +Mc, Bc) preserves H0( ˜ +Mc, mkBc) ⊂ H0( ˜ +Mc, Bc). +Let Φk +c ∈ Jk +B be the k-jet of ˜Φc. Since Φc is Z-invariant, and the space of +k-jets is finite-dimensional, the k-jet Φk +c is G-invariant. Let ˆBc := lim +← Jk +B +be the m-adic completion of Bc, and ˆΦc the image of Φc in this completion. +Since all k-jets Φk +c are G-invariant, the completion ˆΦc is also G-invariant. +Since the ring of germs of holomorphic functions on ˜ +Mc in c is Noetherian +([GN, Chapter II, Theorem B.9]), and Bc, being reflexive, is torsion-free, +the completion map H0( ˜ +Mc, Bc) −→ ˆBc is injective ([AM, Theorem 10.17]). +Therefore, Φ ∈ H0( ˜ +Mc, Bc) is also G-invariant. +4 +Zariski closures and the Chevalley theorem +We recall the following famous theorem due to Chevalley. +– 9 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Let G ⊂ GL(V ) be an algebraic group, and W = V ⊗k ⊗ (V ∗)⊗l a tensor +representation of G. A G-invariant vector v ∈ W is called a tensor invari- +ant of G. A point x ∈ PW is called a projective tensor invariant of G +if it is G-invariant. +Theorem 4.1: (Chevalley theorem, [M]). +An algebraic group is uniquely determined by the set of its projective tensor +invariants. +When an algebraic group is reductive (over C, a group is reductive if and +only if it has a compact real form), a stronger version of Chevalley theorem +is available. +Theorem 4.2: A reductive algebraic group is uniquely determined by the +set of its tensor invariants. +Proof: +[D, Proposition 3.1 (c)]. +We are going to prove the following result, used further on to determine +the Lee field action on a Vaisman manifold. +Theorem 4.3: Let A ∈ GL(n, C) be a diagonal linear operator with the +eigenvalues α1, ..., αn, and A1 an operator which is diagonal in the same +basis, with the eigenvalues |α1|, ..., |αn|. Denote by G the Zariski closure of +the group ⟨A⟩ in GL(n, C). Then G contains A1. +Proof: +Since the operator A is diagonalizable, the algebraic closure +of ⟨A⟩ is a commutative group which contains only semisimple elements. +By [BT, Proposition 1.5], the connected component of G is isomorphic to +(C∗)d, hence G is reductive. Then Theorem 4.2 implies that G is the set of all +g ∈ GL(n, C) which preserve all A-invariant vectors w ∈ W = V ⊗k ⊗(V ∗)⊗l. +The eigenvalues of A on W are products of k instances of αi and l instances +of α−1 +i . The eigenvectors w ∈ W are w = �k +i=1 zmi ⊗�l +i=1 ζni, where zni are +the eigenvectors in V with the eigenvalues αni and ζmi are the eigenvectors +in V ∗ with the eigenvalues α−1 +mi. Then A(w) = �k +i=1 αni +�l +i=1 α−1 +mi. +The +eigenvector w is A-invariant if and only if �k +i=1 αni +�l +i=1 α−1 +mi = 1. +This +implies �k +i=1 |αni| �l +i=1 |α−1 +mi| = 1, hence w is A1-invariant as well. +– 10 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +5 +Holomorphic tensors on Vaisman manifolds +Let M be a Vaisman manifold, and θ♯ its Lee field. Then θ♯ and I(θ♯) are +holomorphic Killing fields (Proposition 2.17). The closure G of the group +et1θ♯+t2I(θ♯) in the group of isometries of M is compact and commutative, +hence it is a compact torus (Lemma 2.19). +Theorem 5.1: Let M be a compact Vaisman manifold, Φ ∈ H0(M, B) +a holomorphic tensor, where B = (Ω1M)⊗k ⊗ TM⊗l, and G the smallest +closed Lie group containing the flows generated by the Lee and the anti-Lee +fields. Then Φ is G-invariant. +Proof. Step 1: +Choose a Vaisman metric on M with LCK rank 1, +and let ˜ +M be the corresponding K¨ahler Z-cover of M, which is considered +as an open algebraic cone. Let G ⊂ Aut( ˜ +M) be the Zariski closure of the +Z-action. Then Φ is G-invariant by Proposition 3.3. To finish the proof of +Theorem 5.1, it would suffice to show that G ⊂ G; since G is closed, this +would follow if we prove that the Lee field θ♯ is tangent to G. +Step 2: Let M −→ H be a holomorphic embedding of M to a diagonal +Hopf manifold. By Theorem 2.18, this embedding commutes with the group +generated by the Lee and the anti-Lee flow, hence it would suffice to show +that θ♯ is tangent to G when M = H. It remains to prove Theorem 5.1 +assuming that M is a Hopf manifold. For this purpose, we compute the +groups G and G for a diagonal Hopf manifold. This can be done explicitly +using the expression for the Vaisman metric on a Hopf manifold obtained in +Theorem 2.14 via the pseudoconvex shells. +Step 3: The rank 2 algebra generated by the Lee and anti-Lee flows on +a Vaisman manifold M is uniquely determined by the complex structure on +M (Theorem 2.18). We express the Lee flow on a Hopf manifold using a +Vaisman structure we shall construct explicitly, and prove that it belongs to +Lie(G). This would imply that the original Lee flow also belongs to Lie(G). +Let H = Cn\0 +⟨A⟩ , where A ∈ GL(n, C) is a diagonal linear contraction with +eigenvalues α1, ..., αn. Let A1 be the matrix which is diagonal in the same +basis as A, and has eigenvalues |αi|. Then ⃗r := log A1 is a holomorphic +contraction vector field, and et⃗r is an A-invariant contraction flow. +The +sphere S2n−1 is a pseudoconvex shell compatible with this contraction flow. +By [OV4], we obtain a et log A1-automorphic plurisubharmonic function ϕ. +We are going to prove that ϕ is A-automorphic. +– 11 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Consider the diagonal U(1)n-action on Cn. The plurisubharmonic func- +tion ϕ is U(1)n-invariant because ⃗r and S2n−1 are U(1)n-invariant. Since +A ∈ U(1)n · A1, and ϕ is A1-automorphic, it is A-automorphic. Therefore, +ddcϕ +ϕ +defines a Vaisman metric on H. By definition, the Lee flow maps the +level sets of ϕ to the level sets of ϕ, with ϕ(etθ♯z) = tϕ(z). By construction +of ϕ, the same is true for e−t log A1, that is, ϕ(e−t log A1z) = tϕ(z). There- +fore, θ♯ = − log A1, where log A1 is the diagonal matrix with the eigenvalues +log |αi| in the same basis. This is the Lee flow for the Vaisman metric ddcϕ +ϕ . +The vector field log A1 belongs to Lie(G), because any power of A1 +fixes all tensor invariants of G by Theorem 4.3, hence the corresponding +1-parametric sugroup also belongs to Lie(G); this gives log A1 ∈ Lie(G). +6 +An application: the Kodaira dimension of Vais- +man manifolds +Definition 6.1: Let M be a compact complex manifold. Its pluricanonical +bundle is the tensor power Kn +M = K⊗n +M , n ⩾ 0. The Kodaira dimension +κ(M) is defined as κ(M) := lim supn +log(dim H0(Kn +M)) +log n +. +Remark 6.2: The Kodaira dimension is equal to −∞ if H0(Kn +M) = 0 for +almost all n > 0, and to 0 if dim H0(Kn +M) is bounded, but non-zero for +infinitely many n. If the function n �→ dim H0(Kn +M) grows as a polynomial +of degree d, the Kodaira dimension of M is d. +Let M be a Vaisman manifold, G the closure of the group generated by +the Lee and the anti-Lee action, ˜ +M a K¨ahler Z-covering of M, and ˜G the +lift of G to ˜ +M. Denote by G the Zariski closure of ˜G in Aut( ˜ +M), considered +as an algebraic variety ([OV8]). +Since ˜G acts on ˜ +M by homotheties, and contains contractions, there is +an open set U ⊂ G in the connected component of G such that for all γ′ ∈ U, +the quotient M′ := +˜ +M +⟨γ′⟩ is LCK ([OV8]). The automorphism γ commutes +with the action of the group generated by the Lee and the anti-Lee field, +hence by [KO] M′ is Vaisman.1 By [BT, Proposition 1.5], the connected +component of G is isomorphic to (C∗)k. It is not hard to see that the union +of all one-dimensional closed subgroups is dense in any algebraic group, +hence U contains a dense subset of γ′ which are contained in a subgroup +1As shown in [KO], an LCK manifold which admits a holomorphic conformal C-action +is Vaisman, assuming that this action cannot be lifted to an isometry of the K¨ahler cover. +– 12 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +C∗ ⊂ G. Since C∗/⟨γ′⟩ is an elliptic curve, the Vaisman manifold M′ = +˜ +M +⟨γ′⟩ +is elliptically fibered.2 As shown in [OV1], the leaf space X is a projective +orbifold. +Theorem 6.3: Let M be a Vaisman manifold, M′ its quasi-regular de- +formation obtained above, and X the corresponding leaf space, which is a +projective orbifold. +Then κ(M) = κ(M′) = κ(X), where κ denotes the +Kodaira dimension. +Proof. +Step 1: +The equality κ(M′) = κ(X) is clear from the ad- +junction formula: +since π : +M′ −→ X is an elliptic fibration, one has +H0(Kk +M′) = H0(Kk +X) for all k. +Step 2: By Theorem 5.1, any section h of the pluricanonical bundle Kk +M +on M is G-invariant. Then, for any γ′ ∈ U, the lift of h to ˜ +M is γ′-invariant, +hence can be obtained as a pullback of a section of Kk +M′ under the quotient +map ˜ +M −→ M′. This defines an injective map Ψ : H0(Kk +M) −→ H0(Kk +M′). +Step 3: To finish the proof of Theorem 6.3 it remains to show that +Ψ : H0(Kk +M) −→ H0(Kk +M′) is surjective. This would follow if we show that +any section h′ ∈ H0(Kk +M′) is G-invariant. Since H0(Kk +M′) = H0(Kk +X) (Step +1), the G-invariance of h′ would follow if we prove that the corresponding +section of Kk +X is invariant under the natural action of G on X. +Recall that the pluricanonical representation is the natural action +of Aut(X) on H0(Kk +X). Since X is an orbifold, its singularities are klt ([P, +Proposition 1.2.1]). By [KK, Theorem 10.61], the image of the pluricanonical +representation is finite; since G is connected, it acts on H0(Kk +X) trivially. +This gives H0(Kk +M) = H0(Kk +M′) = H0(Kk +X) . +As an application, we outline the proof of the deformational stability of +Kodaira dimension for Vaisman manifolds. +Conjecture 6.4: Let Mt, t ∈ R, be a smooth family of compact Vaisman +manifolds. Then the Kodaira dimension of Mt is constant. +Sketch of a proof: We sketch a proof of this conjecture, reducing it to +Conjecture 6.5 below. +2Ellipticaly fibered Vaisman manifolds are called quasi-regular. +– 13 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +Let +˜ +Mt be the family of open algebraic cones obtained from each Mt +by taking a K¨ahler Z-cover; this is possible to do by [OV1]. Each +˜ +Mt is +equipped with a natural algebraic structure by [OV8]. +What follows is a conjecture, which most likely can be proven by the +methods used in [OV8]. +Let ˜ +Mt, t ∈ R, be a smooth family of open algebraic cones. As shown +in [OV8], for an appropriate choice of the C∗-action on +˜ +Mt the quotient +˜ +Mt/C∗ is isomorphic to a complex projective orbifold Xt; this isomorphism +is extended to an isomorphism between +˜ +Mt and Tot◦(Lt), where Lt is an +ample line bundle over Xt. +Conjecture 6.5: Let ˜ +Mt, t ∈ R, be a smooth family of open algebraic cones. +Then for any sufficiently small open subset U ⊂ R, the choice of C∗-action +as above can be performed in such a way that the family (Xt, Lt) depends +smoothly on t ∈ U. +Now, by Theorem 6.3, the Kodaira dimension of Mt is equal to the Ko- +daira dimension of Xt, which is constant, as shown by Siu in [Si]. +Acknowledgements: We are grateful to Nicolina Istrati for her interest +and valuable advice. +References +[AS] +A. Andreotti, Y. T. Siu, Projective embeddings of pseudoconcave spaces, Ann. Scuola +Norm. Sup. Pisa 24, 231-278 (1970). (Cited on page 8.) +[AM] M. F. Atiyah, I. G. 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(Cited on page 2.) +– 15 – +version 1.0, Jan. 3, 2022 + +L. Ornea, M. Verbitsky +Holomorphic tensors on LCK manifolds with potential +[T3] +K. Tsukada, The canonical foliation of a compact generalized Hopf manifold, Differ- +ential Geom. Appl. 11 (1999), no. 1, 13-28. (Cited on page 7.) +[Vai1] I Vaisman, Locally conformal K¨ahler manifolds with parallel Lee form, Rend. Mat. +(6) 12 (1979), no. 2, 263-284. (Cited on page 7.) +[Vai2] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata, 13 (1982), 231-255. +(Cited on page 7.) +[Ve1] M. Verbitsky, Theorems on the vanishing of cohomology for locally conformally +hyper-K¨ahler manifolds, Proc. Steklov Inst. Math. no. 3 246 (2004), 54-78. (Cited +on pages 6 and 7.) +[Ve2] M. Verbitsky, Stable bundles on positive principal elliptic fibrations, Math. Res. Lett. +12 (2005), no. 2-3, 251-264. (Cited on page 2.) +[VVO] M. Verbitsky, V. Vuletescu, L. Ornea Classification of non-K¨ahler surfaces and +locally conformally K¨ahler geometry, Russian Math. Surv. 76 (2021), 261-290. +arxiv:1810.05768. (Cited on page 6.) +Liviu Ornea +University of Bucharest, Faculty of Mathematics and Informatics, +14 Academiei str., 70109 Bucharest, Romania, and: +Institute of Mathematics “Simion Stoilow” of the Romanian Academy, +21, Calea Grivitei Str. 010702-Bucharest, Romania +lornea@fmi.unibuc.ro, liviu.ornea@imar.ro +Misha Verbitsky +Instituto Nacional de Matem´atica Pura e Aplicada (IMPA) +Estrada Dona Castorina, 110 +Jardim Botˆanico, CEP 22460-320 +Rio de Janeiro, RJ - Brasil +also: +Laboratory of Algebraic Geometry, +Faculty of Mathematics, National Research University Higher School +of Economics, 6 Usacheva Str. Moscow, Russia +verbit@verbit.ru, verbit@impa.br +– 16 – +version 1.0, Jan. 3, 2022 + diff --git a/SdAzT4oBgHgl3EQfJPus/content/tmp_files/load_file.txt b/SdAzT4oBgHgl3EQfJPus/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..47ceec14f9539777f1abcbd2cbb9bb3151fd99e8 --- /dev/null +++ b/SdAzT4oBgHgl3EQfJPus/content/tmp_files/load_file.txt @@ -0,0 +1,725 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf,len=724 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='01077v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='AG] 3 Jan 2023 Holomorphic tensors on Vaisman manifolds Liviu Ornea1, Misha Verbitsky2 Dedicated to Professor Valentin Poenaru at his ninetieth birthday Abstract An LCK (locally conformally K¨ahler) manifold is a com- plex manifold admitting a Hermitian form ω which sat- isfies dω = ω ∧ θ, where θ is a closed 1-form, called the Lee form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' An LCK manifold is called Vaisman if the Lee form is parallel with respect to the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The dual vector field, called the Lee field, is holomorphic and Killing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We prove that any holomorphic tensor on a Vaisman manifold is invariant with respect to the Lee field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This is used to compute the Kodaira dimension of Vais- man manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We prove that the Kodaira dimension of a Vaisman manifold obtained as a Z-quotient of an algebraic cone over a projective manifold X is equal to the Kodaira dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This can be applied to prove the defor- mational stability of the Kodaira dimension of Vaisman manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 Locally conformally K¨ahler manifolds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 LCK manifolds with potential .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 Vaisman manifolds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 6 3 Holomorphic tensors on LCK manifolds with potential 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 Extension of reflexive coherent sheaves .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 Invariance of holomorphic tensor fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 8 1Liviu Ornea is partially supported by Romanian Ministry of Education and Research, Program PN-III, Project number PN-III-P4-ID-PCE-2020-0025, Contract 30/04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2021 2Misha Verbitsky is partially supported by the HSE University Basic Research Pro- gram, FAPERJ E-26/202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='912/2018 and CNPq - Process 313608/2017-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Keywords: Locally conformally K¨ahler, LCK potential, algebraic group, Zariski closure 2010 Mathematics Subject Classification: 53C55, 32G05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential 4 Zariski closures and the Chevalley theorem 9 5 Holomorphic tensors on Vaisman manifolds 11 6 An application: the Kodaira dimension of Vaisman mani- folds 12 1 Introduction Locally conformally K¨ahler geometry is probably the simplest complex ge- ometry beyond the K¨ahler which admits many examples, explicit and non- explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A manifold is called LCK (locally conformally K¨ahler) if it admits a K¨ahler covering, with the deck group acting by holomorphic homo- theties (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Every Hopf manifold is LCK, because its universal cover admits a K¨ahler potential, with the deck transform multiplying the potential by a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A manifold with this property is called an LCK manifold with potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' an LCK manifold admits a potential if and only if it can be realized as a complex submanifold in a Hopf manifold ([OV3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Recall that a diagonal Hopf manifold is a manifold of form Cn\\0 ⟨A⟩ , where A is a linear contraction which can be diagonalized in appropriate basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A more special class of LCK manifolds with potential are Vais- man manifolds, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2, which can be characterized as complex manifolds admitting a holomorphic embedding to a diagonal Hopf manifold ([OV9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Every Vaisman manifold M is equipped with an action of the Lee and anti-Lee fields, which are Killing and holomorphic (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' In [Ve2], it was shown that any stable holomorphic bundle on M is equivariant with respect to this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='18, it follows that any complex subvariety is invariant under this action as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let G be the group of holomorphic isometries of a Vaisman manifold M obtained as the closure of the group generated by the Lee and the anti-Lee flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' In this paper, we prove that any holomorphic tensor field on M is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' For holomorphic vector fields and differential forms this result was ob- tained by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Tsukada in [T1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We prove a similar result for LCK manifolds with potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let M be an LCK manifold with potential, and ˜ M its K¨ahler Z-cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We assume that ˜ M is an open algebraic cone (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' this is true automatically when dimC M ⩾ 3 ([OV8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' When dimC M = 2, this would follow if we assume the GSS conjecture ([OV7, Section 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since ˜ M is algebraic, it makes – 2 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential sense to speak of the Zariski closure of the Z-action on ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We consider the Zariski closure G of the Z-action as the smallest algebraic subgroup of Aut( ˜ M) containing this Z-action;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' this group is clearly commutative and positive-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since the action of G commutes with the action of the deck transform group Z ⊂ G on ˜ M, the quotient G/Z naturally acts on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We prove that any holomorphic tensor field on an LCK manifold with potential is G-invariant (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let H = Cn\\0 ⟨A⟩ be a diagonal Hopf manfold, and G the Zariski closure of ⟨A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let αi be the eigenvalues of A, and A1 an operator which has the eigenvalues |αi| in the same basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3, we show that the linear field logA1 belongs to the Zariski closure of ⟨A⟩, and in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1, step 3, we prove that log A1 can be obtained as the Lee field of a Vaisman structure on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This implies that the Lee and anti-Lee field of a Vaisman manifold belong to the Lie algebra of G (the Zariski closure of the Z-action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Therefore, any holomorphic tensor field is Lee and anti-Lee invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 2 Preliminaries For the material in this section, we refer the reader to [OV7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 Locally conformally K¨ahler manifolds Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1: A complex manifold (M, I) is called locally conformally K¨ahler (LCK, for short) if it admits a covering ( ˜ M, I) equipped with a K¨ahler metric ˜ω such that the deck group of the cover acts on ( ˜ M, ˜ω) by holomorphic homotheties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' An LCK metric on an LCK manifold is an Hermitian metric on (M, I) such that its pullback to ˜ M is conformal with ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2: A Hermitian manifold (M, I, g) is LCK if and only if there exists a closed 1-form θ (called the Lee form) such that the fundamental form ω(x, y) := g(Ix, y) satisfies dω = θ ∧ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3: An LCK structure (ω, θ) on a complex manifold (M, I) is called strict if the Lee form is not exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' When the Lee form is exact, the structure (ω, θ) is called globally conformally K¨ahler (GCK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='4: Let H := (Cn \\ 0)/⟨A⟩, A = λ Id, |λ| > 1, be a classical – 3 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Hopf manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' It is LCK because the flat K¨ahler metric ˜g0 = � dzi ⊗dzi is multiplied by 2 by the deck group Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The Lee form on the covering Cn \\0 is θ = −d log |z|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since the Hopf manifold is diffeomorphic with S1 ×S2n−1, the LCK structure is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='5: All complex submanifolds of an LCK manifold are still LCK (but not necessarily strict).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 LCK manifolds with potential Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='6: An LCK manifold has LCK potential if it admits a K¨ahler covering on which the K¨ahler metric has a global and positive potential function ψ such that the deck group multiplies ψ by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 In this case, M is called LCK manifold with potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='7: The diagonal Hopf manifold in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='4 is LCK with potential because ˜g0 has the global automorphic potential ψ := � |zi|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By [OV3], compact LCK manifolds with potential are stable to small deformations of the complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This implies that all linear Hopf manifolds (Cn \\ 0)/⟨A⟩, A ∈ GL(n, C), with eigenvalues of absolute value > 1, are LCK with potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='8: One can easily prove that all complex submanifolds of an LCK manifold with potential are still LCK with potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' One of the most important properties of LCK manifolds with potential is the following Kodaira-type embedding result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='9: ([OV3]) Let M be a compact LCK manifold with potential of complex dimension at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then M admits a holomorphic embedding into a linear Hopf manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The idea of the proof is that the K¨ahler cover of an LCK manifold with potential is not a cone in the Riemannian sense (it lacks the homothety), but is very close to the idea of cone, because it can be completed with just one (singular) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This completion can be realized using a theorem by Rossi and Andreotti-Siu which only works in dimension greater than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' In dimension 2, the result is true, if one assumes the GSS conjecture ([OV6, 1In the sequel, a differential form which is multiplied by a constant factor by the action of the deck group is called automorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 4 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='6]), which is a long-standing conjecture, often assumed to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Below, we give a precise description of the K¨ahler cover of an LCK manifold with potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 Algebraic cones Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='10: ([OV4]) A closed algebraic cone is an affine variety C admitting a C∗-action ρ with a unique fixed point x0, called the origin or the apex, and satisfying the following: (i) C is smooth outside of x0, and C∗ acts freely on C\\x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' (ii) ρ acts on the Zariski tangent space Tx0C with all eigenvalues |αi| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' An open algebraic cone is a complex manifold which is biholomorphic to a closed algebraic cone without the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='11: ([OV8]) Let M be an LCK manifold with proper potential, and ˜ M its K¨ahler Z-cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then ˜ M is an open algebraic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='12: The restriction to proper potentials is not significant, since any LCK metric with potential can be approximated in the C∞-topology (while keeping the complex structure fixed) by an LCK metric with proper potential ([OV2, OV5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' As a topological space, the closed algebraic cone ˜ Mc is obtained as a one-point completion of an open algebraic cone ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' However, the complex structure on a closed algebraic cone is not uniquely determined by ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We refer the reader to [OV8] for this intricate and fascinating subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The manifold ˜ M does not define the LCK manifold M uniquely: M = ˜ M Z is determined by the choice of the Z-action which may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' In [OV8], we proved that an algebraic cone of an LCK manifold with potential is isomorphic to an affine cone over a projective orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This nearly transforms the geometry of LCK manifolds with potential into a chapter of algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We end these preliminaries by citing a result which gives a method of constructing LCK manifolds with potential once the algebraic cone which will be the covering is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We first need the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 5 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='13: Let ˜ M be an open algebraic cone, ˜ Mc the corresponding closed cone, and ⃗r ∈ T ˜ Mc a holomorphic vector field such that for all t > 0 the diffeomorphism et⃗r is a holomorphic contraction of ˜ Mc to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' In this situation, the 1-parameter family et⃗r is called the flow of contractions (or contraction flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A strictly pseudoconvex hypersurface S ⊂ ˜ M is called a pseudoconvex shell if S intersects each orbit of et⃗r, t ∈ R, exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Pseudoconvex shells are used to obtain automorphic plurisubharmonic functions, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='14: ([OV4]) Let ˜ M be an algebraic cone, et⃗r a contraction flow, and S ⊂ ˜ M a pseudoconvex shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then for each λ ∈ R there exists a unique function ϕλ such that Lie⃗r ϕ = λϕ and ϕλ ���S = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Moreover, such ϕλ is strictly plurisubharmonic when λ ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 Vaisman manifolds Among the LCK manifolds with potential, the best understood subclass is formed by the Vaisman manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' These are LCK manifolds with the Lee form parallel with respect to the Levi-Civita connection of the LCK metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let (M, I, ω, θ) be a Vaisman manifold and π : ˜ M → M a K¨ahler cover on which π∗θ is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then it can be seen that the ˜ω-squared norm of π∗θ is a positive, automorphic potential for the K¨ahler metric ˜ω ([OV3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='15: Some examples of Vaisman manifolds: (i) Diagonal Hopf manifolds (Cn\\0)/⟨A⟩ where A is semi-simple and with eigenvalues αi of absolute value > 1, [GO, OV4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' (ii) Elliptic complex surfaces (see [B] for the complete classification of Vais- man compact surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' see also [VVO]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' (iii) All compact submanifolds of a Vaisman manifold ([Ve1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='16: The class of Vaisman manifolds is strict: neither the LCK Inoue surfaces, nor the non-diagonal Hopf manifolds can bear Vaisman met- rics ([B], [OV4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 6 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential The metric dual θ♯ of the Lee form is called the Lee field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='17: ([Vai1, Vai2]) Let (M, g, I, θ) be a Vaisman manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then the Lee field θ♯ is Killing and holomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' moreover, it commutes with Iθ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Denote by Σ the holomorphic 1-dimensional foliation generated by θ♯ and Iθ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' It is called the canonical foliation (the motivation is given in the next theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='18: Let M be a compact Vaisman manifold, and Σ ⊂ TM its canonical foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then: (i) Σ is independent from the choice of the Vaisman metric ([T3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' (ii) dcθ = ω − θ ∧ Iθ ([Vai2]) and the exact (1,1)-form ω0 := dcθ is semi- positive ([Ve1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Therefore, Σ = ker ω0, and ω0 is transversally K¨ahler with respect to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The following result is almost obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We give its proof for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='19: Let M be a compact Vaisman manifold, and G the closure of the Lie group generated by the action of the Lee and the anti-Lee fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then G is a compact torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof: Since the Lee and anti-Lee fields are acting on M by isometries (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='17), the group G is a closed subgroup of the isometry group of M, and the latter is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then G is compact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' since it is the closure of an abelian group, it is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3 Holomorphic tensors on LCK manifolds with potential The purpose of this section is to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' For its proof, we shall recall some facts related to coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 Extension of reflexive coherent sheaves Given a coherent sheaf F over X, let F∗ := Hom(F, OX) be the dual sheaf, that is, the sheaf of module homomorphisms to the ring of regular – 7 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The natural morphism of sheaves F −→ F∗∗ does not need to be an isomorphism: for example, its kernel contains the torsion of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A coherent sheaf is called reflexive if the natural map F −→ F∗∗ is an iso- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' For an introduction to the reflexive sheaves, see [OSS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Here we state some results which are relevant to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' First of all, notice that the sheaf F∗ is already reflexive ([OSS], Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' II, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Moreover, the singular set of a reflexive sheaf over a normal variety has codimension ⩾ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' in particular, a reflexive sheaf over a complex surface is locally free ([OSS], Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' II, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Also, a reflexive sheaf of rank one over a smooth manifold is locally free ([OSS], Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' II, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The most important property of reflexive sheaves is normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let Z ⊂ X be a subvariety of a normal complex variety, codimX Z ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Consider the open embedding map j : X\\Z ֒→ X, and let j∗ and j∗ be the sheaf pullback and pushforward, For any sheaf F, there exists a natural sheaf morphism F −→ j∗j∗F taking a section of F to its restriction to X\\Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A coherent sheaf F is normal if the natural map F −→ j∗j∗F is an isomorphism, for any subvariety Z ⊂ X of codimension ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The following theorem can be understood as a sheaf version of the Har- togs extension theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1: A coherent sheaf F over a normal complex variety is normal if and only if it is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof: [Se, Proposition 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2: ([OV7, Theorem 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='6]) Let M be a normal complex vari- ety with isolated singularities, dimC M ⩾ 3, x ∈ M a point, and M0 := M\\{x} j֒→ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Consider a reflexive sheaf F on M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then j∗F is a reflexive coherent sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Moreover, j∗F is the unique coherent reflexive sheaf on M which is isomorphic to F on M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof: Uniqueness of a reflexive extension follows from the fact that a coherent sheaf F over a normal variety is normal if and only if it is reflexive (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The existence of coherent extension of F follows from [AS, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 Invariance of holomorphic tensor fields We are now ready to prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 8 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3: Let M be a compact LCK manifold with potential, ˜ M its K¨ahler Z-cover, considered as an open algebraic cone, and Φ ∈ H0(M, B) be a holomorphic tensor field on M, where B = (Ω1M)⊗k ⊗ TM⊗l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Denote by ˜Φ its lift to ˜ M, and let G be the Zariski closure of the Z-action on ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then ˜Φ is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 1: Consider a Z-action on a finite-dimensional space W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then any Z-invariant vector w ∈ W is invariant under the Zariski closure of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This is why the statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3 is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' However, the space of tensor fields on ˜ M is not finite-dimensional, which makes Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3 non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Denote by m the maximal ideal of the origin in the closed algebraic cone ˜ Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We think of the quotients O ˜ Mc/mk as of the spaces of jets of holomorphic (or algebraic) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' As in [OV8], we choose ˜ Mc normal, so that the Z-action is extended to ˜ Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then any Z-invariant jet u ∈ O ˜ Mc/mk is also G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 2: From now on, we use the same letter Φ to denote the lift of Φ to ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Consider Φ as a Z-invariant section of the appropriate tensor bundle B = (Ω1M)⊗k ⊗ TM⊗l, which is by construction Z-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2, we extend B to a reflexive coherent sheaf Bc on ˜ Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since coherent reflexive sheaves are normal ([OSS, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' II, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='12], [Se, Proposition 7]), the section Φ admits a holomorphic extension to ˜ Mc, denoted as Φc ∈ H0( ˜ Mc, Bc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Consider the space Jk B := Bc mk+1Bc of k-jets of Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The Z-action on H0( ˜ Mc, Bc) preserves H0( ˜ Mc, mkBc) ⊂ H0( ˜ Mc, Bc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let Φk c ∈ Jk B be the k-jet of ˜Φc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since Φc is Z-invariant, and the space of k-jets is finite-dimensional, the k-jet Φk c is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let ˆBc := lim ← Jk B be the m-adic completion of Bc, and ˆΦc the image of Φc in this completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since all k-jets Φk c are G-invariant, the completion ˆΦc is also G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since the ring of germs of holomorphic functions on ˜ Mc in c is Noetherian ([GN, Chapter II, Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='9]), and Bc, being reflexive, is torsion-free, the completion map H0( ˜ Mc, Bc) −→ ˆBc is injective ([AM, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Therefore, Φ ∈ H0( ˜ Mc, Bc) is also G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 4 Zariski closures and the Chevalley theorem We recall the following famous theorem due to Chevalley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 9 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Let G ⊂ GL(V ) be an algebraic group, and W = V ⊗k ⊗ (V ∗)⊗l a tensor representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A G-invariant vector v ∈ W is called a tensor invari- ant of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' A point x ∈ PW is called a projective tensor invariant of G if it is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1: (Chevalley theorem, [M]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' An algebraic group is uniquely determined by the set of its projective tensor invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' When an algebraic group is reductive (over C, a group is reductive if and only if it has a compact real form), a stronger version of Chevalley theorem is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2: A reductive algebraic group is uniquely determined by the set of its tensor invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof: [D, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We are going to prove the following result, used further on to determine the Lee field action on a Vaisman manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3: Let A ∈ GL(n, C) be a diagonal linear operator with the eigenvalues α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=', αn, and A1 an operator which is diagonal in the same basis, with the eigenvalues |α1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=', |αn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Denote by G the Zariski closure of the group ⟨A⟩ in GL(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then G contains A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof: Since the operator A is diagonalizable, the algebraic closure of ⟨A⟩ is a commutative group which contains only semisimple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By [BT, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='5], the connected component of G is isomorphic to (C∗)d, hence G is reductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 implies that G is the set of all g ∈ GL(n, C) which preserve all A-invariant vectors w ∈ W = V ⊗k ⊗(V ∗)⊗l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The eigenvalues of A on W are products of k instances of αi and l instances of α−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The eigenvectors w ∈ W are w = �k i=1 zmi ⊗�l i=1 ζni, where zni are the eigenvectors in V with the eigenvalues αni and ζmi are the eigenvectors in V ∗ with the eigenvalues α−1 mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then A(w) = �k i=1 αni �l i=1 α−1 mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The eigenvector w is A-invariant if and only if �k i=1 αni �l i=1 α−1 mi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This implies �k i=1 |αni| �l i=1 |α−1 mi| = 1, hence w is A1-invariant as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 10 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential 5 Holomorphic tensors on Vaisman manifolds Let M be a Vaisman manifold, and θ♯ its Lee field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then θ♯ and I(θ♯) are holomorphic Killing fields (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The closure G of the group et1θ♯+t2I(θ♯) in the group of isometries of M is compact and commutative, hence it is a compact torus (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1: Let M be a compact Vaisman manifold, Φ ∈ H0(M, B) a holomorphic tensor, where B = (Ω1M)⊗k ⊗ TM⊗l, and G the smallest closed Lie group containing the flows generated by the Lee and the anti-Lee fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then Φ is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 1: Choose a Vaisman metric on M with LCK rank 1, and let ˜ M be the corresponding K¨ahler Z-cover of M, which is considered as an open algebraic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let G ⊂ Aut( ˜ M) be the Zariski closure of the Z-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then Φ is G-invariant by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' To finish the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1, it would suffice to show that G ⊂ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' since G is closed, this would follow if we prove that the Lee field θ♯ is tangent to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 2: Let M −→ H be a holomorphic embedding of M to a diagonal Hopf manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='18, this embedding commutes with the group generated by the Lee and the anti-Lee flow, hence it would suffice to show that θ♯ is tangent to G when M = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' It remains to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 assuming that M is a Hopf manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' For this purpose, we compute the groups G and G for a diagonal Hopf manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This can be done explicitly using the expression for the Vaisman metric on a Hopf manifold obtained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='14 via the pseudoconvex shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 3: The rank 2 algebra generated by the Lee and anti-Lee flows on a Vaisman manifold M is uniquely determined by the complex structure on M (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We express the Lee flow on a Hopf manifold using a Vaisman structure we shall construct explicitly, and prove that it belongs to Lie(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This would imply that the original Lee flow also belongs to Lie(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let H = Cn\\0 ⟨A⟩ , where A ∈ GL(n, C) is a diagonal linear contraction with eigenvalues α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=', αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let A1 be the matrix which is diagonal in the same basis as A, and has eigenvalues |αi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then ⃗r := log A1 is a holomorphic contraction vector field, and et⃗r is an A-invariant contraction flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The sphere S2n−1 is a pseudoconvex shell compatible with this contraction flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By [OV4], we obtain a et log A1-automorphic plurisubharmonic function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' We are going to prove that ϕ is A-automorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 11 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Consider the diagonal U(1)n-action on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The plurisubharmonic func- tion ϕ is U(1)n-invariant because ⃗r and S2n−1 are U(1)n-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since A ∈ U(1)n · A1, and ϕ is A1-automorphic, it is A-automorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Therefore, ddcϕ ϕ defines a Vaisman metric on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By definition, the Lee flow maps the level sets of ϕ to the level sets of ϕ, with ϕ(etθ♯z) = tϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By construction of ϕ, the same is true for e−t log A1, that is, ϕ(e−t log A1z) = tϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' There- fore, θ♯ = − log A1, where log A1 is the diagonal matrix with the eigenvalues log |αi| in the same basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This is the Lee flow for the Vaisman metric ddcϕ ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The vector field log A1 belongs to Lie(G), because any power of A1 fixes all tensor invariants of G by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3, hence the corresponding 1-parametric sugroup also belongs to Lie(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' this gives log A1 ∈ Lie(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 6 An application: the Kodaira dimension of Vais- man manifolds Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1: Let M be a compact complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Its pluricanonical bundle is the tensor power Kn M = K⊗n M , n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The Kodaira dimension κ(M) is defined as κ(M) := lim supn log(dim H0(Kn M)) log n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2: The Kodaira dimension is equal to −∞ if H0(Kn M) = 0 for almost all n > 0, and to 0 if dim H0(Kn M) is bounded, but non-zero for infinitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' If the function n �→ dim H0(Kn M) grows as a polynomial of degree d, the Kodaira dimension of M is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let M be a Vaisman manifold, G the closure of the group generated by the Lee and the anti-Lee action, ˜ M a K¨ahler Z-covering of M, and ˜G the lift of G to ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Denote by G the Zariski closure of ˜G in Aut( ˜ M), considered as an algebraic variety ([OV8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since ˜G acts on ˜ M by homotheties, and contains contractions, there is an open set U ⊂ G in the connected component of G such that for all γ′ ∈ U, the quotient M′ := ˜ M ⟨γ′⟩ is LCK ([OV8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' The automorphism γ commutes with the action of the group generated by the Lee and the anti-Lee field, hence by [KO] M′ is Vaisman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1 By [BT, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='5], the connected component of G is isomorphic to (C∗)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' It is not hard to see that the union of all one-dimensional closed subgroups is dense in any algebraic group, hence U contains a dense subset of γ′ which are contained in a subgroup 1As shown in [KO], an LCK manifold which admits a holomorphic conformal C-action is Vaisman, assuming that this action cannot be lifted to an isometry of the K¨ahler cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 12 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential C∗ ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since C∗/⟨γ′⟩ is an elliptic curve, the Vaisman manifold M′ = ˜ M ⟨γ′⟩ is elliptically fibered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2 As shown in [OV1], the leaf space X is a projective orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3: Let M be a Vaisman manifold, M′ its quasi-regular de- formation obtained above, and X the corresponding leaf space, which is a projective orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then κ(M) = κ(M′) = κ(X), where κ denotes the Kodaira dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 1: The equality κ(M′) = κ(X) is clear from the ad- junction formula: since π : M′ −→ X is an elliptic fibration, one has H0(Kk M′) = H0(Kk X) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 2: By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1, any section h of the pluricanonical bundle Kk M on M is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then, for any γ′ ∈ U, the lift of h to ˜ M is γ′-invariant, hence can be obtained as a pullback of a section of Kk M′ under the quotient map ˜ M −→ M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This defines an injective map Ψ : H0(Kk M) −→ H0(Kk M′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Step 3: To finish the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3 it remains to show that Ψ : H0(Kk M) −→ H0(Kk M′) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This would follow if we show that any section h′ ∈ H0(Kk M′) is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since H0(Kk M′) = H0(Kk X) (Step 1), the G-invariance of h′ would follow if we prove that the corresponding section of Kk X is invariant under the natural action of G on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Recall that the pluricanonical representation is the natural action of Aut(X) on H0(Kk X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Since X is an orbifold, its singularities are klt ([P, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' By [KK, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='61], the image of the pluricanonical representation is finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' since G is connected, it acts on H0(Kk X) trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' This gives H0(Kk M) = H0(Kk M′) = H0(Kk X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' As an application, we outline the proof of the deformational stability of Kodaira dimension for Vaisman manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='4: Let Mt, t ∈ R, be a smooth family of compact Vaisman manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then the Kodaira dimension of Mt is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Sketch of a proof: We sketch a proof of this conjecture, reducing it to Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 2Ellipticaly fibered Vaisman manifolds are called quasi-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' – 13 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Verbitsky Holomorphic tensors on LCK manifolds with potential Let ˜ Mt be the family of open algebraic cones obtained from each Mt by taking a K¨ahler Z-cover;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' this is possible to do by [OV1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Each ˜ Mt is equipped with a natural algebraic structure by [OV8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' What follows is a conjecture, which most likely can be proven by the methods used in [OV8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Let ˜ Mt, t ∈ R, be a smooth family of open algebraic cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' As shown in [OV8], for an appropriate choice of the C∗-action on ˜ Mt the quotient ˜ Mt/C∗ is isomorphic to a complex projective orbifold Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' this isomorphism is extended to an isomorphism between ˜ Mt and Tot◦(Lt), where Lt is an ample line bundle over Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='5: Let ˜ Mt, t ∈ R, be a smooth family of open algebraic cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Then for any sufficiently small open subset U ⊂ R, the choice of C∗-action as above can be performed in such a way that the family (Xt, Lt) depends smoothly on t ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Now, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='3, the Kodaira dimension of Mt is equal to the Ko- daira dimension of Xt, which is constant, as shown by Siu in [Si].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Acknowledgements: We are grateful to Nicolina Istrati for her interest and valuable advice.' metadata={'source': 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Vuletescu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Ornea Classification of non-K¨ahler surfaces and locally conformally K¨ahler geometry, Russian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 76 (2021), 261-290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' arxiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='05768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' (Cited on page 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=') Liviu Ornea University of Bucharest, Faculty of Mathematics and Informatics, 14 Academiei str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=', 70109 Bucharest, Romania, and: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 010702-Bucharest, Romania lornea@fmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='unibuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='ro, liviu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='ornea@imar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='ro Misha Verbitsky Instituto Nacional de Matem´atica Pura e Aplicada (IMPA) Estrada Dona Castorina, 110 Jardim Botˆanico, CEP 22460-320 Rio de Janeiro, RJ - Brasil also: Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' Moscow, Russia verbit@verbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='ru, verbit@impa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='br – 16 – version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content='0, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} +page_content=' 3, 2022' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQfJPus/content/2301.01077v1.pdf'} diff --git a/T9FKT4oBgHgl3EQflC6k/content/tmp_files/2301.11852v1.pdf.txt b/T9FKT4oBgHgl3EQflC6k/content/tmp_files/2301.11852v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a410eea9e0f7d41867e8f1cf991e2c3717ee8504 --- /dev/null +++ b/T9FKT4oBgHgl3EQflC6k/content/tmp_files/2301.11852v1.pdf.txt @@ -0,0 +1,2791 @@ +A Sequential Global Programming Approach for Two-scale +Optimization of Homogenized Multiphysics Problems with +Application to Biot Porous Media +Bich Ngoc Vu1*, Vladimir Lukeˇs2, Michael Stingl1 and Eduard Rohan2* +1Competence Unit for Scientific Computing, Friedrich-Alexander-Universit¨at +Erlangen-N¨urnberg, Martenstrasse 5a, Erlangen, 91058, Germany. +2Department of Mechanics & NTIS New Technologies for Information Society, University +of West Bohemia in Pilsen, Univerzitn´ı 22, Plzeˇn, 30614, Czech Republic. +*Corresponding author(s). E-mail(s): bich.ngoc.vu@fau.de; rohan@kme.zcu.cz; +Contributing authors: vlukes@kme.zcu.cz; michael.stingl@fau.de; +Abstract +We present a new approach and an algorithm for optimizing the material configuration and behaviour +of a fluid saturated porous medium in a two-scale setting. The state problem is governed by the Biot +model describing the fluid-structure interaction in homogenized poroelastic structures. However, the +approach is widely applicable to multiphysics problems involving several macroscopic fields where +homogenization provides the relationship between the microconfigurations and the macroscopic math- +ematical model. The optimization variables describe the local microstructure design by virtue of the +pore shape which determines the effective medium properties – the material coefficients – computed +by the homogenization method. The main idea of the numerical optimization strategy consists in a) +employing a precomputed database of the material coefficients associated to the geometric parameters +and b) applying the sequential global programming (SGP) method for solving the problem of macro- +scopically optimized distribution of material coefficients. Although there are similarities with the free +material optimization (FMO) approach, only effective material coefficients are considered admissible, +for which a well-defined set of corresponding configurable microstructures exist. Due to the flexibility +of the SGP approach, different types of microstructures with fully independent parametrizations can +easily be handled. The efficiency of the concept is demonstrated by a series of numerical experiments. +We show that the SGP method can handle simultaneously multiple types of microstructures with +nontrivial parametrizations using a considerably low and stable number of state problems to be solved. +Keywords: multi-material optimization; sequential global programming; homogenization; Biot model; +poroelasticity; sensitivity analysis +1 Introduction +The design of fluid-saturated poroelastic media +(FSPM) present a gradually increasing topic of +research interest due to its mathematical complex- +ity and a great application potential. Although +the theory of FSPM has been developed in the +context of geomechanics and civil engineering, +nowadays theses types of materials are abundant +1 +arXiv:2301.11852v1 [cs.CE] 27 Jan 2023 + +2 +Sequential Global Programming Applied to Fluid-saturated Porous Media +in many engineering applications. A convenient +design of microstructures can provide a metama- +terial property related to controllable fluid trans- +port, or elasticity. In particular, soft robots can +be designed as inflatable porous structures gen- +erating a motion and force due to variable fluid +content, e.g., [1]. To this aim, the behaviour of the +fluid-saturated porous materials is described by +the Biot model [2], within the small strain theory, +which was postulated using a phenomenological +approach. The homogenization method enabled +the derivation of the quasistatic Biot’s equations +[3]. Since then, a number of works extended the +results for the dynamic case, which is important +for treating wave propagations, see e.g., [4]. As +an extension beyond the linear theory, a modified +Biot model with strain-dependent poroelastic and +permeability coefficients was proposed in [5]. +Topology optimization of microstructures con- +stituting the FSPM was treated in [6] and [1]. +Therein, the fluid-structure interaction problem +was handled in the homogenization framework and +an approximation towards computational simpli- +fication was proposed. +In this paper, we aim at a two-scale approach +optimization allowing for a spatial grading of +the microstructure design. Two-scale optimization +problems have been already extensively discussed +in literature before. The whole idea started with +the seminal paper of Bendsøe and Kikuchi [7], +in which the following concept was suggested: for +a given parametrization of the unit cell, carry +out the homogenization procedure on a fixed +parameter grid in a preprocessing step. Then, +in every step of the optimization, first retrieve, +for each design element, (approximate) effec- +tive material coefficients by interpolation. Next, +plug these coefficients into the state equation, +solve the latter and evaluate the cost. The other +way round, sensitivities are computed by the +chain rule, i.e. first differentiate the quantity of +interest with respect to the material coefficients +and then differentiate the material coefficients +with respect to to the design parametrization. +This procedure opens the way for the application +of any suitable gradient based optimization solver, +like, e.g., OCM [8], MMA [9] or SnOpt [10], to +name only those, which are most prominently used +in structural topology and material optimization. +While this concept essentially carries over to +other classes of problems, as it is done by [11–13] +for thermomechanical settings, we opted to fol- +low a slightly different avenue in this paper. There +are several reasons: First, the concept depends, +by its nature, to a large extent on the cho- +sen parametrization. If the parameters enter the +homogenized properties in a substantially non- +convex way (as it is the case, if, e.g., rotations +of the base cells are allowed), many local min- +ima might be introduced and additional measures +must be taken to avoid getting trapped in one of +them. Second, it is not easy to extend the original +concept with respect to the use of completely inde- +pendent types of unit cells, either characterized +by different geometries or material configurations. +In this case, specifying a smooth parametriza- +tion is non-trivial. The typical idea would be +to first introduce an independent parametriza- +tion for either cell types (for example using sizing +variables) and then add on top a smooth inter- +polation scheme for the effective tensors as used, +for instance, in multi-material optimization (see +[14]). The problems with that is however, that the +second level of interpolation introduces material +coefficients, for which typically no interpreta- +tion in terms of a microstructure exists. Thus, +an additional penalization strategy is required, +which ensures that those unphysical choices do not +remain in the optimal solution. Such an approach +was successfully demonstrated in the recent work +[15]. In another recent article, [16] chose two unit +cell types, described via level-set functions, such +that the mixture of their geometric parameters +can be directly interpreted as a third unit cell +type. [17] also opted for level-set functions to +describe the geometry of the microstructures. But, +with respect to the handling of multiple material +classes, the authors defined floating patches, where +each patch is a subdomain of the design domain +and only occupied by one microstructure type. +Then, the layout of these patches are optimized +on the macroscopic level and their overlaps are +combined via a differentiable maximum operator. +In our paper, we describe, how these disadvan- +tages can be circumvented using the SGP concept. +The basic idea has been already introduced in [18] +and is now generalized to a multiphysics, two-scale +setting. This involves an extension of an MMA- +type block-separable model function (see [19]) to + +Sequential Global Programming Applied to Fluid-saturated Porous Media +3 +the poroelastic setting, a split of the computa- +tions into an offline and an online phase, which +is particularly suited for homogenization based +problems, and a numerical solution scheme for the +nearly global optimization of block-separable sub- +problems. We would like to note here that the term +block-separable implies that the minimization can +be carried out separately for each design element, +however a design element itself can be described +by multiple design degrees of freedom. For a fur- +ther motivation of the SGP method, we refer to +the first paragraph in section 3. Here, we just like +to add that, in the whole optimization process, +two different types of sensitivities are relevant. +First, there are the sensitivities of constraint or +cost functions with respect to the effective mate- +rial coefficients. These constitute a substantial +ingredient of the block-separable model used in +the heart of the SGP method. Second, there +are the sensitivities of the material coefficients +with respect to the chosen parametrization. In +the context of the suggested two-scale SGP frame- +work, the latter ones are not strictly required, but +can help to come up with an improved interpola- +tion model used in the offline phase. In any case, +the derivation of sensitivities presented in this +paper, for the particular context of fluid saturated +porous media, relies on derivations in [20], where +also the sensitivity of the homogenized coefficients +were reported, see also [5]. +Finally, we would like to comment on the gen- +erality of the presented approach. Although the +SGP concept outlined in our paper can be applied +to a large range of multiphysics two-scale mate- +rial optimization problems, the Biot model of fluid +saturated porous media provides an ideal test bed +for the method. This is for several reasons: first, +the physical coupling is non-trivial. Second, it is +very natural to set up competing objective func- +tions, such as the structural compliance on the +one hand and the enhanced fluid flow through an +outflow boundary, on the other hand. And third, +configurable types of microstructures supporting +either the first or the second goal can be deduced +in a straightforward manner. +The structure of the remainder of this paper +is as follows: In section 2 all ingredients of the +two-scale problem are described. To these belong +a brief repetition of the constitutive laws for the +Biot model (section 2.1), the poroelastic state +problem in variational form (section 2.2), a generic +sketch of the two-scale problem constrained by +the poroelasticity equations (section 2.3) and +an adjoint analysis providing sensitivities with +respect to effective material coefficients, as used +later by the SGP method (section 2.4). Finally, +two types microstructures are suggested in form +of configurable unit cells (section 2.5). In section 3 +the SGP concept for the solution of two-scale opti- +mization problems is introduced in greater detail. +For this, the two-scale problem is discretized and +extended for the use of multiple types of unit cells +(section 3.1). Then, a separable sequential approx- +imation concept is suggested (section 3.2) and last +the SGP method is presented in an algorithmic +form (section 3.3). In section 4, the advantages of +the SGP algorithm will be discussed using various +types of two-scale problems. +2 Formulation of the two-scale +optimization problem +In this section, we explain our optimization strat- +egy. Although it can be applied to similar prob- +lems involving several physical fields or multi- +physics problems, in this paper, we consider the +fluid saturated porous media represented by the +Biot model which can be derived using the homog- +enization of the fluid-structure interaction prob- +lem restricted to small deformation kinematics, +see e.g., [3, 21, 22]. In the next section we report +the homogenization result presented +Notation +We employ the following notation. Since we +deal +with +a +two-scale +problem, +we +distin- +guish +the +“macroscopic” +and +“microscopic” +coordinates, +x +and +y, +respectively. +We +use +∇x = (∂x +i ) and ∇y = (∂y +i ) when differentiation +with respect to coordinate +x +and +y +is +used, +respectively, whereby ∇ +≡ +∇x. By e(u) += +1/2[(∇u)T + ∇u], we denote the strain of a vec- +torial function u, where the transpose operator +is indicated by the superscript T . The Lebesgue +spaces of 2nd-power integrable functions on an +open bounded domain D ⊂ R3 is denoted by +L2(D), the Sobolev space W 1,2(D) of the square +integrable vector-valued functions on D including +the first order generalized derivative, is abbrevi- +ated by H1(D). Further, H1 +#(Ym) is the Sobolev + +4 +Sequential Global Programming Applied to Fluid-saturated Porous Media +space of vector-valued Y-periodic functions (the +subscript #). +2.1 The homogenized Biot – Darcy +model +We report the homogenization result presented +e.g., in [5], cf. [20], where the problem of locally +optimized microstructures has been described. +The homogenized model of the porous elastic +medium incorporates local problems for character- +istic responses which are employed to compute the +effective material coefficients of the Biot model. +The local problems specified below, related +to +the +homogenized +model, +are +defined +at +the +microscopic +representative +unit +cell +Y = Π3 +i=1]0, ℓi[⊂ R3. +which +splits +into +the +solid +part +occupying +domain +Ym +and +the +complementary channel part Yc. Thus, +Y = Ym ∪ Yc ∪ ΓY , +Yc = Y \ Ym , +ΓY = Ym ∩ Yc , +(1) +where by Yd for d = m, c, we denote the closure of +the open bounded domain Yd. By ∼� +Yd = |Y |−1 � +Yd, +with Yd ⊂ Y for d = m, c, we denote the local aver- +age (|Y | is the volume of domain Y ). Obviously, +the unit volume |Y | = 1 can always be chosen. We +employ the usual elasticity bilinear form, involving +two vector fields w and v, that reads +am +Y (w, v) =∼ +� +Ym +(IDey(w)) : ey(v) , +(2) +where ID = (Dijkl) is the elasticity tensor satisfy- +ing the usual symmetries, Dijkl = Dklij = Djikl, +and ey(v) = 1 +2(∇yv + (∇yv)T ) is the linear strain +tensor associated with the displacement field v. +In what follows, by the microstructure Y(x), +we mean the decomposition eq. (1) of the repre- +sentative cell Y and the material properties, as +represented by the elasticity ID only in our case. If +the structure is perfectly periodic, microstructures +Y ≡ Y(x) are independent of the macroscopic +position x ∈ Ω. Otherwise, the local problems +must be considered at any macroscopic posi- +tion, i.e. for almost any x ∈ Ω, see e.g., [21] +in the context of slowly varying “quasi-periodic” +microstructures. It should be pointed out, that +this issue is of a special importance when deal- +ing with homogenization-based material design +optimization; as will be explained below, a regular- +ization is required to control the design variation +within Ω. +The local microstructural response is obtained +by solving the following decoupled problems: +• Find ωij +∈ H1 +#(Ym) for any i, j += 1, 2, 3 +satisfying +am +Y +� +ωij + Πij, v +� += 0 , ∀v ∈ H1 +#(Ym) , +(3) +where Πij = (Πij +k ), i, j, k = 1, 2, 3 with compo- +nents Πij +k = yjδik. +• Find ωP ∈ H1 +#(Ym) satisfying +am +Y +� +ωP , v +� +=∼ +� +ΓY +v · n[m] dSy, ∀v ∈ H1 +#(Ym) . +(4) +• Find (ψi, πi) ∈ H1 +#(Yc) × L2(Yc) for i = 1, 2, 3 +such that +� +Yc +∇yψk : ∇yv − +� +Yc +πk∇ · v = +� +Yc +vk , +� +Yc +q∇y · ψk = 0 , +(5) +∀v ∈ H1 +#(Yc) and ∀q ∈ L2(Yc). +Effective material properties of the homoge- +nized deformable fluid-saturated porous medium +are described in terms of homogenized poroelas- +tic coefficients: the drained elasticity AA, the stress +coupling C and the compressibility N, all being +related to the solid skeleton. All these coefficients +including the intrinsinc hydraulic permeability K +are computed using the characteristic microscopic +responses eqs. (3) to (5) substituted in following +expressions: +Aijkl = am +Y +� +ωij + Πij, ωkl + Πkl� +, +Cij = − ∼ +� +Ym +divyωij = am +Y +� +ωP , Πij� +, +N = am +Y +� +ωP , ωP � +=∼ +� +ΓY +ωP · n dSy , +Kij =∼ +� +Yc +ψj +i =∼ +� +Yc +∇yψi : ∇yψi . +(6) + +Sequential Global Programming Applied to Fluid-saturated Porous Media +5 +Obviously, the tensors AA = (Aijkl), C = (Cij) +and K = (Kij) are symmetric, AA adheres all the +symmetries of ID; moreover AA is positive definite +and N > 0. The hydraulic permeability K is, in +general, positive semi-definite. It is positive def- +inite whenever the channels constitute a simply +connected domain generated as the periodic lattice +by Yc; for this, denoting by Γk +Y ⊂ ∂Y , k = 1, . . . , 6 +the faces of Y , it must hold that Γk +Y ∩ ∂Yc ̸= ∅ for +all k = 1, . . . , 6. +Coupled flow deformation problem +The Biot–Darcy model of poroelastic media for +quasi-static, evolutionary problems imposed in Ω +is constituted by the following equations involv- +ing stress σ, displacement u, strain e(u), fluid +pressure p and the seepage velocity w: +−∇ · σ = f s, +σ = AAe(u) − Bp, +−∇ · w = B : e( ˙u) + M ˙p, +w = −K +¯η +� +∇p − f f� +, +(7) +where the homogenized coefficients are given by +eq. (6) and +B := C + φI , +M := N + φγ . +(8) +Above, ¯η is the relative fluid viscosity, γ is the +fluid compressibility and φ = |Yc|/|Y | is the poros- +ity (volume fraction of the fluid-filled channels). +The effective volume forces in eq. (7), acting in +the solid and fluid phases, are denoted by f s +and f f, respectively. It is important to note that +¯η = ηphys/ε2 +0 is defined for a given fluid (ηphys) +and microstructures scale: ε0 = ℓ0/L where L is +a characteristic macroscopic length, and ℓ0 is the +characteristic microstructure size, typically given +by the “pore diameter”. Thus, for a given fluid, +the effective permeability K/¯η is proportional to +ε2 +0, i.e. reflecting the microstructure size. In con- +trast, all other coefficients are scale-independent +(when the scale separation holds, i.e. ε0 being +small enough). +Remark 1. In this paper, we only consider steady +state problems for the Biot medium, such that all +time derivatives in eq. (7) vanish. Consequently, +the Biot compressibility M is not involved, as +far as the porous phase, generated as a peri- +odic lattice by channels Yc, is connected. For any +microstructure with disconnected pores, such that +Yc ⊂ Y , thus, Yc constitute one, or more inclu- +sions with one cell Y , see [22], the permeability +vanishes. Then, the time integration in eq. (7) +leads to the mass conservation equation in the +form B : e(u) + Mp = 0, assuming an unde- +formed initial configuration with the zero pressure +in the inclusions. In the optimization problem, +besides microstructures with nondegenerate per- +meabilities, we shall consider also microstructures +with spherical, thus, disconnected pores, constitut- +ing impermeable material. For this case, one can +choose either fluid filled pores, or empty pores; the +only difference is the use of the so-called undrained +material elasticity, AAU = AA + M −1B ⊗ B, or the +elasticity AA describing effective elasticity of the +“drained” skeleton, with empty pores. +2.2 State problem formulation +Let Ω ⊂ R3 be an open bounded domain. Its +boundary ∂Ω splits, as follows: ∂Ω = ΓD ∪ ΓN +and also ∂Ω = Γp ∪ Γw, where ΓD ∩ ΓN = ∅ +and Γp ∩ Γw = ∅. Assume Γp consists of two dis- +connected, non-overlapping parts Γk +p, k = 1, 2, +Γp = Γ1 +p ∪ Γ2 +p, and Γ1 +p ∩ Γ2 +p = ∅. +We consider the steady state problems for the +linear Biot continuum occupying domain Ω. The +poroelastic material parameters and the hydraulic +permeability referred to as the homogenized coef- +ficients, in general, are given by the locally defined +microstructures Y(x) which can vary with x ∈ Ω. +The two-scale optimization approach proposed +in this paper enables to combine microstruc- +tures characterized by connected and disconnected +pores, the latter characterized by a vanishing per- +meability. To this aim, the domain Ω = Ω0 ∪ Ω+ +is decomposed into in two parts: the permeable +Ω+ and the impermeable Ω0, which may not con- +stitute connected domains, being split into more +disconnected subparts. Consequently, the inter- +face Γ+ = ∂Ω+ ∩ ∂Ω0 is impermeable. Regard- +ing the boundary decomposition, we assume that +Γk +p+ := Γk +p ∩ ∂Ω+ ̸= ∅, for k = 1, 2, so that +the porous structure permits the fluid transport +through domain Ω+, if this one connects Γ1 +p+ and +Γ2 +p+. +We consider the following macroscopic prob- +lem: Given the traction surface forces g, and + +6 +Sequential Global Programming Applied to Fluid-saturated Porous Media +pressures ¯pk on boundaries Γk +p, find displacements +u and the hydraulic pressure P which satisfy +−∇ · (AAe(u) − PB) = 0 +in Ω , +u = 0 +in ΓD , +(AAe(u) − PB) · n = g +in ΓN , +(9) +where P = 0 in Ω0. Whereas, in Ω+, P satisfies +−∇ · K∇P = 0 +in Ω+ , +P = ¯pk +on Γk +p+ , +k = 1, 2 , +n · K∇P = 0 +on Γw ∪ Γ+ . +(10) +For the steady state problem the set of +equations eq. (7) yields the two problems eq. (9) +and eq. (10) as a decoupled system: first, eq. (10) +can be solved for P, then eq. (9) is solved for u. +Moreover, for the considered type of the bound- +ary conditions and since volume forces are not +involved, the solutions are independent of the +viscosity ¯η, see eq. (7). +Further, we consider an extension of ¯pk from +boundary Γk +p to the whole domain Ω, such that +¯pk = 0 on Γl +p (in the sense of traces) for l ̸= k. +Then P = p + � +k ¯pk in Ω+, such that p = 0 +on Γp+. Note that p can be simply extended by +0 in Ω0. For the sake of notational simplicity, we +introduce ¯p = � +k ¯pk. By virtue of the Dirichlet +boundary conditions for u and p, we introduce the +following spaces: +V0 = {v ∈ H1(Ω) | v = 0 on ΓD} , +Q0 = {q ∈ L2(Ω) ∩ H1(Ω+) | q = 0 on Γp+} . +(11) +We employ the bilinear forms and the linear +functional g, +aΩ (u, v) = +� +Ω +(AAe(u)) : e(v) , +bΩ+ (p, v) = +� +Ω+ +pB : e(v) , +cΩ+ (p, q) = +� +Ω+ +∇q · K∇p , +g(v) = +� +ΓN +g · v . +(12) +In order to define the state problem in the +context of two-scale optimization, we employ the +weak formulation which reads, as follows: Find +u ∈ V0 and p ∈ Q0, such that, for all v ∈ V0 and +q ∈ Q0, +aΩ (u, v) − bΩ+ (p, v) = g(v) + bΩ+ (¯p, v) , +cΩ+ (p, q) = −cΩ+ (¯p, q) . +(13) +To define p uniquely in Ω, p ≡ 0 in Ω0 = Ω \ Ω+. +Since the two fields are decoupled, first p is solved +from eq. (13)2, then u is solved from eq. (13)1, +where p is already known. +Remark 2. In the context of the undrained poros- +ity defined by fluid-filled closed pores Yc ⊂ Y , +see Remark 1, formulation eq. (13) is consistent +also with this microstructure class type Y□ +0 +with +AAU replacing AA in the elasticity bilinear form +eq. (12)1. Pressure is then defined pointwise in Ω0 +by P := −B : e(u)/M. +By α(x) we denote an abstract optimization +variable which determines the homogenized coeffi- +cients for any position x ∈ Ω. Below we consider α +representing several geometrical parameters char- +acterizing microstructures Y(x) of a given type. +Although, in this section, we disregard some par- +ticular details related to the treatment of multiple +types of Y, we bear in mind the existence of +two microstructure classes, Y□ ++ and Y□ +0 , associ- +ated with the pore connectivity type, as discussed +above. The “permeable” domain Ω+ is occupied +by the material given pointwise by Y(x) ∈ Y□ ++ for +all x ∈ Ω+. Hence, both the subdomains of Ω are +defined implicitly by the microstructure type: Ωi +is the set of x ∈ Ω, such that Y(x) ∈ Y□ +i , where +i = +, 0. +In the next section, we shall consider a two- +scale optimization problem which is characterized +by the following features: +• Geometrical restrictions are stated in respective +definitions of the admissibility designs sets for +a chosen type of microstructure. For the sake of +brevity, let A be the set of admissible designs, +further we consider α(x) ∈ A for any x ∈ Ω. +• We +consider +multiple +optimization +criteria +which perform as the objective functions, or +equality constraints. Without loss of generality, +we confine ourselves to the two criteria Φα(u) + +Sequential Global Programming Applied to Fluid-saturated Porous Media +7 +and Ψα(p) that are defined, as follows: +Φα(u) = g(u) , +Ψα(p) = − +� +Γ2p +K∇(p + ¯p) · n . +(14) +While Φα(u) expresses the structural compli- +ance, criterion function Ψα(p) expresses the +amount of the fluid flow through surface Γ2 +p +due to the pressure difference ¯p1 − ¯p2, see the +boundary condition eq. (10)2. These two criteria +are antagonist: the pore volume reduction leads +naturally to stiffening the structure, but reduces +the permeability. Hence, for the objective func- +tion Φα, function Ψα serves as a constraint and +vice versa. +2.3 Two-scale optimization problem +Here, for the ease of notation, we restrict to one +microstructure type only, namely Y(x) ∈ Y□ ++, so +that we may consider Ω ≡ Ω+. Hence, all the bilin- +ear forms in eq. (12) are defined by integration in +Ω. Later, in section 3, we will consider microstruc- +tures characterized by different unit cell types of +classes Y□ ++ and Y□ +0 , however, the formulations +introduced below can be adapted easily. +We first define the direct optimization problem +to find design α(Ω) that minimizes a cost func- +tional based on the criteria defined in eq. (14). +Further, we introduce the set T = S6 × S3 × S3 × +R × R and denote by IH = (AA, B, K, ρm, R) ∈ T +the (local) material parameters involing the effec- +tive (homogenized) material coefficients, the solid +part volume ρm = 1−φ = |Ym|/|Y |, and a regular- +ization parameter R, which typically depends only +on the design. We note that the dimension of the +regularization label R is, for ease of notation, cho- +sen as 1 for now, although later in section 4.3 more +general regularization labels are used. Obviously, +IH is given uniquely by the local admissible design +α(x) ∈ A, x ∈ Ω, whereby for a suitably cho- +sen parametrization, the admissibility set is given +simply by +A = [a, a] ⊂ Rn. +Examples for such parametrizations along with a +description of the lower and upper bounds a, a ∈ +Rn are presented in section 2.5. +For a given admissible design α(Ω), the state +z = (u, p) is the solution of eq. (13), where +the homogenized coefficients IH(α) are given in +eq. (6) using the characteristic responses W (α) := +(ωij, ωP , ψk, πk). W (α) are the solutions of +eqs. (3) to (5), which depend on α(x) in terms of +the microconfigurations Y(x). In this way, map- +ping S : α(Ω) �→ z(Ω) introduces the admissible +state. +It can be defined by a composition map, S = +Z ◦ E ◦ W, where W represents the resolvents of +the characteristic problems imposed on the local +microconfigurations, E provides the homogenized +material, and Z is the resolvent of the macroscopic +state problem, so that +W : α �→ W , +E : (α, W) �→ IH , +Z : IH(Ω) �→ z(Ω) . +(15) +Further, we employ the mapping +H : α �→ IH, +such that H = E ◦ W is the composition map +defined for any admissible design α(x) ∈ A, for +a.a. x ∈ Ω. +The macroscopic state problem is the implicit +form of the mapping Z : IH �→ z, such that +z ∈ S0 = V0 × Q0 satisfies +ϕIH(z, v) = 0 +∀v ∈ S0 , +(16) +where S0 is the space of admissible state problem +solutions. For the Biot medium problem, eq. (16) +is identified with eq. (13). +2.3.1 Direct two-scale optimization +problem +For the given two functions of interest Φ and +Ψ, both depending on the material distribution +IH(x) and the state z(x), the two-scale abstract +optimization problem reads: +min +α∈A Φ(IH, z) + ΛΞΞ(IH) +s.t. Ψ(IH, z) = Ψ0 , +z = S(α), +IH = H(α), +� +Ω +ρm ≤ ¯ρm|Ω| , +(17) + +8 +Sequential Global Programming Applied to Fluid-saturated Porous Media +where the term Ξ(IH) in the objective is related +to the design regularization, namely to param- +eter R, and ΛΞ +∈ R+ is a penalty parame- +ter. Recall the chain mapping H : α(x) �→ IH(x) +for any x +∈ +Ω, then z += +Z(Ω). Below, +we +abbreviate +Φα(z) =: Φ(H(α), z) +and +also +Ψα(z) =: Ψ(H(α), z). In eq. (14), specific exam- +ples relevant for the Biot medium optimization +were given. +Optimization problem eq. (17) is associated +with the following inf-sup problem, +min +α∈A inf +z∈S0 +sup +Λ∈R2,˜z∈S0 +L(α, z, Λ, ˜z) , +(18) +with the Lagrangian function, +L(α, z, Λ, ˜z) = ΛΦΦα(z) ++ ΛΞΞ(H(α)) ++ ΛΨ(Ψα(z) − Ψ0) ++ ϕIH(α)(z, ˜z) , +(19) +where Λ = (ΛΦ, ΛΨ) ∈ R2 are the Lagrange multi- +pliers associated with the objective and constraint +functionals Φ and Ψ, and ˜z ∈ S0 are Lagrange +multipliers – the adjoint variables — associated +with the constraints of the problem eq. (17). +For a while, we may consider material coeffi- +cients IH as the optimization variables (although +they are parameterized by α ∈ A). Further, let +us assume a given value Λ ∈ R2; note that the +entries of Λ can be positive or negative depending +on the desired flow augmentation, or reduction. In +the numerical examples, we chose ΛΦ > 0, whereas +ΛΨ < 0 indicates the constraint effect of Ψ rela- +tive to Φ. Upon denoting by Im(H) = H(A), the +image space of all admissible designs, and defining +Uad = {IH ∈ L∞(Ω; T ) | IH(x) ∈ Im(H) +for a.a. x ∈ Ω} , +the +optimization +problem +eq. +(17) +can +be +rephrased as the two-criteria minimization prob- +lem, +min +IH ∈ Uad +F(IH, z) , +s.t. z = Z(IH) +� +Ω +ρm ≤ ¯ρm|Ω| , +(20) +where +F(IH, z) = ΛΦΦ(IH, z) + ΛΨΨ(IH, z) + ΛΞΞ(IH) . +For the Biot medium optimization, where the +two criterion functions Φα and Ψα are given in +eq. (14), the Lagrangian function attains the form +L(α, (u, p), Λ, (˜v, ˜q)) += ΛΦΦα(u) + ΛΨ(Ψα(p) − Ψ0) + ΛΞΞα(IH) ++ aΩ (u, ˜v) − bΩ (p + ¯p, ˜v) +− g(˜v) + cΩ (p + ¯p, ˜q) . +(21) +2.4 Adjoint responses and the +sensitivity analysis +In this section, we provide details concerning +the sensitivity analysis employed in the preceding +section. We consider α to represent a general opti- +mization variable which is related to the effective +medium parameters IH. It is worth to note that +one may also consider α ≡ IH in the context of the +free material optimization (FMO). +To obtain the adjoint equation, we consider the +optimality condition for (u, p). Thus, from eq. (21) +it follows that +δ(u,p)L(α, (u, p), Λ, (˜v, ˜q)) ◦ (v, q) += ΛΦδuΦα(u; v) + ΛΨδpΨα(p; q) ++ aΩ (v, ˜v) − bΩ (q, ˜v) + cΩ (q, ˜q) , +(22) +where +δuΦα(u; v) = g(v), +δpΨα(p; q) = − +� +Γ2p +K∇q · n. +(23) +To avoid computation of the gradient ∇q on +Γ2 +p ⊂ ∂Ω, we consider ˜p ∈ H1(Ω) such that ˜p = 0 +on Γ \ Γ2 +p, while ˜p = 1 on Γ2 +p, then it is easy to see +that +−Ψα(p) = r(p) := cΩ (p + ¯p, ˜p) , +−δpΨα(p; q) = δpr(p; q) = cΩ (q, ˜p) . +(24) +The optimality conditions eq. (22), related to +the state admissibility, yield the adjoint state + +Sequential Global Programming Applied to Fluid-saturated Porous Media +9 +(˜v, ˜q) ∈ V0 × Q0 which satisfies the following iden- +tities: +∀v ∈ V0 : +aΩ (v, ˜v) = −ΛΦδuΦα(u; v) , +∀q ∈ Q0 : +cΩ (q, ˜q) = bΩ (q, ˜v) − ΛΨδpΨα(p; q). +(25) +These equations can be rewritten using eq. (23) +and eq. (24), as follows for all (˜v, ˜q) ∈ V0 × Q0: +∀v ∈ V0 : +aΩ (v, ˜v) = −ΛΦg(v) +, +∀q ∈ Q0 : +cΩ (q, ˜q) = bΩ (q, ˜v) + ΛΨcΩ (q, ˜p) . +(26) +To allow for the independence of the state adjoint +on Λ, we define the split +˜v = ΛΦ˜ϑ , +˜q = ΛΦ˜q1 + ΛΨ˜q2 , +(27) +where +˜ϑ +and +˜qk, +k += +1, 2 +satisfy +for +all +(˜v, ˜q, ˜q) ∈ V0 × Q0 +∀v ∈ V0 : +aΩ +� +v, ˜ϑ +� += −g(v), +∀q ∈ Q0 : +cΩ (q, ˜q1) = bΩ +� +q, ˜ϑ +� +, +∀q ∈ Q0 : +cΩ (q, ˜q2) = cΩ (q, ˜p) . +(28) +We can compute the total variation of the +Lagrangian with +δtot +α L = ΛΦδug(u; δαu) − ΛΨδpr(p; δαp) ++ ΛΦδαg(u) − ΛΨδαr(p) + ΛΞδαΞα(IH) ++ aΩ (δαu, ˜v) − bΩ (δαp, ˜v) + cΩ (δαp, ˜q) ++ δαaΩ (u, ˜v) − δαbΩ (p + ¯p, ˜v) ++ δαcΩ (p + ¯p, ˜q) . +(29) +If the pair (u, p) solves the state problem and +(˜v, ˜q) is its adjoint state, eq. (29) is equivalent to +the following expression: +δtot +α L = ΛΦδαg(u) − ΛΨδαr(p) + ΛΞδαΞα(IH) ++ δαaΩ (u, ˜v) − δαbΩ (p + ¯p, ˜v) ++ δαcΩ (p + ¯p, ˜q) . +(30) +Above, the shape derivatives δα of the bilinear +forms can be rewritten in terms of the sensitivity +of the homogenized coefficients. Besides the obvi- +ously vanishing derivative δαg(u) = 0, it holds +that +δαaΩ (u, ˜v) ◦ δαAA = +� +Ω +δαAAe(u) : e(˜v) , +δαbΩ (p + ¯p, ˜v) ◦ δαB = +� +Ω +(p + ¯p)δαB : e(˜v) , +δαcΩ (p + ¯p, ˜q) ◦ δαK = +� +Ω +∇˜q · δαK∇(p + ¯p) , +δαr(p) = δαcΩ (p + ¯p, ˜p) ◦ δαK += +� +Ω +∇˜p · δαK∇(p + ¯p) . +(31) +Using the “total pressure” P := p+¯p, the following +tensors are employed to evaluate the expression in +eq. (31): +e(u) ⊗ e(˜ϑ) , +Pe(˜ϑ) , +∇P ⊗ ∇˜q1 , +∇P ⊗ ∇˜q2 , +∇˜p ⊗ ∇P . +(32) +Now, using these tensors, eq. (29) is computed, as +follows: +δtot +α L = −ΛΨδαr(p) ++ ΛΦ +� +δαaΩ +� +u, ˜ϑ +� +− δαbΩ +� +P, ˜ϑ +� ++ δαcΩ (P, ˜q1) +� ++ ΛΨδαcΩ (P, ˜q2) + ΛΞ∂IHΞ(IH)δαIH . +(33) +Hence the variations of L with respect to AA, B +and K are given by the following formulae +δtot +A +A L = ΛΦ +� +Ω +δAAe : e(u) ⊗ e(˜ϑ) , +δtot +B L = −ΛΦ +� +Ω +δBe : Pe(˜ϑ) , +δtot +K L = +� +Ω +δKe : (ΛΦ∇P ⊗ ∇˜q1 ++ΛΨ (∇P ⊗ ∇˜q2 − ∇˜p ⊗ ∇P)) +(34) +As Ξ(IH) solely depends on the regularization +parameter R, see eq. (47), we get +∂IHΞ(IH)δαIH = +� +Ω +(R−F(R)·(δR−∂RF(R)◦δR) + +10 +Sequential Global Programming Applied to Fluid-saturated Porous Media +for the regularization term in eq. (33). In the +context of the finite element discretization intro- +duced in section 3, the homogenized coefficients +are supplied as constants in each element Ωe of the +partitioned domain Ω. Accordingly, the expres- +sions in eq. (32) are supplied elementwise at the +Gauss integration points. +2.5 Design parametrization +The design of the cell Y , that is the decompo- +sition into the solid skeleton Ym and the pores +Yc, can be parameterized in a number of ways. In +[20], we employed a so-called spline-box structure +parameterized by design variables defining posi- +tions of the spline control polyhedron. This kind +of parametrization is convenient due to its gener- +ality to handle quite arbitrary design, but leads +to complicated formulations of design constraints +which are needed to preserve essential geometrical +requirements (e.g., positivity of channel crosssec- +tions). +In this paper, we employ two specific types +of microstructures illustrated in fig. 1, where the +channels are shaped as a 3D cross (type 1), or a +sphere (type 2). Hence, the latter microstructure +is featured by zero permeability and therefore, +we consider dry pores (voids) in the mechani- +cal model. Due to these specific geometries, we +can use a rather simple parametrization, which is +listed in table 1. For a unit cell of type 1, rx and +ry refer to the radii of the cylinders pointing in x- +and y-direction respectively. The third parameter +ϕ describes the cell rotation, about axis z. For the +unit cell type 2, the spherical voids, whose radii are +described by rs, provide an orthotropic material +with nearly isotropic elastic properties. Therefore, +rotations are not enabled for this cell type. Impor- +tantly, box constraints can be imposed on rx, ry +and rs straightforwardly to guarantee geometric +feasibility. +microstructure # +cell parameters +1 +rx +ry +ϕ +2 +rs +- +- +Table 1: The parametrization of the pore geom- +etry for the two types of the microstructures: 1: +the 3D cross, 2: the sphere. +Fig. 1: Parametrization of unit cells: unit cell type +1 is parameterized by radii rx and ry, both ranging +from 0.08 to 0.22, rz = 0.15 and rs = 0.25 are +kept constant; unit cell type 2 is parameterized by +radius rs ranging from 0.1 to 0.4. +To illustrate a sensitivity of the material prop- +erties determined by the homogenized coefficients +IH, In fig. 2, for unit cell type 2, the elasticity as +the only relevant material property is displayed as +function of rs. In fig. 3, for unit cell type 1, selected +components of the poroelastic tensors and of the +permeability are reported as functions of ry. +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +rs [-] +2.5 +3.0 +3.5 +4.0 +4.5 +coefficients A [GPa] +A1111 +Fig. 2: Unit cell type 2: dependence of A1111 on +parameter rs. + +Sequential Global Programming Applied to Fluid-saturated Porous Media +11 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +ry [-] +2.2 +2.4 +2.6 +2.8 +3.0 +coefficients A [GPa] +A1111 +A2222 +A3333 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +ry [-] +0.40 +0.45 +0.50 +0.55 +0.60 +coefficients B [-] +B11 +B22 +B33 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +ry [-] +0.0000 +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +coefficients K [m2 / (Pa · s)] +K11 +K22 +K33 +Fig. 3: Unit cell type 1: dependence of homoge- +nized coefficients AA, B, and K on ry; rx = 0.15 +is fixed. +3 A Sequential Global +Programming formulation +The basic description of the Sequential Global +Programming algorithm along with convergence +aspects were presented in [18], where SGP was +applied to a multi-material optimization based on +a two-dimensional time harmonic Helmholtz state +equation. The setting and procedure described in +this manuscript differs from the one in [18] in the +following major points: first, in [18] a selection +of finitely many fixed materials was considered +as admissible set. In this paper, each admissi- +ble material is computed by homogenizing unit +cell, which itself is configurable by a number +of geometric parameters. Thus, the designer can +choose in each point of the design domain from +M different unit cell types and adjust the geo- +metric parameters for the latter. Second, the SGP +approach is extended to a multi-physics setting +using a slightly different separable approxima- +tion and third, a different solution strategy is +employed for the subproblems arising from this. +This strategy does not impose any assumption on +the parametrization. In particular, parametriza- +tions can be non-analytical and non-differentiable. +This leads to a greater design flexibility. Despite +these differences, there is also an important fea- +ture, the approach presented here has in common +with the one outlined in [18]: separable models +are established in terms of (effective) material +tensors IH rather than their parameterization α. +Then, the parametrization is directly treated at +the level of sub-problems without further convex- +ification. Thanks to the separable character of the +chosen first order model the resulting generally +non-convex sub-problems can - in principal - still +be solved to global optimality. +The advantages of this approach are twofold: +first, due to the separable model functions being +able to capture also non-convex features of the +original cost function typically a low number of +outer iterations, equivalently to the number of +state problems to be solved, is required; and sec- +ond, due to the good fit of the separable models +with the cost function as well as the fact that +non-convex sub-problems are solved to global opti- +mality the overall algorithm is less start value +dependent and less prone to be trapped in poor +local minima. This is in contrast to traditional +approaches, where a local model is established +directly based on the sensitivity of cost functions +with respect to the design parameterization α. +In the following we first derive a fullly dis- +cretized counterpart for a slightly generalized of +problem eq. (20). Then we describe in detail how +the separable first order approximations can be +constructed and finally present a practical out- +line of the full SGP algorithm including a generic +sub-solver allowing to compute near globally opti- +mal solutions for sub-problems using a brute-force +strategy. +3.1 A fully discretized 2-scale +design problem +For the sake of simplicity, the definitions of sets +and functions were introduced in sections 2.2 + +12 +Sequential Global Programming Applied to Fluid-saturated Porous Media +and 2.3 based on the assumption that there is +only one type of unit cell such that M = 1. +Here, for a more general setting, we consider M +unit cell types, each one with ni design param- +eters, and introduce index set I := {1, . . . , M}. +For each unit cell type i ∈ I, the admissibility set +is defined in terms of box constraints and other +purely geometrical constraints. By choosing a suit- +able parameterization, we can identify these with +(geometric) parameter sets +Ai = [ai, ai] ⊂ Rni, +(35) +with ai, ai ∈ Rni being lower and upper bound +vectors constraining the corresponding parameter +vector αi ∈ Rni. +Remark 3. We note that, while in this manuscript +the parameters in eq. (35) are always used to vary +the geometrical properties of the unit cell, varia- +tions in the material parameters could be described +in the same way. Thus, SGP can handle both of +these situations. +We further define for all i ∈ I map +Hi : +� +Ai +→ T +αi +�→ (AA, B, K, ρm, R), +(36) +where Hi(α) performs the homogenization proce- +dure described in section 2.3. fig. 4 illustrates the +components of Hi(αi). +We denote the union of the ranges of all Hi by +H := +M +� +i=1 +Hi(Ai) +(37) +and with that generalize the set of admissible +design functions to become +Uad = {IH ∈ L∞(Ω; T ) | IH(x) ∈ H +for a.e. x ∈ Ω} . +Now the state problem operator +Z : +� +Uad +→ R3 × R +IH +�→ z = (u, p), +(38) +with displacement function u(IH) and hydraulic +pressure function p(IH) reads exactly as before. +We finally use a slightly more general resource +function than in sections 2.2 and 2.3 as follows: +ρ : +� +Uad +→ R +IH +�→ ρ. +(39) +A concretization could be the total volume frac- +tion of a specific material phase (see description +of ¯ρm in section 2.3). +Based on these definitions, we then formulate +an FMO-type problem +min +IH∈Uad +F(IH, z) :=ΛΦΦ(IH, z) + ΛΨΨ(IH, z) ++ ΛΞΞ(IH) +s.t. +z =Z(IH), +ρ(IH) ≤¯ρm, +(40) +where ¯ρm ∈ R is the resource constraint value +and cost functions and Φ, Ψ, Ξ and their weights +ΛΨ, ΛΦ, ΛΞ +have +been +already +introduced +in +section 2.3). +Although +problem +eq. +(40) +is +formulated +directly in the tensor variable IH, a realization of +the feasibility condition IH ∈ Uad would force us +to evaluate the homogenization maps Hi (i ∈ I). +This has the consequence that for each evaluation +of the cost function, a homogenization procedure, +which contains a series of cell problems, has to be +conducted. To alleviate this situation, we follow [7] +and carry out the homogenization procedure only +for discrete samples of the design parameter space. +For each unit cell type i, we introduce a grid with +nodes Anodes +i +⊆ Ai and effective material coeffi- +cients are only computed, via homogenization, at +the sampled nodes of this grid. In addition, we +define a piecewise cubic Hermite interpolator for +these samples to realize the continuous mapping +˜Hi : +� +Ai → T +αi �→ (AA, B, K, ρm, R), +(41) +for all i ∈ I. We denominate this procedure as the +offline phase of a two-scale optimization approach, +as it can be performed independent from the +online optimization procedure that is subject to +constraints, that go beyond the box constraints on +the parameter sets as in eq. (35). + +Sequential Global Programming Applied to Fluid-saturated Porous Media +13 +Fig. 4: Collection of materials: each material, represented by a unit cell object, comes along with a +collection of data such as geometric parameters, physical properties and further labels. +For the case M = 1, the conventional approach +would be now, to perform the optimization based +on the interpolated functions +˜H1 over the full +parameter set A1. This is not directly possible +for M > 1. One way to get around this would +be to introduce another interpolation between the +different unit cell types similar as it is done in dis- +crete material optimization (DMO) [14]. Rather +than that we introduce design grids +Agrid +i +⊂ Ai, i ∈ I, +(42) +for all unit cell types. Only elements of Agrid +i +, i ∈ +I will be considered in the optimization process +later. This way, in general, only an approximate +solution of the design problem can be computed. +However it will turn out that this strategy com- +bines well with the separable non-convex model +introduced later in section 3.2. Moreover the +resulting error can be easily controlled by the dis- +tance and number of samples in Agrid +i +, i ∈ I. The +relation of different grids and mappings for the +material coefficients are visualized and elaborated +in fig. 5. +As we only optimize on Agrid +i +, i ∈ I, eq. (37) +is approximated by +˜H := +M +� +i=1 +˜Hi(Agrid +i +). +(43) +We note that elements of ˜H can be precomputed +already in the offline phase. In general, this leads +to a higher memory requirement, but additionally +reduces online computation time. +Finally, we briefly introduce a finite element +approximation, with nel finite elements, and there- +fore introduce element index set E := {1, . . . , nel} +to indicate a finite element distinctively by its +index e ∈ E. We further assume that the design +is constant on each element and can thus be +represented by +IH ∈ ˜Hnel +We remark that through the definition of ˜H in +eq. (43) this condition already states that only +material tensors are eligible, for which a unit cell +type i and a parameter vector αi in Agrid +i +exists. +Moreover, we replace physical functions Φ and Ψ, +regularization function Ξ and solution operator Z +by their discretized counterparts, e.g., +Zh : +� ˜Hnel → Rndof +IH �→ (u, p) +, +(44) +where ndof is the dimension of the discrete state +solution space. The discretized version of resource +function ρ eq. (39) is +ρh : +� ˜Hnel → R +IH �→ ρh. +(45) +The optimization problem, fully discretized in +design and state space, then reads +min +IH∈ ˜ +Hnel max +λρ∈R+ +Fh(IH, z, λρ) +s.t. +z = Zh(IH), +(46) + +14 +Sequential Global Programming Applied to Fluid-saturated Porous Media +Fig. 5: Left: Sketch of parameter set Ai and samples from its subsets Anodes +i +(blue dots), that serves +as a construction basis of interpolated ˜Hi, and Agrid +i +(red squares), on which the optimization process +is performed. In general, Anodes +i +and Agrid +i +can be fully independent from each other. Right: Simplified +sketch of the original effective material coefficients spaces Hi(Ai) (yellow surface) and the the images of +interpolated ˜Hi(Ai) (red surface). The blue dots and red squares represent the images of the parameters +from respectively Anodes +i +or Agrid +i +. +with +Fh(IH, z, λρ) :=ΛΦΦh(IH, z) + ΛΨΨh(IH, z) ++ λρ (ρh(IH) − ¯ρm) + ΛΞΞh(IH). +We note that we have eliminated the resource +constraint by the Lagrange formalism. Later we +will suggest to use a bisection strategy as intro- +duced in [8] for the framework of the well known +OCM method. We finally specialize the regular- +ization term to become +Ξh(IH) = 1 +2∥R − F(R)∥2, +(47) +where F denotes a standard density filter function +(see, e.g., [23]) with +F : Rnel → Rnel. +(48) +and R is the vector of regularization labels asso- +ciated with all finite elements e ∈ E. +3.2 Construction of subproblems +For any sequential programming algorithm first a +sequence of subproblems has to be defined. Here, +in each iteration k, we construct separable first +order approximations, about an expansion point +IHk ∈ ˜Hnel, for the components of cost function +J (IH, λρ) := Fh(IH, z, λρ) +(49) +of the original optimization problem in eq. (46). +The model problem is +min +IH max +λρ∈R +Jsep +� +IH, λρ; IHk� +(50) +where our model function is defined as +Jsep +� +IH, λρ; IHk� +:= +� +e∈E +Jsep,e +� +IHe, λρ; IHk +e +� += +� +e∈E +� +Jphys +� +IDe; IDk +e +� ++ λρ � +Jvol((ρm)e) ++ ΛΞ � +Jreg,e(Re; Rk +e) + Λg � +Jglob(AAe; AAk +e) +(51) +with +IDe := (AAe, Be, Ke) ∈ S6 × S3 × S3, +IDk +e := (AAk +e, Bk +e, Kk +e) ∈ S6 × S3 × S3, +IH, IHk ∈ ˜Hnel. +In the following, we describe each component of +Jsep in more details. +For this, we split J (IH, λρ) as +J (IH, λρ) = Jphys(IH)+λρJvol(IH)+ΛΞJreg(IH) +with +Jphys(IH) := ΛΦΦh(IH, z) + ΛΨΨh(IH, z), (52) +Jvol(IH) := ρh(IH) − ¯ρm, +(53) +Jreg(IH) := Ξh(IH). +(54) + +Sequential Global Programming Applied to Fluid-saturated Porous Media +15 +From tuple IH, only the effective material +coefficients AA, B and K, are relevant for Jphys. +Consequently, for Jphys, we define a separable +approximation of type +� +e∈E +� +Jphys +� +IDe; IDk +e +� +, +(55) +where � +Jphys is the following generalization of the +first-order MMA-like model suggested in [19] for +functions defined in tensor variables: +� +Jphys +� +IDe; IDk� += Cphys − +� +AAk +e +� +∂Jphys(IDk) +∂AA +� +e +AAk +e, AA−1 +e +� +S6 +− +� +Bk +e +� +∂Jphys(IDk) +∂B +� +e +Bk +e, B−1 +e +� +S3 +− +� +Kk +e +� +∂Jphys(IDk) +∂K +� +e +Kk +e, K−1 +e +� +S3 +. +(56) +Here Cphys is a constant that is chosen to estab- +lish the zeroth order correctness of the model and +< ·, · >{S6,S3} denotes the Frobenius inner prod- +ucts for matrices from S6 and S3, respectively. It +is further mentioned that in contrast to the model +in [19], we refrain from working with flexible gen- +eralized asymptotes LA +A +e ∈ S6, LB +e , LK +e +∈ S3, but +simply choose all of them to be zero matrices. The +partial derivatives of Jphys with respect to the +material coefficients AA, B and K can be easily +extracted from the expressions in eq. (34). +The function Jvol that describes the fraction of +utilized matrix material, is separable by definition, +and depends solely on ρm. We accordingly choose +� +Jvol((ρm)e; ρk +m) = (ρm)e. +(57) +The function Jreg given in eq. (54) solely +depends on the regularization label R ∈ Rnel, +which is a component of tuple IH ∈ ˜Hnel. The +separable approximation of Jreg is thus of the form +� +e∈E +� +Jreg,e(Re; Rk), +(58) +where +� +Jreg,e(Re; Rk) +(59) += 1 +2 +���� �Re +� +Re; Rk� +− +� +F +� +�Re +� +Re; Rk��� +e +���� +2 +. +In eq. (59), we further employ function +�Re +� +R; Rk� +:= +� +Rk +1, . . . , Rk +e−1, R, Rk +e+1, . . . , Rk +nel +� +, +in which the regularization label is varied only in +the e-th entry by value R, and contributions of +expansion point Rk are used in the neighboring +entries. Is is noted that eq. (59) can be reduced to +a convex quadratic function of type +aeR2 +e + beRe + ce, +by precomputing ae, be, ce ∈ R, which are inde- +pendent from Re. +Finally, we implement a step size control for +the design from one iteration to the next one by +adding +� +e∈E +� +Jglob +� +AAe, AAk +e +� += +� +e∈E +1 +2 +���AAe − AAk +e +��� +2 +(60) +with a positive factor Λg to the model cost func- +tion. Alternatively, a more general globalization +strategy, similar to the regularization approach +with regularization label R in eq. (59), could be +pursued by introducing particular globalization +labels. Here, we assume that evaluating the design +step size based on the stiffness tensor AAe and AAk +e +is sufficient, and, in particular, the uniqueness of +the globalization labels, such that +AAe = AA′ +e ⇒ αe = α′ +e, +(61) +is satisfied. +3.3 The SGP algorithm with a +brute-force sub-solver +Having at hand the separable first-order approxi- +mations of the objective function and penalization +terms, we are now able to formulate the iterative +scheme that is described by algorithm 1. We make +extensively use of the separable structure of +Jsep +� +IH, λρ; IHk� += +� +e∈E +Jsep,e +� +IHe, λρ; IHk +e +� + +16 +Sequential Global Programming Applied to Fluid-saturated Porous Media +and solve the subproblems, of each iteration k, +for each finite element e ∈ E individually. This is +done by evaluating Jsep,e for all (finitely many) +IHe ∈ ˜H and, based on these evaluations, iden- +tifying a global minimizer IH∗ +e. Note that, with +each IHe, a unique geometric cell label αe is asso- +ciated and thus, by determining IH∗ +e, we also +determine respective α∗ +e and material class index +i∗. As mentioned already earlier a bisection strat- +egy is applied to treat the resource constraint, see +algorithm 2 for the details. To keep things sim- +ple, it is assumed that the resource constraint is +always active at a minimizer. If no resource con- +straint is applied, the outer loop in algorithm 2 is +simply omitted. +After each iteration, the original cost func- +tion J is evaluated with the current solution +of the subproblems IH∗ +e. If a descent in J was +achieved, we continue the iterative process. If not, +we employ the step width control, by increasing +multiplier Λg of globalization term eq. (60), and +resolve the subproblems using algorithm 2. +Algorithm 1 Sequential Global Programming for +parametrized multi-material optimization +1: k ← 0 +2: initialize IH0 ∈ ˜Hnel +3: Jdiff ← ∞ +4: while Jdiff > 0 and k ≤ kmax do +5: +initialize Λg ∈ R +6: +while Jdiff < 0 do +7: +IH∗ +Λg ← solve eq. (50) to global +optimality using algorithm 2 +8: +increase Λg +9: +end while +10: +IH∗ ← IH∗ +Λg +11: +Jdiff ← J (IHk) − J (IH∗) +12: +k ← k + 1 +13: end while +4 Numerical results +In this section, we demonstrate the abilities of +SGP by means of numerical examples. It is build +up successively by first increasing the design free- +dom to the two-scale optimization problem, while +observing the respective optimized designs and +then studying the effect of regularization. +Algorithm 2 Solve subproblems via brute force +strategy +1: initialize λρ ∈ R for volume bisection +2: while volume constraint is not satisfied do +3: +for all finite element e ∈ E do +4: +for all unit cell types i ∈ I do +5: +α∗ +i ← minimizer on Agrid +i +6: +end for +7: +α∗ ← minimizer among all α∗ +i (i ∈ I) +8: +i∗ ← unit cell type index of α∗ +9: +IH∗ +e ← evaluate ˜Hi∗(α∗) (see eq. (41)) +10: +end for +11: +ρ ← evaluate ρh(IH∗ +e) (see eq. (45)); +12: +if ρ > ¯ρm then +13: +increase λρ +14: +else +15: +decrease λρ +16: +end if +17: end while +In section 4.1, we start with the unit cell that +is constructed by three intersection fluid channels, +visualized in the top row of fig. 1, and study the +impact of the micro-structure’s local orientation +on the performance of the optimized designs. It +will be seen that, thanks to the strength of our +model, we do neither have to use smart initial ori- +entations, as proposed e.g., in [24, 25] by aligning +the anisotropic material with respect to principal +directions of the stress tensor, nor we have to +enforce artificially a regular design. +Then, we present a pareto front and investigate +the influence of different weightings of compliance +and fluid flux, in the cost function, on the result- +ing designs. When we proceed from one point on +the Pareto front to the next one, we intentionally +refrain from using the previous design as a warm +start. Nevertheless and despite the non-convex +character of our weighted cost function, Pareto +curves are obtained, in which none of the points +is dominated by another one. We trace this obser- +vation back to the ability of the SGP method to +avoid poor local solutions. +In section 4.2, we proceed to demonstrate the +ability of SGP to handle more than one unit cell +type. We again compute a Pareto curve for this +case. It will be observed that the new Pareto front +is, due to the increase in the design freedom, is +strictly dominating the previous one. It will be +observed that the more complex parametrization + +Sequential Global Programming Applied to Fluid-saturated Porous Media +17 +does on average not lead to an increase in the +number of state problems to be solved per opti- +mization run. Note that for the settings presented +in section 4.1 and section 4.2, it was not neces- +sary to employ a globalization strategy to control +design changes from one iteration to the next one. +Thus, we set the globalization parameter Λg = 0. +In the end, in section 4.3, we apply a filtering +technique onto the design parameters to both con- +trol the speed of variation of local orientation, as +well as the interface length between the two unit +cell types. Here, we also employ the globalization +term described in eq. (60). +The setting of the poroelastic problem is +depicted in fig. 6. It is a recapitulation of the +macroscopic problem setting from [20], where the +authors selected a finite element from the macro- +scopic domain and optimized the shape of the +local microstructure via a spline box approach. +In the present paper we provide an extension to +this example by solving the two-scale optimiza- +tion problem with the SGP method described in +section 3. We note that we work with a rather +coarse discretization of the macroscopic domain. +The reason is that such a discretization is suffi- +cient to demonstrate the capabilities of SGP as +described above. On the other hand, it is readily +seen in algorithm 2 that the number of macro- +scopic elements enters the computational com- +plexity for SGP linearly. Thus, in principle there +is no obstacle to work with finer discretizations. +4.1 Optimization with one unit cell +type +In this section, we employ unit cell type 1, +depicted in fig. 1. The geometry consists of three +joint cylindrical fluid channels, filled with Glyc- +erine (Young’s modulus 4.35 GPa, dynamic vis- +cosity 0.95 Pa s), that are perpendicular to each +other and intersect a hollow sphere in the middle +of the cell domain. These channels are embed- +ded in matrix material made of Polystyrene with +Young’s modulus of 3.9 GPa and dynamic vis- +cosity of 0.34 Pa s. The feasible range for the +geometric design parameters is A1 = [0.08, 0.22]2. +Thus, in each finite element e ∈ E, we have the +design parameters α1 = (rx, ry)⊤ ∈ A1 to steer +the radii of the channels pointing in y- and x- +direction. The radius of the fluid channel that +Fig. +6: +Setup +of +the +macroscopic +problem: +mechanical traction force f =(0, −1, 0)⊤ acts on a +part of the body’s surface (red) while support is +provided on ΓD and pressure values p1 = 1.0 and +p2 = 0.5 are prescribed on Γp1 and Γp2. The design +domain is discretized by 15 x 10 x 2 hexahedra. +points in z-direction (out-of-plane) is kept con- +stant. At the boundaries of the design parameter +space, the volume fractions of the stiff mate- +rial phase are ρ +� +H1 +� +[0.08, 0.08]⊤�� += 0.7154 and +ρ +� +H1 +� +[0.22, 0.22]⊤�� += 0.879. The directional +stiffness of the softest version of this unit cell +is visualized in fig. 7 by means of a polar plot. +The interpolation of H1 is based on Anodes +1 +. Here, +Fig. 7: Visualization of directional stiffness of unit +cell with maximally opened fluid channels (rx = +0.22, ry = 0.22). This spherical plot was generated +by drawing the entry A1111 of the rotated material +tensor AA ∈ S6 for varying rotation angles (θ, φ) ∈ +[0, 2π]2 about z- and y-axes. For instance, the +sketched arrow points to (π/2, 0) and its length +of 1.9457 comes from first entry of the material +tensor that is rotated by π/2 about the z-axis. +Anodes +1 +is the parameter grid spanned by the com- +ponents of α1, and for each component we chose + +1.9457 +yY18 +Sequential Global Programming Applied to Fluid-saturated Porous Media +11 equally spaced samples. The subproblems of +the SGP algorithm are solved based on the dis- +crete parameter grid Agrid +1 +. For this grid, we chose +a sample size of 28 for each of the two channel +radii; again the samples are equally spaced. +For the following optimization results with the +weighted sum formulation of structural compli- +ance and fluid flux, we employ an initial design +guess, visualized in fig. 8, that is neither particu- +larly favorable for the mechanical nor for the fluid +flow state. +Fig. +8: +Homogeneous +initial +design +with +rx = ry = 0.15 and no cell rotation and physical +performance Φinit = 28.9 and Ψinit = 0.135. +For the described setting, we choose ΛΨ = −10 +and obtain the optimized design shown in fig. 9a. +Note that the design domain is discretized by two +finite element layers in z-direction. We made the +experience that, for all numerical results presented +in this paper, the differences of optimized designs +at layer z = 0 and layer z = 1 are so small such +that they cannot be visually discernible. For this +reason, we will only show optimized designs for +layer z = 0 in the rest of the paper. +SGP stopped after 19 iterations, because the +difference between the objective values of the old +and new design was found to be 0. We note +that this comparably low number of iterations is +related to the fineness of the design discretiza- +tion. Thus, using more grid points could lead to a +slightly larger number of iterations. On the other +hand, in those experiments that we performed in +this direction, the visualizations of the obtained +result could be hardly distinguished, see fig. 10. +This is why we do not report results for differ- +ent choices of Agrid +i +, i ∈ I. A second observation +we can make is that the fluid channels in result- +ing designs are fully connected. This is due to the +fact that no rotational design degrees of freedom +were used. On the other hand we will see next that +(a) Optimized design (z = 0) +(b) Optimized design (z = 1) +(c) Mechanical state +(d) Pressure field +(e) Velocity field +Fig. 9: Optimization result for ΛΨ = −10 and +fixed local micro-structure orientation (no rota- +tion) with Φopt = 27.25 and Ψopt = 0.275 for the +optimized design in (a),(b). The initial guess is the +design shown in fig. 8. In (c) the mechanical state +of the optimized design is visualized by deform- +ing the domain by the physical displacements. The +strain energy is shown in colors. In (e), the flow +direction is visualized by equally scaled arrows and +the colors indicate magnitude of the flow field. + +strain energy +1.0e-05 +0.1 +0.2 +0.3 +4.0e-01pressure +5.0e-01 +0.6 +0.7 +0.8 +0.9 +1.0e+00velocitymagnifude +0.0e+00 +5 +1.0e+01Sequential Global Programming Applied to Fluid-saturated Porous Media +19 +(a) +(b) +Fig. 10: Two optimized designs for different sam- +ple sizes of Agrid +1 +. (a) 10 samples each for rx and ry +and 180 samples for ϕ. (b) 28 samples each for rx +and ry and 180 samples for ϕ. Here, ΛΨ = 1 and +ΛΨ = −10. The visual differences are barely per- +ceptible, although (b) has a 1.5% lower compliance +and a 1.7% higher flux than (a). +the performance is getting way better, if also local +rotations of the micro-structures are allowed. +4.1.1 Optimized local in-plane rotation +of micro-structure +We introduce angle variable ϕ ∈ [0, π] to allow +in-plane rotation, about the z-axis, of the micro- +structure. The effective material coefficients are +rotated by ϕ with the following analytical expres- +sions: +AArot(rx, ry, ϕ) = Q6(ϕ)AA(rx, ry)Q6(ϕ)T , +Brot(rx, ry, ϕ) = Q3(ϕ)B(rx, ry)Q3(ϕ)T , +Krot(rx, ry, ϕ) = Q3(ϕ)K(rx, ry)Q3(ϕ)T , +(62) +where Q6 ∈ R6×6 are rotation matrices for the +stiffness tensor AA in Voigt notation and Q3 ∈ +R3×3 are rotation matrices for the Biot coupling +and permeability tensor. We note that no addi- +tional evaluation of the homogenization operators +are required, as, instead of the micro-structure, +the effective material tensors are rotated. ϕ is +discretized with 180 steps for the brute force +approach to solve the SGP subproblem with algo- +rithm 2. +Let us again set ΛΦ = 1 and ΛΨ = −10, as +in fig. 9, and observe in figs. 11a and 11b how +the design evolves as both physical models coun- +teract each other: the mechanical model strives +for as much material as possible to minimize the +compliance while the fluid flux is maximized when +there is less material in the design domain. The +convergence plot for the merit function J and +(a) Design after one iteration +(b) Optimized design +(c) Pressure field +(d) Velocity field +(e) Mechanical strain +Fig. 11: Optimized design with rotational design +degrees of freedom and respective physical state +for ΛΦ = 1 and ΛΨ = −10, with Φopt = 27.1 and +Ψopt = 0.413. +compliance function Φ, displayed in fig. 12, shows +that the compliance drops in the first iteration, +then increases a bit and finally settles around +the value of 27.0. In general, we observed in our +numerical studies, that the largest design changes +occur within a few iterations in the beginning. +Afterwards, minor changes are made to further +tweak the objective. This behavior shows the good +quality of the SGP model and its approximations, + +XX +XXXare +5.0e-01 +0.6 +0.7 +0.8 +0.9 +1.0e+00velocity magnitude +0.0e+00 +5 +1.0e+01strain energy +1.0e-05 +0.1 +0.2 +0.3 +4.0e-0120 +Sequential Global Programming Applied to Fluid-saturated Porous Media +Fig. 12: Convergence plots for design shown in +fig. 11. +described in section 3. Let us have a closer look +into the intermediate designs shown in fig. 11a. +Again, the initial guess is neither particularly +favorable for the mechanical nor for the fluid flow +state. After the first iteration, we see in fig. 11a +that some channels, close to the outflow region, +are opened widely and cells closer to the mechan- +ical support were adjusted to have narrower fluid +channels to improve the mechanical performance +of the design. In comparison to the solution in +fig. 9, where the orientation was fixed, this solu- +tion has a 1% smaller compliance and a fluid flux +which is about 47% higher. +We would like to emphasize that local orien- +tation field looks rather smooth although we have +neither applied a stress based warm start for the +rotation variable, as proposed by [24, 25], nor we +have employed a regularization technique. We also +can observe that the total number of iterations +required did not increase after addition of the +additional design degrees of freedom. +We conclude this subsection by presenting a +Pareto front for this type of bicriterial weighted +sum formulation in fig. 13. All optimizations were +based on the initial guess that is shown in fig. 8. +This implies that again, no warm starting tech- +nique was employed to proceed from one point +to the next on the Pareto curve. Nevertheless a +Pareto curve is obtained, in which none of the +points is dominated by another one. This again +is a hint that the SGP method is able to avoid +poor local solutions. The number of outer itera- +tions required to solve the problems corresponding +to all points on the Pareto curve varied between +3 and 31. The rather low number of 3 iterations +was obtained for the extreme case, where ΛΨ = 0. +The optimized designs for various choices of ΛΨ +25 +26 +27 +28 +29 +30 +0 +0.25 +0.5 +ΛΨ = −3 +−5 +−10 +−15 +−30 +−60 +compliance Φ +flux Ψ +Fig. 13: Pareto front for varying ΛΨ in weighted- +sum formulation Fphys = Φ+ΛΨΨ. The optimiza- +tion was based on cells of type 1 and the initial +design was always [0.15, 0.15, 0]nel. As we are mini- +mizing Φ and maximizing Ψ, a point P = (PΦ, Pψ) +in the image space of Φ and Ψ is dominating a +point Q = (QΦ, Qψ) if PΦ ≤ QΦ and PΨ ≥ QΨ. +are visualized in fig. 14. It is observed that the +with decreasing ΛΨ the compliance minimized is +design (fig. 14a) is almost smoothly transformed +into a fully flux based design (fig. 14h). +4.2 Optimization with two unit cell +types +We want to study the ability of SGP to handle +more than one unit cell type. For this purpose, +we add unit cell type 2 that comprises of a void +sphere surrounded by matrix material (see second +row of fig. 1). The only design parameter is the +radius rs ∈ [0.1, 0.4] of the void sphere in this +case. The smaller the void sphere, the higher the +volume fraction of the matrix phase and therefore +the stiffer the cell. Thus, cells of type 2 are par- +ticularly favorable for the mechanical part of the +objective. When only optimizing the compliance, +we obtain the trivial solution shown in fig. 15. + +30 +compliance +fmerit +28 +26 +24 +22 +20 +0 +5 +10 +15 +20 +iteration0.25 +0.5 +0.2 +volume fraction +0.4 +0.15 +flux +0.3 +0.1 +0.2 +0.05 +flux +0.1 +total channel volume +0 +0 +5 +10 +15 +20 +iterationSequential Global Programming Applied to Fluid-saturated Porous Media +21 +(a) Compliance minimized design +(b) ΛΨ = −3 +(c) ΛΨ = −5 +(d) ΛΨ = −10 +(e) ΛΨ = −15 +(f) ΛΨ = −30 +(g) ΛΨ = −60 +(h) Flux maximized design +Fig. 14: Visualization of optimized designs associated with the labeled points in fig. 13. +(a) Only cells of type 1 +(b) Cells of type 2 Optimized +design with Φopt = 19.62 +Fig. 15: Compliance minimized designs: (a) only +allowing cells of type 1 and (b) allowing choices of +type 1 and 2. The red dots visualize the void inclu- +sions of cells of type 2. The optimized compliance +of design (b) is 24% better than compliance of the +optimized design (a). +For the fluid flow, cells of type 2 are futile +as they are not permeable. However, for numeri- +cal reasons, we set the permeability of the latter +cells to 0.001. Cells of type 1 have orthotropic +mechanical properties and transversal isotropic +permeability tensors, whereas cells of type 2 have +isotropic mechanical properties and no perme- +ability. Although cell types 1 and 2 are dis- +junct in their parameter spaces, the corresponding +ranges of volume fractions, of the stiff matrix +material, overlap. We have ρ (H1([0.08, 0.08])) = +87.9%, +ρ (H1([0.22, 0.22])) += +71.54% +and +ρ (H2(0.4)) += +73.19%, +ρ (H2(0.1)) += +99.6%. +Anodes +2 +, the basis for the interpolation of H2, con- +sisted of 30 uniformly distributed samples for rs ∈ +[0.1, 0.4] and the optimization procedure was per- +formed on Agrid +2 +with 60 samples, again uniformly +distributed. +Next, we present the updated Pareto front for +compliance minimization and fluid flux maximiza- +tion with both unit cell types in fig. 16. +We again +19 20 21 22 23 24 25 26 27 28 29 30 +0 +0.25 +0.5 +ΛΨ = −2 +−5 +−60 +compliance Φ +flux Ψ +Fig. 16: Comparison of Pareto curves for varying +ΛΨ. Blue: optimization with cells of type 1 and 2. +Red: optimization with only cells of type 1. The +blue curve clearly dominates the red curve. +stress that we did not use enhanced initial designs +for the computation of the points on the Pareto +curve. The comparison of the new (blue) curve +with the old (red) curve shows that consistently +better designs are obtained. Points on the blue +curve strictly dominate points on the red curve in +the Pareto sense. This is not surprising as, with + +XXXX22 +Sequential Global Programming Applied to Fluid-saturated Porous Media +the addition of a new unit cell type, the design +freedom is increased. Still it is worth to mention +that the fact that we do not observe any outliers +in this respect again underlines the stability of +our SGP method. The numbers of required outer +iterations varied between 4 and 40, which means +that no significant increase in the number of iter- +ations is observed, although a second cell type has +been added. In fig. 17, we can observe how the +number of cells of type 2, in the optimized design, +decreases with decreasing ΛΨ. This is expected, as +cell type 2 is completely useless for a flux favored +design. +We note that so far all results presented have +been computed without employing a resource con- +straint. Just to demonstrate that SGP can also +easily handle problems, where a resource con- +straint is added, we briefly discuss a selected result +in fig. 18. +4.3 Optimization with both cell +types and regularization of +design labels and interface +We introduce a regularization of the optimiza- +tion problem by applying a weighted-sum filter F +(e.g., [23, 26]), that is often used in the context +of topology optimization, on regularization labels +that are directly related to the unit cells’ geomet- +ric parameters. For this we introduce mappings +l1 : +� +A1 +→ R3 +(rx, ry, ϕ) +�→ R1 +(63) +where +R1 = +�rx − 0.08 +0.14 +, ry − 0.08 +0.14 +, cos +� +2ϕ +π − π +2 +��⊤ +, +and +l2 : +� +A2 +→ R3 +rs +�→ R2 = (−1, −1, −1)⊤. +(64) +This choice of labeling has the following effects: +Within type 1, the maximal distance from lower +to upper label bound is 1. This is the same dis- +tance required to jump from the stiffest cell of +type 1, with rx = ry = 0.08, to any cell of type +2. Therefore, the interface between cells of type +(a) +Compliance +minimized +design +(b) ΛΨ = −2 +(c) ΛΨ = −5 +(d) ΛΨ = −60 +(e) Flux maximized design +Fig. 17: Results of bicriterial optimization with +cells from both type 1 and 2 for varying ΛΨ. The +designs visualized here corresponds to the labeled +data points of the pareto curve in fig. 16. +1 and 2 is also penalized. The most expensive +change is a jump from type 1, which is preferred +by the compliance, to any cell of type 2, which +is most beneficial for the fluid flux. The shifted +cosine function appearing in the expression for + +Sequential Global Programming Applied to Fluid-saturated Porous Media +23 +(a) Optimized design at z = 0 +(b) +Mechanical +state +with +Φopt = 23.78 +Fig. 18: Result of pure compliance minimization +when allowing unit cells of type 1 and 2 with an +active volume fraction constraint setting ¯ρm = 0.8 +on the stiff material phase. Comparing to fig. 17a, +it is observed that only now also cells of type 1 +appear in the design. Moreover, the resource con- +straint leads to a variation of the parameter rs for +cell type 2. +(R1)3 is employed to circumvent disambiguities for +the angular variable. +Employing these regularization labels, Jreg +from eq. (59) changes to +Jreg(R) = 1 +2 +3 +� +ℓ=1 +∥Rℓ − F(Rℓ)∥2, +(65) +where Rℓ ∈ Rnel collects the ℓ-the components +of the regularization label assigned to each finite +element, which is defined by formula eq. (63) or +eq. (64), if cell type 1 or cell type 2 is chosen for +the corresponding finite element e, respectively. +Next, we study the influence of regularization with +the optimized result for the particular choice ΛΨ = +−3. The result displayed in fig. 19 displays the +changes in design with increasing regularization +parameter pfilt. The respective objective values are +listed in table 2. The regularization of fluid chan- +nel radii can be observed well when comparing +the designs in the right lower corner of fig. 19b +and fig. 19c. With increasing pfilt, the interface +(a) Initial design +(b) No regularization +(c) ΛΞ = 0.01 +(d) ΛΞ = 0.02 +(e) ΛΞ = 0.025 +Fig. 19: Results for varying ΛΞ with filter radius +of 1.3 elements and ΛΨ = −3. +between unit cell types 1 and 2, at the right upper +corner of the design domain, vanishes and the +design is dominated by cells of type 1. +ΛΞ +Jmer,opt +Jreg,opt +Φopt +Ψopt +0 +21.34 +11.5 +21.57426 +0.0765 +0.01 +21.65 +0.0389 +21.65041 +0.0140 +0.011 +21.67 +0.0484 +21.66846 +0.0142 +0.015 +21.99 +0.0747 +21.95892 +0.0139 +0.02 +22.40 +0.092 +22.34912 +0.0135 +0.025 +22.70 +0.0712 +22.66730 +0.0136 +Table 2: Performance of designs shown in fig. 19 +with Jmer,opt(ΛΞ) = Jreg,opt(ΛΞ)+Φopt +ΛΨΨopt +5 Conclusion and Outlook +We presented an Sequential Global Programming +(SGP) approach to homogenization-based struc- +tural optimization which can be viewed as an free +material optimization constrained by the set of +admissible geometric material parameters. + +X +Xstrain energy +1.0e-05 +0.1 +0.2 +0.3 +4.0e-0124 +Sequential Global Programming Applied to Fluid-saturated Porous Media +By means of numerical examples, where we +successively added more ingredients to the opti- +mization problem, we demonstrated that the pro- +posed SGP approach, with its first-order approxi- +mations, provides good and reasonable optimized +designs without the necessity of particular design +initialization or the employment of a regulariza- +tion strategy for purposes of convergence. Fur- +thermore, SGP is able to handle several material +classes with disjunct parameter sets without addi- +tional interpolation and penalization strategies. +We further observed that optimizing the local +orientation of the microstructure brings along a +significant improvement, up to 48%, of the fluid +flux. We have not actively addressed the subject +of connectivity within the microstructure, that +is to ensure connectivity of the fluid saturated +channels. However, the regularization approach +presented in section 4.3 can be used to control +the degree of variation of the local microstruc- +ture rotation and we have seen, by means of the +presented numerical examples, that only a mild +regularization has already a fair impact on the +design. +Although the resolution of the finite element +approximation, and thus the number of design +elements, of the examples in section section 4 +was chosen rather coarsely, it served the pur- +pose of demonstrating the presented features of +SGP. With regard to finer resolutions: the algo- +rithm can be well parallelized with respect to the +design elements due to the block-separability of +the first-order approximations. +The brute-force approach in the subproblem +solver, described in algorithm 2, can further be +speeded up by employing a hierarchical scanning +of the design grids Agrid +i +: Start with a rather coarse +number of samples and determine the minimizer +among those. In the next level, consider only the +current minimizer and its neighbors and perform +the same search within this subset of Agrid +i +, for all +i ∈ I. Repeat this step until the maximum desired +number of levels or some accuracy is achieved. +Note that, with this strategy, the quality of the +design depends on the number of samples on the +coarsest grid level. An alternative would be to +apply a Lipschitz optimization solver, see [27], +to each design element and type in a black box +manner. +Further research will focus on extending the +SGP approach for homogenization-based opti- +mization to transient problems and, in partic- +ular, to dynamic metamaterial design. Another +challenge is to extend the proposed optimiza- +tion approach for an approximate treatment of +nonlinear two-scale problems with the homoge- +nized coefficients depending on the macroscopic +response by virtue of the sensitivity analysis as +discussed in [5]. +6 Acknowledgments +The authors B. N. Vu and M. Stingl gratefully +acknowledge the financial support by the Ger- +man Federal Ministry for Economic Affairs and +Climate Action (BMWK) in the course of the +FIONA (LuFo VI-1, FKZ: 20W1913F) project. +The research conducted by E. Rohan and V. +Lukeˇs was supported by the grant projects GACR +19-04956S and GACR 22-00863K of the Czech +Scientific Foundation. +7 Statements and +Declarations +The authors declare that they have no known com- +peting financial interests or personal relationships +that could have appeared to influence the work +reported in this paper. +8 Replication of results +The +algorithm +of +the +proposed +optimization +approach was described in algorithm 1 and algo- +rithm 2. Its implementation, as well as exemplary +problem settings and respective data to reproduce +the numerical results presented in section 4, are +publicly available on https://gitlab.com/bnvu/ +sgp-poroel. +References +[1] Andreasen, C.S., Sigmund, O.: Topology +optimization +of +fluid–structure-interaction +problems in poroelasticity. 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Springer, Boston, MA (1995) + diff --git a/T9FKT4oBgHgl3EQflC6k/content/tmp_files/load_file.txt b/T9FKT4oBgHgl3EQflC6k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4afafa9d3b9f2033359e5d351719198934a4b15f --- /dev/null +++ b/T9FKT4oBgHgl3EQflC6k/content/tmp_files/load_file.txt @@ -0,0 +1,1060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf,len=1059 +page_content='A Sequential Global Programming Approach for Two-scale Optimization of Homogenized Multiphysics Problems with Application to Biot Porous Media Bich Ngoc Vu1*, Vladimir Lukeˇs2, Michael Stingl1 and Eduard Rohan2* 1Competence Unit for Scientific Computing, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Martenstrasse 5a, Erlangen, 91058, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2Department of Mechanics & NTIS New Technologies for Information Society, University of West Bohemia in Pilsen, Univerzitn´ı 22, Plzeˇn, 30614, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' E-mail(s): bich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='ngoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='vu@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' rohan@kme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='cz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Contributing authors: vlukes@kme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='cz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='stingl@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Abstract We present a new approach and an algorithm for optimizing the material configuration and behaviour of a fluid saturated porous medium in a two-scale setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The state problem is governed by the Biot model describing the fluid-structure interaction in homogenized poroelastic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' However, the approach is widely applicable to multiphysics problems involving several macroscopic fields where homogenization provides the relationship between the microconfigurations and the macroscopic math- ematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The optimization variables describe the local microstructure design by virtue of the pore shape which determines the effective medium properties – the material coefficients – computed by the homogenization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The main idea of the numerical optimization strategy consists in a) employing a precomputed database of the material coefficients associated to the geometric parameters and b) applying the sequential global programming (SGP) method for solving the problem of macro- scopically optimized distribution of material coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although there are similarities with the free material optimization (FMO) approach, only effective material coefficients are considered admissible, for which a well-defined set of corresponding configurable microstructures exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Due to the flexibility of the SGP approach, different types of microstructures with fully independent parametrizations can easily be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The efficiency of the concept is demonstrated by a series of numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We show that the SGP method can handle simultaneously multiple types of microstructures with nontrivial parametrizations using a considerably low and stable number of state problems to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Keywords: multi-material optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' sequential global programming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' homogenization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Biot model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' poroelasticity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' sensitivity analysis 1 Introduction The design of fluid-saturated poroelastic media (FSPM) present a gradually increasing topic of research interest due to its mathematical complex- ity and a great application potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although the theory of FSPM has been developed in the context of geomechanics and civil engineering, nowadays theses types of materials are abundant 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='11852v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='CE] 27 Jan 2023 2 Sequential Global Programming Applied to Fluid-saturated Porous Media in many engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' A convenient design of microstructures can provide a metama- terial property related to controllable fluid trans- port, or elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In particular, soft robots can be designed as inflatable porous structures gen- erating a motion and force due to variable fluid content, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To this aim, the behaviour of the fluid-saturated porous materials is described by the Biot model [2], within the small strain theory, which was postulated using a phenomenological approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The homogenization method enabled the derivation of the quasistatic Biot’s equations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Since then, a number of works extended the results for the dynamic case, which is important for treating wave propagations, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' As an extension beyond the linear theory, a modified Biot model with strain-dependent poroelastic and permeability coefficients was proposed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Topology optimization of microstructures con- stituting the FSPM was treated in [6] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Therein, the fluid-structure interaction problem was handled in the homogenization framework and an approximation towards computational simpli- fication was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In this paper, we aim at a two-scale approach optimization allowing for a spatial grading of the microstructure design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Two-scale optimization problems have been already extensively discussed in literature before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The whole idea started with the seminal paper of Bendsøe and Kikuchi [7], in which the following concept was suggested: for a given parametrization of the unit cell, carry out the homogenization procedure on a fixed parameter grid in a preprocessing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then, in every step of the optimization, first retrieve, for each design element, (approximate) effec- tive material coefficients by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Next, plug these coefficients into the state equation, solve the latter and evaluate the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The other way round, sensitivities are computed by the chain rule, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' first differentiate the quantity of interest with respect to the material coefficients and then differentiate the material coefficients with respect to to the design parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This procedure opens the way for the application of any suitable gradient based optimization solver, like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', OCM [8], MMA [9] or SnOpt [10], to name only those, which are most prominently used in structural topology and material optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' While this concept essentially carries over to other classes of problems, as it is done by [11–13] for thermomechanical settings, we opted to fol- low a slightly different avenue in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' There are several reasons: First, the concept depends, by its nature, to a large extent on the cho- sen parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' If the parameters enter the homogenized properties in a substantially non- convex way (as it is the case, if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', rotations of the base cells are allowed), many local min- ima might be introduced and additional measures must be taken to avoid getting trapped in one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Second, it is not easy to extend the original concept with respect to the use of completely inde- pendent types of unit cells, either characterized by different geometries or material configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In this case, specifying a smooth parametriza- tion is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The typical idea would be to first introduce an independent parametriza- tion for either cell types (for example using sizing variables) and then add on top a smooth inter- polation scheme for the effective tensors as used, for instance, in multi-material optimization (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The problems with that is however, that the second level of interpolation introduces material coefficients, for which typically no interpreta- tion in terms of a microstructure exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, an additional penalization strategy is required, which ensures that those unphysical choices do not remain in the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Such an approach was successfully demonstrated in the recent work [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In another recent article, [16] chose two unit cell types, described via level-set functions, such that the mixture of their geometric parameters can be directly interpreted as a third unit cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' [17] also opted for level-set functions to describe the geometry of the microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' But, with respect to the handling of multiple material classes, the authors defined floating patches, where each patch is a subdomain of the design domain and only occupied by one microstructure type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then, the layout of these patches are optimized on the macroscopic level and their overlaps are combined via a differentiable maximum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In our paper, we describe, how these disadvan- tages can be circumvented using the SGP concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The basic idea has been already introduced in [18] and is now generalized to a multiphysics, two-scale setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This involves an extension of an MMA- type block-separable model function (see [19]) to Sequential Global Programming Applied to Fluid-saturated Porous Media 3 the poroelastic setting, a split of the computa- tions into an offline and an online phase, which is particularly suited for homogenization based problems, and a numerical solution scheme for the nearly global optimization of block-separable sub- problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We would like to note here that the term block-separable implies that the minimization can be carried out separately for each design element, however a design element itself can be described by multiple design degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For a fur- ther motivation of the SGP method, we refer to the first paragraph in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, we just like to add that, in the whole optimization process, two different types of sensitivities are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' First, there are the sensitivities of constraint or cost functions with respect to the effective mate- rial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' These constitute a substantial ingredient of the block-separable model used in the heart of the SGP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Second, there are the sensitivities of the material coefficients with respect to the chosen parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the context of the suggested two-scale SGP frame- work, the latter ones are not strictly required, but can help to come up with an improved interpola- tion model used in the offline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In any case, the derivation of sensitivities presented in this paper, for the particular context of fluid saturated porous media, relies on derivations in [20], where also the sensitivity of the homogenized coefficients were reported, see also [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Finally, we would like to comment on the gen- erality of the presented approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although the SGP concept outlined in our paper can be applied to a large range of multiphysics two-scale mate- rial optimization problems, the Biot model of fluid saturated porous media provides an ideal test bed for the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is for several reasons: first, the physical coupling is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Second, it is very natural to set up competing objective func- tions, such as the structural compliance on the one hand and the enhanced fluid flow through an outflow boundary, on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' And third, configurable types of microstructures supporting either the first or the second goal can be deduced in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The structure of the remainder of this paper is as follows: In section 2 all ingredients of the two-scale problem are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To these belong a brief repetition of the constitutive laws for the Biot model (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1), the poroelastic state problem in variational form (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2), a generic sketch of the two-scale problem constrained by the poroelasticity equations (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3) and an adjoint analysis providing sensitivities with respect to effective material coefficients, as used later by the SGP method (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Finally, two types microstructures are suggested in form of configurable unit cells (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In section 3 the SGP concept for the solution of two-scale opti- mization problems is introduced in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this, the two-scale problem is discretized and extended for the use of multiple types of unit cells (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then, a separable sequential approx- imation concept is suggested (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2) and last the SGP method is presented in an algorithmic form (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In section 4, the advantages of the SGP algorithm will be discussed using various types of two-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2 Formulation of the two-scale optimization problem In this section, we explain our optimization strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although it can be applied to similar prob- lems involving several physical fields or multi- physics problems, in this paper, we consider the fluid saturated porous media represented by the Biot model which can be derived using the homog- enization of the fluid-structure interaction prob- lem restricted to small deformation kinematics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', [3, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the next section we report the homogenization result presented Notation We employ the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Since we deal with a two-scale problem, we distin- guish the “macroscopic” and “microscopic” coordinates, x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We use ∇x = (∂x i ) and ∇y = (∂y i ) when differentiation with respect to coordinate x and y is used, respectively, whereby ∇ ≡ ∇x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' By e(u) = 1/2[(∇u)T + ∇u], we denote the strain of a vec- torial function u, where the transpose operator is indicated by the superscript T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The Lebesgue spaces of 2nd-power integrable functions on an open bounded domain D ⊂ R3 is denoted by L2(D), the Sobolev space W 1,2(D) of the square integrable vector-valued functions on D including the first order generalized derivative, is abbrevi- ated by H1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Further, H1 #(Ym) is the Sobolev 4 Sequential Global Programming Applied to Fluid-saturated Porous Media space of vector-valued Y-periodic functions (the subscript #).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 The homogenized Biot – Darcy model We report the homogenization result presented e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', in [5], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' [20], where the problem of locally optimized microstructures has been described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The homogenized model of the porous elastic medium incorporates local problems for character- istic responses which are employed to compute the effective material coefficients of the Biot model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The local problems specified below, related to the homogenized model, are defined at the microscopic representative unit cell Y = Π3 i=1]0, ℓi[⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' which splits into the solid part occupying domain Ym and the complementary channel part Yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, Y = Ym ∪ Yc ∪ ΓY , Yc = Y \\ Ym , ΓY = Ym ∩ Yc , (1) where by Yd for d = m, c, we denote the closure of the open bounded domain Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' By ∼� Yd = |Y |−1 � Yd, with Yd ⊂ Y for d = m, c, we denote the local aver- age (|Y | is the volume of domain Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Obviously, the unit volume |Y | = 1 can always be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We employ the usual elasticity bilinear form, involving two vector fields w and v, that reads am Y (w, v) =∼ � Ym (IDey(w)) : ey(v) , (2) where ID = (Dijkl) is the elasticity tensor satisfy- ing the usual symmetries, Dijkl = Dklij = Djikl, and ey(v) = 1 2(∇yv + (∇yv)T ) is the linear strain tensor associated with the displacement field v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In what follows, by the microstructure Y(x), we mean the decomposition eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (1) of the repre- sentative cell Y and the material properties, as represented by the elasticity ID only in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' If the structure is perfectly periodic, microstructures Y ≡ Y(x) are independent of the macroscopic position x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Otherwise, the local problems must be considered at any macroscopic posi- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' for almost any x ∈ Ω, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', [21] in the context of slowly varying “quasi-periodic” microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It should be pointed out, that this issue is of a special importance when deal- ing with homogenization-based material design optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' as will be explained below, a regular- ization is required to control the design variation within Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The local microstructural response is obtained by solving the following decoupled problems: Find ωij ∈ H1 #(Ym) for any i, j = 1, 2, 3 satisfying am Y � ωij + Πij, v � = 0 , ∀v ∈ H1 #(Ym) , (3) where Πij = (Πij k ), i, j, k = 1, 2, 3 with compo- nents Πij k = yjδik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Find ωP ∈ H1 #(Ym) satisfying am Y � ωP , v � =∼ � ΓY v · n[m] dSy, ∀v ∈ H1 #(Ym) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (4) Find (ψi, πi) ∈ H1 #(Yc) × L2(Yc) for i = 1, 2, 3 such that � Yc ∇yψk : ∇yv − � Yc πk∇ · v = � Yc vk , � Yc q∇y · ψk = 0 , (5) ∀v ∈ H1 #(Yc) and ∀q ∈ L2(Yc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Effective material properties of the homoge- nized deformable fluid-saturated porous medium are described in terms of homogenized poroelas- tic coefficients: the drained elasticity AA, the stress coupling C and the compressibility N, all being related to the solid skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' All these coefficients including the intrinsinc hydraulic permeability K are computed using the characteristic microscopic responses eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (3) to (5) substituted in following expressions: Aijkl = am Y � ωij + Πij, ωkl + Πkl� , Cij = − ∼ � Ym divyωij = am Y � ωP , Πij� , N = am Y � ωP , ωP � =∼ � ΓY ωP · n dSy , Kij =∼ � Yc ψj i =∼ � Yc ∇yψi : ∇yψi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (6) Sequential Global Programming Applied to Fluid-saturated Porous Media 5 Obviously, the tensors AA = (Aijkl), C = (Cij) and K = (Kij) are symmetric, AA adheres all the symmetries of ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' moreover AA is positive definite and N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The hydraulic permeability K is, in general, positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is positive def- inite whenever the channels constitute a simply connected domain generated as the periodic lattice by Yc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' for this, denoting by Γk Y ⊂ ∂Y , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' , 6 the faces of Y , it must hold that Γk Y ∩ ∂Yc ̸= ∅ for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Coupled flow deformation problem The Biot–Darcy model of poroelastic media for quasi-static, evolutionary problems imposed in Ω is constituted by the following equations involv- ing stress σ, displacement u, strain e(u), fluid pressure p and the seepage velocity w: −∇ · σ = f s, σ = AAe(u) − Bp, −∇ · w = B : e( ˙u) + M ˙p, w = −K ¯η � ∇p − f f� , (7) where the homogenized coefficients are given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (6) and B := C + φI , M := N + φγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (8) Above, ¯η is the relative fluid viscosity, γ is the fluid compressibility and φ = |Yc|/|Y | is the poros- ity (volume fraction of the fluid-filled channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The effective volume forces in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (7), acting in the solid and fluid phases, are denoted by f s and f f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is important to note that ¯η = ηphys/ε2 0 is defined for a given fluid (ηphys) and microstructures scale: ε0 = ℓ0/L where L is a characteristic macroscopic length, and ℓ0 is the characteristic microstructure size, typically given by the “pore diameter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, for a given fluid, the effective permeability K/¯η is proportional to ε2 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' reflecting the microstructure size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In con- trast, all other coefficients are scale-independent (when the scale separation holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' ε0 being small enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In this paper, we only consider steady state problems for the Biot medium, such that all time derivatives in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (7) vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Consequently, the Biot compressibility M is not involved, as far as the porous phase, generated as a peri- odic lattice by channels Yc, is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For any microstructure with disconnected pores, such that Yc ⊂ Y , thus, Yc constitute one, or more inclu- sions with one cell Y , see [22], the permeability vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then, the time integration in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (7) leads to the mass conservation equation in the form B : e(u) + Mp = 0, assuming an unde- formed initial configuration with the zero pressure in the inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the optimization problem, besides microstructures with nondegenerate per- meabilities, we shall consider also microstructures with spherical, thus, disconnected pores, constitut- ing impermeable material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this case, one can choose either fluid filled pores, or empty pores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' the only difference is the use of the so-called undrained material elasticity, AAU = AA + M −1B ⊗ B, or the elasticity AA describing effective elasticity of the “drained” skeleton, with empty pores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 State problem formulation Let Ω ⊂ R3 be an open bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Its boundary ∂Ω splits, as follows: ∂Ω = ΓD ∪ ΓN and also ∂Ω = Γp ∪ Γw, where ΓD ∩ ΓN = ∅ and Γp ∩ Γw = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Assume Γp consists of two dis- connected, non-overlapping parts Γk p, k = 1, 2, Γp = Γ1 p ∪ Γ2 p, and Γ1 p ∩ Γ2 p = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We consider the steady state problems for the linear Biot continuum occupying domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The poroelastic material parameters and the hydraulic permeability referred to as the homogenized coef- ficients, in general, are given by the locally defined microstructures Y(x) which can vary with x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The two-scale optimization approach proposed in this paper enables to combine microstruc- tures characterized by connected and disconnected pores, the latter characterized by a vanishing per- meability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To this aim, the domain Ω = Ω0 ∪ Ω+ is decomposed into in two parts: the permeable Ω+ and the impermeable Ω0, which may not con- stitute connected domains, being split into more disconnected subparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Consequently, the inter- face Γ+ = ∂Ω+ ∩ ∂Ω0 is impermeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Regard- ing the boundary decomposition, we assume that Γk p+ := Γk p ∩ ∂Ω+ ̸= ∅, for k = 1, 2, so that the porous structure permits the fluid transport through domain Ω+, if this one connects Γ1 p+ and Γ2 p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We consider the following macroscopic prob- lem: Given the traction surface forces g, and 6 Sequential Global Programming Applied to Fluid-saturated Porous Media pressures ¯pk on boundaries Γk p, find displacements u and the hydraulic pressure P which satisfy −∇ · (AAe(u) − PB) = 0 in Ω , u = 0 in ΓD , (AAe(u) − PB) · n = g in ΓN , (9) where P = 0 in Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Whereas, in Ω+, P satisfies −∇ · K∇P = 0 in Ω+ , P = ¯pk on Γk p+ , k = 1, 2 , n · K∇P = 0 on Γw ∪ Γ+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (10) For the steady state problem the set of equations eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (7) yields the two problems eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (9) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (10) as a decoupled system: first, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (10) can be solved for P, then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (9) is solved for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Moreover, for the considered type of the bound- ary conditions and since volume forces are not involved, the solutions are independent of the viscosity ¯η, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Further, we consider an extension of ¯pk from boundary Γk p to the whole domain Ω, such that ¯pk = 0 on Γl p (in the sense of traces) for l ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then P = p + � k ¯pk in Ω+, such that p = 0 on Γp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Note that p can be simply extended by 0 in Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the sake of notational simplicity, we introduce ¯p = � k ¯pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' By virtue of the Dirichlet boundary conditions for u and p, we introduce the following spaces: V0 = {v ∈ H1(Ω) | v = 0 on ΓD} , Q0 = {q ∈ L2(Ω) ∩ H1(Ω+) | q = 0 on Γp+} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (11) We employ the bilinear forms and the linear functional g, aΩ (u, v) = � Ω (AAe(u)) : e(v) , bΩ+ (p, v) = � Ω+ pB : e(v) , cΩ+ (p, q) = � Ω+ ∇q · K∇p , g(v) = � ΓN g · v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (12) In order to define the state problem in the context of two-scale optimization, we employ the weak formulation which reads, as follows: Find u ∈ V0 and p ∈ Q0, such that, for all v ∈ V0 and q ∈ Q0, aΩ (u, v) − bΩ+ (p, v) = g(v) + bΩ+ (¯p, v) , cΩ+ (p, q) = −cΩ+ (¯p, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (13) To define p uniquely in Ω, p ≡ 0 in Ω0 = Ω \\ Ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Since the two fields are decoupled, first p is solved from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (13)2, then u is solved from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (13)1, where p is already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the context of the undrained poros- ity defined by fluid-filled closed pores Yc ⊂ Y , see Remark 1, formulation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (13) is consistent also with this microstructure class type Y□ 0 with AAU replacing AA in the elasticity bilinear form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (12)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Pressure is then defined pointwise in Ω0 by P := −B : e(u)/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' By α(x) we denote an abstract optimization variable which determines the homogenized coeffi- cients for any position x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Below we consider α representing several geometrical parameters char- acterizing microstructures Y(x) of a given type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although, in this section, we disregard some par- ticular details related to the treatment of multiple types of Y, we bear in mind the existence of two microstructure classes, Y□ + and Y□ 0 , associ- ated with the pore connectivity type, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The “permeable” domain Ω+ is occupied by the material given pointwise by Y(x) ∈ Y□ + for all x ∈ Ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Hence, both the subdomains of Ω are defined implicitly by the microstructure type: Ωi is the set of x ∈ Ω, such that Y(x) ∈ Y□ i , where i = +, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the next section, we shall consider a two- scale optimization problem which is characterized by the following features: Geometrical restrictions are stated in respective definitions of the admissibility designs sets for a chosen type of microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the sake of brevity, let A be the set of admissible designs, further we consider α(x) ∈ A for any x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We consider multiple optimization criteria which perform as the objective functions, or equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Without loss of generality, we confine ourselves to the two criteria Φα(u) Sequential Global Programming Applied to Fluid-saturated Porous Media 7 and Ψα(p) that are defined, as follows: Φα(u) = g(u) , Ψα(p) = − � Γ2p K∇(p + ¯p) · n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (14) While Φα(u) expresses the structural compli- ance, criterion function Ψα(p) expresses the amount of the fluid flow through surface Γ2 p due to the pressure difference ¯p1 − ¯p2, see the boundary condition eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (10)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' These two criteria are antagonist: the pore volume reduction leads naturally to stiffening the structure, but reduces the permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Hence, for the objective func- tion Φα, function Ψα serves as a constraint and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 Two-scale optimization problem Here, for the ease of notation, we restrict to one microstructure type only, namely Y(x) ∈ Y□ +, so that we may consider Ω ≡ Ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Hence, all the bilin- ear forms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (12) are defined by integration in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Later, in section 3, we will consider microstruc- tures characterized by different unit cell types of classes Y□ + and Y□ 0 , however, the formulations introduced below can be adapted easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We first define the direct optimization problem to find design α(Ω) that minimizes a cost func- tional based on the criteria defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Further, we introduce the set T = S6 × S3 × S3 × R × R and denote by IH = (AA, B, K, ρm, R) ∈ T the (local) material parameters involing the effec- tive (homogenized) material coefficients, the solid part volume ρm = 1−φ = |Ym|/|Y |, and a regular- ization parameter R, which typically depends only on the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that the dimension of the regularization label R is, for ease of notation, cho- sen as 1 for now, although later in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 more general regularization labels are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Obviously, IH is given uniquely by the local admissible design α(x) ∈ A, x ∈ Ω, whereby for a suitably cho- sen parametrization, the admissibility set is given simply by A = [a, a] ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Examples for such parametrizations along with a description of the lower and upper bounds a, a ∈ Rn are presented in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For a given admissible design α(Ω), the state z = (u, p) is the solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (13), where the homogenized coefficients IH(α) are given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (6) using the characteristic responses W (α) := (ωij, ωP , ψk, πk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' W (α) are the solutions of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (3) to (5), which depend on α(x) in terms of the microconfigurations Y(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In this way, map- ping S : α(Ω) �→ z(Ω) introduces the admissible state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It can be defined by a composition map, S = Z ◦ E ◦ W, where W represents the resolvents of the characteristic problems imposed on the local microconfigurations, E provides the homogenized material, and Z is the resolvent of the macroscopic state problem, so that W : α �→ W , E : (α, W) �→ IH , Z : IH(Ω) �→ z(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (15) Further, we employ the mapping H : α �→ IH, such that H = E ◦ W is the composition map defined for any admissible design α(x) ∈ A, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The macroscopic state problem is the implicit form of the mapping Z : IH �→ z, such that z ∈ S0 = V0 × Q0 satisfies ϕIH(z, v) = 0 ∀v ∈ S0 , (16) where S0 is the space of admissible state problem solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the Biot medium problem, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (16) is identified with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 Direct two-scale optimization problem For the given two functions of interest Φ and Ψ, both depending on the material distribution IH(x) and the state z(x), the two-scale abstract optimization problem reads: min α∈A Φ(IH, z) + ΛΞΞ(IH) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Ψ(IH, z) = Ψ0 , z = S(α), IH = H(α), � Ω ρm ≤ ¯ρm|Ω| , (17) 8 Sequential Global Programming Applied to Fluid-saturated Porous Media where the term Ξ(IH) in the objective is related to the design regularization, namely to param- eter R, and ΛΞ ∈ R+ is a penalty parame- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Recall the chain mapping H : α(x) �→ IH(x) for any x ∈ Ω, then z = Z(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Below, we abbreviate Φα(z) =: Φ(H(α), z) and also Ψα(z) =: Ψ(H(α), z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (14), specific exam- ples relevant for the Biot medium optimization were given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Optimization problem eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (17) is associated with the following inf-sup problem, min α∈A inf z∈S0 sup Λ∈R2,˜z∈S0 L(α, z, Λ, ˜z) , (18) with the Lagrangian function, L(α, z, Λ, ˜z) = ΛΦΦα(z) + ΛΞΞ(H(α)) + ΛΨ(Ψα(z) − Ψ0) + ϕIH(α)(z, ˜z) , (19) where Λ = (ΛΦ, ΛΨ) ∈ R2 are the Lagrange multi- pliers associated with the objective and constraint functionals Φ and Ψ, and ˜z ∈ S0 are Lagrange multipliers – the adjoint variables — associated with the constraints of the problem eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For a while, we may consider material coeffi- cients IH as the optimization variables (although they are parameterized by α ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Further, let us assume a given value Λ ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' note that the entries of Λ can be positive or negative depending on the desired flow augmentation, or reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the numerical examples, we chose ΛΦ > 0, whereas ΛΨ < 0 indicates the constraint effect of Ψ rela- tive to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Upon denoting by Im(H) = H(A), the image space of all admissible designs, and defining Uad = {IH ∈ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' T ) | IH(x) ∈ Im(H) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' x ∈ Ω} , the optimization problem eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (17) can be rephrased as the two-criteria minimization prob- lem, min IH ∈ Uad F(IH, z) , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' z = Z(IH) � Ω ρm ≤ ¯ρm|Ω| , (20) where F(IH, z) = ΛΦΦ(IH, z) + ΛΨΨ(IH, z) + ΛΞΞ(IH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the Biot medium optimization, where the two criterion functions Φα and Ψα are given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (14), the Lagrangian function attains the form L(α, (u, p), Λ, (˜v, ˜q)) = ΛΦΦα(u) + ΛΨ(Ψα(p) − Ψ0) + ΛΞΞα(IH) + aΩ (u, ˜v) − bΩ (p + ¯p, ˜v) − g(˜v) + cΩ (p + ¯p, ˜q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (21) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4 Adjoint responses and the sensitivity analysis In this section, we provide details concerning the sensitivity analysis employed in the preceding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We consider α to represent a general opti- mization variable which is related to the effective medium parameters IH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is worth to note that one may also consider α ≡ IH in the context of the free material optimization (FMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To obtain the adjoint equation, we consider the optimality condition for (u, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (21) it follows that δ(u,p)L(α, (u, p), Λ, (˜v, ˜q)) ◦ (v, q) = ΛΦδuΦα(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' v) + ΛΨδpΨα(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' q) + aΩ (v, ˜v) − bΩ (q, ˜v) + cΩ (q, ˜q) , (22) where δuΦα(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' v) = g(v), δpΨα(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' q) = − � Γ2p K∇q · n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (23) To avoid computation of the gradient ∇q on Γ2 p ⊂ ∂Ω, we consider ˜p ∈ H1(Ω) such that ˜p = 0 on Γ \\ Γ2 p, while ˜p = 1 on Γ2 p, then it is easy to see that −Ψα(p) = r(p) := cΩ (p + ¯p, ˜p) , −δpΨα(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' q) = δpr(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' q) = cΩ (q, ˜p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (24) The optimality conditions eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (22), related to the state admissibility, yield the adjoint state Sequential Global Programming Applied to Fluid-saturated Porous Media 9 (˜v, ˜q) ∈ V0 × Q0 which satisfies the following iden- tities: ∀v ∈ V0 : aΩ (v, ˜v) = −ΛΦδuΦα(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' v) , ∀q ∈ Q0 : cΩ (q, ˜q) = bΩ (q, ˜v) − ΛΨδpΨα(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (25) These equations can be rewritten using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (23) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (24), as follows for all (˜v, ˜q) ∈ V0 × Q0: ∀v ∈ V0 : aΩ (v, ˜v) = −ΛΦg(v) , ∀q ∈ Q0 : cΩ (q, ˜q) = bΩ (q, ˜v) + ΛΨcΩ (q, ˜p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (26) To allow for the independence of the state adjoint on Λ, we define the split ˜v = ΛΦ˜ϑ , ˜q = ΛΦ˜q1 + ΛΨ˜q2 , (27) where ˜ϑ and ˜qk, k = 1, 2 satisfy for all (˜v, ˜q, ˜q) ∈ V0 × Q0 ∀v ∈ V0 : aΩ � v, ˜ϑ � = −g(v), ∀q ∈ Q0 : cΩ (q, ˜q1) = bΩ � q, ˜ϑ � , ∀q ∈ Q0 : cΩ (q, ˜q2) = cΩ (q, ˜p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (28) We can compute the total variation of the Lagrangian with δtot α L = ΛΦδug(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' δαu) − ΛΨδpr(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' δαp) + ΛΦδαg(u) − ΛΨδαr(p) + ΛΞδαΞα(IH) + aΩ (δαu, ˜v) − bΩ (δαp, ˜v) + cΩ (δαp, ˜q) + δαaΩ (u, ˜v) − δαbΩ (p + ¯p, ˜v) + δαcΩ (p + ¯p, ˜q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (29) If the pair (u, p) solves the state problem and (˜v, ˜q) is its adjoint state, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (29) is equivalent to the following expression: δtot α L = ΛΦδαg(u) − ΛΨδαr(p) + ΛΞδαΞα(IH) + δαaΩ (u, ˜v) − δαbΩ (p + ¯p, ˜v) + δαcΩ (p + ¯p, ˜q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (30) Above, the shape derivatives δα of the bilinear forms can be rewritten in terms of the sensitivity of the homogenized coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Besides the obvi- ously vanishing derivative δαg(u) = 0, it holds that δαaΩ (u, ˜v) ◦ δαAA = � Ω δαAAe(u) : e(˜v) , δαbΩ (p + ¯p, ˜v) ◦ δαB = � Ω (p + ¯p)δαB : e(˜v) , δαcΩ (p + ¯p, ˜q) ◦ δαK = � Ω ∇˜q · δαK∇(p + ¯p) , δαr(p) = δαcΩ (p + ¯p, ˜p) ◦ δαK = � Ω ∇˜p · δαK∇(p + ¯p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (31) Using the “total pressure” P := p+¯p, the following tensors are employed to evaluate the expression in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (31): e(u) ⊗ e(˜ϑ) , Pe(˜ϑ) , ∇P ⊗ ∇˜q1 , ∇P ⊗ ∇˜q2 , ∇˜p ⊗ ∇P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (32) Now, using these tensors, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (29) is computed, as follows: δtot α L = −ΛΨδαr(p) + ΛΦ � δαaΩ � u, ˜ϑ � − δαbΩ � P, ˜ϑ � + δαcΩ (P, ˜q1) � + ΛΨδαcΩ (P, ˜q2) + ΛΞ∂IHΞ(IH)δαIH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (33) Hence the variations of L with respect to AA, B and K are given by the following formulae δtot A A L = ΛΦ � Ω δAAe : e(u) ⊗ e(˜ϑ) , δtot B L = −ΛΦ � Ω δBe : Pe(˜ϑ) , δtot K L = � Ω δKe : (ΛΦ∇P ⊗ ∇˜q1 +ΛΨ (∇P ⊗ ∇˜q2 − ∇˜p ⊗ ∇P)) (34) As Ξ(IH) solely depends on the regularization parameter R, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (47), we get ∂IHΞ(IH)δαIH = � Ω (R−F(R)·(δR−∂RF(R)◦δR) 10 Sequential Global Programming Applied to Fluid-saturated Porous Media for the regularization term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the context of the finite element discretization intro- duced in section 3, the homogenized coefficients are supplied as constants in each element Ωe of the partitioned domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Accordingly, the expres- sions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (32) are supplied elementwise at the Gauss integration points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 Design parametrization The design of the cell Y , that is the decompo- sition into the solid skeleton Ym and the pores Yc, can be parameterized in a number of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In [20], we employed a so-called spline-box structure parameterized by design variables defining posi- tions of the spline control polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This kind of parametrization is convenient due to its gener- ality to handle quite arbitrary design, but leads to complicated formulations of design constraints which are needed to preserve essential geometrical requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', positivity of channel crosssec- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In this paper, we employ two specific types of microstructures illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 1, where the channels are shaped as a 3D cross (type 1), or a sphere (type 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Hence, the latter microstructure is featured by zero permeability and therefore, we consider dry pores (voids) in the mechani- cal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Due to these specific geometries, we can use a rather simple parametrization, which is listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For a unit cell of type 1, rx and ry refer to the radii of the cylinders pointing in x- and y-direction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The third parameter ϕ describes the cell rotation, about axis z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the unit cell type 2, the spherical voids, whose radii are described by rs, provide an orthotropic material with nearly isotropic elastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Therefore, rotations are not enabled for this cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Impor- tantly, box constraints can be imposed on rx, ry and rs straightforwardly to guarantee geometric feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' microstructure # cell parameters 1 rx ry ϕ 2 rs Table 1: The parametrization of the pore geom- etry for the two types of the microstructures: 1: the 3D cross, 2: the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 1: Parametrization of unit cells: unit cell type 1 is parameterized by radii rx and ry, both ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22, rz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15 and rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='25 are kept constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' unit cell type 2 is parameterized by radius rs ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To illustrate a sensitivity of the material prop- erties determined by the homogenized coefficients IH, In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2, for unit cell type 2, the elasticity as the only relevant material property is displayed as function of rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 3, for unit cell type 1, selected components of the poroelastic tensors and of the permeability are reported as functions of ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='40 rs [-] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 coefficients A [GPa] A1111 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 2: Unit cell type 2: dependence of A1111 on parameter rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Sequential Global Programming Applied to Fluid-saturated Porous Media 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22 ry [-] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0 coefficients A [GPa] A1111 A2222 A3333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22 ry [-] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='60 coefficients B [-] B11 B22 B33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22 ry [-] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0010 coefficients K [m2 / (Pa · s)] K11 K22 K33 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 3: Unit cell type 1: dependence of homoge- nized coefficients AA, B, and K on ry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' rx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 3 A Sequential Global Programming formulation The basic description of the Sequential Global Programming algorithm along with convergence aspects were presented in [18], where SGP was applied to a multi-material optimization based on a two-dimensional time harmonic Helmholtz state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The setting and procedure described in this manuscript differs from the one in [18] in the following major points: first, in [18] a selection of finitely many fixed materials was considered as admissible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In this paper, each admissi- ble material is computed by homogenizing unit cell, which itself is configurable by a number of geometric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, the designer can choose in each point of the design domain from M different unit cell types and adjust the geo- metric parameters for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Second, the SGP approach is extended to a multi-physics setting using a slightly different separable approxima- tion and third, a different solution strategy is employed for the subproblems arising from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This strategy does not impose any assumption on the parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In particular, parametriza- tions can be non-analytical and non-differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This leads to a greater design flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Despite these differences, there is also an important fea- ture, the approach presented here has in common with the one outlined in [18]: separable models are established in terms of (effective) material tensors IH rather than their parameterization α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then, the parametrization is directly treated at the level of sub-problems without further convex- ification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thanks to the separable character of the chosen first order model the resulting generally non-convex sub-problems can - in principal - still be solved to global optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The advantages of this approach are twofold: first, due to the separable model functions being able to capture also non-convex features of the original cost function typically a low number of outer iterations, equivalently to the number of state problems to be solved, is required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' and sec- ond, due to the good fit of the separable models with the cost function as well as the fact that non-convex sub-problems are solved to global opti- mality the overall algorithm is less start value dependent and less prone to be trapped in poor local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is in contrast to traditional approaches, where a local model is established directly based on the sensitivity of cost functions with respect to the design parameterization α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the following we first derive a fullly dis- cretized counterpart for a slightly generalized of problem eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then we describe in detail how the separable first order approximations can be constructed and finally present a practical out- line of the full SGP algorithm including a generic sub-solver allowing to compute near globally opti- mal solutions for sub-problems using a brute-force strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 A fully discretized 2-scale design problem For the sake of simplicity, the definitions of sets and functions were introduced in sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 12 Sequential Global Programming Applied to Fluid-saturated Porous Media and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 based on the assumption that there is only one type of unit cell such that M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, for a more general setting, we consider M unit cell types, each one with ni design param- eters, and introduce index set I := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For each unit cell type i ∈ I, the admissibility set is defined in terms of box constraints and other purely geometrical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' By choosing a suit- able parameterization, we can identify these with (geometric) parameter sets Ai = [ai, ai] ⊂ Rni, (35) with ai, ai ∈ Rni being lower and upper bound vectors constraining the corresponding parameter vector αi ∈ Rni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that, while in this manuscript the parameters in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (35) are always used to vary the geometrical properties of the unit cell, varia- tions in the material parameters could be described in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, SGP can handle both of these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We further define for all i ∈ I map Hi : � Ai → T αi �→ (AA, B, K, ρm, R), (36) where Hi(α) performs the homogenization proce- dure described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 4 illustrates the components of Hi(αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We denote the union of the ranges of all Hi by H := M � i=1 Hi(Ai) (37) and with that generalize the set of admissible design functions to become Uad = {IH ∈ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' T ) | IH(x) ∈ H for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' x ∈ Ω} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Now the state problem operator Z : � Uad → R3 × R IH �→ z = (u, p), (38) with displacement function u(IH) and hydraulic pressure function p(IH) reads exactly as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We finally use a slightly more general resource function than in sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 as follows: ρ : � Uad → R IH �→ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (39) A concretization could be the total volume frac- tion of a specific material phase (see description of ¯ρm in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Based on these definitions, we then formulate an FMO-type problem min IH∈Uad F(IH, z) :=ΛΦΦ(IH, z) + ΛΨΨ(IH, z) + ΛΞΞ(IH) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' z =Z(IH), ρ(IH) ≤¯ρm, (40) where ¯ρm ∈ R is the resource constraint value and cost functions and Φ, Ψ, Ξ and their weights ΛΨ, ΛΦ, ΛΞ have been already introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although problem eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (40) is formulated directly in the tensor variable IH, a realization of the feasibility condition IH ∈ Uad would force us to evaluate the homogenization maps Hi (i ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This has the consequence that for each evaluation of the cost function, a homogenization procedure, which contains a series of cell problems, has to be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To alleviate this situation, we follow [7] and carry out the homogenization procedure only for discrete samples of the design parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For each unit cell type i, we introduce a grid with nodes Anodes i ⊆ Ai and effective material coeffi- cients are only computed, via homogenization, at the sampled nodes of this grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In addition, we define a piecewise cubic Hermite interpolator for these samples to realize the continuous mapping ˜Hi : � Ai → T αi �→ (AA, B, K, ρm, R), (41) for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We denominate this procedure as the offline phase of a two-scale optimization approach, as it can be performed independent from the online optimization procedure that is subject to constraints, that go beyond the box constraints on the parameter sets as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Sequential Global Programming Applied to Fluid-saturated Porous Media 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 4: Collection of materials: each material, represented by a unit cell object, comes along with a collection of data such as geometric parameters, physical properties and further labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the case M = 1, the conventional approach would be now, to perform the optimization based on the interpolated functions ˜H1 over the full parameter set A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is not directly possible for M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' One way to get around this would be to introduce another interpolation between the different unit cell types similar as it is done in dis- crete material optimization (DMO) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rather than that we introduce design grids Agrid i ⊂ Ai, i ∈ I, (42) for all unit cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Only elements of Agrid i , i ∈ I will be considered in the optimization process later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This way, in general, only an approximate solution of the design problem can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' However it will turn out that this strategy com- bines well with the separable non-convex model introduced later in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Moreover the resulting error can be easily controlled by the dis- tance and number of samples in Agrid i , i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The relation of different grids and mappings for the material coefficients are visualized and elaborated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' As we only optimize on Agrid i , i ∈ I, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (37) is approximated by ˜H := M � i=1 ˜Hi(Agrid i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (43) We note that elements of ˜H can be precomputed already in the offline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In general, this leads to a higher memory requirement, but additionally reduces online computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Finally, we briefly introduce a finite element approximation, with nel finite elements, and there- fore introduce element index set E := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' , nel} to indicate a finite element distinctively by its index e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We further assume that the design is constant on each element and can thus be represented by IH ∈ ˜Hnel We remark that through the definition of ˜H in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (43) this condition already states that only material tensors are eligible, for which a unit cell type i and a parameter vector αi in Agrid i exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Moreover, we replace physical functions Φ and Ψ, regularization function Ξ and solution operator Z by their discretized counterparts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', Zh : � ˜Hnel → Rndof IH �→ (u, p) , (44) where ndof is the dimension of the discrete state solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The discretized version of resource function ρ eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (39) is ρh : � ˜Hnel → R IH �→ ρh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (45) The optimization problem, fully discretized in design and state space, then reads min IH∈ ˜ Hnel max λρ∈R+ Fh(IH, z, λρ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' z = Zh(IH), (46) 14 Sequential Global Programming Applied to Fluid-saturated Porous Media Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 5: Left: Sketch of parameter set Ai and samples from its subsets Anodes i (blue dots), that serves as a construction basis of interpolated ˜Hi, and Agrid i (red squares), on which the optimization process is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In general, Anodes i and Agrid i can be fully independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Right: Simplified sketch of the original effective material coefficients spaces Hi(Ai) (yellow surface) and the the images of interpolated ˜Hi(Ai) (red surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The blue dots and red squares represent the images of the parameters from respectively Anodes i or Agrid i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' with Fh(IH, z, λρ) :=ΛΦΦh(IH, z) + ΛΨΨh(IH, z) + λρ (ρh(IH) − ¯ρm) + ΛΞΞh(IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that we have eliminated the resource constraint by the Lagrange formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Later we will suggest to use a bisection strategy as intro- duced in [8] for the framework of the well known OCM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We finally specialize the regular- ization term to become Ξh(IH) = 1 2∥R − F(R)∥2, (47) where F denotes a standard density filter function (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', [23]) with F : Rnel → Rnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (48) and R is the vector of regularization labels asso- ciated with all finite elements e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 Construction of subproblems For any sequential programming algorithm first a sequence of subproblems has to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, in each iteration k, we construct separable first order approximations, about an expansion point IHk ∈ ˜Hnel, for the components of cost function J (IH, λρ) := Fh(IH, z, λρ) (49) of the original optimization problem in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The model problem is min IH max λρ∈R Jsep � IH, λρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IHk� (50) where our model function is defined as Jsep � IH, λρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IHk� := � e∈E Jsep,e � IHe, λρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IHk e � = � e∈E � Jphys � IDe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IDk e � + λρ � Jvol((ρm)e) + ΛΞ � Jreg,e(Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rk e) + Λg � Jglob(AAe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' AAk e) (51) with IDe := (AAe, Be, Ke) ∈ S6 × S3 × S3, IDk e := (AAk e, Bk e, Kk e) ∈ S6 × S3 × S3, IH, IHk ∈ ˜Hnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the following, we describe each component of Jsep in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this, we split J (IH, λρ) as J (IH, λρ) = Jphys(IH)+λρJvol(IH)+ΛΞJreg(IH) with Jphys(IH) := ΛΦΦh(IH, z) + ΛΨΨh(IH, z), (52) Jvol(IH) := ρh(IH) − ¯ρm, (53) Jreg(IH) := Ξh(IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (54) Sequential Global Programming Applied to Fluid-saturated Porous Media 15 From tuple IH, only the effective material coefficients AA, B and K, are relevant for Jphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Consequently, for Jphys, we define a separable approximation of type � e∈E � Jphys � IDe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IDk e � , (55) where � Jphys is the following generalization of the first-order MMA-like model suggested in [19] for functions defined in tensor variables: � Jphys � IDe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IDk� = Cphys − � AAk e � ∂Jphys(IDk) ∂AA � e AAk e, AA−1 e � S6 − � Bk e � ∂Jphys(IDk) ∂B � e Bk e, B−1 e � S3 − � Kk e � ∂Jphys(IDk) ∂K � e Kk e, K−1 e � S3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (56) Here Cphys is a constant that is chosen to estab- lish the zeroth order correctness of the model and < ·, · >{S6,S3} denotes the Frobenius inner prod- ucts for matrices from S6 and S3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is further mentioned that in contrast to the model in [19], we refrain from working with flexible gen- eralized asymptotes LA A e ∈ S6, LB e , LK e ∈ S3, but simply choose all of them to be zero matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The partial derivatives of Jphys with respect to the material coefficients AA, B and K can be easily extracted from the expressions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The function Jvol that describes the fraction of utilized matrix material, is separable by definition, and depends solely on ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We accordingly choose � Jvol((ρm)e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' ρk m) = (ρm)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (57) The function Jreg given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (54) solely depends on the regularization label R ∈ Rnel, which is a component of tuple IH ∈ ˜Hnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The separable approximation of Jreg is thus of the form � e∈E � Jreg,e(Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rk), (58) where � Jreg,e(Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rk) (59) = 1 2 ���� �Re � Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rk� − � F � �Re � Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rk��� e ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (59), we further employ function �Re � R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rk� := � Rk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' , Rk e−1, R, Rk e+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' , Rk nel � , in which the regularization label is varied only in the e-th entry by value R, and contributions of expansion point Rk are used in the neighboring entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Is is noted that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (59) can be reduced to a convex quadratic function of type aeR2 e + beRe + ce, by precomputing ae, be, ce ∈ R, which are inde- pendent from Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Finally, we implement a step size control for the design from one iteration to the next one by adding � e∈E � Jglob � AAe, AAk e � = � e∈E 1 2 ���AAe − AAk e ��� 2 (60) with a positive factor Λg to the model cost func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Alternatively, a more general globalization strategy, similar to the regularization approach with regularization label R in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (59), could be pursued by introducing particular globalization labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, we assume that evaluating the design step size based on the stiffness tensor AAe and AAk e is sufficient, and, in particular, the uniqueness of the globalization labels, such that AAe = AA′ e ⇒ αe = α′ e, (61) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 The SGP algorithm with a brute-force sub-solver Having at hand the separable first-order approxi- mations of the objective function and penalization terms, we are now able to formulate the iterative scheme that is described by algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We make extensively use of the separable structure of Jsep � IH, λρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IHk� = � e∈E Jsep,e � IHe, λρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' IHk e � 16 Sequential Global Programming Applied to Fluid-saturated Porous Media and solve the subproblems, of each iteration k, for each finite element e ∈ E individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is done by evaluating Jsep,e for all (finitely many) IHe ∈ ˜H and, based on these evaluations, iden- tifying a global minimizer IH∗ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Note that, with each IHe, a unique geometric cell label αe is asso- ciated and thus, by determining IH∗ e, we also determine respective α∗ e and material class index i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' As mentioned already earlier a bisection strat- egy is applied to treat the resource constraint, see algorithm 2 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' To keep things sim- ple, it is assumed that the resource constraint is always active at a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' If no resource con- straint is applied, the outer loop in algorithm 2 is simply omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' After each iteration, the original cost func- tion J is evaluated with the current solution of the subproblems IH∗ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' If a descent in J was achieved, we continue the iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' If not, we employ the step width control, by increasing multiplier Λg of globalization term eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (60), and resolve the subproblems using algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Algorithm 1 Sequential Global Programming for parametrized multi-material optimization 1: k ← 0 2: initialize IH0 ∈ ˜Hnel 3: Jdiff ← ∞ 4: while Jdiff > 0 and k ≤ kmax do 5: initialize Λg ∈ R 6: while Jdiff < 0 do 7: IH∗ Λg ← solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (50) to global optimality using algorithm 2 8: increase Λg 9: end while 10: IH∗ ← IH∗ Λg 11: Jdiff ← J (IHk) − J (IH∗) 12: k ← k + 1 13: end while 4 Numerical results In this section, we demonstrate the abilities of SGP by means of numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is build up successively by first increasing the design free- dom to the two-scale optimization problem, while observing the respective optimized designs and then studying the effect of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Algorithm 2 Solve subproblems via brute force strategy 1: initialize λρ ∈ R for volume bisection 2: while volume constraint is not satisfied do 3: for all finite element e ∈ E do 4: for all unit cell types i ∈ I do 5: α∗ i ← minimizer on Agrid i 6: end for 7: α∗ ← minimizer among all α∗ i (i ∈ I) 8: i∗ ← unit cell type index of α∗ 9: IH∗ e ← evaluate ˜Hi∗(α∗) (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (41)) 10: end for 11: ρ ← evaluate ρh(IH∗ e) (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (45));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 12: if ρ > ¯ρm then 13: increase λρ 14: else 15: decrease λρ 16: end if 17: end while In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1, we start with the unit cell that is constructed by three intersection fluid channels, visualized in the top row of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 1, and study the impact of the micro-structure’s local orientation on the performance of the optimized designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It will be seen that, thanks to the strength of our model, we do neither have to use smart initial ori- entations, as proposed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', in [24, 25] by aligning the anisotropic material with respect to principal directions of the stress tensor, nor we have to enforce artificially a regular design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Then, we present a pareto front and investigate the influence of different weightings of compliance and fluid flux, in the cost function, on the result- ing designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' When we proceed from one point on the Pareto front to the next one, we intentionally refrain from using the previous design as a warm start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Nevertheless and despite the non-convex character of our weighted cost function, Pareto curves are obtained, in which none of the points is dominated by another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We trace this obser- vation back to the ability of the SGP method to avoid poor local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2, we proceed to demonstrate the ability of SGP to handle more than one unit cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We again compute a Pareto curve for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It will be observed that the new Pareto front is, due to the increase in the design freedom, is strictly dominating the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It will be observed that the more complex parametrization Sequential Global Programming Applied to Fluid-saturated Porous Media 17 does on average not lead to an increase in the number of state problems to be solved per opti- mization run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Note that for the settings presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 and section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2, it was not neces- sary to employ a globalization strategy to control design changes from one iteration to the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, we set the globalization parameter Λg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the end, in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3, we apply a filtering technique onto the design parameters to both con- trol the speed of variation of local orientation, as well as the interface length between the two unit cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, we also employ the globalization term described in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The setting of the poroelastic problem is depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is a recapitulation of the macroscopic problem setting from [20], where the authors selected a finite element from the macro- scopic domain and optimized the shape of the local microstructure via a spline box approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the present paper we provide an extension to this example by solving the two-scale optimiza- tion problem with the SGP method described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that we work with a rather coarse discretization of the macroscopic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The reason is that such a discretization is suffi- cient to demonstrate the capabilities of SGP as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' On the other hand, it is readily seen in algorithm 2 that the number of macro- scopic elements enters the computational com- plexity for SGP linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, in principle there is no obstacle to work with finer discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 Optimization with one unit cell type In this section, we employ unit cell type 1, depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The geometry consists of three joint cylindrical fluid channels, filled with Glyc- erine (Young’s modulus 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='35 GPa, dynamic vis- cosity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='95 Pa s), that are perpendicular to each other and intersect a hollow sphere in the middle of the cell domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' These channels are embed- ded in matrix material made of Polystyrene with Young’s modulus of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9 GPa and dynamic vis- cosity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='34 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The feasible range for the geometric design parameters is A1 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, in each finite element e ∈ E, we have the design parameters α1 = (rx, ry)⊤ ∈ A1 to steer the radii of the channels pointing in y- and x- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The radius of the fluid channel that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 6: Setup of the macroscopic problem: mechanical traction force f =(0, −1, 0)⊤ acts on a part of the body’s surface (red) while support is provided on ΓD and pressure values p1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0 and p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 are prescribed on Γp1 and Γp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The design domain is discretized by 15 x 10 x 2 hexahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' points in z-direction (out-of-plane) is kept con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' At the boundaries of the design parameter space, the volume fractions of the stiff mate- rial phase are ρ � H1 � [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08]⊤�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='7154 and ρ � H1 � [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22]⊤�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The directional stiffness of the softest version of this unit cell is visualized in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 7 by means of a polar plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The interpolation of H1 is based on Anodes 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 7: Visualization of directional stiffness of unit cell with maximally opened fluid channels (rx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22, ry = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This spherical plot was generated by drawing the entry A1111 of the rotated material tensor AA ∈ S6 for varying rotation angles (θ, φ) ∈ [0, 2π]2 about z- and y-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For instance, the sketched arrow points to (π/2, 0) and its length of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9457 comes from first entry of the material tensor that is rotated by π/2 about the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Anodes 1 is the parameter grid spanned by the com- ponents of α1, and for each component we chose 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9457 yY18 Sequential Global Programming Applied to Fluid-saturated Porous Media 11 equally spaced samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The subproblems of the SGP algorithm are solved based on the dis- crete parameter grid Agrid 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this grid, we chose a sample size of 28 for each of the two channel radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' again the samples are equally spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the following optimization results with the weighted sum formulation of structural compli- ance and fluid flux, we employ an initial design guess, visualized in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 8, that is neither particu- larly favorable for the mechanical nor for the fluid flow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 8: Homogeneous initial design with rx = ry = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15 and no cell rotation and physical performance Φinit = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9 and Ψinit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the described setting, we choose ΛΨ = −10 and obtain the optimized design shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Note that the design domain is discretized by two finite element layers in z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We made the experience that, for all numerical results presented in this paper, the differences of optimized designs at layer z = 0 and layer z = 1 are so small such that they cannot be visually discernible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this reason, we will only show optimized designs for layer z = 0 in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' SGP stopped after 19 iterations, because the difference between the objective values of the old and new design was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that this comparably low number of iterations is related to the fineness of the design discretiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, using more grid points could lead to a slightly larger number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' On the other hand, in those experiments that we performed in this direction, the visualizations of the obtained result could be hardly distinguished, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is why we do not report results for differ- ent choices of Agrid i , i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' A second observation we can make is that the fluid channels in result- ing designs are fully connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is due to the fact that no rotational design degrees of freedom were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' On the other hand we will see next that (a) Optimized design (z = 0) (b) Optimized design (z = 1) (c) Mechanical state (d) Pressure field (e) Velocity field Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 9: Optimization result for ΛΨ = −10 and fixed local micro-structure orientation (no rota- tion) with Φopt = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='25 and Ψopt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='275 for the optimized design in (a),(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The initial guess is the design shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In (c) the mechanical state of the optimized design is visualized by deform- ing the domain by the physical displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The strain energy is shown in colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In (e), the flow direction is visualized by equally scaled arrows and the colors indicate magnitude of the flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' strain energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-01pressure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e+00velocitymagnifude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e+00 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e+01Sequential Global Programming Applied to Fluid-saturated Porous Media 19 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 10: Two optimized designs for different sam- ple sizes of Agrid 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (a) 10 samples each for rx and ry and 180 samples for ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (b) 28 samples each for rx and ry and 180 samples for ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Here, ΛΨ = 1 and ΛΨ = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The visual differences are barely per- ceptible, although (b) has a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5% lower compliance and a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='7% higher flux than (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' the performance is getting way better, if also local rotations of the micro-structures are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 Optimized local in-plane rotation of micro-structure We introduce angle variable ϕ ∈ [0, π] to allow in-plane rotation, about the z-axis, of the micro- structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The effective material coefficients are rotated by ϕ with the following analytical expres- sions: AArot(rx, ry, ϕ) = Q6(ϕ)AA(rx, ry)Q6(ϕ)T , Brot(rx, ry, ϕ) = Q3(ϕ)B(rx, ry)Q3(ϕ)T , Krot(rx, ry, ϕ) = Q3(ϕ)K(rx, ry)Q3(ϕ)T , (62) where Q6 ∈ R6×6 are rotation matrices for the stiffness tensor AA in Voigt notation and Q3 ∈ R3×3 are rotation matrices for the Biot coupling and permeability tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that no addi- tional evaluation of the homogenization operators are required, as, instead of the micro-structure, the effective material tensors are rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' ϕ is discretized with 180 steps for the brute force approach to solve the SGP subproblem with algo- rithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Let us again set ΛΦ = 1 and ΛΨ = −10, as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 9, and observe in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 11a and 11b how the design evolves as both physical models coun- teract each other: the mechanical model strives for as much material as possible to minimize the compliance while the fluid flux is maximized when there is less material in the design domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The convergence plot for the merit function J and (a) Design after one iteration (b) Optimized design (c) Pressure field (d) Velocity field (e) Mechanical strain Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 11: Optimized design with rotational design degrees of freedom and respective physical state for ΛΦ = 1 and ΛΨ = −10, with Φopt = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 and Ψopt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' compliance function Φ, displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 12, shows that the compliance drops in the first iteration, then increases a bit and finally settles around the value of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In general, we observed in our numerical studies, that the largest design changes occur within a few iterations in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Afterwards, minor changes are made to further tweak the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This behavior shows the good quality of the SGP model and its approximations, XX XXXare 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e+00velocity magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e+00 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e+01strain energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-0120 Sequential Global Programming Applied to Fluid-saturated Porous Media Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 12: Convergence plots for design shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Let us have a closer look into the intermediate designs shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Again, the initial guess is neither particularly favorable for the mechanical nor for the fluid flow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' After the first iteration, we see in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 11a that some channels, close to the outflow region, are opened widely and cells closer to the mechan- ical support were adjusted to have narrower fluid channels to improve the mechanical performance of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In comparison to the solution in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 9, where the orientation was fixed, this solu- tion has a 1% smaller compliance and a fluid flux which is about 47% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We would like to emphasize that local orien- tation field looks rather smooth although we have neither applied a stress based warm start for the rotation variable, as proposed by [24, 25], nor we have employed a regularization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We also can observe that the total number of iterations required did not increase after addition of the additional design degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We conclude this subsection by presenting a Pareto front for this type of bicriterial weighted sum formulation in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' All optimizations were based on the initial guess that is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This implies that again, no warm starting tech- nique was employed to proceed from one point to the next on the Pareto curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Nevertheless a Pareto curve is obtained, in which none of the points is dominated by another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This again is a hint that the SGP method is able to avoid poor local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The number of outer itera- tions required to solve the problems corresponding to all points on the Pareto curve varied between 3 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The rather low number of 3 iterations was obtained for the extreme case, where ΛΨ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The optimized designs for various choices of ΛΨ 25 26 27 28 29 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 ΛΨ = −3 −5 −10 −15 −30 −60 compliance Φ flux Ψ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 13: Pareto front for varying ΛΨ in weighted- sum formulation Fphys = Φ+ΛΨΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The optimiza- tion was based on cells of type 1 and the initial design was always [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15, 0]nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' As we are mini- mizing Φ and maximizing Ψ, a point P = (PΦ, Pψ) in the image space of Φ and Ψ is dominating a point Q = (QΦ, Qψ) if PΦ ≤ QΦ and PΨ ≥ QΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' are visualized in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' It is observed that the with decreasing ΛΨ the compliance minimized is design (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 14a) is almost smoothly transformed into a fully flux based design (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 14h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 Optimization with two unit cell types We want to study the ability of SGP to handle more than one unit cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this purpose, we add unit cell type 2 that comprises of a void sphere surrounded by matrix material (see second row of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The only design parameter is the radius rs ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4] of the void sphere in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The smaller the void sphere, the higher the volume fraction of the matrix phase and therefore the stiffer the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Thus, cells of type 2 are par- ticularly favorable for the mechanical part of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' When only optimizing the compliance, we obtain the trivial solution shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 30 compliance fmerit 28 26 24 22 20 0 5 10 15 20 iteration0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 volume fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='15 flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='05 flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 total channel volume 0 0 5 10 15 20 iterationSequential Global Programming Applied to Fluid-saturated Porous Media 21 (a) Compliance minimized design (b) ΛΨ = −3 (c) ΛΨ = −5 (d) ΛΨ = −10 (e) ΛΨ = −15 (f) ΛΨ = −30 (g) ΛΨ = −60 (h) Flux maximized design Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 14: Visualization of optimized designs associated with the labeled points in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (a) Only cells of type 1 (b) Cells of type 2 Optimized design with Φopt = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='62 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 15: Compliance minimized designs: (a) only allowing cells of type 1 and (b) allowing choices of type 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The red dots visualize the void inclu- sions of cells of type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The optimized compliance of design (b) is 24% better than compliance of the optimized design (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For the fluid flow, cells of type 2 are futile as they are not permeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' However, for numeri- cal reasons, we set the permeability of the latter cells to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Cells of type 1 have orthotropic mechanical properties and transversal isotropic permeability tensors, whereas cells of type 2 have isotropic mechanical properties and no perme- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although cell types 1 and 2 are dis- junct in their parameter spaces, the corresponding ranges of volume fractions, of the stiff matrix material, overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We have ρ (H1([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08])) = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='9%, ρ (H1([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='22])) = 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='54% and ρ (H2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4)) = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='19%, ρ (H2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1)) = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Anodes 2 , the basis for the interpolation of H2, con- sisted of 30 uniformly distributed samples for rs ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='4] and the optimization procedure was per- formed on Agrid 2 with 60 samples, again uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Next, we present the updated Pareto front for compliance minimization and fluid flux maximiza- tion with both unit cell types in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We again 19 20 21 22 23 24 25 26 27 28 29 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 ΛΨ = −2 −5 −60 compliance Φ flux Ψ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 16: Comparison of Pareto curves for varying ΛΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Blue: optimization with cells of type 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Red: optimization with only cells of type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The blue curve clearly dominates the red curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' stress that we did not use enhanced initial designs for the computation of the points on the Pareto curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The comparison of the new (blue) curve with the old (red) curve shows that consistently better designs are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Points on the blue curve strictly dominate points on the red curve in the Pareto sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is not surprising as, with XXXX22 Sequential Global Programming Applied to Fluid-saturated Porous Media the addition of a new unit cell type, the design freedom is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Still it is worth to mention that the fact that we do not observe any outliers in this respect again underlines the stability of our SGP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The numbers of required outer iterations varied between 4 and 40, which means that no significant increase in the number of iter- ations is observed, although a second cell type has been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 17, we can observe how the number of cells of type 2, in the optimized design, decreases with decreasing ΛΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is expected, as cell type 2 is completely useless for a flux favored design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We note that so far all results presented have been computed without employing a resource con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Just to demonstrate that SGP can also easily handle problems, where a resource con- straint is added, we briefly discuss a selected result in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 Optimization with both cell types and regularization of design labels and interface We introduce a regularization of the optimiza- tion problem by applying a weighted-sum filter F (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=', [23, 26]), that is often used in the context of topology optimization, on regularization labels that are directly related to the unit cells’ geomet- ric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' For this we introduce mappings l1 : � A1 → R3 (rx, ry, ϕ) �→ R1 (63) where R1 = �rx − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='14 , ry − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='14 , cos � 2ϕ π − π 2 ��⊤ , and l2 : � A2 → R3 rs �→ R2 = (−1, −1, −1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (64) This choice of labeling has the following effects: Within type 1, the maximal distance from lower to upper label bound is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' This is the same dis- tance required to jump from the stiffest cell of type 1, with rx = ry = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='08, to any cell of type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Therefore, the interface between cells of type (a) Compliance minimized design (b) ΛΨ = −2 (c) ΛΨ = −5 (d) ΛΨ = −60 (e) Flux maximized design Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 17: Results of bicriterial optimization with cells from both type 1 and 2 for varying ΛΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The designs visualized here corresponds to the labeled data points of the pareto curve in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 1 and 2 is also penalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The most expensive change is a jump from type 1, which is preferred by the compliance, to any cell of type 2, which is most beneficial for the fluid flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The shifted cosine function appearing in the expression for Sequential Global Programming Applied to Fluid-saturated Porous Media 23 (a) Optimized design at z = 0 (b) Mechanical state with Φopt = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='78 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 18: Result of pure compliance minimization when allowing unit cells of type 1 and 2 with an active volume fraction constraint setting ¯ρm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='8 on the stiff material phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Comparing to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 17a, it is observed that only now also cells of type 1 appear in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Moreover, the resource con- straint leads to a variation of the parameter rs for cell type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (R1)3 is employed to circumvent disambiguities for the angular variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Employing these regularization labels, Jreg from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (59) changes to Jreg(R) = 1 2 3 � ℓ=1 ∥Rℓ − F(Rℓ)∥2, (65) where Rℓ ∈ Rnel collects the ℓ-the components of the regularization label assigned to each finite element, which is defined by formula eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (63) or eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' (64), if cell type 1 or cell type 2 is chosen for the corresponding finite element e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Next, we study the influence of regularization with the optimized result for the particular choice ΛΨ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The result displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 19 displays the changes in design with increasing regularization parameter pfilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The respective objective values are listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The regularization of fluid chan- nel radii can be observed well when comparing the designs in the right lower corner of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 19b and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 19c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' With increasing pfilt, the interface (a) Initial design (b) No regularization (c) ΛΞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='01 (d) ΛΞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='02 (e) ΛΞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='025 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 19: Results for varying ΛΞ with filter radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 elements and ΛΨ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' between unit cell types 1 and 2, at the right upper corner of the design domain, vanishes and the design is dominated by cells of type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' ΛΞ Jmer,opt Jreg,opt Φopt Ψopt 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='57426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='01 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0389 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='65041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='011 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0484 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='66846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='015 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0747 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='95892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='02 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='092 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='34912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='025 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0712 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='66730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0136 Table 2: Performance of designs shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 19 with Jmer,opt(ΛΞ) = Jreg,opt(ΛΞ)+Φopt +ΛΨΨopt 5 Conclusion and Outlook We presented an Sequential Global Programming (SGP) approach to homogenization-based struc- tural optimization which can be viewed as an free material optimization constrained by the set of admissible geometric material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' X Xstrain energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='0e-0124 Sequential Global Programming Applied to Fluid-saturated Porous Media By means of numerical examples, where we successively added more ingredients to the opti- mization problem, we demonstrated that the pro- posed SGP approach, with its first-order approxi- mations, provides good and reasonable optimized designs without the necessity of particular design initialization or the employment of a regulariza- tion strategy for purposes of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Fur- thermore, SGP is able to handle several material classes with disjunct parameter sets without addi- tional interpolation and penalization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We further observed that optimizing the local orientation of the microstructure brings along a significant improvement, up to 48%, of the fluid flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' We have not actively addressed the subject of connectivity within the microstructure, that is to ensure connectivity of the fluid saturated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' However, the regularization approach presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='3 can be used to control the degree of variation of the local microstruc- ture rotation and we have seen, by means of the presented numerical examples, that only a mild regularization has already a fair impact on the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Although the resolution of the finite element approximation, and thus the number of design elements, of the examples in section section 4 was chosen rather coarsely, it served the pur- pose of demonstrating the presented features of SGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' With regard to finer resolutions: the algo- rithm can be well parallelized with respect to the design elements due to the block-separability of the first-order approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The brute-force approach in the subproblem solver, described in algorithm 2, can further be speeded up by employing a hierarchical scanning of the design grids Agrid i : Start with a rather coarse number of samples and determine the minimizer among those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' In the next level, consider only the current minimizer and its neighbors and perform the same search within this subset of Agrid i , for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Repeat this step until the maximum desired number of levels or some accuracy is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Note that, with this strategy, the quality of the design depends on the number of samples on the coarsest grid level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' An alternative would be to apply a Lipschitz optimization solver, see [27], to each design element and type in a black box manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Further research will focus on extending the SGP approach for homogenization-based opti- mization to transient problems and, in partic- ular, to dynamic metamaterial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Another challenge is to extend the proposed optimiza- tion approach for an approximate treatment of nonlinear two-scale problems with the homoge- nized coefficients depending on the macroscopic response by virtue of the sensitivity analysis as discussed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 6 Acknowledgments The authors B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Vu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Stingl gratefully acknowledge the financial support by the Ger- man Federal Ministry for Economic Affairs and Climate Action (BMWK) in the course of the FIONA (LuFo VI-1, FKZ: 20W1913F) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' The research conducted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Rohan and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Lukeˇs was supported by the grant projects GACR 19-04956S and GACR 22-00863K of the Czech Scientific Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 7 Statements and Declarations The authors declare that they have no known com- peting financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' 8 Replication of results The algorithm of the proposed optimization approach was described in algorithm 1 and algo- rithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' Its implementation, as well as exemplary problem settings and respective data to reproduce the numerical results presented in section 4, are publicly available on https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content='com/bnvu/ sgp-poroel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FKT4oBgHgl3EQflC6k/content/2301.11852v1.pdf'} +page_content=' References [1] Andreasen, C.' 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a/TNAyT4oBgHgl3EQf8Pq6/content/tmp_files/2301.00854v1.pdf.txt b/TNAyT4oBgHgl3EQf8Pq6/content/tmp_files/2301.00854v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef7066dc4aa65fc20433c551a78474599a1d2e7e --- /dev/null +++ b/TNAyT4oBgHgl3EQf8Pq6/content/tmp_files/2301.00854v1.pdf.txt @@ -0,0 +1,1334 @@ +Draft version January 4, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +PHANGS-JWST First Results: Mapping the 3.3 µm Polycyclic Aromatic Hydrocarbon Vibrational +Band in Nearby Galaxies with NIRCam Medium Bands +Karin M. Sandstrom +,1 J´er´emy Chastenet +,2 Jessica Sutter +,3 Adam K. Leroy +,4 Oleg V. Egorov +,5 +Thomas G. Williams +,6, 7 Alberto D. Bolatto +,8 M´ed´eric Boquien +,9 Yixian Cao +,10 Daniel A. Dale +,11 +Janice C. Lee +,12, 13 Erik Rosolowsky +,14 Eva Schinnerer +,7 Ashley. T. Barnes +,15 +Francesco Belfiore +,16 F. Bigiel +,15 M´elanie Chevance +,17, 18 Kathryn Grasha +,19, 20 Brent Groves +,21 +Hamid Hassani +,14 Annie Hughes +,22 Ralf S. Klessen +,17, 23 J. M. Diederik Kruijssen +,18 +Kirsten L. Larson +,24 Daizhong Liu +,10 Laura A. Lopez +,4, 25, 26 Sharon E. Meidt +,2 Eric J. Murphy +,27 +Mattia C. Sormani +,17 David A. Thilker +,28 and Elizabeth J. Watkins +5 +(Received Oct 21, 2022; Revised Dec 29, 2022) +Submitted to The Astrophysical Journal Letters +ABSTRACT +We present maps of the 3.3 µm polycyclic aromatic hydrocarbon (PAH) emission feature in NGC 628, +NGC 1365, and NGC 7496 as observed with the Near-Infrared Camera (NIRCam) imager on JWST +from the PHANGS-JWST Cycle 1 Treasury project. We create maps that isolate the 3.3 µm PAH +feature in the F335M filter (F335MPAH) using combinations of the F300M and F360M filters for removal +of starlight continuum. This continuum removal is complicated by contamination of the F360M by +PAH emission and variations in the stellar spectral energy distribution slopes between 3.0 and 3.6 µm. +We modify the empirical prescription from Lai et al. (2020) to remove the starlight continuum in our +highly resolved galaxies, which have a range of starlight- and PAH-dominated lines-of-sight. Analyzing +radially binned profiles of the F335MPAH emission, we find that between 5−65% of the F335M intensity +comes from the 3.3 µm feature within the inner 0.5 r25 of our targets. This percentage systematically +varies from galaxy to galaxy, and shows radial trends within the galaxies related to each galaxy’s +distribution of stellar mass, interstellar medium, and star formation. The 3.3 µm emission is well +correlated with the 11.3 µm PAH feature traced with the MIRI F1130W filter, as is expected, since +both features arise from C-H vibrational modes. The average F335MPAH/F1130W ratio agrees with +the predictions of recent models by Draine et al. (2021) for PAHs with size and charge distributions +shifted towards larger grains with normal or higher ionization. +Keywords: Polycyclic aromatic hydrocarbons (1280), Interstellar dust (836), Medium band photometry +(1021), James Webb Space Telescope (2291) +1. INTRODUCTION +Polycyclic aromatic hydrocarbon (PAH) emission is +observed ubiquitously in the interstellar medium (ISM) +of most massive, star-forming galaxies, carrying 10-20% +of the total infrared emission (Draine et al. 2007; Smith +et al. 2007; Tielens 2008; Li 2020). Consequently, these +PAH vibrational bands are detectable in galaxies out +Corresponding author: Karin Sandstrom +kmsandstrom@ucsd.edu +to high redshift (e.g. Riechers et al. 2014). In the era +of JWST, PAH emission will be a widely observed and +valuable tracer of the ISM across a large range of red- +shifts. In particular, 3.3 µm PAH emission – the short- +est wavelength PAH feature of significant strength – will +potentially be detectable in galaxies out to z ∼ 7 with +JWST MIRI spectroscopy. +The 3.3 µm PAH feature arises from a C-H stretch- +ing vibration (Schutte et al. 1993; van Diedenhoven +et al. 2004). Laboratory and theoretical studies suggest +that the feature is dominated by small, neutral PAHs +(Maragkoudakis et al. 2020; Draine et al. 2021; Kerkeni +arXiv:2301.00854v1 [astro-ph.GA] 2 Jan 2023 + +ID2 +Sandstrom et al. +et al. 2022). The longer wavelength C-H bending mode +at 11.3 µm is also dominated by neutral PAHs but is +expected to arise from larger grains, and the ratio of +3.3 µm to 11.3 µm is therefore expected to be a rela- +tively clean diagnostic of the average size of the PAHs +(Maragkoudakis et al. 2020; Lai et al. 2020; Draine et al. +2021). +The 3.3 µm PAH feature was not covered by the spec- +troscopic instruments on Spitzer. Due to the wide wave- +length coverage of the Spitzer IRAC 3.6 µm filter, it was +difficult to isolate the 3.3 µm feature contribution to +the broadband emission (though principal component +approaches have found a signal correlated with longer +wavelength PAH emission; Meidt et al. 2012; Querejeta +et al. 2015). +Observations with ISO and Akari mea- +sured the 3.3 µm feature spectroscopically (e.g. Ver- +straete et al. 2001; Imanishi et al. 2010; Lee et al. 2012; +Lai et al. 2020), as have some ground-based studies (e.g. +Sloan et al. 1997), but these efforts have generally fo- +cused on very bright targets—Milky Way photodissocia- +tion regions, ultraluminous infrared galaxies (ULIRGs), +and galaxy nuclei, for example. JWST observations pro- +vide the first opportunity to map 3.3 µm emission in +nearby galaxies, including both star-forming regions and +the diffuse ISM. Moreover, the NIRCam medium-band +filter set of F300M, F335M, and F360M (see Figure 1 +for an illustration) provides one of the first opportuni- +ties to make large area maps of 3.3 µm emission, and use +the comparison to 11.3 µm PAH emission traced by the +MIRI F1130W filter to map PAH size variations. The +NIRCam point spread function at F300M, F335M, and +F360M has FWHM ∼ 0.10 − 0.12′′, yielding 5 − 10 pc +resolution in galaxies at distances of 10 − 20 Mpc. +There are some challenges in isolating the 3.3 µm PAH +feature using the NIRCam medium bands. The varia- +tion of the stellar spectral energy distribution appears +to be large enough that scaling F300M with a single +stellar color does not provide a clean subtraction (see +Section 3). While using a linear interpolation between +F300M and F360M can address this issue and remove +stellar continuum with a varying slope, contamination +of the F360M band by the 3.3 µm feature itself and +the nearby 3.4 µm “aliphatic” feature (thought to arise +from PAHs with an aliphatic sub-group, which encom- +passes a variety of non-aromatic hydrocarbon structures +that may be attached to the PAHs; Yang et al. 2017) +and the 3.47 µm “plateau” features also attributed to +PAHs (Hammonds et al. 2015; Lai et al. 2020) compli- +cates the linear interpolation in places with bright PAH +emission. Both features are substantially weaker than +the 3.3 µm PAH (for instance, the 3.4 µm feature is +on average ∼ 8% of the strength of 3.3 µm feature; Lai +Figure 1. The NIRCam F300M, F335M, and F360M wave- +length coverage overlaid on the 1C template spectrum from +Lai et al. (2020) (square points) with their PAHFIT-based +decomposition (Smith et al. 2007). +The continuum (in- +cluding starlight, hot dust, and the effects of attenuation) +is shown in blue. +Orange lines show the individual dust +features, including the 3.3 µm PAH feature, the 3.4 µm +“aliphatic” emission feature and the 3.47 µm plateau, which +are labeled with arrows. The 3.74 µm Pfund γ emission line +is shown in magenta. The combined fit too all components +is shown in green. +PAH (or aliphatic) dust emission fea- +tures contribute both to F335M and F360M, as well as longer +wavelength dust emission in the F360M filter. These contri- +butions complicate continuum subtraction in our highly re- +solved galaxies, where individual lines of sight may be dom- +inated by PAH emission in all three filters. +et al. 2020), but they fall in the F360M band, potentially +complicating the continuum subtraction. The degree to +which the 3.3 and 3.4 µm features vary relative to each +other is not yet well characterized in normal star-forming +galaxies (though Akari observations have shown varia- +tions in starbursts and luminous infrared galaxies; Lai +et al. 2020). Spectroscopic observations with NIRSpec +in nearby galaxies are well suited to addressing this ques- +tion in the near future. The 3.3 µm feature itself can +significantly contribute to both of the nearby medium +bands in cases where the PAH emission is bright com- +pared to the underlying continuum. For nearby galaxies, +where individual stars and diffuse emission are highly +resolved, we expect to find regions with different rela- +tive contributions from PAHs and starlight, making it +necessary to carefully test any continuum subtraction +procedure in both regimes. +In this Letter, we describe the production of the +first 3.3 µm PAH maps of NGC 628, NGC 1365, and +NGC 7496 from the PHANGS survey. +Our goal is +to demonstrate this key capability of JWST to map + +1.2 +F300M +F335M +F360M +1.0 +N +(MJy/sr) +0.8 +Normalized I, +0.6 +0.4 +- +3.3 +3.4 3.47 +0.2 +0.0 +2.8 +3.0 +3.2 +3.4 +3.6 +3.8 +4.0 +Wavelength (μum)PAH 3.3 Map +3 +Table 1. Galaxy Properties +Target +Distance +Inclination +P.A. +r25 +(Mpc) +(◦) +(◦) +(kpc) +NGC 628 +9.8 +9 +21 +14.1 +NGC 1365 +19.6 +55 +201 +34.2 +NGC 7496 +18.7 +36 +194 +9.1 +Note—Properties adopted from Leroy et al. (2021) +following Lee et al. (2022a), which draws orientations +from Lang et al. (2020) and distances from Anand +et al. (2021). +the ISM at high angular resolution (∼ 0.1′′) with the +NIRCam medium bands. The PHANGS-JWST Trea- +sury program will eventually cover 19 galaxies from the +PHANGS sample (see Lee et al. 2022a for more details). +These targets have deep, high-resolution, ancillary in- +formation from ALMA (Leroy et al. 2021), VLT-MUSE +(Emsellem et al. 2022), Hubble (Lee et al. 2022b), As- +troSat (Hassani et al. in prep), and more. Combining +NIRCam imaging of the 3.3 µm PAH feature with MIRI +imaging of the 7.7 and 11.3 µm features enables one of +the first studies of the variation of both PAH size and +charge over large regions of nearby galaxies (Chastenet +et al. 2022), in Hii regions (Egorov et al. 2022), and +near young star clusters and associations (Dale et al. +2022; Rodriguez et al. 2022). +As part of this effort +we found it was necessary to adjust the current best +NIRCam medium band continuum removal prescription +in the literature, from Lai et al. (2020), to account for +PAH emission (or related dust emission) contamination +of F360M filter (Section 3). While we expect the optimal +recipe may evolve, especially informed by future JWST +NIRSpec spectral mapping, our results already demon- +strate that the NIRCam medium band filter set is one +of the most powerful available tools to map the ISM at +high resolution and offers a new window into PAH prop- +erties. Our work here also highlights the need for future +spectroscopic calibration of the empirical medium-band +PAH continuum subtraction recipes. +2. OBSERVATIONS +We use observations of the first three galaxies from +the PHANGS-JWST survey (Lee et al. 2022a) observed +with NIRCam—NGC 628, NGC 1365, and NGC 7496. +The properties we adopt for the galaxies are listed in +Table 1. The three galaxies were observed with the long +wavelength (LW) channel of the NIRCam instrument on +JWST with the F300M, F335M, and F360M filters be- +tween July and August 2022. The targets were covered +with small 2- or 4-pointing mosaics. The total integra- +tion for each mosaic tile was 386.5 seconds in the F300M +and F335M filters and 429.4 seconds in F360M. The +resulting uncertainties in the images are ∼0.05, 0.045, +and 0.06 MJy sr−1 for the F300M, F335M, and F360M +bands. +These observations were obtained simultane- +ously with deep F200W observations in the SW (short +wavelength) channel. After the NIRCam observations, a +similar region of the galaxy was observed with the MIRI +instrument, using the F770W, F1000W, F1130W, and +F2100W filters. The observations and data processing +are described in detail in Lee et al. in this Issue. The +data processing included a step of correcting for astro- +metric offsets between the NIRCam images. In the fol- +lowing analysis, in addition to the F300M, F335M, and +F360M observations, we also make use of the F200W +and F1130W maps1. +F200W primarily samples stel- +lar continuum, providing a template for stellar emis- +sion that is relatively uncontaminated by strong PAH +or hot dust emission. The F1130W filter, on the other +hand, is quite narrowly centered on the 11.3 µm PAH +feature, yielding a measurement that in essentially all +cases is dominated by PAH emission. +The contribu- +tion of starlight at 11.3 µm is minimal and over the +0.7 µm wavelength filter width of F1130W, PAH emis- +sion greatly exceeds the contribution from other small, +stochastically heated dust grains. +Spectroscopy from +the “Spitzer Infrared Nearby Galaxies Survey” (SINGS) +shows that wavelengths around 11.3 µm are dominated +by PAH emission in ∼Z⊙ galaxies like our targets (Smith +et al. 2007; Whitcomb et al. 2022). +To correctly trace the faint diffuse emission in the +images, we made small adjustments to the background +level of the images (∼0.1 MJy sr−1). In the pipeline- +produced data products, the NIRCam maps have a small +but significant negative offset that is evident in faint re- +gions of the map. In each of the galaxies, we found ap- +proximately empty sky regions outside the galaxy to de- +termine the background level. We measured the average +values in these regions using a biweight mean algorithm +to reject outliers. +We found the average background +level from all empty sky regions and subtracted it from +the map. +After adding back in the offsets, all maps +reach ∼ 0 MJy sr−1 in their outskirts, without becom- +ing significantly negative. The origin of the background +offsets is not yet known and may be resolved with fu- +1 The filter widths of the F200W, F300M, F335M, F360M, F1130W +filters are ∆µm = 0.461, 0.318, 0.347, 0.372, and 0.7 µm. + +4 +Sandstrom et al. +ture updates to the NIRCam processing and calibration +pipeline. +To investigate the optimal continuum subtraction for +the 3.3 µm feature, in Section 3 we explore use of the +F300M and F360M filters to predict the continuum in +the F335M image. For this comparison, we do not con- +volve the images to matched resolution at this time, +since the small differences in FWHM mean that kernel +generation is subject to artifacts and noise amplification. +Our tests indicate that the FWHM of the PSF is similar +across these three bands to within ∼ 10%. We use the +F1130W to select regions with strong PAH emission in +the images. We do not convolve to match F1130W since +it is used only to select coarse regions of the image (i.e. +F1130W < 1 or > 10 MJy sr−1). We regrid the maps to +the common astrometric grid of the F335M data with +pixel scale 0.063′′ using bilinear interpolation. +We note that the PHANGS-JWST NIRCam imaging +is affected by 1/f noise. +As described in Lee et al. +(2022a), we have applied a correction that minimizes +the striping. However, in the difference images between +F300M, F335M, and F360M used in this Letter, the +stripes are prominent even after correction. Future work +or updates to the calibration files and pipeline process- +ing may be able to remove this noise. +3. SUBTRACTING 3.3 µm CONTINUUM +In Figure 2 we show images of a representative 1.5×1.5 +kpc region in NGC 628 in all the photometric bands +that we consider in this analysis. We will use this re- +gion to visualize the results of our continuum subtrac- +tion. The F300M, F335M, and F360M images are shown +with identical intensity scaling. Diffuse emission can be +seen clearly in the F335M and F360M filters. In bright +star-forming regions, diffuse emission is also evident in +F300M and F200W, potentially arising from hot dust +emission and/or nebular emission lines covered by those +filters. The diffuse emission present in the F360M filter +could be the result of a combination of 3.3 µm emis- +sion, other nearby emission lines, hot dust continuum, +and/or the 3.4 µm aliphatic feature and faint 3.47 µm +PAH “plateau” feature. +3.1. Lai et al. (2020) Prescription for 3.3 µm +Continuum Subtraction +Previous work by Lai et al. (2020) using a combination +of Akari and Spitzer spectroscopy determined a set of +coefficients for a linear combination of the F300M and +F360M bands to remove the continuum, following the +form: +F335MPAH = F335M − F335Mcont. +(1) +F335Mcont = A × F300M + B × F360M +(2) +Their recommended coefficients are ALai = 0.35 and +BLai = 0.65. +For the current investigation, we work +in surface brightness units (MJy sr−1) and do not in- +tegrate over the filter width or convert to νFν. There- +fore, our F335MPAH and F335Mcont maps are in units +of MJy sr−1. +Following Lai et al. (2020), F335MPAH +can be converted to an integrated 3.3 µm band flux in +units of 10−14 erg s−1 cm−2 by multiplying with a fac- +tor of 10.78 (appropriate for nearby galaxies at z = 0). +The Lai et al. (2020) prescription predicts colors for the +F335M continuum described by: +F335Mcont +F300M += ALai + BLai +F360M +F300M +(3) +This calibration was derived for moderately obscured +galaxies with high star formation rates. Lai et al. (2020) +find through synthetic photometry that their targets +are typically continuum dominated in the F335M band, +with ∼15% of the flux being due to the PAH feature. Lai +et al. (2020) note that water ice absorption at 3.05 µm +in the F300M band may influence their results for the +more heavily obscured targets. Our sample has minimal +obscuration on average (median E(B−V) ∼ 0.2−0.3 for +HII regions in these targets; Groves et al. submitted) +and lower star formation surface densities than most of +the Lai et al. (2020) sample. In the following, we in- +vestigate continuum subtraction with the three medium +bands, but we note that unlike Lai et al. (2020) we are +unable to spectroscopically constrain any contamination +from the 3.4 µm “aliphatic” or 3.47 µm “plateau” fea- +tures. These features are significantly fainter than the +3.3 µm band, so may not dominate the diffuse signal in +F360M. +3.2. Observed F300M-F335M-F360M Colors in PAH- +and Continuum-Dominated Regions +The PHANGS-JWST observations resolve individual +stars and ISM emission at 5−40 pc scales in the F300M +to F1130W bands. From Figure 2, it is clear that there +is a wide variation in the degree to which any given line +of sight is continuum or PAH dominated. In addition, it +is clear that in places where the emission is PAH domi- +nated, the F360M band traces PAH related emission as +well. This suggests that a simple linear interpolation in +the continuum following Equation 3 will not suffice. +To +investigate +these +issues, +we +measure +the +F300M/F335M/F360M colors in regions of the images +we expect to be continuum dominated versus PAH dom- +inated using F1130W as a guide. We first select all re- +gions where F1130W < 1 MJy sr−1 to represent a low + +PAH 3.3 Map +5 +Figure 2. A representative 1.5 ×1.5 kpc region of NGC 628 shown in the F200W, F300M, F335M, F360M, and F1130W bands +with an asinh color table at each filter’s native resolution. The F1130W filter traces primarily PAH emission from the 11.3 µm +feature. The F200W filter traces primarily stellar continuum. The F335M filter is centered on the 3.3 µm PAH feature and +includes both stellar continuum and PAH emission. Diffuse emission is visible in the F335M and F360M bands. In the F1130W +panel, we highlight faint PAH emission with F1130W < 1 MJy sr−1 with a red contour and bright PAH emission with F1130W +> 10 MJy sr−1 with a green contour. These faint and bright selections are used in Figure 3. +PAH emission region (shown with a red contour in the +bottom right panel of Figure 2). We also select regions +where F1130W > 10 MJy sr−1, representing highly PAH +emission dominated regions (green contour in bottom +right panel of Figure 2). In Figure 3, we show the re- +sults of this selection for pixels above a 5σ detection +threshold in the F300M, F335M, and F360M bands. +On Figure 3 we also include synthetic photometry in +the NIRCam medium bands for the 1A, 1B, and 1C +template spectra from Lai et al. (2020). These template +spectra show only moderate attenuation, making them +the most comparable templates to our targets. The tem- +plate spectra, which represent the spectra used to formu- +late the Lai et al. (2020) continuum prescription cover a +much narrower range of F360M/F300M colors than our +observations. +For the low PAH emission regions (F1130W < 1 MJy +sr−1), we find 3.0−3.6 µm colors in good agreement with +the Lai et al. (2020) prediction. These measurements +also suggest that starlight continuum at these wave- +lengths is not well described by a single color. This can +be seen in the extension of the both the F335M/F300M +and F360M/F300M colors along the Lai et al. (2020) +slope. If a single stellar color would suffice, these points +would be expected to be clustered around a single value +of F335M/F300M and F360M/F300M. The range of col- +ors suggests we are seeing contributions from a vari- +ety of stellar populations, including red supergiants and +asymptotic giant branch stars which show a wider range +of near- to mid-IR spectral shapes, some related to cir- +cumstellar dust (Meidt et al. 2012). +From the strongly PAH-dominated regions identified +by high F1130W surface brightness (F1130W > 10 MJy +sr−1) in Figure 3, we see that PAH emission appears in +the color-color diagram with a linear slope BPAH = 1.6 +and offset APAH = −0.2, where we describe the linear +relation in the colors with: +F335MPAH +F300M += APAH + BPAH +F360M +F300M +(4) + +F300M +F335M +F360M +MJy sr-1 +2.00 +48'0" +1.43 +55" +0.98 +50" +0.62 +45" +0.30 +40" +0.01 +15°47'35" +01h36m40.5s 40.0s +39.5s +39.0s +01h36m40.5s 40.0s +39.5s +39.0s +01h36m40.5s 40.0s +39.5s +39.0sMJy sr-1 +F200W +F1130W +MJy sr-1 +4.00 +50.0 +48'0" +2.58 +19.9 +55" +1.62 +6L +50" +0.95 +3.1 +45" +0.44 +1.1 +40" +0.01 +15°4735" +0.0 +01h36m40.5s 40.0s +39.5s +39.0s +01h36m40.5s 40.0s +39.5s +39.0s6 +Sandstrom et al. +Figure 3. F335M/F300M versus F360M/F300M color in our representative region of NGC 628 selected by F1130W surface +brightness. On the left, the PAH emission at F1130W is faint (F1130W < 1 MJy sr−1), so the 3.0-3.6 µm colors should be +dominated by stars. This region is highlighted with a red contour in Figure 2. On the right, we select bright regions in F1130W +(> 10 MJy sr−1), for which we expect the colors to be dominated by PAH emission. The PAH-bright region is highlighted with +a green contour in Figure 2. The red line in each panel shows the Lai et al. (2020) prescription for the stellar continuum. In +starlight dominated regions, this prescription does well at predicting the continuum at F335M. However, in regions dominated +by PAHs, the F360M/F300M color is also responding to the PAH emission, leading to overestimates of the F335M continuum. +We include synthetic photometry for the 1A, 1B, and 1C template spectra from Lai et al. (2020) with red symbols, illustrating +that our observed colors span a much wider range than the spectra used to create the continuum subtraction prescription. +We found the same slope using all pixels in the three +target galaxies where F1130W > 10MJy sr−1, suggest- +ing that the BPAH = 1.6 slope is a good representa- +tion of the colors of PAH dominated emission across our +sample. This comparison shows that in regions domi- +nated by PAH emission, the F360M/F300M color and +the F335M/F300M color are both tracing the PAH fea- +ture(s) in this wavelength range. Performing a linear in- +terpolation with the Lai et al. (2020) slope would lead to +subtracing off 3.3 µm PAH emission, because the F360M +filter is capturing emission from this feature. +3.3. An Optimized Continuum Subtraction Recipe for +Highly Resolved Galaxies +To avoid oversubtraction of PAH emission, we deter- +mine a correction for the predicted F335Mcont based on +the observed F335M/F300M and F360M/F300M colors. +This approach represents a first order correction to the +continuum subtraction approach from Lai et al. (2020), +and future effort on spectroscopic 3.3 µm PAH obser- +vations will be necessary to develop a more rigorous +procedure for highly resolved targets like ours. +The +nature of this correction is shown in Figure 4. +The +basic approach is to use the observed F335M/F300M +color as an indication of how PAH-dominated the emis- +sion is in a given location. +Using the observed slope +of BPAH = 1.6 for PAH-dominated emission, we can +scale the F360M/F300M color to where it intersects +the relationship from Lai et al. (2020), obtaining a cor- +rected F360M/F300M color with which to predict the +F335Mcont. The intersection of these two lines is where +the PAH contribution to both colors is assumed to be +zero. +For each point, we scale the measured F360M/F300M +value (xm) and F335M/F300M value (ym) along the +PAH slope to where it intersects with the continuum +relationship described by Lai et al. (2020). This yields a +corrected F360M/F300M ratio (xc) and F335M/F300M +ratio (yc). The PAH slope is described as: +y = APAH + BPAHx, +(5) +and the Lai et al. (2020) slope is given by: +y = ALai + BLaix. +(6) +Using Equation 5, we can write the relationship between +xm, ym and xc, yc as: +ym − yc +xm − xc += BPAH. +(7) + +F1130W < 1 MJy sr-1 +F1130w > 10 MJy sr-1 +5 +4 +F335M/F300M +3 +2 +1C +Lai et al. 2020 +:B +1 +0 +2 +30 +1 +1 +2 +3 +F360M/F300M +F360M/F300MPAH 3.3 Map +7 +Figure +4. +Illustration +of +the +correction +to +the +F360M/F300M color to minimize over-subtraction of PAH +emission from the F335M filter. +The correction works by +scaling the colors along the PAH-dominated color trend +which has a slope of BPAH = 1.6. The black points show +the pixel-by-pixel colors derived in PAH-faint regions of the +map (F1130W < 1 MJy sr−1). The gray points show colors +in the PAH-bright regions (F1130W > 10 MJy sr−1). Two +examples of the correction are shown with yellow and cyan +stars. The measured values xm, ym are scaled along the PAH +slope till it intersects the Lai et al. (2020) relation for stellar +continuum colors to obtain corrected values xc, yc. +The corrected values will lie on Equation 6, so: +yc = ALai + BLaixc +(8) +We can then write xc in terms of the measured colors +(xm, ym) as follows: +xc = BPAHxm − ym + ALai +BPAH − BLai +. +(9) +Putting this back into the Lai et al. (2020) formula, we +can then obtain a prediction for yc, which represents the +F335M/F300M color appropriate for continuum, and +subsequently the F335M continuum as follows: +yc = BLai +�BPAHxm − ym + ALai +BPAH − BLai +� ++ ALai +(10) +F335Mcont = yc × F300M. +(11) +In Figure 5 we show a comparison of the F335M con- +tinuum derived from the above method and that from +Lai et al. (2020). The right panel shows the difference +in the continuum predicted by the two techniques. It +is clear that the corrected formula more cleanly isolates +stellar continuum, minimizing over-subtraction of PAH +emission, while maintaining the success of the Lai et al. +(2020) formula at subtracting starlight. While this em- +pirical correction can be fine-tuned in the future, this +approach provides a straightforward technique to mea- +sure the 3.3 µm feature for these first science applica- +tions with NIRCam medium band imaging. We proceed +in the following Section to interpret the resulting maps +of PAH emission. +In the future, we plan to investigate more sophisti- +cated approaches to determining the PAH contamina- +tion of the various NIRCam medium bands. The ba- +sic assumptions in this approach are that the F300M +is relatively uncontaminated by diffuse emission, and +the F335M/F300M ratio therefore gives a reasonable +estimation of the degree of PAH emission. +However, +since there is a range of colors appropriate for the stars, +a single scaled stellar spectral energy distribution tied +to F300M does not appear to work well at removing +the continuum. +Instead, we interpret the spread in +F360M/F300M at a fixed F335M/F300M as variations +in the stellar SED. This may not be the case, as vari- +ous other effects can alter the 3.0-3.6 µm colors, includ- +ing variations in the 3.4 µm feature, extinction, ice ab- +sorption features, other diffuse emission from hot dust +and/or nebular emission. Future work comparing the +3.0 − 3.6 µm stellar colors with stellar population mod- +eling based on the PHANGS-MUSE observations (Em- +sellem et al. 2022) may allow development of more pre- +cise stellar continuum recipes. Future spectroscopic cal- +ibration of F335M continuum subtraction recipes will be +critical to fully exploit the capability to map PAH emis- +sion with medium band filters on NIRCam and move +beyond the first-order correction presented here. +4. RESULTS +In Figures 6, 7, and 8 we show the F335MPAH +and F335Mcont maps for all three targets, using our +continuum subtraction scheme described above. +The +F335MPAH maps are the highest resolution view of the +PAH emission in these galaxies, with linear resolution +between 5-10 pc. +This resolution is similar to what +can be achieved with Hubble optical imaging. +We +find typical uncertainties of σ ∼ 0.07 MJy sr−1 in the +F335MPAH maps for these galaxies, as measured in faint +regions of the map with minimal emission (this value +also matches expectations from propagating measured +errors in F300M, F335M, and F360M through the cor- +rection formulae). Given that our observations required +only ∼ 400 seconds of integration per field, the sensi- +tivity of the maps is impressive. In the future, deeper + +3 +F335M/F300M +Measured Xm,ym +2 +Lai et al. (2020) +Corrected Xc,yo +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +F360M/F300M8 +Sandstrom et al. +Figure 5. The predicted F335Mcont from the Lai et al. (2020) formula in Equation 3 (left), the empirical prescription presented +in this work in Equations 10 and +11 (middle), and the difference between them (right) for our representative region from +NGC 628 shown in Figure 2. The left and middle panels show the same 1.5×1.5 kpc region of NGC 628 shown in Figure 2 with +the same asinh color table. The right panel shows that significant amounts of diffuse emission are included in the F335Mcont +from the Lai et al. (2020) prescription because of the fact that F360M also includes some PAH-related emission. Our prescription +minimizes this contamination by correcting the F360M/F300M colors while still cleanly subtracting the stellar continuum. +observations with medium bands to map the 3.3 µm +PAH feature will be straightforward with NIRCam. +4.1. PAH-to-Continuum Ratios +As a result of our continuum subtraction, we can +measure the fraction of the F335M band that is due +to PAH emission (i.e. +F335MPAH/F335M). We show +radial profiles of this fraction in Figure 9 (left). +To +create these profiles, we binned the F335MPAH and +F335M intensities in bins of 0.01r25, including all pix- +els where the emission in both filters was detected at +> 3σ. +We measure the median of the F335MPAH +and F335M in these bins and then divide to obtain +the ratio as a function of radius. +This value is low- +est in the galaxy centers, where high stellar mass sur- +face density leads to starlight dominating the F335M +band. In all three galaxies the fraction increases rela- +tively smoothly with radius outside the central 0.1r25. +Values range between 5 − 65% , with NGC 628 span- +ning both the lowest and highest part of that range over +the range of radii we cover (∼ 0.3r25). Both NGC 1365 +and 7496 are barred galaxies with high central gas sur- +face densities associated with circumnuclear star form- +ing rings. These regions produce strong PAH emission, +which causes the upturn in the F335MPAH/F335M in +their centers. +NGC 628, on the other hand, shows +a monotonic increase in F335MPAH/F335M with ra- +dius, suggesting that starlight surface brightness falls +more rapidly than PAH surface brightness as a func- +tion of radius, which may reflect varying scale-lengths +for the stellar mass and ISM distributions. The vary- +ing behavior even among these first three targets from +PHANGS-JWST emphasizes the need for continuum re- +moval recipes that work for both continuum and PAH- +dominated sight-lines, since starlight makes up a highly +variable fraction of the emission in the NIRCam medium +bands. +4.2. Typical 3.3/11.3 PAH Feature Ratios and +Comparison to Draine et al. (2021) +Both the 3.3 and 11.3 µm PAH features are thought to +arise from vibrations in C-H bonds, which are strongest +from neutral grains (Schutte et al. 1993; van Dieden- +hoven et al. 2004; Kerkeni et al. 2022). Because smaller +PAHs gain more energy per vibrational mode upon ab- +sorbing a UV photon of a given energy, they are able to +more effectively excite the shorter wavelength emission +at 3.3 µm compared to larger PAHs. Thus, the 3.3/11.3 +ratio for a fixed radiation field spectrum is expected to +trace the average size of the grains (Maragkoudakis et al. +2020; Draine et al. 2021; Rigopoulou et al. 2021). Recent +models from Draine et al. (2021) predict the 3.3/11.3 ra- +tio for a range of PAH size and charge distributions, in +radiation fields of varying intensity and hardness. +In the right panel of Figure 9, we show the radial +profile of the F335MPAH/F1130W ratio for each galaxy. +The data are radially binned as described in Section 4.1, +with the addition of masking out radial ranges affected +by saturation for the F1130W filter. +We masked the +inner 7.5′′ radius region, which extends to the radius +where the point spread function for the bright central +source drops to 10−3 of its peak value (see Hassani et al. +2022, for further details). This corresponds to a cut at +< 0.04r25 (< 1.4 kpc) for NGC 1365 and < 0.08r25 +(< 0.7 kpc) for NGC 7496. We additionally mask the +very central region of NGC 628 (< 0.02r25, or < 0.3 +kpc) where evolved stellar populations contribute to the +F1130W emission in a hole in the ISM distribution near + +MJy sr-1 +F335M. +This Work +Lai - This Work +2.00 +0.2 +0.1 +48'0" +1.43 +0.1 +55" +0.98 +50" +0.0 +0.62 +45" +-0.1 +0.30 +40" +-0.1 +0.01 +15°47'35" +-0.2 +01h36m40.5°40.0s +39.5s +39.0s +01h36m40.5°40.0s +39.5s +39.0s +01h36m40.5*40.0s +39.5s +39.0sPAH 3.3 Map +9 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +F335M +(MJy sr-1) +500 pc +500 pc ++15° 46′ 00″ +47′ 00″ +48′ 00″ +49′ 00″ ++1h 45m 00s ++1h 40m 00s ++1h 45m 00s ++1h 40m 00s +F335MPAH +F335MCont ++15° 46′ 00″ +47′ 00″ +48′ 00″ +49′ 00″ +Figure 6. PAH 3.3 µm emission and continuum for NGC 628. +the nuclear star cluster (see Hoyer et al. 2022). +Er- +ror bars show the error on the mean. Our results show +variations of the ratio between 0.02 − 0.08, with an av- +erage value of ∼ 0.05 across the three targets. These +ratios are investigated in more detail in Chastenet et al. +(2022) and Dale et al. (2022). For comparison, we plot +two ratios from the Draine et al. (2021) models. These +values are derived by filter convolutions of the models +as described in Dale et al. (2022). We use the results +for a stellar population age of 1 Gyr and representative +size (“small”, “standard”, and “large”) and charge dis- +tributions (“low”, “standard”, and “high”). For most of +these combinations, the predicted F335MPAH/F1130W +predictions are ouside our plot range, but our results are +consistent with emission from a population of PAHs with +“large” characteristic size and “standard” or “high” ion- +ization. This is in agreement with the results from Dale +et al. (2022) who also find the best alignment with the +“large” and “high” ionization Draine et al. (2021) model +grid results, although they use much younger stellar pop- +ulation ages to represent the environments of embedded +clusters. +5. CONCLUSIONS +The capability of NIRCam medium bands on JWST +to map the 3.3 µm PAH feature at high angular resolu- +tion and sensitivity provides an invaluable new tool for +studying PAHs. Combined with the longer wavelength +MIRI imaging, the 3.3/11.3 PAH feature ratio (traced by +F335MPAH/F1130W) presents one of the cleanest diag- +nostics of PAH size, helping to interpret a range of other +band ratio variations (e.g. 7.7/11.3) which can have both +size and charge dependence. In addition, the 3.3 µm fea- +ture can be mapped with NIRCam at 2 − 3 times finer +angular resolution than the 7.7 µm or 11.3 µm bands, + +1.8 +1.6 +1.4 +1.2 +0.8 +0.6 +0.4 +0.21.8 +1.6 +1.4 +1.2 +0.8 +0.6 +0.4 +0.210 +Sandstrom et al. +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +F335M +(MJy sr-1) +500 pc +500 pc ++3h 33m 40s ++3h 33m 35s ++3h 33m 30s ++3h 33m 45s ++3h 33m 25s ++3h 33m 40s ++3h 33m 35s ++3h 33m 30s ++3h 33m 45s ++3h 33m 25s +-36° 09′ 00″ +8′ 30″ +8′ 00″ +7′ 30″ +F335MPAH +F335MCont +-36° 09′ 00″ +8′ 30″ +8′ 00″ +7′ 30″ +Figure 7. PAH 3.3 µm emission and continuum for NGC 1365. +yielding 5–10 pc resolution in our targets. This allows +measurements of the sizes of H II regions and bubbles +(see Watkins et al. 2022; Barnes et al. 2022), the iden- +tification of filamentary structure (Thilker et al. subm.; +Meidt et al. 2022), the identification of embedded clus- +ters Rodriguez et al. (2022), and potentially tracing the +gas column at higher resolution than is routinely possi- +ble with any millimeter or radio facilities (Leroy et al. +2022; Sandstrom et al. subm.). +In this Letter, we have presented a first approach to +using the NIRCam medium bands F300M, F335M, and +F360M to create a map of the 3.3 µm PAH feature. +We find a key consideration for highly resolved galax- +ies like our targets is to correct the F360M/F300M col- +ors to account for contamination by PAH emission (the +3.3 µm feature itself, the 3.4 µm aliphatic, and 3.47 +µm plateau features) in F360M. We develop an empir- +ical first-order correction to the Lai et al. (2020) pre- +scription, which combines the successes of the Lai et al. +(2020) formula at removing starlight with a scaling using +the F335M/F300M colors to correct for PAH contami- +nation in F360M. We demonstrate that this approach +succeeds in mitigating over-subtraction of PAH emis- +sion from F335M that would result from a simple linear + +1.8 +1.6 +1.4 +1.2 +0.8 +0.6 +0.4 +0.21.8 +1.6 +1.4 +1.2 +0.8 +0.6 +0.4 +0.21.8 +1.6 +1.4 +1.2 +0.8 +0.6 +0.4 +0.2PAH 3.3 Map +11 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +F335M +(MJy sr-1) +F335MPAH +F335MCont +500 pc +500 pc +-43° 26′ 30″ +26′ 00″ +25′ 30″ +25′ 00″ +-43° 26′ 30″ +26′ 00″ +25′ 30″ +25′ 00″ ++23h 09m 52s ++23h 09m 48s ++23h 09m 44s ++23h 09m 52s ++23h 09m 48s ++23h 09m 44s +Figure 8. PAH 3.3 µm emission and continuum for NGC 7496. +Figure 9. (left) Fraction of the F335M band from the 3.3µm PAH feature as a function of galactocentric radius in units +of r25. Error bars show the error on the mean, which is very small given the large number of measurements that contribute +in each radial bin. The fraction of the F335M filter emission that traces PAHs varies systematically from the inner, stellar +continuum dominated regions to the fainter outskirts of the disk. (right) The ratio of the 3.3 and 11.3 µm PAH features (traced +by F335MPAH/F1130W) as a function of galactocentric radius (error bars show error on the mean). Due to saturated sources +in the centers of NGC 1365 and 7496 at F1130W, we have masked the inner r = 7.5′′ region, which corresponds to the inner +0.04 r25 for NGC 1365, and the inner 0.08 r25 for NGC 7496. We have also masked the inner 0.02 r25 for NGC 628 due to +contamination by evolved stellar populations at F1130W. Significant variations in the median F335MPAH/F1130W ratios as a +function of radius exist in the galaxies. Our observed ratios are generally consistent with expectations from the Draine et al. +(2021) models for PAH size distributions shifted towards larger grains and charge distributions with either “high” or “standard” +ionization. + +1.8 +1.6 +1.4 +1.2 +1 +0.8 +0.6 +0.4 +0.21.8 +1.6 +1.4 +1.2 +1 +0.8 +0.6 +0.4 +0.21.8 +1.6 +1.4 +1.2 +0.8 +0.6 +0.4 +0.212 +Sandstrom et al. +interpolation across the bands. Future work to calibrate +the continuum subtraction using NIRSpec observations +in highly resolved nearby galaxies will be critical to move +beyond our first-order correction, and deal with effects +such as attenuation, absorption features, stellar popu- +lation variations, hot dust, and/or nebular emission in +the various bands. +ACKNOWLEDGEMENTS +This work is based on observations made with the +NASA/ESA/CSA James Webb Space Telescope. The +data were obtained from the Mikulski Archive for Space +Telescopes at the Space Telescope Science Institute, +which is operated by the Association of Universities for +Research in Astronomy, Inc., under NASA contract NAS +5-03127 for JWST. These observations are associated +with program 2107. The specific observations analyzed +can be accessed via 10.17909/9bdf-jn24. +The authors thank the anonymous referee for feedback +that improved the paper. The authors thank Thomas +Lai for helpful conversations and providing fits to tem- +plate spectra used in Figure 1. KS acknowledges funding +support from grant support by JWST-GO-02107.006- +A. TGW acknowledges funding from the European Re- +search Council (ERC) under the European Union’s Hori- +zon 2020 research and innovation programme (grant +agreement No. 694343). JMDK gratefully acknowledges +funding from the European Research Council (ERC) un- +der the European Union’s Horizon 2020 research and in- +novation programme via the ERC Starting Grant MUS- +TANG (grant agreement number 714907). COOL Re- +search DAO is a Decentralized Autonomous Organiza- +tion supporting research in astrophysics aimed at un- +covering our cosmic origins. +MB acknowledges sup- +port from FONDECYT regular grant 1211000 and by +the ANID BASAL project FB210003. +EJW acknowl- +edges the funding provided by the Deutsche Forschungs- +gemeinschaft (DFG, German Research Foundation) – +Project-ID 138713538 – SFB 881 (“The Milky Way +System”, subproject P1). MC gratefully acknowledges +funding from the DFG through an Emmy Noether Re- +search Group (grant number CH2137/1-1). FB would +like to acknowledge funding from the European Re- +search Council (ERC) under the European Union’s Hori- +zon 2020 research and innovation programme (grant +agreement No.726384/Empire) ER and HH acknowledge +the support of the Natural Sciences and Engineering +Research Council of Canada (NSERC), funding refer- +ence number RGPIN-2022-03499. KG is supported by +the Australian Research Council through the Discov- +ery Early Career Researcher Award (DECRA) Fellow- +ship DE220100766 funded by the Australian Govern- +ment. +KG is supported by the Australian Research +Council Centre of Excellence for All Sky Astrophysics +in 3 Dimensions (ASTRO 3D), through project num- +ber CE170100013. JC acknowledges support from ERC +starting grant #851622 DustOrigin. AKL gratefully ac- +knowledges support by grants 1653300 and 2205628 from +the National Science Foundation, by award JWST-GO- +02107.009-A, and by a Humboldt Research Award from +the Alexander von Humboldt Foundation. 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Bigiel +,15 M´elanie Chevance +,17, 18 Kathryn Grasha +,19, 20 Brent Groves +,21 +Hamid Hassani +,14 Annie Hughes +,22 Ralf S. Klessen +,17, 23 J. M. Diederik Kruijssen +,18 +Kirsten L. Larson +,24 Daizhong Liu +,10 Laura A. Lopez +,4, 25, 26 Sharon E. Meidt +,2 Eric J. Murphy +,27 +Mattia C. Sormani +,17 David A. Thilker +,28 and Elizabeth J. Watkins +5 +1Center for Astrophysics & Space Sciences, Department of Physics, University of California, San Diego, 9500 Gilman Drive, San Diego, +CA 92093, USA +2Sterrenkundig Observatorium, Ghent University, Krijgslaan 281-S9, 9000 Gent, Belgium +3Center for Astrophysics & Space Sciences, University of California, San Diego, 9500 Gilman Drive, San Diego, CA 92093, USA +4Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA +5Astronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg, M¨onchhofstraße 12-14, 69120 Heidelberg, Germany +6Sub-department of Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK +7Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117 Heidelberg, Germany +8Department of Astronomy and Joint Space-Science Institute, University of Maryland, College Park, MD 20742, USA +9Centro de Astronom´ıa (CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta, Chile +10Max-Planck-Institut f¨ur Extraterrestrische Physik (MPE), Giessenbachstraße 1, D-85748 Garching, Germany +11Department of Physics and Astronomy, University of Wyoming, Laramie, WY 82071, USA +12Gemini Observatory/NSF’s NOIRLab, 950 N. Cherry Avenue, Tucson, AZ, USA +13Steward Observatory, University of Arizona, 933 N Cherry Ave,Tucson, AZ 85721, USA +14Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2E1, Canada +15Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germany +16INAF — Arcetri Astrophysical Observatory, Largo E. Fermi 5, I-50125, Florence, Italy +17Universit¨at Heidelberg, Zentrum f¨ur Astronomie, Institut f¨ur Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, +Germany +18Cosmic Origins Of Life (COOL) Research DAO, coolresearch.io +19Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia +20ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia +21International Centre for Radio Astronomy Research, University of Western Australia, 7 Fairway, Crawley, 6009 WA, Australia +22IRAP, Universit´e de Toulouse, CNRS, CNES, UPS, (Toulouse), France +23Universit¨at Heidelberg, Interdisziplin¨ares Zentrum f¨ur Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, +Germany +24AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +25Center for Cosmology and Astroparticle Physics, 191 West Woodruff Avenue, Columbus, OH 43210, USA +26Flatiron Institute, Center for Computational Astrophysics, NY 10010, USA +27National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA +28Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA + +ID \ No newline at end of file diff --git a/TNAyT4oBgHgl3EQf8Pq6/content/tmp_files/load_file.txt b/TNAyT4oBgHgl3EQf8Pq6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65e3284312cca1c53903c7a98af604482c11fe57 --- /dev/null +++ b/TNAyT4oBgHgl3EQf8Pq6/content/tmp_files/load_file.txt @@ -0,0 +1,1197 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf,len=1196 +page_content='Draft version January 4, 2023 Typeset using LATEX twocolumn style in AASTeX631 PHANGS-JWST First Results: Mapping the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm Polycyclic Aromatic Hydrocarbon Vibrational Band in Nearby Galaxies with NIRCam Medium Bands Karin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Sandstrom ,1 J´er´emy Chastenet ,2 Jessica Sutter ,3 Adam K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Leroy ,4 Oleg V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Egorov ,5 Thomas G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Williams ,6, 7 Alberto D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Bolatto ,8 M´ed´eric Boquien ,9 Yixian Cao ,10 Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Dale ,11 Janice C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lee ,12, 13 Erik Rosolowsky ,14 Eva Schinnerer ,7 Ashley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Barnes ,15 Francesco Belfiore ,16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Bigiel ,15 M´elanie Chevance ,17, 18 Kathryn Grasha ,19, 20 Brent Groves ,21 Hamid Hassani ,14 Annie Hughes ,22 Ralf S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Klessen ,17, 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Diederik Kruijssen ,18 Kirsten L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Larson ,24 Daizhong Liu ,10 Laura A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lopez ,4, 25, 26 Sharon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Meidt ,2 Eric J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Murphy ,27 Mattia C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Sormani ,17 David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Thilker ,28 and Elizabeth J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Watkins 5 (Received Oct 21, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Revised Dec 29, 2022) Submitted to The Astrophysical Journal Letters ABSTRACT We present maps of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm polycyclic aromatic hydrocarbon (PAH) emission feature in NGC 628, NGC 1365, and NGC 7496 as observed with the Near-Infrared Camera (NIRCam) imager on JWST from the PHANGS-JWST Cycle 1 Treasury project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We create maps that isolate the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature in the F335M filter (F335MPAH) using combinations of the F300M and F360M filters for removal of starlight continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This continuum removal is complicated by contamination of the F360M by PAH emission and variations in the stellar spectral energy distribution slopes between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We modify the empirical prescription from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) to remove the starlight continuum in our highly resolved galaxies, which have a range of starlight- and PAH-dominated lines-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Analyzing radially binned profiles of the F335MPAH emission, we find that between 5−65% of the F335M intensity comes from the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature within the inner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 r25 of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This percentage systematically varies from galaxy to galaxy, and shows radial trends within the galaxies related to each galaxy’s distribution of stellar mass, interstellar medium, and star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emission is well correlated with the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature traced with the MIRI F1130W filter, as is expected, since both features arise from C-H vibrational modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The average F335MPAH/F1130W ratio agrees with the predictions of recent models by Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) for PAHs with size and charge distributions shifted towards larger grains with normal or higher ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Keywords: Polycyclic aromatic hydrocarbons (1280), Interstellar dust (836), Medium band photometry (1021), James Webb Space Telescope (2291) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' INTRODUCTION Polycyclic aromatic hydrocarbon (PAH) emission is observed ubiquitously in the interstellar medium (ISM) of most massive, star-forming galaxies, carrying 10-20% of the total infrared emission (Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Tielens 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Li 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Consequently, these PAH vibrational bands are detectable in galaxies out Corresponding author: Karin Sandstrom kmsandstrom@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='edu to high redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Riechers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the era of JWST, PAH emission will be a widely observed and valuable tracer of the ISM across a large range of red- shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In particular, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH emission – the short- est wavelength PAH feature of significant strength – will potentially be detectable in galaxies out to z ∼ 7 with JWST MIRI spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature arises from a C-H stretch- ing vibration (Schutte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' van Diedenhoven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Laboratory and theoretical studies suggest that the feature is dominated by small, neutral PAHs (Maragkoudakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Kerkeni arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='00854v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='GA] 2 Jan 2023 ID2 Sandstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The longer wavelength C-H bending mode at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm is also dominated by neutral PAHs but is expected to arise from larger grains, and the ratio of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm is therefore expected to be a rela- tively clean diagnostic of the average size of the PAHs (Maragkoudakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature was not covered by the spec- troscopic instruments on Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Due to the wide wave- length coverage of the Spitzer IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 µm filter, it was difficult to isolate the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature contribution to the broadband emission (though principal component approaches have found a signal correlated with longer wavelength PAH emission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Meidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Observations with ISO and Akari mea- sured the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature spectroscopically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Ver- straete et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Imanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020), as have some ground-based studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 1997), but these efforts have generally fo- cused on very bright targets—Milky Way photodissocia- tion regions, ultraluminous infrared galaxies (ULIRGs), and galaxy nuclei, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' JWST observations pro- vide the first opportunity to map 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emission in nearby galaxies, including both star-forming regions and the diffuse ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Moreover, the NIRCam medium-band filter set of F300M, F335M, and F360M (see Figure 1 for an illustration) provides one of the first opportuni- ties to make large area maps of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emission, and use the comparison to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH emission traced by the MIRI F1130W filter to map PAH size variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The NIRCam point spread function at F300M, F335M, and F360M has FWHM ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='10 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='12′′, yielding 5 − 10 pc resolution in galaxies at distances of 10 − 20 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' There are some challenges in isolating the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature using the NIRCam medium bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The varia- tion of the stellar spectral energy distribution appears to be large enough that scaling F300M with a single stellar color does not provide a clean subtraction (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' While using a linear interpolation between F300M and F360M can address this issue and remove stellar continuum with a varying slope, contamination of the F360M band by the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature itself and the nearby 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm “aliphatic” feature (thought to arise from PAHs with an aliphatic sub-group, which encom- passes a variety of non-aromatic hydrocarbon structures that may be attached to the PAHs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2017) and the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='47 µm “plateau” features also attributed to PAHs (Hammonds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020) compli- cates the linear interpolation in places with bright PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Both features are substantially weaker than the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH (for instance, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm feature is on average ∼ 8% of the strength of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The NIRCam F300M, F335M, and F360M wave- length coverage overlaid on the 1C template spectrum from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) (square points) with their PAHFIT-based decomposition (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The continuum (in- cluding starlight, hot dust, and the effects of attenuation) is shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Orange lines show the individual dust features, including the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm “aliphatic” emission feature and the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='47 µm plateau, which are labeled with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='74 µm Pfund γ emission line is shown in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The combined fit too all components is shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' PAH (or aliphatic) dust emission fea- tures contribute both to F335M and F360M, as well as longer wavelength dust emission in the F360M filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These contri- butions complicate continuum subtraction in our highly re- solved galaxies, where individual lines of sight may be dom- inated by PAH emission in all three filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020), but they fall in the F360M band, potentially complicating the continuum subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The degree to which the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm features vary relative to each other is not yet well characterized in normal star-forming galaxies (though Akari observations have shown varia- tions in starbursts and luminous infrared galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Spectroscopic observations with NIRSpec in nearby galaxies are well suited to addressing this ques- tion in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature itself can significantly contribute to both of the nearby medium bands in cases where the PAH emission is bright com- pared to the underlying continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For nearby galaxies, where individual stars and diffuse emission are highly resolved, we expect to find regions with different rela- tive contributions from PAHs and starlight, making it necessary to carefully test any continuum subtraction procedure in both regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In this Letter, we describe the production of the first 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH maps of NGC 628, NGC 1365, and NGC 7496 from the PHANGS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our goal is to demonstrate this key capability of JWST to map 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 F300M F335M F360M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 N (MJy/sr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 Normalized I, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 Wavelength (μum)PAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 Map 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Galaxy Properties Target Distance Inclination P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' r25 (Mpc) (◦) (◦) (kpc) NGC 628 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 9 21 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 NGC 1365 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 55 201 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 NGC 7496 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7 36 194 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 Note—Properties adopted from Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) following Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022a), which draws orientations from Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) and distances from Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' the ISM at high angular resolution (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1′′) with the NIRCam medium bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The PHANGS-JWST Trea- sury program will eventually cover 19 galaxies from the PHANGS sample (see Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022a for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These targets have deep, high-resolution, ancillary in- formation from ALMA (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2021), VLT-MUSE (Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022), Hubble (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022b), As- troSat (Hassani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' in prep), and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Combining NIRCam imaging of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature with MIRI imaging of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm features enables one of the first studies of the variation of both PAH size and charge over large regions of nearby galaxies (Chastenet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022), in Hii regions (Egorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022), and near young star clusters and associations (Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' As part of this effort we found it was necessary to adjust the current best NIRCam medium band continuum removal prescription in the literature, from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020), to account for PAH emission (or related dust emission) contamination of F360M filter (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' While we expect the optimal recipe may evolve, especially informed by future JWST NIRSpec spectral mapping, our results already demon- strate that the NIRCam medium band filter set is one of the most powerful available tools to map the ISM at high resolution and offers a new window into PAH prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our work here also highlights the need for future spectroscopic calibration of the empirical medium-band PAH continuum subtraction recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' OBSERVATIONS We use observations of the first three galaxies from the PHANGS-JWST survey (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022a) observed with NIRCam—NGC 628, NGC 1365, and NGC 7496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The properties we adopt for the galaxies are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The three galaxies were observed with the long wavelength (LW) channel of the NIRCam instrument on JWST with the F300M, F335M, and F360M filters be- tween July and August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The targets were covered with small 2- or 4-pointing mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The total integra- tion for each mosaic tile was 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 seconds in the F300M and F335M filters and 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 seconds in F360M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The resulting uncertainties in the images are ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='045, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='06 MJy sr−1 for the F300M, F335M, and F360M bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These observations were obtained simultane- ously with deep F200W observations in the SW (short wavelength) channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' After the NIRCam observations, a similar region of the galaxy was observed with the MIRI instrument, using the F770W, F1000W, F1130W, and F2100W filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The observations and data processing are described in detail in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' in this Issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The data processing included a step of correcting for astro- metric offsets between the NIRCam images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the fol- lowing analysis, in addition to the F300M, F335M, and F360M observations, we also make use of the F200W and F1130W maps1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' F200W primarily samples stel- lar continuum, providing a template for stellar emis- sion that is relatively uncontaminated by strong PAH or hot dust emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The F1130W filter, on the other hand, is quite narrowly centered on the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature, yielding a measurement that in essentially all cases is dominated by PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The contribu- tion of starlight at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm is minimal and over the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7 µm wavelength filter width of F1130W, PAH emis- sion greatly exceeds the contribution from other small, stochastically heated dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Spectroscopy from the “Spitzer Infrared Nearby Galaxies Survey” (SINGS) shows that wavelengths around 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm are dominated by PAH emission in ∼Z⊙ galaxies like our targets (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Whitcomb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' To correctly trace the faint diffuse emission in the images, we made small adjustments to the background level of the images (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 MJy sr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the pipeline- produced data products, the NIRCam maps have a small but significant negative offset that is evident in faint re- gions of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In each of the galaxies, we found ap- proximately empty sky regions outside the galaxy to de- termine the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We measured the average values in these regions using a biweight mean algorithm to reject outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We found the average background level from all empty sky regions and subtracted it from the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' After adding back in the offsets, all maps reach ∼ 0 MJy sr−1 in their outskirts, without becom- ing significantly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The origin of the background offsets is not yet known and may be resolved with fu- 1 The filter widths of the F200W, F300M, F335M, F360M, F1130W filters are ∆µm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='461, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='318, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='347, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='372, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 4 Sandstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' ture updates to the NIRCam processing and calibration pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' To investigate the optimal continuum subtraction for the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature, in Section 3 we explore use of the F300M and F360M filters to predict the continuum in the F335M image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For this comparison, we do not con- volve the images to matched resolution at this time, since the small differences in FWHM mean that kernel generation is subject to artifacts and noise amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our tests indicate that the FWHM of the PSF is similar across these three bands to within ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We use the F1130W to select regions with strong PAH emission in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We do not convolve to match F1130W since it is used only to select coarse regions of the image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' F1130W < 1 or > 10 MJy sr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We regrid the maps to the common astrometric grid of the F335M data with pixel scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='063′′ using bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We note that the PHANGS-JWST NIRCam imaging is affected by 1/f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' As described in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022a), we have applied a correction that minimizes the striping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' However, in the difference images between F300M, F335M, and F360M used in this Letter, the stripes are prominent even after correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Future work or updates to the calibration files and pipeline process- ing may be able to remove this noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' SUBTRACTING 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm CONTINUUM In Figure 2 we show images of a representative 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 kpc region in NGC 628 in all the photometric bands that we consider in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We will use this re- gion to visualize the results of our continuum subtrac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The F300M, F335M, and F360M images are shown with identical intensity scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Diffuse emission can be seen clearly in the F335M and F360M filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In bright star-forming regions, diffuse emission is also evident in F300M and F200W, potentially arising from hot dust emission and/or nebular emission lines covered by those filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The diffuse emission present in the F360M filter could be the result of a combination of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emis- sion, other nearby emission lines, hot dust continuum, and/or the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm aliphatic feature and faint 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='47 µm PAH “plateau” feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) Prescription for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm Continuum Subtraction Previous work by Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) using a combination of Akari and Spitzer spectroscopy determined a set of coefficients for a linear combination of the F300M and F360M bands to remove the continuum, following the form: F335MPAH = F335M − F335Mcont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (1) F335Mcont = A × F300M + B × F360M (2) Their recommended coefficients are ALai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='35 and BLai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For the current investigation, we work in surface brightness units (MJy sr−1) and do not in- tegrate over the filter width or convert to νFν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' There- fore, our F335MPAH and F335Mcont maps are in units of MJy sr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Following Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020), F335MPAH can be converted to an integrated 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm band flux in units of 10−14 erg s−1 cm−2 by multiplying with a fac- tor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='78 (appropriate for nearby galaxies at z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) prescription predicts colors for the F335M continuum described by: F335Mcont F300M = ALai + BLai F360M F300M (3) This calibration was derived for moderately obscured galaxies with high star formation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) find through synthetic photometry that their targets are typically continuum dominated in the F335M band, with ∼15% of the flux being due to the PAH feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) note that water ice absorption at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='05 µm in the F300M band may influence their results for the more heavily obscured targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our sample has minimal obscuration on average (median E(B−V) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 for HII regions in these targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' submitted) and lower star formation surface densities than most of the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the following, we in- vestigate continuum subtraction with the three medium bands, but we note that unlike Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) we are unable to spectroscopically constrain any contamination from the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm “aliphatic” or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='47 µm “plateau” fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These features are significantly fainter than the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm band, so may not dominate the diffuse signal in F360M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Observed F300M-F335M-F360M Colors in PAH- and Continuum-Dominated Regions The PHANGS-JWST observations resolve individual stars and ISM emission at 5−40 pc scales in the F300M to F1130W bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' From Figure 2, it is clear that there is a wide variation in the degree to which any given line of sight is continuum or PAH dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In addition, it is clear that in places where the emission is PAH domi- nated, the F360M band traces PAH related emission as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This suggests that a simple linear interpolation in the continuum following Equation 3 will not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' To investigate these issues, we measure the F300M/F335M/F360M colors in regions of the images we expect to be continuum dominated versus PAH dom- inated using F1130W as a guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We first select all re- gions where F1130W < 1 MJy sr−1 to represent a low PAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 Map 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' A representative 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 kpc region of NGC 628 shown in the F200W, F300M, F335M, F360M, and F1130W bands with an asinh color table at each filter’s native resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The F1130W filter traces primarily PAH emission from the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The F200W filter traces primarily stellar continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The F335M filter is centered on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature and includes both stellar continuum and PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Diffuse emission is visible in the F335M and F360M bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the F1130W panel, we highlight faint PAH emission with F1130W < 1 MJy sr−1 with a red contour and bright PAH emission with F1130W > 10 MJy sr−1 with a green contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These faint and bright selections are used in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' PAH emission region (shown with a red contour in the bottom right panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We also select regions where F1130W > 10 MJy sr−1, representing highly PAH emission dominated regions (green contour in bottom right panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In Figure 3, we show the re- sults of this selection for pixels above a 5σ detection threshold in the F300M, F335M, and F360M bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' On Figure 3 we also include synthetic photometry in the NIRCam medium bands for the 1A, 1B, and 1C template spectra from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These template spectra show only moderate attenuation, making them the most comparable templates to our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The tem- plate spectra, which represent the spectra used to formu- late the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) continuum prescription cover a much narrower range of F360M/F300M colors than our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For the low PAH emission regions (F1130W < 1 MJy sr−1), we find 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 µm colors in good agreement with the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These measurements also suggest that starlight continuum at these wave- lengths is not well described by a single color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This can be seen in the extension of the both the F335M/F300M and F360M/F300M colors along the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' If a single stellar color would suffice, these points would be expected to be clustered around a single value of F335M/F300M and F360M/F300M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The range of col- ors suggests we are seeing contributions from a vari- ety of stellar populations, including red supergiants and asymptotic giant branch stars which show a wider range of near- to mid-IR spectral shapes, some related to cir- cumstellar dust (Meidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' From the strongly PAH-dominated regions identified by high F1130W surface brightness (F1130W > 10 MJy sr−1) in Figure 3, we see that PAH emission appears in the color-color diagram with a linear slope BPAH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 and offset APAH = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2, where we describe the linear relation in the colors with: F335MPAH F300M = APAH + BPAH F360M F300M (4) F300M F335M F360M MJy sr-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='00 48\'0" 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='43 55" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='98 50" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='62 45" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='30 40" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='01 15°47\'35" 01h36m40.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='01 15°4735" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 01h36m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 01h36m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s6 Sandstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' F335M/F300M versus F360M/F300M color in our representative region of NGC 628 selected by F1130W surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' On the left, the PAH emission at F1130W is faint (F1130W < 1 MJy sr−1), so the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 µm colors should be dominated by stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This region is highlighted with a red contour in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' On the right, we select bright regions in F1130W (> 10 MJy sr−1), for which we expect the colors to be dominated by PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The PAH-bright region is highlighted with a green contour in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The red line in each panel shows the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) prescription for the stellar continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In starlight dominated regions, this prescription does well at predicting the continuum at F335M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' However, in regions dominated by PAHs, the F360M/F300M color is also responding to the PAH emission, leading to overestimates of the F335M continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We include synthetic photometry for the 1A, 1B, and 1C template spectra from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) with red symbols, illustrating that our observed colors span a much wider range than the spectra used to create the continuum subtraction prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We found the same slope using all pixels in the three target galaxies where F1130W > 10MJy sr−1, suggest- ing that the BPAH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 slope is a good representa- tion of the colors of PAH dominated emission across our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This comparison shows that in regions domi- nated by PAH emission, the F360M/F300M color and the F335M/F300M color are both tracing the PAH fea- ture(s) in this wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Performing a linear in- terpolation with the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) slope would lead to subtracing off 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH emission, because the F360M filter is capturing emission from this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' An Optimized Continuum Subtraction Recipe for Highly Resolved Galaxies To avoid oversubtraction of PAH emission, we deter- mine a correction for the predicted F335Mcont based on the observed F335M/F300M and F360M/F300M colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This approach represents a first order correction to the continuum subtraction approach from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020), and future effort on spectroscopic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH obser- vations will be necessary to develop a more rigorous procedure for highly resolved targets like ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The nature of this correction is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The basic approach is to use the observed F335M/F300M color as an indication of how PAH-dominated the emis- sion is in a given location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Using the observed slope of BPAH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 for PAH-dominated emission, we can scale the F360M/F300M color to where it intersects the relationship from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020), obtaining a cor- rected F360M/F300M color with which to predict the F335Mcont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The intersection of these two lines is where the PAH contribution to both colors is assumed to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For each point, we scale the measured F360M/F300M value (xm) and F335M/F300M value (ym) along the PAH slope to where it intersects with the continuum relationship described by Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This yields a corrected F360M/F300M ratio (xc) and F335M/F300M ratio (yc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The PAH slope is described as: y = APAH + BPAHx, (5) and the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) slope is given by: y = ALai + BLaix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (6) Using Equation 5, we can write the relationship between xm, ym and xc, yc as: ym − yc xm − xc = BPAH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (7) F1130W < 1 MJy sr-1 F1130w > 10 MJy sr-1 5 4 F335M/F300M 3 2 1C Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020 :B 1 0 2 30 1 1 2 3 F360M/F300M F360M/F300MPAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 Map 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Illustration of the correction to the F360M/F300M color to minimize over-subtraction of PAH emission from the F335M filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The correction works by scaling the colors along the PAH-dominated color trend which has a slope of BPAH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The black points show the pixel-by-pixel colors derived in PAH-faint regions of the map (F1130W < 1 MJy sr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The gray points show colors in the PAH-bright regions (F1130W > 10 MJy sr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Two examples of the correction are shown with yellow and cyan stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The measured values xm, ym are scaled along the PAH slope till it intersects the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) relation for stellar continuum colors to obtain corrected values xc, yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The corrected values will lie on Equation 6, so: yc = ALai + BLaixc (8) We can then write xc in terms of the measured colors (xm, ym) as follows: xc = BPAHxm − ym + ALai BPAH − BLai .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (9) Putting this back into the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) formula, we can then obtain a prediction for yc, which represents the F335M/F300M color appropriate for continuum, and subsequently the F335M continuum as follows: yc = BLai �BPAHxm − ym + ALai BPAH − BLai � + ALai (10) F335Mcont = yc × F300M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (11) In Figure 5 we show a comparison of the F335M con- tinuum derived from the above method and that from Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The right panel shows the difference in the continuum predicted by the two techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' It is clear that the corrected formula more cleanly isolates stellar continuum, minimizing over-subtraction of PAH emission, while maintaining the success of the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) formula at subtracting starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' While this em- pirical correction can be fine-tuned in the future, this approach provides a straightforward technique to mea- sure the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature for these first science applica- tions with NIRCam medium band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We proceed in the following Section to interpret the resulting maps of PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the future, we plan to investigate more sophisti- cated approaches to determining the PAH contamina- tion of the various NIRCam medium bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The ba- sic assumptions in this approach are that the F300M is relatively uncontaminated by diffuse emission, and the F335M/F300M ratio therefore gives a reasonable estimation of the degree of PAH emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' However, since there is a range of colors appropriate for the stars, a single scaled stellar spectral energy distribution tied to F300M does not appear to work well at removing the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Instead, we interpret the spread in F360M/F300M at a fixed F335M/F300M as variations in the stellar SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This may not be the case, as vari- ous other effects can alter the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 µm colors, includ- ing variations in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm feature, extinction, ice ab- sorption features, other diffuse emission from hot dust and/or nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Future work comparing the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 µm stellar colors with stellar population mod- eling based on the PHANGS-MUSE observations (Em- sellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022) may allow development of more pre- cise stellar continuum recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Future spectroscopic cal- ibration of F335M continuum subtraction recipes will be critical to fully exploit the capability to map PAH emis- sion with medium band filters on NIRCam and move beyond the first-order correction presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' RESULTS In Figures 6, 7, and 8 we show the F335MPAH and F335Mcont maps for all three targets, using our continuum subtraction scheme described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The F335MPAH maps are the highest resolution view of the PAH emission in these galaxies, with linear resolution between 5-10 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This resolution is similar to what can be achieved with Hubble optical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We find typical uncertainties of σ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='07 MJy sr−1 in the F335MPAH maps for these galaxies, as measured in faint regions of the map with minimal emission (this value also matches expectations from propagating measured errors in F300M, F335M, and F360M through the cor- rection formulae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Given that our observations required only ∼ 400 seconds of integration per field, the sensi- tivity of the maps is impressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the future, deeper 3 F335M/F300M Measured Xm,ym 2 Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) Corrected Xc,yo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 F360M/F300M8 Sandstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The predicted F335Mcont from the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) formula in Equation 3 (left), the empirical prescription presented in this work in Equations 10 and 11 (middle), and the difference between them (right) for our representative region from NGC 628 shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The left and middle panels show the same 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5 kpc region of NGC 628 shown in Figure 2 with the same asinh color table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The right panel shows that significant amounts of diffuse emission are included in the F335Mcont from the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) prescription because of the fact that F360M also includes some PAH-related emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our prescription minimizes this contamination by correcting the F360M/F300M colors while still cleanly subtracting the stellar continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' observations with medium bands to map the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature will be straightforward with NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' PAH-to-Continuum Ratios As a result of our continuum subtraction, we can measure the fraction of the F335M band that is due to PAH emission (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' F335MPAH/F335M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We show radial profiles of this fraction in Figure 9 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' To create these profiles, we binned the F335MPAH and F335M intensities in bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='01r25, including all pix- els where the emission in both filters was detected at > 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We measure the median of the F335MPAH and F335M in these bins and then divide to obtain the ratio as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This value is low- est in the galaxy centers, where high stellar mass sur- face density leads to starlight dominating the F335M band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In all three galaxies the fraction increases rela- tively smoothly with radius outside the central 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1r25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Values range between 5 − 65% , with NGC 628 span- ning both the lowest and highest part of that range over the range of radii we cover (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3r25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Both NGC 1365 and 7496 are barred galaxies with high central gas sur- face densities associated with circumnuclear star form- ing rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These regions produce strong PAH emission, which causes the upturn in the F335MPAH/F335M in their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' NGC 628, on the other hand, shows a monotonic increase in F335MPAH/F335M with ra- dius, suggesting that starlight surface brightness falls more rapidly than PAH surface brightness as a func- tion of radius, which may reflect varying scale-lengths for the stellar mass and ISM distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The vary- ing behavior even among these first three targets from PHANGS-JWST emphasizes the need for continuum re- moval recipes that work for both continuum and PAH- dominated sight-lines, since starlight makes up a highly variable fraction of the emission in the NIRCam medium bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Typical 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 PAH Feature Ratios and Comparison to Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) Both the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH features are thought to arise from vibrations in C-H bonds, which are strongest from neutral grains (Schutte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' van Dieden- hoven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Kerkeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Because smaller PAHs gain more energy per vibrational mode upon ab- sorbing a UV photon of a given energy, they are able to more effectively excite the shorter wavelength emission at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm compared to larger PAHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Thus, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 ratio for a fixed radiation field spectrum is expected to trace the average size of the grains (Maragkoudakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Rigopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Recent models from Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) predict the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 ra- tio for a range of PAH size and charge distributions, in radiation fields of varying intensity and hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In the right panel of Figure 9, we show the radial profile of the F335MPAH/F1130W ratio for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The data are radially binned as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1, with the addition of masking out radial ranges affected by saturation for the F1130W filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We masked the inner 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5′′ radius region, which extends to the radius where the point spread function for the bright central source drops to 10−3 of its peak value (see Hassani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022, for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This corresponds to a cut at < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='04r25 (< 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 kpc) for NGC 1365 and < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='08r25 (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7 kpc) for NGC 7496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We additionally mask the very central region of NGC 628 (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='02r25, or < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 kpc) where evolved stellar populations contribute to the F1130W emission in a hole in the ISM distribution near MJy sr-1 F335M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This Work Lai - This Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 48\'0" 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 55" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='98 50" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='62 45" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='30 40" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='01 15°47\'35" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 01h36m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5°40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 01h36m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5°40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 01h36m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5*40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0sPAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 Map 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 F335M (MJy sr-1) 500 pc 500 pc +15° 46′ 00″ 47′ 00″ 48′ 00″ 49′ 00″ +1h 45m 00s +1h 40m 00s +1h 45m 00s +1h 40m 00s F335MPAH F335MCont +15° 46′ 00″ 47′ 00″ 48′ 00″ 49′ 00″ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' PAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emission and continuum for NGC 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' the nuclear star cluster (see Hoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Er- ror bars show the error on the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our results show variations of the ratio between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='08, with an av- erage value of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='05 across the three targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These ratios are investigated in more detail in Chastenet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022) and Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For comparison, we plot two ratios from the Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These values are derived by filter convolutions of the models as described in Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We use the results for a stellar population age of 1 Gyr and representative size (“small”, “standard”, and “large”) and charge dis- tributions (“low”, “standard”, and “high”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' For most of these combinations, the predicted F335MPAH/F1130W predictions are ouside our plot range, but our results are consistent with emission from a population of PAHs with “large” characteristic size and “standard” or “high” ion- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This is in agreement with the results from Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022) who also find the best alignment with the “large” and “high” ionization Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) model grid results, although they use much younger stellar pop- ulation ages to represent the environments of embedded clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' CONCLUSIONS The capability of NIRCam medium bands on JWST to map the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature at high angular resolu- tion and sensitivity provides an invaluable new tool for studying PAHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Combined with the longer wavelength MIRI imaging, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 PAH feature ratio (traced by F335MPAH/F1130W) presents one of the cleanest diag- nostics of PAH size, helping to interpret a range of other band ratio variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3) which can have both size and charge dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In addition, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm fea- ture can be mapped with NIRCam at 2 − 3 times finer angular resolution than the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='7 µm or 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm bands, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 F335M (MJy sr-1) 500 pc 500 pc +3h 33m 40s +3h 33m 35s +3h 33m 30s +3h 33m 45s +3h 33m 25s +3h 33m 40s +3h 33m 35s +3h 33m 30s +3h 33m 45s +3h 33m 25s 36° 09′ 00″ 8′ 30″ 8′ 00″ 7′ 30″ F335MPAH F335MCont 36° 09′ 00″ 8′ 30″ 8′ 00″ 7′ 30″ Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' PAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emission and continuum for NGC 1365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' yielding 5–10 pc resolution in our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' This allows measurements of the sizes of H II regions and bubbles (see Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022), the iden- tification of filamentary structure (Thilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' subm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Meidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022), the identification of embedded clus- ters Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2022), and potentially tracing the gas column at higher resolution than is routinely possi- ble with any millimeter or radio facilities (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Sandstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' subm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' In this Letter, we have presented a first approach to using the NIRCam medium bands F300M, F335M, and F360M to create a map of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We find a key consideration for highly resolved galax- ies like our targets is to correct the F360M/F300M col- ors to account for contamination by PAH emission (the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm feature itself, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 µm aliphatic, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='47 µm plateau features) in F360M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We develop an empir- ical first-order correction to the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) pre- scription, which combines the successes of the Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2020) formula at removing starlight with a scaling using the F335M/F300M colors to correct for PAH contami- nation in F360M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We demonstrate that this approach succeeds in mitigating over-subtraction of PAH emis- sion from F335M that would result from a simple linear 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2PAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 Map 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 F335M (MJy sr-1) F335MPAH F335MCont 500 pc 500 pc 43° 26′ 30″ 26′ 00″ 25′ 30″ 25′ 00″ 43° 26′ 30″ 26′ 00″ 25′ 30″ 25′ 00″ +23h 09m 52s +23h 09m 48s +23h 09m 44s +23h 09m 52s +23h 09m 48s +23h 09m 44s Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' PAH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm emission and continuum for NGC 7496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (left) Fraction of the F335M band from the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3µm PAH feature as a function of galactocentric radius in units of r25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Error bars show the error on the mean, which is very small given the large number of measurements that contribute in each radial bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The fraction of the F335M filter emission that traces PAHs varies systematically from the inner, stellar continuum dominated regions to the fainter outskirts of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (right) The ratio of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='3 µm PAH features (traced by F335MPAH/F1130W) as a function of galactocentric radius (error bars show error on the mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Due to saturated sources in the centers of NGC 1365 and 7496 at F1130W, we have masked the inner r = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='5′′ region, which corresponds to the inner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='04 r25 for NGC 1365, and the inner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='08 r25 for NGC 7496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' We have also masked the inner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='02 r25 for NGC 628 due to contamination by evolved stellar populations at F1130W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Significant variations in the median F335MPAH/F1130W ratios as a function of radius exist in the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Our observed ratios are generally consistent with expectations from the Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' (2021) models for PAH size distributions shifted towards larger grains and charge distributions with either “high” or “standard” ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='212 Sandstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' interpolation across the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Future work to calibrate the continuum subtraction using NIRSpec observations in highly resolved nearby galaxies will be critical to move beyond our first-order correction, and deal with effects such as attenuation, absorption features, stellar popu- lation variations, hot dust, and/or nebular emission in the various bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' These observations are associated with program 2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The specific observations analyzed can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='17909/9bdf-jn24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The authors thank the anonymous referee for feedback that improved the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' The authors thank Thomas Lai for helpful conversations and providing fits to tem- plate spectra used in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' KS acknowledges funding support from grant support by JWST-GO-02107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='006- A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' TGW acknowledges funding from the European Re- search Council (ERC) under the European Union’s Hori- zon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' 694343).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' JMDK gratefully acknowledges funding from the European Research Council (ERC) un- der the European Union’s Horizon 2020 research and in- novation programme via the ERC Starting Grant MUS- TANG (grant agreement number 714907).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' COOL Re- search DAO is a Decentralized Autonomous Organiza- tion supporting research in astrophysics aimed at un- covering our cosmic origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' MB acknowledges sup- port from FONDECYT regular grant 1211000 and by the ANID BASAL project FB210003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' EJW acknowl- edges the funding provided by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”, subproject P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' MC gratefully acknowledges funding from the DFG through an Emmy Noether Re- search Group (grant number CH2137/1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' FB would like to acknowledge funding from the European Re- search Council (ERC) under the European Union’s Hori- zon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='726384/Empire) ER and HH acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding refer- ence number RGPIN-2022-03499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' KG is supported by the Australian Research Council through the Discov- ery Early Career Researcher Award (DECRA) Fellow- ship DE220100766 funded by the Australian Govern- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' KG is supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project num- ber CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' JC acknowledges support from ERC starting grant #851622 DustOrigin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' AKL gratefully ac- knowledges support by grants 1653300 and 2205628 from the National Science Foundation, by award JWST-GO- 02107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content='009-A, and by a Humboldt Research Award from the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' OE gratefully acknowledges funding from the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) in the form of an Emmy Noether Research Group (grant num- ber KR4598/2-1, PI Kreckel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' Facilities: JWST REFERENCES Anand, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=', Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAyT4oBgHgl3EQf8Pq6/content/2301.00854v1.pdf'} +page_content=' C.' metadata={'source': 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b/UtAzT4oBgHgl3EQfJvuy/content/tmp_files/2301.01086v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..453a1f1aea1835bcddc9279bb16f015253b0df0a --- /dev/null +++ b/UtAzT4oBgHgl3EQfJvuy/content/tmp_files/2301.01086v1.pdf.txt @@ -0,0 +1,1491 @@ +Prepared for submission to JHEP +Cavendish-HEP-22/11, P3H-22-129, TTK-22-49 +NNLO QCD corrections to event shapes at the LHC +Manuel Alvarez,c Josu Cantero,d Michal Czakon,a Javier Llorente,e Alexander Mitov,b +Rene Ponceletb +aInstitut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, +D-52056 Aachen, Germany +bCavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom +cDepartamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid. Madrid, Spain. +dInstituto de F´ısica Corpuscular. Valencia, Spain. +eDepartment of Physics, Simon Fraser University. Burnaby, Canada. +E-mail: mczakon@physik.rwth-aachen.de, adm74@cam.ac.uk, +poncelet@hep.phy.cam.ac.uk, josu.cantero.garcia@cern.ch, +manuel.alvarez.estevez@cern.ch, javier.llorente.merino@cern.ch +Abstract: In this work we perform the first ever calculation of jet event shapes at hadron +colliders at next-to-next-to leading order (NNLO) in QCD. The inclusion of higher order +corrections removes the shape difference observed between data and next-to-leading order +predictions. +The theory uncertainty at NNLO is comparable to, or slightly larger than, +existing measurements. Except for narrow kinematical ranges where all-order resummation +becomes important, the NNLO predictions for the event shapes considered in the present +work are reliable. As a prime application of the results derived in this work we provide a +detailed investigation of the prospects for the precision determination of the strong coupling +constant and its running through TeV scales from LHC data. +arXiv:2301.01086v1 [hep-ph] 3 Jan 2023 + +Contents +1 +Introduction +1 +2 +Definitions and computational details +3 +2.1 +Definition of event shapes +5 +3 +Event-shapes at the LHC +6 +3.1 +Perturbative QCD corrections +7 +3.2 +Strong coupling dependence +11 +3.3 +Estimation of non-perturbative corrections +16 +4 +Conclusions +16 +1 +Introduction +One of the most direct ways of exhibiting the QCD dynamics at hadron colliders is by studying +multi-jet events. Such events are special since they are not very sensitive to the electroweak +(EW) sector of the Standard Model (SM), contain almost exclusively QCD partons, and are +driven by QCD’s dynamics, color structure and single coupling constant, αS. +The characterization of multi-jet events is often performed in terms of so-called event +shapes. These are single-valued observables which encode the events’ topology and energy +flow. Event shapes are constructed in such a way that configurations with back-to-back jets +can naturally be singled out from ones with more isotropic energy distribution in the final +state. Event shapes are sensitive to the probability of additional emissions, i.e., the QCD +dynamics and the strong coupling constant. +First measurements of multi-jet event shapes were performed at LEP [1–4] and were used +to extract αS [5]. Event shapes and multi-jet production at hadron colliders have been studied +extensively, starting with the Tevatron [6]. At the LHC, the ATLAS and CMS experiments +have measured generic multi-jet differential cross-sections [7–10], event shapes at 7 and 13 +TeV [11–16], azimuthal decorrelations at 8 TeV [17] and transverse-energy-energy-correlations +at 13 TeV [18]. +Theoretical work on jet production in hadron-hadron collisions also has a long history. +Fixed-order, next-to-leading-order (NLO) QCD corrections for dijet events have been first +obtained in refs. [19, 20]. These have been complemented by NLO EW corrections [21, 22] +and parton-shower effects [23, 24]. Dijet cross-sections with NLO accuracy are also available +from general-purpose event generators [25–27]. The computation of next-to-next-to-leading- +order (NNLO) QCD corrections to dijet events has been achieved relatively recently [28–30]. +– 1 – + +Based on simple energy-momentum considerations it is clear that dijet events contribute +trivially to event shapes. Therefore, only events with three or more jets in the final state +can span non-trivial values of event shapes while dijet events tend to only have endpoint +contributions. Three-jet cross-sections have been studied at NLO QCD [31, 32] and NLO +EW, including parton-shower effects [33]. +Very recently, the first computation of NNLO +QCD corrections to three-jet production has been performed by some of the authors of the +present article [34], mainly focusing on three-to-two jet ratios. Higher jet multiplicities have +also been investigated in the literature with up to five jet production known through NLO in +QCD [35, 36]. +Fixed-order calculations do not always adequately describe multi-jet event shapes. For +certain kinematic configurations one has to resum contributions from QCD emissions to all +orders in αS in order to have sensible theoretical predictions. Resummation with Next-to- +leading-logarithmic (NLL) accuracy matched to NLO QCD predictions has been achieved +for global event shapes [37, 38]. +As their name suggests, global event shapes feature the +important property that they are defined in the full phase space. The presence of phase space +restriction like phase space cuts, radiation gaps, etc., renders an event shape non-global and +gives rise to associated logarithms, so-called non-global logarithms [39, 40]. The resummation +of non-global logarithms is notoriously difficult and has been achieved only in special cases, +see for example, refs. [41–47]. +The full phase space is never accessible at hadron colliders due to the missing forward +coverage of the detector. This limits the implementation of global event shapes experimen- +tally. A possible work-around is to re-define event shapes by exponentially suppressing recoil +terms, which regulates the contribution from the unavailable phase space [38]. +However, +such an implementation would require the measurement of event’s hadronic recoil which is +accompanied by a significant experimental uncertainty. +It also requires the measurement +of individual particle momenta which increases the sensitivity to hadronisation and other +non-perturbative effects that are difficult to control. +An alternative approach to event shapes is to define event shapes in terms of reconstructed +jets instead of particles. Such an approach has the advantage that non-perturbative effects +from hadronisation and multi-parton interactions (MPI) are suppressed. +Experimentally, +working with jets instead of calorimeter clusters, which are the proxies of the particles at +the detector level, is also advantageous since detector-level jets are well-defined and well- +calibrated objects. The calibration of the jet energy scale and resolution together with their +corresponding uncertainties is typically well under control, whereas this not the case for +calorimeter clusters in general. +In addition, it is easier to associate detector-level jets to +particle-level jets, in contrast to the case of a calorimeter cluster which typically cannot be +uniquely associated to a single particle. +Event shapes are a powerful tool for testing parton-shower models in state-of-the-art +Monte Carlo simulations and for tuning shower parameters. For example, differences in the +matching to fixed-order matrix elements and in the ordering of the emissions (pT , θ, etc) can +lead to important differences in the description of these observables [16]. +– 2 – + +In this work, we compute the NNLO QCD corrections to a set of event shapes based on +reconstructed jets. As explained above, jet-based event shapes allow for precise experimental +measurements, i.e. an increased precision of theoretical predictions will make it possible to +closely scrutinize our ability to precisely describe QCD at hadron colliders and to extract αS +and its running through TeV scales with unprecedented precision. While in this study we do +not use MC or resummed predictions, we identify regions where such corrections might be +needed. +This article is organised as follows: in sec. 2 we give details about the computational +setup and the definitions of the event shapes studied in the present work. The main results +are presented in sec. 3. Specifically, perturbative corrections and PDF effects are discussed +in sec. 3.1, the dependence on αS and its extraction in sec. 3.2, while sec. 3.3 contains an +estimation of non-perturbative effects. Our conclusions can be found in sec. 4. +2 +Definitions and computational details +In this work we compute the NNLO QCD corrections to the set of multi-jet event shape +observables specified in sec. 2.1 below. To span the full kinematic ranges of these observables, +final states containing at least three jets are required. For the calculation of the NNLO QCD +corrections to the inclusive three-jet cross section we employ the sector-improved residue +subtraction framework [48, 49], which has already been applied to inclusive jet [30], two jet- +as well three-jet production [34]. Tree-level matrix elements are evaluated with the help of +the AvH library [50]. All contributing one-loop amplitudes are taken from the OpenLoops +2 library [51]. The two-loop amplitudes for three-jet production are only available in the +leading-colour approximation [52]. In the present work we utilize these approximate double +virtual corrections to three-jet production along the lines of ref. [34]. Specifically, the two-loop +finite remainder +R(2)(µ2 +R) = 2 Re +� +M†(0)F(2)� +(µ2 +R) + +��F(1)��2(µ2 +R) +≡ R(2)(s12) + +4 +� +i=1 +ci lni +�µ2 +R +s12 +� +, +(2.1) +is approximated by replacing the function R(2)(s12) with its leading-colour approximation +provided with the software described in ref. [53]. It is implemented in terms of the so-called +“pentagon functions” [54] and rational functions of the kinematic invariants [52]. We work in +the nf = 5 scheme, i.e. we consistently neglect contributions from top-quarks in the partonic +cross sections and in the running of αS. +The αS expansion of the n-jet differential cross section reads: +dσn(µR, µF , PDF, αS(µR, αS,0)) = +� +i +αSn+i(µR, αS,0)dσ(i) +n (µR, µF , PDF) , +(2.2) +– 3 – + +where µR,F are the renormalisation and factorisation scales and PDF labels the parton distri- +bution set. The dependence on the initial value of the strong coupling αS,0 ≡ αS(µR = mZ) +is also made explicit for later convenience. +The strong coupling constant satisfies the following renormalisation group equation (RGE): +µ2 +R +dαS +dµ2 +R += β(αS) = −(b0αS2 + b1αS3 + b2αS4 + . . . ) , +(2.3) +where b0,1,2 are the 1-, 2- and 3-loop β-function coefficients. The strong coupling constant +at a scale µR, together with the corresponding PDF set, is obtained from the LHAPDF library +[55]. Specifically, αS(µR) is derived from the numerical solution of the RGE eq. (2.3) starting +from the reference scale µR = mZ. In the present work we are interested in the three lowest +orders in perturbation theory and define them for convenience as +dσLO +n += αSndσ(0) +n , +dσNLO +n += αSndσ(0) +n ++ αSn+1dσ(1) +n , +dσNNLO +n += αSndσ(0) +n ++ αSn+1dσ(1) +n ++ αSn+2dσ(2) +n . +(2.4) +All multi-jet observables in this work are computed as ratios +Ri(µR, µF , PDF, αS,0) = dσi +3(µR, µF , PDF, αS,0) +dσi +2(µR, µF , PDF, αS,0) , +(2.5) +for i ∈ {LO, NLO, NNLO}. We study the NLO and NNLO QCD K-factors: +KNLO = RNLO +RLO +and +KNNLO = RNNLO +RNLO . +(2.6) +To estimate missing higher-order contributions, we take the envelope of a 7-point variation +of µR and µF by a factor of 2 with the constraint 1 +2 ≤ µR +µF ≤ 2. For all predictions presented +in this work, we use as a central scale choice +µR = µF = ˆHT = +� +i∈partons +pT,i , +(2.7) +where the sum runs over all partons in the final state. +The PDF uncertainties are estimated by following the prescriptions of the individual PDF +sets. Due to the high computational costs, PDF member variations have been evaluated only +at NLO QCD using the NNLO version of the corresponding PDF set, and then extrapolated +to NNLO QCD by using the following K-factor approach +RNNLO(PDFi) ≈ RNLO(PDFi) +RNLO(PDF0) RNNLO(PDF0) , +(2.8) +where PDFi denotes the i-th member of the PDF set (0 corresponds to the central mem- +ber). The justification for this approximation is that the ratios have a reduced dependence +on the PDF due to cancellation between the two- and three-jet cross sections. Indeed the +PDF uncertainties for the ratios are typically small and dwarfed by the remaining numerical +integration uncertainties. +– 4 – + +2.1 +Definition of event shapes +This section provides the definitions of all event shapes computed in section 3. +A classical event shape observable is the transverse thrust T⊥ (or rather τ⊥ = 1 − T⊥) +[56, 57] and its minor component Tm, defined by +T⊥ = +� +i |⃗pT,i · ˆn⊥| +� +i |⃗pT,i| +, +and +Tm = +� +i |⃗pT,i × ˆn⊥| +� +i |⃗pT,i| +. +(2.9) +The thrust axis n⊥ maximises the projection of all jets to this axis. As explained in the +Introduction, we define the thrust and all other event shapes in terms of reconstructed jets, +i.e. the sum i in eq. (2.9) above runs over the list of all reconstructed jets passing selection +criteria. The thrust separates topologies that are of back-to-back type (small τ⊥) from ones +that are isotropic (large τ⊥). Another event shape that characterises the anisotropy of an +event is the linearised sphericity tensor [58, 59]: +Mxyz = +1 +� +i |⃗pi| +� +i +1 +|⃗pi| +� +� +� +p2 +x,i +px,ipy,i px,ipz,i +py,ipx,i +p2 +y,i +py,ipz,i +pz,ipx,i pz,ipy,i +p2 +z,i +� +� +� . +(2.10) +In this work, we study certain combinations of the eigenvalues λ1, λ2, λ3 of Mxyz. They are +denoted A (the so-called aplanarity), C, and D [60]: +A = 3 +2λ3 , +C = 3(λ1λ2 + λ1λ3 + λ2λ3) , +D = 27λ1λ2λ3 . +(2.11) +Furthermore, from the transverse linearised sphericity tensor +Mxy = +1 +� +i |⃗pT,i| +� +i +1 +|⃗pT,i| +� +p2 +x,i +px,ipy,i +py,ipx,i +p2 +y,i +� +, +(2.12) +with eigenvalues µ1, µ2, one defines the transverse sphericity variable: +S⊥ = +µ2 +µ1 + µ2 +. +(2.13) +The final observable we study in this article is the transverse energy-energy correlator +(TEEC) [18, 61, 62]. For multi-jet events, it is defined as +1 +σ2 +dσ +d cos ∆φ = 1 +σ2 +� +ij +� +dσ x⊥,ix⊥,j +dx⊥,idx⊥,jd cos ∆φij +δ(cos ∆φ − cos ∆φij)dx⊥,idx⊥,jd cos ∆φij , +(2.14) +where x⊥,i = E⊥,i/ � +k E⊥,k, and with E⊥ = +� +E2 − p2z being invariant under boosts along +the z-axis. The angle ∆φij is the azimuthal opening between a pair of jets ij. +Note that in the presence of phase space cut(s), the TEEC picks up an implicit depen- +dence on the corresponding kinematic variable(s). As specified in sec. 3 below, the TEEC is +– 5 – + +measured in bins of the variable HT,2. The denominator in eq. (2.14) is the two-jet cross- +section evaluated in the corresponding bin of HT,2. To be able to generate a contribution +away from the endpoints cos ∆φij = ±1, the numerator in eq. (2.14) requires at least a 2 → 3 +process. A fixed-order 2 → 3 calculation diverges at the endpoints cos ∆φij = ±1 which +correspond to the back-to-back limit. To avoid these infrared sensitive regions, a constraint +| cos ∆φij| ≤ | cos ∆φmax| is imposed. For a sufficiently small value of | cos ∆φmax| contri- +butions from resummation-dominated phase space regions are suppressed and fixed-order +perturbative calculations are expected to be reliable. +3 +Event-shapes at the LHC +In this section we present NNLO QCD predictions for event shape observables measured at +13 TeV by the ATLAS collaboration [16] 1. For the numerator (denominator) of eq. (2.5) we +require at least three (two) R = 0.4 anti-kT [63] jets that fulfill the requirements +• HT,2 = pT,1 + pT,2 ≥ 1 TeV where pT,i is the transverse momentum of the i-th hardest +jet, +• |y(j)| ≤ 2.4 and pT (j) ≥ 100 GeV. +The inputs for the event shapes are all jets that pass the above requirements. +Due to the high computational cost of the three jet cross section, we compute the event +shapes in bins that are coarser than the bins used for the measurement in ref. [16]. +To +facilitate the comparison between the measurement and our predictions we rebin the data as +follows: the systematic uncertainties between bins are considered as fully correlated, whereas +the statistical uncertainties are added in quadrature. All event shapes are measured in three +separate HT,2 regions +1000 GeV ≤ HT,2 < 1500 GeV, +1500 GeV ≤ HT,2 < 2000 GeV, +2000 GeV ≤ HT,2 , +which allows for the separation of different energy scales. +For the computation of the TEEC, see eq. (2.14), we adopt a slightly different phase +space corresponding to the selection in ref. [18]: +• HT,2 = pT,1 + pT,2 ≥ 1 TeV, +• at least 2 anti-kT jets with R = 0.4 and pT ≥ 60 GeV |y(j)| ≤ 2.4. +For this observable we consider a finer HT,2 binning +[1000, 1200, 1400, 1600, 1800, 2000, 2300, 2600, 3000, 3500, ∞) , +together with the “inclusive” case HT,2 ≥ 1000 GeV. +1The ATLAS measurement has been performed in jet-multiplicity bins. We compare our predictions with +the inclusive data presented in appendix A. +– 6 – + +[] +0.6 +0.8 +1.0 +1.2 +1.4 +1000GeV ≤ HT,2 < 1500GeV +NNPDF30 µR = µF = ˆHT +[] +0.6 +0.8 +1.0 +1.2 +1.4 +ratio to data +1500GeV ≤ HT,2 < 2000GeV +LO +NLO +NNLO +ATLAS +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +τ⊥ +0.6 +0.8 +1.0 +1.2 +1.4 +2000GeV ≤ HT,2 +[] +0.6 +0.8 +1.0 +1.2 +1.4 +1000GeV ≤ HT,2 < 1500GeV +NNPDF30 µR = µF = ˆHT +[] +0.6 +0.8 +1.0 +1.2 +1.4 +ratio to data +1500GeV ≤ HT,2 < 2000GeV +LO +NLO +NNLO +ATLAS +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Tm +0.6 +0.8 +1.0 +1.2 +1.4 +2000GeV ≤ HT,2 +Figure 1. +The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins. +The solid lines show fixed order LO (green), NLO (blue), and NNLO (red) predictions normalised to +ATLAS data (black) [16]. The coloured bands show the scale variation. +The central choice for renormalisation and factorisation scales is ˆHT , defined in equation +(2.7). As the default PDF choice, we use the NNPDF30[64] PDF set 2. +3.1 +Perturbative QCD corrections +In fig. 1 to fig. 3, we show the LO, NLO and NNLO QCD corrections to the six event shapes +defined in section 2.1. All predictions are shown as ratios to the data. Coloured bands show +the envelope of the 7-point scale variation, and vertical bars indicate the remaining uncertainty +from Monte Carlo integration. The data is shown with a bar indicating the statistical and +systematical uncertainty added in quadrature. +The results for the transverse thrust τ⊥ are shown on the left-hand side of fig. 1. We +observe significant NLO QCD corrections to the shape of the LO prediction. In particular, +for large thrust values representing events with hard resolved radiation, corrections can be +as large as 50%. The scale uncertainty increases, indicating that the LO scale dependence +does not suffice to estimate missing higher effects for this observable. The NLO result does +agree fairly well with data within the uncertainties, but one can observe the tendency to +undershoot the data in the tail of the distribution. Turning to the NNLO QCD results, we +observe small corrections to the shape at NLO and a sizable reduction of the scale dependence +by a factor of two to four. The NNLO QCD predictions are fully contained within the NLO +2This PDF set has been superseded by NNPDF31 [65] and NNPDF40 [66] which have smaller uncertainties +and are based on better methodology. These differences are not so crucial for the current discussion as the +PDF uncertainties are comparatively small. +– 7 – + +QCD scale uncertainties indicating perturbative convergence. Agreement with data, within +uncertainties, can be observed over the full range of the thrust observable. Importantly, the +NNLO QCD corrections lead to theory/data shape agreement in the tail. +The behavior of the transverse thrust τ⊥ for small values of τ⊥ deserves special attention. +As can be observed in fig. 1 the behavior of the fixed order prediction at all three orders +is different, showing change in slope and increased scale variation, especially at LO. This +behavior is accompanied by an increase in the MC uncertainty of the predictions and the +agreement with data at NNLO is worse than in all other bins. +This is the case for all +three HT,2 slices. The reason behind this behavior is that in this kinematics the fixed order +expansion starts to break down and all-order resummation effects become important. This +behavior is well known from the e+e− thrust [67] for which very high resummation as well +as fixed order accuracy has been achieved. In general, we wish to point out the substantial +progress that has recently been achieved for the resummation and the description of event +shape observables at high energy colliders [68–75]. +On the right-hand side of fig. 1, we show the thrust minor component Tm. The pattern +of higher-order corrections is similar to the thrust observable: large NLO and smaller NNLO +corrections. The reduction in scale dependence when going from NLO to NNLO QCD, partic- +ularly for larger values of Tm, is large and can reach a factor of 10 in the intermediate region. +The NLO QCD predictions for Tm do not describe the shape of the observable very well, +especially around the peak region Tm ∼ 0.15. This situation is improved by the NNLO QCD +calculation, which predicts the shape well. The lowest Tm bin suffers from large statistical +uncertainties on the NNLO QCD predictions, for reasons similar to the ones discussed in the +context of the transverse thrust τ⊥. +We find good agreement between ATLAS data and NNLO QCD predictions for the +transverse sphericity S⊥ and aplanarity A, both shown in figure 2. Similar level of NNLO/data +agreement can also be observed for the D variable shown in figure 3. In terms of the behavior of +perturbative corrections, the three variables behave similarly to the thrust. The second-order +QCD corrections are most significant in the regions described by events with well-separated +and isotropically distributed jets and eliminate the notable shape differences observed at NLO +QCD. +The C variable, shown in the left-hand side of figure 3, has a different behaviour. The +NLO QCD predictions have little scale dependence in the intermediate region for C and larger +scale dependence at the two endpoints C = 0 and C = 1. In the central region, neither the +normalisation nor the shape of the data is well described within uncertainties. The numerical +integration of the NNLO QCD corrections suffers from particularly large cancellations leading +to significant remaining statistical errors. For that reason, it is unclear if the second-order +corrections improve the theory-data agreement or if the seemingly good agreement with data +is simply due to the large statistical errors. +Finally, we turn to the case of the TEEC. In the two top panels in figure 4, we show +perturbative predictions for the inclusive HT,2 selection. A preliminary measurement of this +observable has been presented in ref. [18]. However, the numbers are not yet public and +– 8 – + +[] +0.6 +0.8 +1.0 +1.2 +1.4 +1000GeV ≤ HT,2 < 1500GeV +NNPDF30 µR = µF = ˆHT +[] +0.6 +0.8 +1.0 +1.2 +1.4 +ratio to data +1500GeV ≤ HT,2 < 2000GeV +LO +NLO +NNLO +ATLAS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +S⊥ +0.6 +0.8 +1.0 +1.2 +1.4 +2000GeV ≤ HT,2 +[] +0.6 +0.8 +1.0 +1.2 +1.4 +1000GeV ≤ HT,2 < 1500GeV +NNPDF30 µR = µF = ˆHT +[] +0.6 +0.8 +1.0 +1.2 +1.4 +ratio to data +1500GeV ≤ HT,2 < 2000GeV +LO +NLO +NNLO +ATLAS +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +A +0.6 +0.8 +1.0 +1.2 +1.4 +2000GeV ≤ HT,2 +Figure 2. +As in fig. 1 but for the transverse sphericity S⊥ (left) and the aplanarity A (right). +[] +0.6 +0.8 +1.0 +1.2 +1.4 +1000GeV ≤ HT,2 < 1500GeV +NNPDF30 µR = µF = ˆHT +[] +0.6 +0.8 +1.0 +1.2 +1.4 +ratio to data +1500GeV ≤ HT,2 < 2000GeV +LO +NLO +NNLO +ATLAS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +C +0.6 +0.8 +1.0 +1.2 +1.4 +2000GeV ≤ HT,2 +[] +0.6 +0.8 +1.0 +1.2 +1.4 +1000GeV ≤ HT,2 < 1500GeV +NNPDF30 µR = µF = ˆHT +[] +0.6 +0.8 +1.0 +1.2 +1.4 +ratio to data +1500GeV ≤ HT,2 < 2000GeV +LO +NLO +NNLO +ATLAS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +D +0.6 +0.8 +1.0 +1.2 +1.4 +2000GeV ≤ HT,2 +Figure 3. +As in fig. 1 but for the variables C (left) and D (right). +– 9 – + +therefore we show no theory/data comparison for this observable in this work. +The typical pattern of higher-order corrections familiar from our discussion of even shapes +also emerges for the TEEC. The NLO QCD corrections are sizable relative to the leading order +result, barely touching the uncertainty estimate from scale uncertainties. The NLO QCD +estimate of the scale uncertainty is flat and about 8%. The corrections from NNLO QCD are +smaller, between 0% and 5%, and lie fully within the NLO scale uncertainty band. The scale +variation uncertainty of the NNLO QCD prediction is about 1−3%. The remaining numerical +integration uncertainty of the NNLO QCD prediction is about ∼ 1 − 3%, i.e. it is similar to +the scale uncertainty 3. Unlike the other event shapes discussed so far, the corrections are +relatively flat, and there is no phase space region which receives larger corrections than the +others. Notably, within scale uncertainty, the NNLO QCD prediction is consistent with the +central NLO QCD one. +In the double differential case, shown in figure 5, we see that the above picture persists +also for the individual HT,2 slices. Only for HT,2 > 3 TeV we observe an increase in the +perturbative correction and scale uncertainty. Still, NNLO QCD corrections remain within the +NLO scale uncertainty bands, indicating perturbative stability even at the highest available +energies. +We next turn our attention to PDF uncertainties. In fig. 6 we show the NNLO QCD +predictions for the thrust and the minor component, evaluated with the following PDF sets: +NNPDF30 [64], MMHT2014 [76], CT18 [77] and HERAPDF [78]. The PDF uncertainty +estimate for each set is shown as a colored band around the corresponding central value, +normalised to the central NNPDF30 prediction. The numerical MC uncertainties are not +shown because they are fully correlated between different sets, as the same events have been +used to evaluate them. +Starting with the uncertainties for our default PDF set NNPDF30, for both variables +we observe that the uncertainties around the peak of the distribution are very small, below +1%, and are therefore negligible. The uncertainties increase continuously towards the tails +of the distributions and become as large as 2 − 3%. Such a behaviour is expected since the +cancellation between PDF dependencies in the numerator and denominator is realised best +around the peak of the distribution. The spread between the different PDF sets is about +1 − 2%. All PDFs are compatible within their uncertainty estimates. Furthermore, at NLO +QCD the PDF uncertainties are much smaller than the scale ones. At NNLO QCD the scale +uncertainties are much reduced – reaching about 5% in the tails of the distributions – which +leads to PDF uncertainties becoming comparable to the scale ones. Still, at NNLO the scale +uncertainty remains the dominant one. +The PDF uncertainties of the other event shapes considered in the present work feature +similar behavior. More details can be read off the results that are available in electronic +format [79]. +3Since integration uncertainty might lead to imperfect cancellations, the actual scale uncertainty might be +smaller than indicated. +– 10 – + +[] +0.05 +0.10 +0.15 +dσ/d cos(φ) [pb] +µR = µF = ˆHT +NNPDF30 +LO +NLO +NNLO +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +cos(φ) +0.8 +0.9 +1.0 +1.1 +1.2 +ratio to NLO +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +cos(φ) +0.99 +1.00 +1.01 +ratio to NNLO +NNLO - PDF uncertainty +Figure 4. +The TEEC variable in the inclusive HT,2 ≥ 1 TeV bin. The top panel shows the absolute +differential distribution through LO (light green), NLO (blue) and NNLO (red) QCD. The coloured +bands show the scale uncertainty estimates and vertical bars indicate statistical uncertainties. The +second panel shows the ratio to the central NLO QCD prediction. The third panel shows the PDF +uncertainty estimate from NNPDF30 at NLO QCD. +The PDF uncertainty of the TEEC distribution computed with the NNPDF30 PDF set +is shown in fig. 4 for the inclusive HT,2 > 1 TeV bin. The uncertainty is flat, below 1%, +indicating that there is a strong cancellation between numerator and denominator for this +variable. A similar PDF behavior is also observed in the individual HT,2 bins, which we do +not show here but supply in electronic form [79]. +3.2 +Strong coupling dependence +In this section we quantify the sensitivity of the various event shapes to the value of the +strong coupling constant αS,0. While in the present work we do not attempt to extract a +value of αS,0, the results in this section are an essential step in this direction. To estimate +the dependence of the event shapes on the strong coupling constant αS,0, we evaluate them +through NNLO QCD using PDF sets with different values of αS,0. Each PDF set provides a +different range of αS,0 values, and these have been summarized in table 1. +The αS,0 dependence in each bin can be derived by expressing the event shape in that bin +as a function of αS,0. This can be achieved as follows. First, consider a perturbative solution +– 11 – + +[] +0.9 +1.0 +1.1 +1.2 +1000GeV ≤ HT,2 < 1200GeV +LO +NLO +NNLO +[] +0.9 +1.0 +1.1 +1.2 +1200GeV ≤ HT,2 < 1400GeV +[] +0.9 +1.0 +1.1 +1.2 +1400GeV ≤ HT,2 < 1600GeV +[] +0.9 +1.0 +1.1 +1.2 +1600GeV ≤ HT,2 < 1800GeV +[] +0.9 +1.0 +1.1 +1.2 +1800GeV ≤ HT,2 < 2000GeV +[] +0.9 +1.0 +1.1 +1.2 +2000GeV ≤ HT,2 < 2300GeV +[] +0.9 +1.0 +1.1 +1.2 +2300GeV ≤ HT,2 < 2600GeV +[] +0.9 +1.0 +1.1 +1.2 +2600GeV ≤ HT,2 < 3000GeV +[] +0.9 +1.0 +1.1 +1.2 +3000GeV ≤ HT,2 < 3500GeV +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +cos(φ) +0.9 +1.0 +1.1 +1.2 +3500GeV ≤ HT,2 +Scale: µR = µF = ˆHT PDF: NNPDF30 +ratio to NLO +Figure 5. +The TEEC variable double differentially in HT,2 bins. The panels show LO (light green), +NLO (blue) and NNLO (red) QCD prediction as a ratio to the central NLO QCD prediction. The +coloured bands show the scale uncertainty estimates and vertical bars indicate statistical uncertainties. +– 12 – + +[] +0.90 +0.95 +1.00 +1.05 +1.10 +1000GeV ≤ HT,2 < 1500GeV +µR = µF = ˆHT +NNLO QCD +NNPDF30 +HERAPDF20 +MMHT2014 +CT18 +[] +0.90 +0.95 +1.00 +1.05 +1.10 +ratio to NNPDF30 +1500GeV ≤ HT,2 < 2000GeV +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +τ⊥ +0.90 +0.95 +1.00 +1.05 +1.10 +2000GeV ≤ HT,2 +[] +0.90 +0.95 +1.00 +1.05 +1.10 +1000GeV ≤ HT,2 < 1500GeV +µR = µF = ˆHT +NNLO QCD +NNPDF30 +HERAPDF20 +MMHT2014 +CT18 +[] +0.90 +0.95 +1.00 +1.05 +1.10 +ratio to NNPDF30 +1500GeV ≤ HT,2 < 2000GeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Tm +0.90 +0.95 +1.00 +1.05 +1.10 +2000GeV ≤ HT,2 +Figure 6. +The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins. The +solid lines show fixed-order NNLO QCD predictions for different PDF sets normalised to NNPDF30. +The coloured band show the estimated PDF uncertainties. +PDF set +αS,0 values +NNPDF30 +0.115, 0.117, 0.118, 0.119, 0.121 +CT18 +0.110, 0.111, . . . , 0.123, 0.124 +MMHT2014 +0.108, 0.109, . . . , 0.127, 0.128 +HERAPDF20 +0.110, 0.111, . . . , 0.129, 0.130 +Table 1. The values for αS,0 = αS(µR = mZ) available with each PDF set. +of the RGE for the strong coupling constant, i.e. +αS(µR, αS,0) = αS,0 +� +1 − αS,0b0 ln +� µ2 +R +m2 +Z +� ++ O +� +αS,02�� +. +(3.1) +Setting for simplicity µ = µR = µF , one can then rewrite eq. (2.5) in such a way that the +RGE running of the strong coupling is absorbed into the partonic cross section +RNNLO(µ, αS,0) = dσNNLO +3 +(µ, αS,0) +dσNNLO +2 +(µ, αS,0) += +αS,03 � +d˜σ(0) +3 (µ) + αS,0d˜σ(1) +3 (µ) + αS,02d˜σ(2) +3 (µ) + O(αS,03) +� +αS,02 +� +d˜σ(0) +2 (µ) + αS,0d˜σ(1) +2 (µ) + αS,02d˜σ(2) +2 (µ) + O(αS,03) +� . +(3.2) +Eq. (3.2) makes explicit the non-PDF αS,0 dependence of the event shape RNNLO(µ, αS,0). +It also makes it clear that the leading αS,0 dependence of RNNLO(µ, αS,0) is linear. +– 13 – + +[] +0 +5 +10 +15 +20 +1000GeV ≤ HT,2 < 1500GeV +µR = µF = ˆHT +NNPDF30 +HERAPDF20 +MMHT2014 +CT18 +[] +0 +5 +10 +15 +20 +˜c1 +1500GeV ≤ HT,2 < 2000GeV +NNLO QCD +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +τ⊥ +0 +5 +10 +15 +20 +2000GeV ≤ HT,2 +[] +0 +5 +10 +15 +20 +1000GeV ≤ HT,2 < 1500GeV +µR = µF = ˆHT +NNPDF30 +HERAPDF20 +MMHT2014 +CT18 +[] +0 +5 +10 +15 +20 +˜c1 +1500GeV ≤ HT,2 < 2000GeV +NNLO QCD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Tm +0 +5 +10 +15 +20 +2000GeV ≤ HT,2 +Figure 7. +The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins. The +solid lines show the coefficient ˜c1 at NNLO QCD for different PDF sets. +Based on the above observations, a very practical way of parameterizing the αS,0 depen- +dence of the computed event shape is to assume the functional form +RNNLO,fit(µ, αS,0) = c0 + c1(αS,0 − 0.118) + c2(αS,0 − 0.118)2 + c3(αS,0 − 0.118)3 , +(3.3) +whose coefficients ci in each bin are determined by fitting RNNLO as defined in eq. (2.5) 4. We +stress that the fit encodes the unexpanded in αS,0 value of RNNLO as well as the αS running +obtained directly from the numeric RGE solution as provided by LHAPDF, i.e. one does not +need to utilize the analytic RGE eq. (3.1). +The coefficients c2 and c3 are typically small. Moreover, one is only interested in percent- +level variations of αS,0 around 0.118 (which is a convenient proxy for αS,0’s world average). +Therefore, in order to assess the sensitivity of RNNLO to the value of αS,0 it is sufficient to +focus on the coefficient c1. In practice we consider the rescaled coefficient +˜c1 = +c1 +RNNLO(αS,0 = 0.118) , +(3.4) +which corresponds to the normalised first derivative of RNNLO with respect to αS,0 at αS,0 = +0.118. The interpretation of ˜c1 is that if αS,0 changes by an amount δαS,0 around αS,0 = 0.118, +the ratio RNNLO(αS,0)/RNNLO(αS,0 = 0.118) changes by ˜c1δαS. +The NNLO value of the coefficient ˜c1 for the event shapes τ⊥ and Tm is shown in fig. 7. +We observe that the relative sensitivity of these two observables to αS,0 increases with larger +values of the corresponding kinematic variables, reaching a plateau of ˜c1 ≈ 10. This increase +4The coefficient c0 is not independent from c1,2,3 as follows from the linear αS,0 dependence of RNNLO. +– 14 – + +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +cos(φ) +0 +2 +4 +6 +8 +10 +˜c1 +µR = µF = ˆHT +NNPDF30 +NNLO +Figure 8. As in fig. 7 but for the TEEC. +HT,2 bin +[1000, 1500] GeV +[1500, 2000] GeV +≥ 2000 GeV +⟨ ˆHT ⟩ +1371 ± 7 GeV +1928 ± 13 GeV +2607 ± 15 GeV +Table 2. The average ˆHT in each HT,2 bin computed from the integrated cross section of dσNNLO +2 +in +each bin. The uncertainty indicated above is from MC integration. +in sensitivity is consistent with the observation that the contribution from higher multiplicity +matrix elements becomes more significant for larger thrust values. These come with additional +powers of αS, increasing the sensitivity to αS,0. A value ˜c1 ≈ 10 implies that a 1% shift in +αS,0 leads to ≈ 1% shift in the prediction. This suggests that these event shapes are suitable +for potential future extraction of αS,0 from LHC data. +In fig. 8 we show the dependence of the TEEC on αS,0. The sensitivity of this observable +is largely kinematics-independent, with a value ˜c1 ≈ 6. This implies that a 1% shift in αS,0 +changes the TEEC by about 0.6%, which is similar to the αS,0 sensitivity of the other event +shapes around their peak regions. +Up to this point we considered the case where one uses the measured event shapes in +order to extract αS,0, i.e. the value of αS at the standard reference scale µF = mZ. Besides +this reference value of the strong coupling, there is a long-standing interest in experimentally +verifying its SM running, i.e. the dependence of αS on its energy scale. Keeping in mind that +αS is not an observable, which implies that there is an arbitrariness in the energy scale that +is associated with a given measurement. Technically, αS is ran up to the scale µR, therefore, +a natural choice for the αS scale associated to a binned measurement is the mean value ⟨µR⟩ +of the renormalisation scale in each bin. In this work we will not elaborate on the question of +choosing this scale (see, for example, ref. [80] for a broader discussion on this topic). We will +only remark that if a particular choice of µR results in a perturbatively convergent prediction, +it is reasonable to assume that it represents the relevant physical scale for this process and +observable. +In this work we use the scale choice µR = ˆHT which, indeed, exhibits good +perturbative convergence. From this we conclude that the best choice for the energy scale +in each HT,2 bin is ⟨ ˆHT ⟩ and in the context of our calculation, it should be interpreted as +the energy scale at which αS is extracted. The results for ⟨ ˆHT ⟩ in each HT,2 bin is shown in +table 2. +– 15 – + +[] +0.96 +0.98 +1.00 +1.02 +1000GeV ≤ HT,2 < 1500GeV +µR = µF = ˆHT +Herwig +Pythia A14 +[] +0.96 +0.98 +1.00 +1.02 +MC(NP=ON)/MC(NP=OFF) +1500GeV ≤ HT,2 < 2000GeV +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +τ⊥ +0.96 +0.98 +1.00 +1.02 +2000GeV ≤ HT,2 +[] +0.96 +0.98 +1.00 +1.02 +1000GeV ≤ HT,2 < 1500GeV +µR = µF = ˆHT +Herwig +Pythia A14 +[] +0.96 +0.98 +1.00 +1.02 +MC(NP=ON)/MC(NP=OFF) +1500GeV ≤ HT,2 < 2000GeV +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Tm +0.96 +0.98 +1.00 +1.02 +2000GeV ≤ HT,2 +Figure 9. +The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins. The +solid lines show the estimated non-perturbative effects from Herwig and Pythia. +3.3 +Estimation of non-perturbative corrections +We conclude the discussion of event shape observables with a discussion of non-perturbative +effects. Non-perturbative corrections can have a non-trivial impact on event shapes, see for +example the discussion of thrust [67] and energy-energy correlator [68] in e+e− collisions, and +can impact the accuracy of αS extractions. The observables studied in this work are based +on clustered jets which reduces the sensitivity to such effects. One possibility for assessing +the effects of hadronisation and multi-parton interactions (MPI) is to make use of Monte +Carlo event generators. For this purpose, we evaluate the event shapes with Herwig [81–83] +and Pythia [84, 85] at LO 5, once with active hadronisation and MPI and once without. +The ratio of the two predictions serves as an estimate of the expected non-perturbative +corrections. Numerical results for the thrust and thrust minor are presented in figure 9. The +non-perturbative effects reach 1% which is rather small. This confirms the expectation that +for the event shapes considered in this work the non-perturbative effects are subdominant to +other sources of theory uncertainty. +4 +Conclusions +In this work we perform the first calculation of jet event shapes at hadron colliders at NNLO +in QCD. Specifically, we consider the transverse thrust τ⊥ and its minor component Tm, the +shapes A, C, and D derived from the linearised sphericity tensor, the transverse sphericity +5NLO+PS is not yet readily available for three jet production. The exception is a study with the Sherpa +event generator [33]. +– 16 – + +variable S⊥ and the transverse energy-energy correlator. In order to be able to describe the +full kinematics of these event shapes, one needs to include all contributions with at least three +jets in the final state. Such a calculation became possible at NNLO only very recently [34]. +The immediate goal of this work is to clarify if higher order corrections to jet event shapes +at hadron colliders significantly improve the theory/data comparison for these observables. +Another goal relates to the well-known fact that such observables require all-order resumma- +tion in certain regions of phase space. One would like to clarify if by including NNLO QCD +corrections to these observables, kinematic regions where fixed order perturbation theory is +reliable can be clearly identified. Such regions are important because in e+e− collisions they +are typically used for measuring the strong coupling constant as well as for tuning parameters +of shower Monte Carlo event generators. +In this work we provide predictions for typical ATLAS setups at 13 TeV and compare +them with data where available (only for the TEEC no public numbers are available). Across +all event shapes we observe that NNLO QCD reduces significantly the scale uncertainty of the +predictions, typically by a factor of 2 to 4 relative to NLO. More importantly, the inclusion +of NNLO QCD corrections has a large impact on the shapes of these observables. At NLO +QCD one typically observes a theory/data agreement within the theory scale uncertainty, +however, the shapes of theory and data tend to be rather different. Once NNLO corrections +are included the theory and data shapes tend to “align” well. +Once NNLO QCD corrections are included, one can clearly identify narrow kinematic +regions where fixed order predictions are unreliable. Likely, this is due to missing all-order +resummation. As it might be expected, for all observables that show such a behavior, this +is the limit where a three-jet final state starts to resemble a two-jet one. Outside of this +relatively narrow region, the event shapes are reliably described by fixed order calculations. +For all event shapes we observe that the total experimental uncertainty tends to be +smaller than the theory one at NNLO QCD. The dominant source of theory uncertainty +is scale variation. A second, comparable source of theory uncertainty is the Monte Carlo +integration one. The three jet calculation is extremely computationally expensive and one +cannot expect to improve on it unless very significant computational resources are deployed. +This might be required for future high precision theory/data comparisons, since in places +where the MC uncertainty is large it tends to also blow up the estimated scale variation. A +smaller but not insignificant source of theoretical uncertainty is the PDF one. We estimate +it to be probably about half the scale one or less, which makes it less relevant in immediate +precision applications. +We have also estimated the effect of non-perturbative corrections +which we find to be around or below 1%, and therefore negligible. We have not investigated +the effect of EW corrections. These are expected to be small, partly because the event shapes +are defined as ratios of three- to two-jet cross sections. +Important immediate applications of our results relate to the extraction of the strong +coupling constant. An obvious application is the precision determination of αS(mZ) from +LHC jet data. Although we do not perform such an extraction in this work, we have provided a +detailed investigation of its feasibility and prospects. Our analysis demonstrates that the event +– 17 – + +shapes considered in the present work have sensitivities to the value of αS(mZ) of between +about 0.5% and 1%. This makes them suitable for the extraction of αS in the kinematic regions +where fixed order perturbation theory is reliable. A second, no less important, application is +the measurement of the running of αS. The suitability of the three-to-two jet cross section for +such a measurement has been well known for a very long time, however the readily available +NLO QCD predictions [31, 32] are not precise enough for performing such a measurement. +The calculation of the NNLO QCD corrections, provided in the present work, allows for the +first time to precisely map out the running of the QCD coupling constant to energy scales +as large as several TeV. A measurement with such an unprecedented precision will allow new +ways for searching for physics beyond the SM and for improving our understanding of the +running of the SM coupling constants well above the EW scale. +Acknowledgments +The work of M.C. was supported by the Deutsche Forschungsgemeinschaft under grant +396021762 – TRR 257. The research of A.M. and R.P. has received funding from the European +Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation +Programme (grant agreement no. 683211). A.M. was also supported by the UK STFC grants +ST/L002760/1 and ST/K004883/1. R.P. acknowledges support from the Leverhulme Trust +and the Isaac Newton Trust. +This work was performed using the Cambridge Service for +Data Driven Discovery (CSD3), part of which is operated by the University of Cambridge +Research Computing on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The +DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants +ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1. DiRAC is +part of the National e-Infrastructure. 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Commun. 191 (2015) 159 [1410.3012]. +– 23 – + diff --git a/UtAzT4oBgHgl3EQfJvuy/content/tmp_files/load_file.txt b/UtAzT4oBgHgl3EQfJvuy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9bfcf70a08855a1d44e8ccd696ec115c5dbdc620 --- /dev/null +++ b/UtAzT4oBgHgl3EQfJvuy/content/tmp_files/load_file.txt @@ -0,0 +1,1353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf,len=1352 +page_content='Prepared for submission to JHEP Cavendish-HEP-22/11, P3H-22-129, TTK-22-49 NNLO QCD corrections to event shapes at the LHC Manuel Alvarez,c Josu Cantero,d Michal Czakon,a Javier Llorente,e Alexander Mitov,b Rene Ponceletb aInstitut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany bCavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom cDepartamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' dInstituto de F´ısica Corpuscular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Valencia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' eDepartment of Physics, Simon Fraser University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Burnaby, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' E-mail: mczakon@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='de, adm74@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='uk, poncelet@hep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='uk, josu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='cantero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='garcia@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='ch, manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='alvarez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='estevez@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='ch, javier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='llorente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='merino@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='ch Abstract: In this work we perform the first ever calculation of jet event shapes at hadron colliders at next-to-next-to leading order (NNLO) in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The inclusion of higher order corrections removes the shape difference observed between data and next-to-leading order predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The theory uncertainty at NNLO is comparable to, or slightly larger than, existing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Except for narrow kinematical ranges where all-order resummation becomes important, the NNLO predictions for the event shapes considered in the present work are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As a prime application of the results derived in this work we provide a detailed investigation of the prospects for the precision determination of the strong coupling constant and its running through TeV scales from LHC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='01086v1 [hep-ph] 3 Jan 2023 Contents 1 Introduction 1 2 Definitions and computational details 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 Definition of event shapes 5 3 Event-shapes at the LHC 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 Perturbative QCD corrections 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 Strong coupling dependence 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 Estimation of non-perturbative corrections 16 4 Conclusions 16 1 Introduction One of the most direct ways of exhibiting the QCD dynamics at hadron colliders is by studying multi-jet events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Such events are special since they are not very sensitive to the electroweak (EW) sector of the Standard Model (SM), contain almost exclusively QCD partons, and are driven by QCD’s dynamics, color structure and single coupling constant, αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The characterization of multi-jet events is often performed in terms of so-called event shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' These are single-valued observables which encode the events’ topology and energy flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Event shapes are constructed in such a way that configurations with back-to-back jets can naturally be singled out from ones with more isotropic energy distribution in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Event shapes are sensitive to the probability of additional emissions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=', the QCD dynamics and the strong coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' First measurements of multi-jet event shapes were performed at LEP [1–4] and were used to extract αS [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Event shapes and multi-jet production at hadron colliders have been studied extensively, starting with the Tevatron [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' At the LHC, the ATLAS and CMS experiments have measured generic multi-jet differential cross-sections [7–10], event shapes at 7 and 13 TeV [11–16], azimuthal decorrelations at 8 TeV [17] and transverse-energy-energy-correlations at 13 TeV [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Theoretical work on jet production in hadron-hadron collisions also has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Fixed-order, next-to-leading-order (NLO) QCD corrections for dijet events have been first obtained in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' These have been complemented by NLO EW corrections [21, 22] and parton-shower effects [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Dijet cross-sections with NLO accuracy are also available from general-purpose event generators [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The computation of next-to-next-to-leading- order (NNLO) QCD corrections to dijet events has been achieved relatively recently [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 1 – Based on simple energy-momentum considerations it is clear that dijet events contribute trivially to event shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Therefore, only events with three or more jets in the final state can span non-trivial values of event shapes while dijet events tend to only have endpoint contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Three-jet cross-sections have been studied at NLO QCD [31, 32] and NLO EW, including parton-shower effects [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Very recently, the first computation of NNLO QCD corrections to three-jet production has been performed by some of the authors of the present article [34], mainly focusing on three-to-two jet ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Higher jet multiplicities have also been investigated in the literature with up to five jet production known through NLO in QCD [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Fixed-order calculations do not always adequately describe multi-jet event shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For certain kinematic configurations one has to resum contributions from QCD emissions to all orders in αS in order to have sensible theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Resummation with Next-to- leading-logarithmic (NLL) accuracy matched to NLO QCD predictions has been achieved for global event shapes [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As their name suggests, global event shapes feature the important property that they are defined in the full phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The presence of phase space restriction like phase space cuts, radiation gaps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=', renders an event shape non-global and gives rise to associated logarithms, so-called non-global logarithms [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The resummation of non-global logarithms is notoriously difficult and has been achieved only in special cases, see for example, refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [41–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The full phase space is never accessible at hadron colliders due to the missing forward coverage of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This limits the implementation of global event shapes experimen- tally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A possible work-around is to re-define event shapes by exponentially suppressing recoil terms, which regulates the contribution from the unavailable phase space [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' However, such an implementation would require the measurement of event’s hadronic recoil which is accompanied by a significant experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' It also requires the measurement of individual particle momenta which increases the sensitivity to hadronisation and other non-perturbative effects that are difficult to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' An alternative approach to event shapes is to define event shapes in terms of reconstructed jets instead of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Such an approach has the advantage that non-perturbative effects from hadronisation and multi-parton interactions (MPI) are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Experimentally, working with jets instead of calorimeter clusters, which are the proxies of the particles at the detector level, is also advantageous since detector-level jets are well-defined and well- calibrated objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The calibration of the jet energy scale and resolution together with their corresponding uncertainties is typically well under control, whereas this not the case for calorimeter clusters in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In addition, it is easier to associate detector-level jets to particle-level jets, in contrast to the case of a calorimeter cluster which typically cannot be uniquely associated to a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Event shapes are a powerful tool for testing parton-shower models in state-of-the-art Monte Carlo simulations and for tuning shower parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For example, differences in the matching to fixed-order matrix elements and in the ordering of the emissions (pT , θ, etc) can lead to important differences in the description of these observables [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 2 – In this work, we compute the NNLO QCD corrections to a set of event shapes based on reconstructed jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As explained above, jet-based event shapes allow for precise experimental measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' an increased precision of theoretical predictions will make it possible to closely scrutinize our ability to precisely describe QCD at hadron colliders and to extract αS and its running through TeV scales with unprecedented precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' While in this study we do not use MC or resummed predictions, we identify regions where such corrections might be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This article is organised as follows: in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 2 we give details about the computational setup and the definitions of the event shapes studied in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The main results are presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Specifically, perturbative corrections and PDF effects are discussed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1, the dependence on αS and its extraction in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2, while sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 contains an estimation of non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Our conclusions can be found in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 2 Definitions and computational details In this work we compute the NNLO QCD corrections to the set of multi-jet event shape observables specified in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' To span the full kinematic ranges of these observables, final states containing at least three jets are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For the calculation of the NNLO QCD corrections to the inclusive three-jet cross section we employ the sector-improved residue subtraction framework [48, 49], which has already been applied to inclusive jet [30], two jet- as well three-jet production [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Tree-level matrix elements are evaluated with the help of the AvH library [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' All contributing one-loop amplitudes are taken from the OpenLoops 2 library [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The two-loop amplitudes for three-jet production are only available in the leading-colour approximation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In the present work we utilize these approximate double virtual corrections to three-jet production along the lines of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Specifically, the two-loop finite remainder R(2)(µ2 R) = 2 Re � M†(0)F(2)� (µ2 R) + ��F(1)��2(µ2 R) ≡ R(2)(s12) + 4 � i=1 ci lni �µ2 R s12 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1) is approximated by replacing the function R(2)(s12) with its leading-colour approximation provided with the software described in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' It is implemented in terms of the so-called “pentagon functions” [54] and rational functions of the kinematic invariants [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We work in the nf = 5 scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' we consistently neglect contributions from top-quarks in the partonic cross sections and in the running of αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The αS expansion of the n-jet differential cross section reads: dσn(µR, µF , PDF, αS(µR, αS,0)) = � i αSn+i(µR, αS,0)dσ(i) n (µR, µF , PDF) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2) – 3 – where µR,F are the renormalisation and factorisation scales and PDF labels the parton distri- bution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The dependence on the initial value of the strong coupling αS,0 ≡ αS(µR = mZ) is also made explicit for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The strong coupling constant satisfies the following renormalisation group equation (RGE): µ2 R dαS dµ2 R = β(αS) = −(b0αS2 + b1αS3 + b2αS4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3) where b0,1,2 are the 1-, 2- and 3-loop β-function coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The strong coupling constant at a scale µR, together with the corresponding PDF set, is obtained from the LHAPDF library [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Specifically, αS(µR) is derived from the numerical solution of the RGE eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3) starting from the reference scale µR = mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In the present work we are interested in the three lowest orders in perturbation theory and define them for convenience as dσLO n = αSndσ(0) n , dσNLO n = αSndσ(0) n + αSn+1dσ(1) n , dσNNLO n = αSndσ(0) n + αSn+1dσ(1) n + αSn+2dσ(2) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4) All multi-jet observables in this work are computed as ratios Ri(µR, µF , PDF, αS,0) = dσi 3(µR, µF , PDF, αS,0) dσi 2(µR, µF , PDF, αS,0) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5) for i ∈ {LO, NLO, NNLO}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We study the NLO and NNLO QCD K-factors: KNLO = RNLO RLO and KNNLO = RNNLO RNLO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6) To estimate missing higher-order contributions, we take the envelope of a 7-point variation of µR and µF by a factor of 2 with the constraint 1 2 ≤ µR µF ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For all predictions presented in this work, we use as a central scale choice µR = µF = ˆHT = � i∈partons pT,i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='7) where the sum runs over all partons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The PDF uncertainties are estimated by following the prescriptions of the individual PDF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Due to the high computational costs, PDF member variations have been evaluated only at NLO QCD using the NNLO version of the corresponding PDF set, and then extrapolated to NNLO QCD by using the following K-factor approach RNNLO(PDFi) ≈ RNLO(PDFi) RNLO(PDF0) RNNLO(PDF0) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8) where PDFi denotes the i-th member of the PDF set (0 corresponds to the central mem- ber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The justification for this approximation is that the ratios have a reduced dependence on the PDF due to cancellation between the two- and three-jet cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Indeed the PDF uncertainties for the ratios are typically small and dwarfed by the remaining numerical integration uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 4 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 Definition of event shapes This section provides the definitions of all event shapes computed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A classical event shape observable is the transverse thrust T⊥ (or rather τ⊥ = 1 − T⊥) [56, 57] and its minor component Tm, defined by T⊥ = � i |⃗pT,i · ˆn⊥| � i |⃗pT,i| , and Tm = � i |⃗pT,i × ˆn⊥| � i |⃗pT,i| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9) The thrust axis n⊥ maximises the projection of all jets to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As explained in the Introduction, we define the thrust and all other event shapes in terms of reconstructed jets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' the sum i in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9) above runs over the list of all reconstructed jets passing selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The thrust separates topologies that are of back-to-back type (small τ⊥) from ones that are isotropic (large τ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Another event shape that characterises the anisotropy of an event is the linearised sphericity tensor [58, 59]: Mxyz = 1 � i |⃗pi| � i 1 |⃗pi| � � � p2 x,i px,ipy,i px,ipz,i py,ipx,i p2 y,i py,ipz,i pz,ipx,i pz,ipy,i p2 z,i � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10) In this work, we study certain combinations of the eigenvalues λ1, λ2, λ3 of Mxyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' They are denoted A (the so-called aplanarity), C, and D [60]: A = 3 2λ3 , C = 3(λ1λ2 + λ1λ3 + λ2λ3) , D = 27λ1λ2λ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='11) Furthermore, from the transverse linearised sphericity tensor Mxy = 1 � i |⃗pT,i| � i 1 |⃗pT,i| � p2 x,i px,ipy,i py,ipx,i p2 y,i � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='12) with eigenvalues µ1, µ2, one defines the transverse sphericity variable: S⊥ = µ2 µ1 + µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='13) The final observable we study in this article is the transverse energy-energy correlator (TEEC) [18, 61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For multi-jet events, it is defined as 1 σ2 dσ d cos ∆φ = 1 σ2 � ij � dσ x⊥,ix⊥,j dx⊥,idx⊥,jd cos ∆φij δ(cos ∆φ − cos ∆φij)dx⊥,idx⊥,jd cos ∆φij , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='14) where x⊥,i = E⊥,i/ � k E⊥,k, and with E⊥ = � E2 − p2z being invariant under boosts along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The angle ∆φij is the azimuthal opening between a pair of jets ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Note that in the presence of phase space cut(s), the TEEC picks up an implicit depen- dence on the corresponding kinematic variable(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As specified in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3 below, the TEEC is – 5 – measured in bins of the variable HT,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The denominator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='14) is the two-jet cross- section evaluated in the corresponding bin of HT,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' To be able to generate a contribution away from the endpoints cos ∆φij = ±1, the numerator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='14) requires at least a 2 → 3 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A fixed-order 2 → 3 calculation diverges at the endpoints cos ∆φij = ±1 which correspond to the back-to-back limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' To avoid these infrared sensitive regions, a constraint | cos ∆φij| ≤ | cos ∆φmax| is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For a sufficiently small value of | cos ∆φmax| contri- butions from resummation-dominated phase space regions are suppressed and fixed-order perturbative calculations are expected to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3 Event-shapes at the LHC In this section we present NNLO QCD predictions for event shape observables measured at 13 TeV by the ATLAS collaboration [16] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For the numerator (denominator) of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5) we require at least three (two) R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 anti-kT [63] jets that fulfill the requirements HT,2 = pT,1 + pT,2 ≥ 1 TeV where pT,i is the transverse momentum of the i-th hardest jet, |y(j)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 and pT (j) ≥ 100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The inputs for the event shapes are all jets that pass the above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Due to the high computational cost of the three jet cross section, we compute the event shapes in bins that are coarser than the bins used for the measurement in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' To facilitate the comparison between the measurement and our predictions we rebin the data as follows: the systematic uncertainties between bins are considered as fully correlated, whereas the statistical uncertainties are added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' All event shapes are measured in three separate HT,2 regions 1000 GeV ≤ HT,2 < 1500 GeV, 1500 GeV ≤ HT,2 < 2000 GeV, 2000 GeV ≤ HT,2 , which allows for the separation of different energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For the computation of the TEEC, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='14), we adopt a slightly different phase space corresponding to the selection in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [18]: HT,2 = pT,1 + pT,2 ≥ 1 TeV, at least 2 anti-kT jets with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 and pT ≥ 60 GeV |y(j)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For this observable we consider a finer HT,2 binning [1000, 1200, 1400, 1600, 1800, 2000, 2300, 2600, 3000, 3500, ∞) , together with the “inclusive” case HT,2 ≥ 1000 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1The ATLAS measurement has been performed in jet-multiplicity bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We compare our predictions with the inclusive data presented in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 6 – [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 1000GeV ≤ HT,2 < 1500GeV NNPDF30 µR = µF = ˆHT [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 ratio to data 1500GeV ≤ HT,2 < 2000GeV LO NLO NNLO ATLAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='30 τ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 2000GeV ≤ HT,2 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 1000GeV ≤ HT,2 < 1500GeV NNPDF30 µR = µF = ˆHT [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 ratio to data 1500GeV ≤ HT,2 < 2000GeV LO NLO NNLO ATLAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 Tm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 2000GeV ≤ HT,2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The solid lines show fixed order LO (green), NLO (blue), and NNLO (red) predictions normalised to ATLAS data (black) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The coloured bands show the scale variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The central choice for renormalisation and factorisation scales is ˆHT , defined in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As the default PDF choice, we use the NNPDF30[64] PDF set 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 Perturbative QCD corrections In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1 to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3, we show the LO, NLO and NNLO QCD corrections to the six event shapes defined in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' All predictions are shown as ratios to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Coloured bands show the envelope of the 7-point scale variation, and vertical bars indicate the remaining uncertainty from Monte Carlo integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The data is shown with a bar indicating the statistical and systematical uncertainty added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The results for the transverse thrust τ⊥ are shown on the left-hand side of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We observe significant NLO QCD corrections to the shape of the LO prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In particular, for large thrust values representing events with hard resolved radiation, corrections can be as large as 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The scale uncertainty increases, indicating that the LO scale dependence does not suffice to estimate missing higher effects for this observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NLO result does agree fairly well with data within the uncertainties, but one can observe the tendency to undershoot the data in the tail of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Turning to the NNLO QCD results, we observe small corrections to the shape at NLO and a sizable reduction of the scale dependence by a factor of two to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NNLO QCD predictions are fully contained within the NLO 2This PDF set has been superseded by NNPDF31 [65] and NNPDF40 [66] which have smaller uncertainties and are based on better methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' These differences are not so crucial for the current discussion as the PDF uncertainties are comparatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 7 – QCD scale uncertainties indicating perturbative convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Agreement with data, within uncertainties, can be observed over the full range of the thrust observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Importantly, the NNLO QCD corrections lead to theory/data shape agreement in the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The behavior of the transverse thrust τ⊥ for small values of τ⊥ deserves special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As can be observed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1 the behavior of the fixed order prediction at all three orders is different, showing change in slope and increased scale variation, especially at LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This behavior is accompanied by an increase in the MC uncertainty of the predictions and the agreement with data at NNLO is worse than in all other bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This is the case for all three HT,2 slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The reason behind this behavior is that in this kinematics the fixed order expansion starts to break down and all-order resummation effects become important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This behavior is well known from the e+e− thrust [67] for which very high resummation as well as fixed order accuracy has been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In general, we wish to point out the substantial progress that has recently been achieved for the resummation and the description of event shape observables at high energy colliders [68–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' On the right-hand side of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1, we show the thrust minor component Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The pattern of higher-order corrections is similar to the thrust observable: large NLO and smaller NNLO corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The reduction in scale dependence when going from NLO to NNLO QCD, partic- ularly for larger values of Tm, is large and can reach a factor of 10 in the intermediate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NLO QCD predictions for Tm do not describe the shape of the observable very well, especially around the peak region Tm ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This situation is improved by the NNLO QCD calculation, which predicts the shape well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The lowest Tm bin suffers from large statistical uncertainties on the NNLO QCD predictions, for reasons similar to the ones discussed in the context of the transverse thrust τ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We find good agreement between ATLAS data and NNLO QCD predictions for the transverse sphericity S⊥ and aplanarity A, both shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Similar level of NNLO/data agreement can also be observed for the D variable shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In terms of the behavior of perturbative corrections, the three variables behave similarly to the thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The second-order QCD corrections are most significant in the regions described by events with well-separated and isotropically distributed jets and eliminate the notable shape differences observed at NLO QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The C variable, shown in the left-hand side of figure 3, has a different behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NLO QCD predictions have little scale dependence in the intermediate region for C and larger scale dependence at the two endpoints C = 0 and C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In the central region, neither the normalisation nor the shape of the data is well described within uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The numerical integration of the NNLO QCD corrections suffers from particularly large cancellations leading to significant remaining statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For that reason, it is unclear if the second-order corrections improve the theory-data agreement or if the seemingly good agreement with data is simply due to the large statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Finally, we turn to the case of the TEEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In the two top panels in figure 4, we show perturbative predictions for the inclusive HT,2 selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A preliminary measurement of this observable has been presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' However, the numbers are not yet public and – 8 – [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 1000GeV ≤ HT,2 < 1500GeV NNPDF30 µR = µF = ˆHT [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 ratio to data 1500GeV ≤ HT,2 < 2000GeV LO NLO NNLO ATLAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 S⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 2000GeV ≤ HT,2 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 1000GeV ≤ HT,2 < 1500GeV NNPDF30 µR = µF = ˆHT [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 ratio to data 1500GeV ≤ HT,2 < 2000GeV LO NLO NNLO ATLAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 2000GeV ≤ HT,2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1 but for the transverse sphericity S⊥ (left) and the aplanarity A (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 1000GeV ≤ HT,2 < 1500GeV NNPDF30 µR = µF = ˆHT [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 1000GeV ≤ HT,2 < 1500GeV NNPDF30 µR = µF = ˆHT [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 ratio to data 1500GeV ≤ HT,2 < 2000GeV LO NLO NNLO ATLAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 2000GeV ≤ HT,2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 1 but for the variables C (left) and D (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 9 – therefore we show no theory/data comparison for this observable in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The typical pattern of higher-order corrections familiar from our discussion of even shapes also emerges for the TEEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NLO QCD corrections are sizable relative to the leading order result, barely touching the uncertainty estimate from scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NLO QCD estimate of the scale uncertainty is flat and about 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The corrections from NNLO QCD are smaller, between 0% and 5%, and lie fully within the NLO scale uncertainty band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The scale variation uncertainty of the NNLO QCD prediction is about 1−3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The remaining numerical integration uncertainty of the NNLO QCD prediction is about ∼ 1 − 3%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' it is similar to the scale uncertainty 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Unlike the other event shapes discussed so far, the corrections are relatively flat, and there is no phase space region which receives larger corrections than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Notably, within scale uncertainty, the NNLO QCD prediction is consistent with the central NLO QCD one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In the double differential case, shown in figure 5, we see that the above picture persists also for the individual HT,2 slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Only for HT,2 > 3 TeV we observe an increase in the perturbative correction and scale uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Still, NNLO QCD corrections remain within the NLO scale uncertainty bands, indicating perturbative stability even at the highest available energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We next turn our attention to PDF uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 6 we show the NNLO QCD predictions for the thrust and the minor component, evaluated with the following PDF sets: NNPDF30 [64], MMHT2014 [76], CT18 [77] and HERAPDF [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The PDF uncertainty estimate for each set is shown as a colored band around the corresponding central value, normalised to the central NNPDF30 prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The numerical MC uncertainties are not shown because they are fully correlated between different sets, as the same events have been used to evaluate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Starting with the uncertainties for our default PDF set NNPDF30, for both variables we observe that the uncertainties around the peak of the distribution are very small, below 1%, and are therefore negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The uncertainties increase continuously towards the tails of the distributions and become as large as 2 − 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Such a behaviour is expected since the cancellation between PDF dependencies in the numerator and denominator is realised best around the peak of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The spread between the different PDF sets is about 1 − 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' All PDFs are compatible within their uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Furthermore, at NLO QCD the PDF uncertainties are much smaller than the scale ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' At NNLO QCD the scale uncertainties are much reduced – reaching about 5% in the tails of the distributions – which leads to PDF uncertainties becoming comparable to the scale ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Still, at NNLO the scale uncertainty remains the dominant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The PDF uncertainties of the other event shapes considered in the present work feature similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' More details can be read off the results that are available in electronic format [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3Since integration uncertainty might lead to imperfect cancellations, the actual scale uncertainty might be smaller than indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 10 – [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15 dσ/d cos(φ) [pb] µR = µF = ˆHT NNPDF30 LO NLO NNLO −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 cos(φ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 ratio to NLO −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 cos(φ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='01 ratio to NNLO NNLO - PDF uncertainty Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The TEEC variable in the inclusive HT,2 ≥ 1 TeV bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The top panel shows the absolute differential distribution through LO (light green), NLO (blue) and NNLO (red) QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The coloured bands show the scale uncertainty estimates and vertical bars indicate statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The second panel shows the ratio to the central NLO QCD prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The third panel shows the PDF uncertainty estimate from NNPDF30 at NLO QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The PDF uncertainty of the TEEC distribution computed with the NNPDF30 PDF set is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 4 for the inclusive HT,2 > 1 TeV bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The uncertainty is flat, below 1%, indicating that there is a strong cancellation between numerator and denominator for this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A similar PDF behavior is also observed in the individual HT,2 bins, which we do not show here but supply in electronic form [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 Strong coupling dependence In this section we quantify the sensitivity of the various event shapes to the value of the strong coupling constant αS,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' While in the present work we do not attempt to extract a value of αS,0, the results in this section are an essential step in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' To estimate the dependence of the event shapes on the strong coupling constant αS,0, we evaluate them through NNLO QCD using PDF sets with different values of αS,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Each PDF set provides a different range of αS,0 values, and these have been summarized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The αS,0 dependence in each bin can be derived by expressing the event shape in that bin as a function of αS,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This can be achieved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' First, consider a perturbative solution – 11 – [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1000GeV ≤ HT,2 < 1200GeV LO NLO NNLO [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1200GeV ≤ HT,2 < 1400GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1400GeV ≤ HT,2 < 1600GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1600GeV ≤ HT,2 < 1800GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 1800GeV ≤ HT,2 < 2000GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 2000GeV ≤ HT,2 < 2300GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 2300GeV ≤ HT,2 < 2600GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 2600GeV ≤ HT,2 < 3000GeV [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 3000GeV ≤ HT,2 < 3500GeV −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 cos(φ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 3500GeV ≤ HT,2 Scale: µR = µF = ˆHT PDF: NNPDF30 ratio to NLO Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The TEEC variable double differentially in HT,2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The panels show LO (light green), NLO (blue) and NNLO (red) QCD prediction as a ratio to the central NLO QCD prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The coloured bands show the scale uncertainty estimates and vertical bars indicate statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 12 – [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 1000GeV ≤ HT,2 < 1500GeV µR = µF = ˆHT NNLO QCD NNPDF30 HERAPDF20 MMHT2014 CT18 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 ratio to NNPDF30 1500GeV ≤ HT,2 < 2000GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='30 τ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 2000GeV ≤ HT,2 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 1000GeV ≤ HT,2 < 1500GeV µR = µF = ˆHT NNLO QCD NNPDF30 HERAPDF20 MMHT2014 CT18 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 ratio to NNPDF30 1500GeV ≤ HT,2 < 2000GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 Tm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 2000GeV ≤ HT,2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The solid lines show fixed-order NNLO QCD predictions for different PDF sets normalised to NNPDF30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The coloured band show the estimated PDF uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' PDF set αS,0 values NNPDF30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='115, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='117, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='119, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='121 CT18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='110, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='111, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='123, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='124 MMHT2014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='108, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='109, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='127, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='128 HERAPDF20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='110, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='111, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='129, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='130 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The values for αS,0 = αS(µR = mZ) available with each PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' of the RGE for the strong coupling constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' αS(µR, αS,0) = αS,0 � 1 − αS,0b0 ln � µ2 R m2 Z � + O � αS,02�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1) Setting for simplicity µ = µR = µF , one can then rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5) in such a way that the RGE running of the strong coupling is absorbed into the partonic cross section RNNLO(µ, αS,0) = dσNNLO 3 (µ, αS,0) dσNNLO 2 (µ, αS,0) = αS,03 � d˜σ(0) 3 (µ) + αS,0d˜σ(1) 3 (µ) + αS,02d˜σ(2) 3 (µ) + O(αS,03) � αS,02 � d˜σ(0) 2 (µ) + αS,0d˜σ(1) 2 (µ) + αS,02d˜σ(2) 2 (µ) + O(αS,03) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2) makes explicit the non-PDF αS,0 dependence of the event shape RNNLO(µ, αS,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' It also makes it clear that the leading αS,0 dependence of RNNLO(µ, αS,0) is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 13 – [] 0 5 10 15 20 1000GeV ≤ HT,2 < 1500GeV µR = µF = ˆHT NNPDF30 HERAPDF20 MMHT2014 CT18 [] 0 5 10 15 20 ˜c1 1500GeV ≤ HT,2 < 2000GeV NNLO QCD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='30 τ⊥ 0 5 10 15 20 2000GeV ≤ HT,2 [] 0 5 10 15 20 1000GeV ≤ HT,2 < 1500GeV µR = µF = ˆHT NNPDF30 HERAPDF20 MMHT2014 CT18 [] 0 5 10 15 20 ˜c1 1500GeV ≤ HT,2 < 2000GeV NNLO QCD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 Tm 0 5 10 15 20 2000GeV ≤ HT,2 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The solid lines show the coefficient ˜c1 at NNLO QCD for different PDF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Based on the above observations, a very practical way of parameterizing the αS,0 depen- dence of the computed event shape is to assume the functional form RNNLO,fit(µ, αS,0) = c0 + c1(αS,0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118) + c2(αS,0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118)2 + c3(αS,0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118)3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3) whose coefficients ci in each bin are determined by fitting RNNLO as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We stress that the fit encodes the unexpanded in αS,0 value of RNNLO as well as the αS running obtained directly from the numeric RGE solution as provided by LHAPDF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' one does not need to utilize the analytic RGE eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The coefficients c2 and c3 are typically small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Moreover, one is only interested in percent- level variations of αS,0 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118 (which is a convenient proxy for αS,0’s world average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Therefore, in order to assess the sensitivity of RNNLO to the value of αS,0 it is sufficient to focus on the coefficient c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In practice we consider the rescaled coefficient ˜c1 = c1 RNNLO(αS,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4) which corresponds to the normalised first derivative of RNNLO with respect to αS,0 at αS,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The interpretation of ˜c1 is that if αS,0 changes by an amount δαS,0 around αS,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118, the ratio RNNLO(αS,0)/RNNLO(αS,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='118) changes by ˜c1δαS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The NNLO value of the coefficient ˜c1 for the event shapes τ⊥ and Tm is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We observe that the relative sensitivity of these two observables to αS,0 increases with larger values of the corresponding kinematic variables, reaching a plateau of ˜c1 ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This increase 4The coefficient c0 is not independent from c1,2,3 as follows from the linear αS,0 dependence of RNNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 14 – −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='75 cos(φ) 0 2 4 6 8 10 ˜c1 µR = µF = ˆHT NNPDF30 NNLO Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 7 but for the TEEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' HT,2 bin [1000, 1500] GeV [1500, 2000] GeV ≥ 2000 GeV ⟨ ˆHT ⟩ 1371 ± 7 GeV 1928 ± 13 GeV 2607 ± 15 GeV Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The average ˆHT in each HT,2 bin computed from the integrated cross section of dσNNLO 2 in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The uncertainty indicated above is from MC integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' in sensitivity is consistent with the observation that the contribution from higher multiplicity matrix elements becomes more significant for larger thrust values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' These come with additional powers of αS, increasing the sensitivity to αS,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A value ˜c1 ≈ 10 implies that a 1% shift in αS,0 leads to ≈ 1% shift in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This suggests that these event shapes are suitable for potential future extraction of αS,0 from LHC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 8 we show the dependence of the TEEC on αS,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The sensitivity of this observable is largely kinematics-independent, with a value ˜c1 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This implies that a 1% shift in αS,0 changes the TEEC by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6%, which is similar to the αS,0 sensitivity of the other event shapes around their peak regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Up to this point we considered the case where one uses the measured event shapes in order to extract αS,0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' the value of αS at the standard reference scale µF = mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Besides this reference value of the strong coupling, there is a long-standing interest in experimentally verifying its SM running, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' the dependence of αS on its energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Keeping in mind that αS is not an observable, which implies that there is an arbitrariness in the energy scale that is associated with a given measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Technically, αS is ran up to the scale µR, therefore, a natural choice for the αS scale associated to a binned measurement is the mean value ⟨µR⟩ of the renormalisation scale in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In this work we will not elaborate on the question of choosing this scale (see, for example, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' [80] for a broader discussion on this topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We will only remark that if a particular choice of µR results in a perturbatively convergent prediction, it is reasonable to assume that it represents the relevant physical scale for this process and observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In this work we use the scale choice µR = ˆHT which, indeed, exhibits good perturbative convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' From this we conclude that the best choice for the energy scale in each HT,2 bin is ⟨ ˆHT ⟩ and in the context of our calculation, it should be interpreted as the energy scale at which αS is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The results for ⟨ ˆHT ⟩ in each HT,2 bin is shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 15 – [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='02 1000GeV ≤ HT,2 < 1500GeV µR = µF = ˆHT Herwig Pythia A14 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='02 MC(NP=ON)/MC(NP=OFF) 1500GeV ≤ HT,2 < 2000GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='30 τ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='02 2000GeV ≤ HT,2 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='02 1000GeV ≤ HT,2 < 1500GeV µR = µF = ˆHT Herwig Pythia A14 [] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='02 MC(NP=ON)/MC(NP=OFF) 1500GeV ≤ HT,2 < 2000GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='6 Tm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='02 2000GeV ≤ HT,2 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The transverse thrust τ⊥ (left) and the thrust minor Tm (right) in three HT,2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The solid lines show the estimated non-perturbative effects from Herwig and Pythia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='3 Estimation of non-perturbative corrections We conclude the discussion of event shape observables with a discussion of non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Non-perturbative corrections can have a non-trivial impact on event shapes, see for example the discussion of thrust [67] and energy-energy correlator [68] in e+e− collisions, and can impact the accuracy of αS extractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The observables studied in this work are based on clustered jets which reduces the sensitivity to such effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' One possibility for assessing the effects of hadronisation and multi-parton interactions (MPI) is to make use of Monte Carlo event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For this purpose, we evaluate the event shapes with Herwig [81–83] and Pythia [84, 85] at LO 5, once with active hadronisation and MPI and once without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The ratio of the two predictions serves as an estimate of the expected non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Numerical results for the thrust and thrust minor are presented in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The non-perturbative effects reach 1% which is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This confirms the expectation that for the event shapes considered in this work the non-perturbative effects are subdominant to other sources of theory uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 4 Conclusions In this work we perform the first calculation of jet event shapes at hadron colliders at NNLO in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Specifically, we consider the transverse thrust τ⊥ and its minor component Tm, the shapes A, C, and D derived from the linearised sphericity tensor, the transverse sphericity 5NLO+PS is not yet readily available for three jet production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The exception is a study with the Sherpa event generator [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' – 16 – variable S⊥ and the transverse energy-energy correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In order to be able to describe the full kinematics of these event shapes, one needs to include all contributions with at least three jets in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Such a calculation became possible at NNLO only very recently [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The immediate goal of this work is to clarify if higher order corrections to jet event shapes at hadron colliders significantly improve the theory/data comparison for these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Another goal relates to the well-known fact that such observables require all-order resumma- tion in certain regions of phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' One would like to clarify if by including NNLO QCD corrections to these observables, kinematic regions where fixed order perturbation theory is reliable can be clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Such regions are important because in e+e− collisions they are typically used for measuring the strong coupling constant as well as for tuning parameters of shower Monte Carlo event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' In this work we provide predictions for typical ATLAS setups at 13 TeV and compare them with data where available (only for the TEEC no public numbers are available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Across all event shapes we observe that NNLO QCD reduces significantly the scale uncertainty of the predictions, typically by a factor of 2 to 4 relative to NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' More importantly, the inclusion of NNLO QCD corrections has a large impact on the shapes of these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' At NLO QCD one typically observes a theory/data agreement within the theory scale uncertainty, however, the shapes of theory and data tend to be rather different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Once NNLO corrections are included the theory and data shapes tend to “align” well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Once NNLO QCD corrections are included, one can clearly identify narrow kinematic regions where fixed order predictions are unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Likely, this is due to missing all-order resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' As it might be expected, for all observables that show such a behavior, this is the limit where a three-jet final state starts to resemble a two-jet one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Outside of this relatively narrow region, the event shapes are reliably described by fixed order calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' For all event shapes we observe that the total experimental uncertainty tends to be smaller than the theory one at NNLO QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The dominant source of theory uncertainty is scale variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A second, comparable source of theory uncertainty is the Monte Carlo integration one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The three jet calculation is extremely computationally expensive and one cannot expect to improve on it unless very significant computational resources are deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This might be required for future high precision theory/data comparisons, since in places where the MC uncertainty is large it tends to also blow up the estimated scale variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A smaller but not insignificant source of theoretical uncertainty is the PDF one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We estimate it to be probably about half the scale one or less, which makes it less relevant in immediate precision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We have also estimated the effect of non-perturbative corrections which we find to be around or below 1%, and therefore negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' We have not investigated the effect of EW corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' These are expected to be small, partly because the event shapes are defined as ratios of three- to two-jet cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Important immediate applications of our results relate to the extraction of the strong coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' An obvious application is the precision determination of αS(mZ) from LHC jet data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Although we do not perform such an extraction in this work, we have provided a detailed investigation of its feasibility and prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Our analysis demonstrates that the event – 17 – shapes considered in the present work have sensitivities to the value of αS(mZ) of between about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='5% and 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This makes them suitable for the extraction of αS in the kinematic regions where fixed order perturbation theory is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A second, no less important, application is the measurement of the running of αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The suitability of the three-to-two jet cross section for such a measurement has been well known for a very long time, however the readily available NLO QCD predictions [31, 32] are not precise enough for performing such a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The calculation of the NNLO QCD corrections, provided in the present work, allows for the first time to precisely map out the running of the QCD coupling constant to energy scales as large as several TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A measurement with such an unprecedented precision will allow new ways for searching for physics beyond the SM and for improving our understanding of the running of the SM coupling constants well above the EW scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Acknowledgments The work of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' was supported by the Deutsche Forschungsgemeinschaft under grant 396021762 – TRR 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The research of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' 683211).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' was also supported by the UK STFC grants ST/L002760/1 and ST/K004883/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' acknowledges support from the Leverhulme Trust and the Isaac Newton Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' This work was performed using the Cambridge Service for Data Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' The DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' DiRAC is part of the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfJvuy/content/2301.01086v1.pdf'} +page_content=' Simulations were performed with computing resources granted by RWTH Aachen University under project p0020025.' 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Differential Equation Solvers +Matthew J. H. Wright +December 2022 +Abstract +We motivate the use of neural networks for the construction of numerical solutions to differential +equations. We prove that there exists a feed-forward neural network that can arbitrarily minimise an +objective function that is zero at the solution of Poisson’s equation, allowing us to guarantee that neural +network solution estimates can get arbitrarily close to the exact solutions. +We also show how these +estimates can be appreciably enhanced through various strategies, in particular through the construction +of error correction networks, for which we propose a general method. We conclude by providing numerical +experiments that attest to the validity of all such strategies for variants of Poisson’s equation. The source +code for this project can be found at https://github.com/mjhwright/error-correction. +1 +Introduction +Differential equations are among the most ubiquitous problems in contemporary mathematics. In recent +years, developments in artificial neural networks have prompted new research into their capacity to be +excellent differential equation solvers [1–5]. They are universal approximators [6]; they can circumvent the +curse of dimensionality [7]; and they are continuous. However, practically, their construction and optimisation +costs are enough to deter the discerning user. +In this paper, we explain a method by which neural networks can numerically solve differential equations. +We further this by providing three strategies that can be targeted to improve the efficacy of the solver. +The first two – sinusoidal representation networks [8] and random Fourier features [9] – are well-established +in the field of artificial neural networks and machine learning. The third is a novel technique called error +correction [10–14]. We explain how error correction can be implemented recursively, with little modification +to the original solver, to give enhanced numerical solutions to differential equations, and we present results +that demonstrate this. +This paper is designed to give a flavour of the competence of artificial neural networks in this field, while +also highlighting their certain limitations. +2 +Background +Throughout this paper, we consider differential equations with solution φ : Rd → R. Consequently, our +neural network approximation is a function N : Rd → R. +2.1 +Universal approximation theorems +The realisation of neural networks’ capabilities to learn seemingly any function has brought about numerous +universal approximation theorems. These state that, under certain conditions, neural networks are able to +approximate any function to arbitrary closeness. We recall one of these theorems by Hornik [6]. +First, define the set of all functions represented by a neural network with a single hidden layer of width n +and identity activation on the output layer as +1 +arXiv:2301.13146v1 [math.NA] 28 Dec 2022 + +A n(σ) = +� +N : Rd → R, N(x) = W(1) � +σ +� +W(0)x + b(0)�� ++ b(1)� +where x ∈ Rd, W(0) ∈ Rn×d, W(1) ∈ R1×n, b(0) ∈ Rn, b(1) ∈ R, and σ : R → R is applied element-wise. +Then, +A (σ) = +∞ +� +n=1 +A n(σ) +(1) +is the set of all such functions with any number of neurons. Define also Cm(Rd) as the space of all functions +that, together with their partial derivatives of order |α| ≤ m, are continuous on Rd. +Theorem 1. [6] If σ ∈ Cm(Rd) is nonconstant and bounded, then A (σ) is uniformly m-dense on all compact +sets of Cm(Rd), i.e. for all φ ∈ Cm(Rd), for all compact sets Ω ⊂ Rd and for all ϵ > 0, there exists N ∈ A (σ) +such that +max +|α|≤m sup +x∈Ω +|∂(α) +x N(x) − ∂(α) +x φ(x)| < ϵ +Theorem 1 illustrates the universal approximation quality for single-layer networks of arbitrary width. Ap- +plying results from an earlier paper by Hornik et al. [15], this can be extended to multilayer networks. +Crucially, these theorems tell us that neural networks are dense on certain function spaces, but they do not +tell us how to train a network to realise this. +2.2 +Neural network differential equation solvers +Using neural networks to solve differential equations was introduced in the late 1990s [1], but experienced +a modern resurgence through the publication of two papers [2,3] on physics-informed neural networks. The +deep Galerkin method [4] which we describe below is very similar to the method described in [2] only, +instead of using experimental data, we train a network on points randomly sampled across the domain of +the differential equation. +Consider Poisson’s equation with Dirichlet boundary conditions: +� +∇2φ += f in Ω +φ += g on ∂Ω +(2) +Lemma 2. Let Ω ⊂ Rd be a smooth, compact domain. Then there exists at most one solution φ to (2). +Proof. Suppose φ and ϕ both satisfy the conditions of (2) and let ω = φ − ϕ. Then ω is harmonic in Ω and +zero on ∂Ω. Then, +� +Ω +ω(x)∇2ω(x) dx = +� +∂Ω +ω(x)δnω(x) dx − +� +Ω +||∇ω(x)||2 dx = − +� +Ω +||∇ω(x)||2 dx = 0 +and ∇ω = 0. Thus, ω = 0 and φ = ϕ. +We now seek an approximation N to φ. Define the objective function +J (N) = +� +Ω +|∇2N(x) − f(x)|2ν1(x) dx + +� +∂Ω +|N(x) − g(x)|2ν2(x) dx +2 + +for probability distributions ν1 on Ω and ν2 on ∂Ω. By uniqueness of φ, J (N) = 0 =⇒ N = φ. However, +minimising the objective function directly is impractical. First, we transform the problem into a machine +learning framework. Our approximation N = N(·; θ) becomes a neural network with parameters θ. +Deep Galerkin method +We demonstrate the algorithm for the deep Galerkin method [4] when applied to Poisson’s equation (2): +1. Randomly sample points {xi}M +i=1 from Ω and {yj}N +j=1 from ∂Ω according to respective probability +distributions ν1 and ν2, and propagate them through a feed-forward neural network N(·; θ). +2. Calculate the loss: +L(θ) = 1 +M +M +� +i=1 +� +∇2N(xi; θ) − f(xi) +�2 + 1 +N +N +� +j=1 +(N(yj; θ) − g(yj))2 +3. Update parameters θt+1 = θt − η∇θL(θt) with learning rate η > 0 and t ∈ N0. +4. Repeat until ∇θL(θt) ≈ 0. +This is a minibatch gradient descent implementation, where M and N are the size of the minibatches and +M > N. +Lemma 3. E[∇θL(θt)|θt] = ∇θJ (N(·; θt)) +Proof. Assume L sufficiently smooth and bounded to interchange derivatives and integrals. Then, +E[∇θL(θt)|θt] = ∇θ +� +� 1 +M +M +� +i=1 +E +� +(∇2N(xi; θt) − f(xi))2� ++ 1 +N +N +� +j=1 +E +� +(N(yj; θt) − g(yj))2� +� +� += ∇θ +� +� 1 +M +M +� +i=1 +� +Ω +(∇2N(x; θt) − f(x))2ν1(x) dx + 1 +N +N +� +j=1 +� +∂Ω +(N(y; θt) − g(y))2ν2(y) dy +� +� += ∇θ +�� +Ω +(∇2N(x; θt) − f(x))2ν1(x) dx + +� +∂Ω +(N(y; θt) − g(y))2ν2(y) dy +� += ∇θJ (N(·; θt)) +Therefore, the ∇θL(θt) are unbiased estimates of ∇θJ (N(·; θt)), and we can assume a step in the descent +direction of L is also one in J . Thus, any minimisation of L should translate to a local minimisation of J . +Minimisation of J (N) +We prove the following theorem, adapted from the original deep Galerkin method paper [4]. +Theorem 4. Let A (σ) be given by (1), for nonconstant, bounded σ, and let Ω ∈ Rd be a compact domain +and consider measures ν1, ν2 whose supports are contained in Ω, ∂Ω respectively. Assume further that ∇2φ +is locally Lipschitz with Lipschitz constant that can have at most polynomial growth on ∇φ, uniformly with +respect to x, i.e. +|∇2N − ∇2φ| ≤ +� +||∇N|| +a +2 + ||∇φ|| +b +2 +� +||∇N − ∇φ|| +(3) +for some constants 0 ≤ a, b < ∞. Then, for all ϵ > 0, there exists a constant κ > 0 such that there exists a +function N ∈ A (σ) with +J (N) ≤ κϵ +3 + +Proof. The condition given by (3) implies that +|∇2N − ∇2φ|2 ≤ +� +||∇N|| +a +2 + ||∇φ|| +b +2 +�2 +||∇N − ∇φ||2 +≤ +� +||∇N||a + ||∇φ||b + 2||∇N|| +a +2 ||∇φ|| +b +2 +� +||∇N − ∇φ||2 +≤ 2 +� +||∇N||a + ||∇φ||b� +||∇N − ∇φ||2 +with the last line following from Young’s inequality [16]. Then, +� +Ω +|∇2N(x) − ∇2φ(x)|2 dν1(x) ≤ 2 +� +Ω +� +||∇N(x)||a + ||∇φ(x)||b� +||∇N(x) − ∇φ(x)||2 dν1(x) +≤ 2 +�� +Ω +� +||∇N(x)||a + ||∇φ(x)||b�p dν1(x) +� 1 +p �� +Ω +||∇N(x) − ∇φ(x)||2q dν1(x) +� 1 +q +if we apply H¨older’s inequality [16] for exponents p, q satisfying 1 +p + 1 +q = 1 and 1 ≤ p, q ≤ ∞. Furthermore, +� +Ω +|∇2N(x) − ∇2φ(x)|2 dν1(x) ≤ K +�� +Ω +� +||∇N(x) − ∇φ(x)||a + ||∇φ(x)||max{a,b}�p +dν1(x) +� 1 +p +· +�� +Ω +||∇N(x) − ∇φ(x)||2q dν1(x) +� 1 +q +≤ K(ϵa + sup +x∈Ω +||∇φ(x)||max{a,b})ϵ2 +for some constant K. The last line follows from Theorem 1. Applying this result and Theorem 1 again to +the objective function J , we obtain: +J (N) = +� +Ω +|∇2N(x) − f(x)|2 dν1(x) + +� +∂Ω +|N(x) − g(x)|2 dν2(x) += +� +Ω +|∇2N(x) − ∇2φ(x)|2 dν1(x) + +� +∂Ω +|N(x) − φ(x)|2 dν2(x) +≤ K(ϵa + sup +x∈Ω +||∇φ(x)||max{a,b})ϵ2 + ϵ2 +Finally, a rescaling of ϵ > 0 yields +J (N) ≤ κϵ +for some constant κ > 0 which may depend on sup +x∈Ω +||∇φ(x)||. +Theorem 4 guarantees the existence of a feed-forward neural network N that, under relatively relaxed +conditions, makes the objective function J (N) for Poisson’s equation arbitrarily small. However, neural +network objective functions are highly non-convex. This means they have numerous minima and, while +gradient descent algorithms like the deep Galerkin method are extremely effective at reaching said minima +[17], there is no guarantee of achieving the global minimum i.e., in our case, finding the unique solution. +Many authors research such ideas in non-convex optimisation [18], but we do not touch on them here, and +present only empirical evidence of our solver finding/not finding global minima in the Results section (see +4). +4 + +3 +Methods +We now present three highly accessible methods to enhance the performance of a neural network trained to +solve differential equations via the deep Galerkin method. +3.1 +Sinusoidal representation networks +Consider a neural network that is trained to approximate a function directly. +We need only the first- +order derivatives of the activation functions to backpropagate, and thus ReLU seems a natural choice [19]. +However, our framework requires a network to learn a function via its derivatives. ReLU networks cannot +do this without significant loss of information since they have second derivative zero. They are incapable of +accurately modelling a signal’s higher-order derivatives. +A recent paper [8] highlighting these limitations proposes something the authors call a sinusoidal repre- +sentation network or SIREN. This is a neural network that implicitly defines a function, in our case N, +with sinusoidal activations. Thus, while regular feed-forward networks with, say, ReLU activation may be +excellent function approximators, a SIREN can further accurately fit derivatives of functions φ through its +own derivatives. ReLU networks typically cannot, due to their piecewise linear nature. This idea is hidden +in Theorem 1 since ReLU is continuous but not differentiable, and so a network N with ReLU activation +could only achieve +sup +x |∂(α) +x N(x) − ∂(α) +x φ(x)| < ϵ +for α = 0. By contrast, sin ∈ C∞, so the equivalent statement is true for any |α| < ∞. +Evaluating the gradient of a SIREN scales quadratically in the number of layers of the SIREN [8]. So, fitting +higher-order derivatives is no easy task. However, for simple differential equations like Poisson’s equation, +it is computationally feasible, and the authors of [8] provide experimental results that show SIRENs are +excellent at modelling first and second-order derivatives of complicated signals, as well as the high-frequency +signals themselves. +3.2 +Random Fourier features +Recent works [20, 21] have described a spectral bias inherent to neural networks learning functions. They +prioritise learning the low-frequency modes of the functions and thus, high frequencies are captured much +later in the training procedure. +In many ways, this is a key reason behind the immense success of neural networks. Often, they are over- +parameterised, i.e. +the number of parameters far exceeds the number of training samples yet, counter- +intuitively, they still show remarkable capacity to generalise well [22]. Spectral bias may explain part of this +phenomenon because it suggests, if there is a way to fit data effectively with only low frequencies, then a +neural network will do just this, without needing to resort to high frequencies that overfit the data. +However, this also means that neural networks struggle to learn high frequency functions. Theoretical results +in [23] show that a one-dimensional function of pure frequency ω, e.g. cos(ωx), is learned in time that scales +with ω2. This is ratified experimentally. +A 2020 paper [9] publishes results on the use of a Fourier feature mapping to effectively overcome this +spectral bias, and allow multilayer perceptrons (MLPs) to learn high frequency functions in low-dimensional +domains. The authors motivate such work with neural tangent kernel (NTK) theory. NTKs have been shown +to model the behaviour of MLPs in the infinite-width limit during training [24]. We do not describe them in +detail here, but give a summary of the main idea behind Fourier feature mapping. For two different inputs +x, x′ to the MLP, the corresponding NTK can be given by +5 + +NTK(x, x′) = h(xT x′) +where h is some scalar function [9]. +The mapping +γ(x) = [cos(2πBx), sin(2πBx)]T +(4) +is a Gaussian random Fourier feature mapping for x ∈ Rd, where each entry in B ∈ Rn×d is sampled from a +normal distribution with mean zero and variance Σ2. Therefore, +NTK(γ(x), γ(x′)) = h(γ(x)T γ(x′)) += h (cos(2πBx) cos(2πBx′) + sin(2πBx) sin(2πBx′)) += h(cos(2πB(x − x′))) +Crucially, this defines a kernel function with width controlled by the random matrix B. Kernel functions are +used to fit data, and their width directly influences whether they overfit (with high frequencies) or underfit +(with low frequencies). So, given that this function characterises the evolution of the MLP during training, +we can tune the network towards learning particular frequencies by simply changing Σ: +• A small Σ gives a wide kernel that will underfit a high-frequency function. +• A large Σ gives a narrow kernel that will overfit a low-frequency function. +In our framework, Σ is now just another hyperparameter, and we can find the optimal Σ through a simple +sweep of values. We choose the value that gives the fastest convergence. The authors of [9] also advise that +n, the number of Fourier features, improves performance with size. Of course, there is a computational cost +associated with increasing n, so it is best taken ‘as small as gives good enough results.’ +3.3 +Error correction +We introduce the main work of this paper; the novel technique error correction [10–14] is designed to increase +the efficacy of any neural network differential equation solver. This method is general and can be applied to +all differential equations, in combination with any such similar strategies, such as Koopman boosting [25] or +those presented above. Much of the work here was proposed in [10] and formalised in [11], which the reader +should refer to as supplement. +When dealing with neural networks, we bank on the idea that a ‘small enough’ loss implies a ‘good enough’ +accuracy. Now, in many scenarios, this ideology fails because zero loss would represent drastic overfitting. +Conveniently, this does not concern us as we want our network to fit the (training) data as accurately as +possible. Still, the original problem remains; how can we know how close we are to the true solution φ? +It turns out analysis and estimation of the unknown error between φ and N is possible. Indeed, in [12], the +author shows how you can obtain specific bounds on this error, without knowledge of φ. In this section, we +provide a correction method (based on this error) to enhance neural network differential equation solvers, by +overcoming performance saturation when the network settles around a local minimum of the loss function. +Here, we also make use of differential equation operators which send true solutions to zero. Consider this +for Poisson’s equation (2): +F[·] = ∇2[·] − f +6 + +Define φϵ = φ − N as the error between the unknown solution φ and a fixed approximation N. Clearly, +F[N] = ∇2N − f += ∇2[φ − φϵ] − f += ∇2φ − f − ∇2φϵ += −∇2φϵ +since F[φ] = ∇2φ − f = 0. Thus, F[N] + ∇2φϵ = 0 and, given that F[N] is completely independent to φϵ, +we have defined a new Poisson’s equation. Our general strategy now will be to train a neural network Nϵ to +approximate φϵ through the conditions of this new differential equation. Then, N + Nϵ ≈ N + φϵ = φ. +Before we formalise and evaluate this method, note that it applies also to differential equations with non- +linear terms. Consider the Poisson-Boltzmann equation with Dirichlet boundary conditions: +� +∇2φ + sinh φ += f in Ω +φ += g on ∂Ω +Define the operator +G[·] = ∇2[·] + sinh[·] − f +and, once again, have φϵ = φ − N. Then, +G[N] = ∇2N + sinh N − f += ∇2[φ − φϵ] + sinh N + sinh φ − sinh φ − f += ∇2φ + sinh φ − f − ∇2φϵ + sinh N − sinh φ += −∇2φϵ + sinh N − sinh(N + φϵ) +since G[φ] = ∇2φ + sinh φ − f = 0. A clever trick of adding and subtracting sinh φ allows the G[φ] term to +be removed from the equation. In the last line, we simply seek to keep the equation explicit in N and φϵ. +Theoretical results +Now, we formalise this idea of error correction, adapting the approach from [11]. Consider a differential +equation over Ω in operator form: +F0[φ] = A[φ] + B[φ] + C = 0 +(5) +where A represents the terms that depend linearly on φ, B represents those that depend non-linearly on φ, +and C is independent of φ. The solution φ may also admit some constraints on the boundary ∂Ω but, for +now, these are not of interest. Assume also that φ is unique. +We first prove a result that follows from the inverse function theorem [26]: +Theorem 5. (Inverse function theorem). Suppose that F : Rn → Rn is continuously differentiable in some +open set containing x∗, and suppose moreover that the Jacobian DF(x∗) is invertible. Then there exists +open sets U, V ⊂ Rn with x∗ ∈ U and F(x∗) ∈ V such that F : U → V is a bijection, and F −1 : V → U is +continuously differentiable for all y ∈ V with +DF −1(y) = +� +DF(F −1(y)) +�−1 +7 + +Corollary 6. Suppose that F0 : R → R in (5) is continuously differentiable in some open set containing φ∗, +that DF0[φ∗] is invertible, and F0[φ∗] = 0. Then, there is a neighbourhood of 0 small enough such that +F0[N] → 0 =⇒ N → φ∗ +Proof. By Theorem 5, choose neighbourhoods U, V ⊂ R with φ∗ ∈ U, 0 ∈ V such that F0 : U → V is a +bijection and F0 +−1 : V → U is continuous differentiable for all y ∈ V . For N ∈ U, the continuity of F0 +−1 +implies that +F0[N] → 0 =⇒ N → φ∗ +Thus, assuming we can minimise the loss function for some neural network N such that F0[N] → 0 at all +points, then N → φ at all points. So, let us train such a network N0 to approximate φ via (5). Define also +φ1 = φ − N0. +F0[N0] = A[N0] + B[N0] + C += A[φ − φ1] + B[N0] + B[φ] − B[φ] + C += A[φ] + B[φ] + C − A[φ1] + B[N0] − B[φ] += −A[φ1] + B[N0] − B[N0 + φ1] +since F0[φ] = A[φ] + B[φ] + C = 0 by definition. We have defined a new differential equation in operator +form: +F1[φ1] = F0[N0] + A[φ1] − B[N0] + B[N0 + φ1] = 0 +φ1 solves the above equation exactly and, given the uniqueness of φ, is also unique. Now, train some other +neural network N1 to approximate φ1, and define φ2 = φ1 − N1. Once again, +F1[N1] = F0[N0] + A[N1] − B[N0] + B[N0 + N1] += F0[N0] + A[φ1 − φ2] − B[N0] + B[N0 + N1] + B[N0 + φ1] − B[N0 + φ1] += F0[N0] + A[φ1] − B[N0] + B[N0 + φ1] − A[φ2] + B[N0 + N1] − B[N + φ1] += −A[φ2] + B[N0 + N1] − B[N0 + φ1] += −A[φ2] + B[N0 + N1] − B[N0 + N1 + φ2] +since F1[φ1] = F0[N0] + A[φ1] − B[N0] + B[N0 + φ1] = 0, and φ1 = N1 + φ2. We define a further differential +equation in operator form: +F2[φ2] = F1[N1] + A[φ2] − B[N0 + N1] + B[N0 + N1 + φ2] = 0 +Now, repeat the process. This algorithm can continue indefinitely, and we summarise the steps below. The +idea is that our error-corrected approximation N0 +N1 +N2 +... will be more accurate than the once-trained +approximation N0. This strategy is not unseen in the field of numerical methods to differential equations, +we just apply it here to neural network solvers. +Let us define a recursive differential equation for the kth error correction. At this point, we have trained the +initial network N0, and also a further k − 1 residual networks N1, N2, ..., Nk−1. Our current error-corrected +8 + +approximation is N (k−1) = N0 +N1 +N2 +...+Nk−1. Define φk = φk−1 − Nk−1. Now, train a new network +Nk to approximate φk through the following differential equation: +Fk[φk] = Fk−1[Nk−1] + A[φk] − B[N (k−1)] + B[N (k−1) + φk] = 0 +(6) +Remark. Fk[Nk] ≡ F0[N (k)]. +Corollary 7. Suppose that Fk : R → R in (6) is continuously differentiable in some open set containing φ∗ +k, +that DFk[φ∗ +k] is invertible, and Fk[φ∗ +k] = 0. Then, there is a neighbourhood of 0 small enough such that +Fk[Nk] → 0 =⇒ Nk → φ∗ +k +Furthermore, +|N (k) − φ| = O (|Fk[Nk]|) +Proof. The first result follows analogously from the inverse function theorem as in Corollary 6. +By Theorem 5, F0 +−1 is continuously differentiable on some open set around 0. +Thus, it is also locally +Lipschitz continuous around 0, meaning there exists some constant α ≥ 0 such that +|N (k) − φ| = |F0 +−1[F0[N (k)] − F0 +−1[F0[φ]]| +≤ α|F0[N (k)] − F0[φ]| +≤ α|F0[N (k)]| +≤ α|Fk[Nk]| +and therefore, +|N (k) − φ| = O (|Fk[Nk]|) +Finally, given Dirichlet boundary conditions φ = g on ∂Ω, any φk is known exactly over ∂Ω since φk = +φ − N (k−1). Thus, the loss function for the kth error correction can be defined as +Lk(θ(k)) = 1 +M +M +� +i=1 +� +Fk +� +Nk +� +xi; θ(k)���2 ++ 1 +N +N +� +j=1 +� +Nk +� +yj; θ(k)� +− φk(yj) +�2 +(7) +for some randomly sampled points {xi}M +i=1 from Ω and {yj}N +j=1 from ∂Ω. +Algorithm +The error correction algorithm to order K proceeds as follows: +1. Train a neural network N0 to satisfy the conditions of a differential equation given by (5) and constraint +conditions. Once the loss has converged, stop training and freeze the parameters of N0. +2. Initiate and train new neural networks {Nk}K +k=1 in sequence to satisfy differential equations given by +(6), via loss functions (7). Once the loss has converged, stop training, freeze the parameters of Nk, +and proceed with Nk+1. +9 + +3. The solution to (5) is approximated by N := N (K) = +K +� +k=0 +Nk. +This is given above for Dirichlet boundary conditions, but works generally if you incorporate the constraint +conditions into all loss functions. +Poisson’s equation +For Poisson’s equation (2), B ≡ 0 since the Laplacian is linear, so we can write the kth differential equation +as +Fk[φk] = Fk−1[Nk−1] + ∇2[φk] = 0 +which is a Poisson’s equation with our usual f = −Fk−1[Nk−1]. Thus, we can apply Theorem 4 to guarantee +that there are neural networks out there that can get very, very close to the φk. In the next section, we +provide evidence that shows, if we can train just two or three of these networks to reasonably approximate +their true solutions, our error-corrected approximation will be a more accurate numerical solution to the +original differential equation. +4 +Results +We present results for a variety of different Poisson’s equations (2). Our choice of Poisson’s equation is +motivated by its immense application in many areas of theoretical physics, including electrostatics and fluid +dynamics. It is also the simplest second-order, linear PDE, making for a concise yet insightful demonstration +of the power of error correction in neural network differential equation solvers. +To achieve this, we choose the function f on the RHS to force a particular solution φ that we want to +capture. For example, f(x) = 1 would force the solution φ(x) = 1 +2x2 + c1x + c0. However, in general, f can +be anything, particularly something which does not admit a closed-form solution to (2), and we do this for +ease of visualising φ. +Knowing the ground truth solution φ in closed form also allows us to compute the relative error +� +x∈S +(φ(x) − N(x))2 +� +x∈S +φ(x)2 +at each epoch (iteration) of the training procedure, so we have an understanding of the success of our solver. +It is important to note that, while we know φ and the relative error associated with our approximation, the +neural network does not, and is solely trained via the loss function. +All neural networks used are SIRENs with 5 hidden layers and 128 hidden units per layer. They are trained +on batches of 256, using the stochastic gradient descent variant Adam [27], and learning rates are manually +tuned for each case of Poisson’s equation. All experiments are run on a 1.8 GHz Dual-Core Intel Core i5 +CPU. +10 + +4.1 +3D Poisson’s Equation +Figure 1: φ(x, y, z) = sin(5x) sin(5y) sin(5z) at z = π +10 +Figure 1 shows the solution to Poisson’s equation: +� +∇2φ += −75 sin(5x) sin(5y) sin(5z) in Ω = [−π, π] × [−π, π] × [−π, π] +φ += 0 on ∂Ω +(8) +Figure 2 shows our numerical solutions, with N (0) = N0 on the left, N (1) = N0 + N1 in the centre, and +N (2) = N0 + N1 + N2 on the right. We refer to these as Error Correction 0, 1 and 2, respectively. +Visually, all error corrections seem to capture the solution well. Furthermore, each correction decreases the +relative error (printed at the bottom of of Figure 2). Error Correction 1 does so significantly, while the +improvement in accuracy from Error Correction 2 is marginal. +This is further captured in Figure 3, which is a plot of the loss and relative error per epoch. After finding +a local minimum in Error Correction 0, the loss fluctuates erratically until we initialise Error Correction 1. +The improvement is truly appreciable, and felt across the trends in relative error too. +11 + +Error Correction 0 +Error Correction 1 +Error Correction 2 +Ground Truth +1.00 +3： + 0.75 +2 + 0.50 +1 +0.25 +0 +0.00 +0.25 +-1 +0.50 +0.75 +1.00 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +Relative error: 0.156 +Relative error: 0.028 +Relative error: 0.020Figure 2: Numerical solutions N (0), N (1) and N (2) to (8) +Figure 3: Per-epoch loss and relative errors for numerical solutions to (8) +12 + +ORDER 0 +ORDER 1 +ORDER 2 +102 +LosS +101 +0 +10000 +20000 +30000 +40000 +50000 +60000 +Epochs +ORDER0 +ORDER1 +ORDER 2 +101 +Relative error +100 +10-1 +0 +10000 +20000 +30000 +40000 +50000 +60000 +EpochsError Correction 0 +Error Correction 1 +Error Correction 2 +Ground Truth +1.00 +3： + 0.75 +2 + 0.50 +1 +0.25 +0 +0.00 +0.25 +-1 +0.50 +0.75 +1.00 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +Relative error: 0.156 +Relative error: 0.028 +Relative error: 0.0204.2 +2D Poisson’s Equation +(i) +Figure 4: φ(x, y) = sin(20x) sin(20y) +Figure 4 shows the solution to Poisson’s equation: +� +∇2φ += −800 sin(5x) sin(5y) in Ω = [−π, π] × [−π, π] +φ += 0 on ∂Ω +(9) +Due to the highly oscillatory nature of the solution, a neural network will struggle to accurately capture its +structure. This is demonstrated in Figure 5, where the approximation cannot account for so many peaks +and troughs in the solution. +Figure 5: Naive attempt at a numerical solution to (9) +To obtain a realistic solution, we apply a Gaussian random Fourier feature mapping to the input, before +passing it through the network. After a simple sweep of values, we take Σ = 1 and n = 256, as defined in +(4). Figures 6 and 7 show similar trends to those in the previous experiment. +13 + +3 +2 +1 +0 +-1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3ErrorCorrection o +ErrorCorrection1 +ErrorCorrection2 +Ground Truth +1.00 +0.75 +0.50 +0.25 +0 +0.00 +0.25 +0.50 +0.75 +-1.00 +-2 +0 +2 +-2 +0 +2 +0 +2 +-2 +0 +2 +Relative error: 0.063 +Relative error: 0.040 +Relative error: 0.033Figure 6: Numerical solutions N (0), N (1) and N (2) to (9), trained using random Fourier features with Σ = 1 +and n = 256 +Figure 7: Per-epoch loss and relative errors for numerical solutions to (9) +14 + +ORDER 0 +ORDER 1 +ORDER 2 +105 +104 +LosS +103. +102 +h +0 +10000 +20000 +30000 +40000 +50000 +60000 +Epochs +ORDER0 +ORDER1 +ORDER 2 +Relative error +100 +10-1 +10000 +20000 +30000 +40000 +50000 +60000 +EpochsErrorCorrection o +ErrorCorrection1 +ErrorCorrection2 +Ground Truth +1.00 +0.75 +0.50 +0.25 +0 +0.00 +0.25 +0.50 +0.75 +-1.00 +-2 +0 +2 +-2 +0 +2 +0 +2 +-2 +0 +2 +Relative error: 0.063 +Relative error: 0.040 +Relative error: 0.033(ii) +Figure 8: φ(x, y) = (π2 − y2) sin(10x) +Figure 8 shows the solution to Poisson’s equation: +� +∇2φ += (100y2 − 100π2 − 2) sin(10x) in Ω = [−π, π] × [−π, π] +φ += 0 on ∂Ω +(10) +In Figure 9, we train a neural network N (0) to approximate the solution to (10) for 211 epochs, but we save +its parameter states after 210 epochs. These define a new network which we call N0. The fully-trained N (0) +achieves a reasonable relative error. Roughness is clearly visible in the plot. +In Figure 10, we plot the half-trained N0 on the left. As expected, it has not yet reached the accuracy of +N (0). However, we also initiate an error correction N1, of N0, that trains for another 210 epochs. Thus, we +produce an approximation N (1) = N0+N1 that has also trained for a total of 211 epochs. This is significantly +more accurate than N (0), and the plot is visibly smoother. Figures 9 and 10 provide a clear exemplification +of the immediate fruitfulness of a single error correction. +15 + +Groundtruth +0 +-5 +m +2 +L +-3 +0 +-2 +-1 +0 +2 +3 +3Figure 9: Numerical solution N (0) to (10), trained for 211 epochs +Figure 10: Numerical solutions N0 and N (1) to (10), trained for a total of 211 epochs +16 + +Error Correction O +10 +5 +0 +-5 +-10 +15 +2 +3 +1 +32-1 +0 +Relative Error = 0.244 +-1 +0 +1 +2 +-3 +3ErrorCorrection +ErrorCorrection1 +10 +10 +5 +5 +0 +0 +-5 +-10 +-5 +-15 +-10 +3 +3 +2 +2 +1 +1 +0 +3 +0 +Relative Error = 0.423 +-1 +Relative Error = 0.035 +-1 +-1 +0 +1 +-2 +1 +~2 +2 +3 +2 +3 +3 +35 +Discussion +Our results do not endorse error correction as a tool to marginally reduce error across tens of corrections. +Instead, they suggest training a network for half the allotted time, and devoting the other half to a single +error correction. This can yield significantly more accurate results. +Error correction is not without cost however. In our implementation, we train correction networks on newly +sampled points. This means that to obtain Fk[Nk], we must first make k forward passes of the new data +through N0, N1, ..., Nk−1 and differentiate these to compute Fk−1[Nk−1]. The time complexity of producing +a kth order approximation N (k), assuming the number of epochs E and batch size B per correction, and +optimisation costs, are kept constant across all corrections, is O +� +EB(k + 1)2� +. If we instead pass identical +batches through each correction network, storing the Fk−1[Nk−1] in memory, we can have a time complexity +of O(EB(k + 1)), however the space complexity would be substantially increased. +6 +Further Work +Over time, this study of neural network differential equation solvers naturally lent itself to hot-off-the-press +topics in machine learning like sinusoidal representation networks [8] and random Fourier features [9], for the +simple reason that such concepts are inextricably linked through their applications. Outside of differential +equations, neural networks as continuous parameterisations of discrete signals have immense potential in 3D +shape representation, but also in image, video and audio representation and reconstruction. These problems +may utilise neural networks as function approximators or, as we did, derivative approximators. There is no +reason to suggest why the ideas of error correction cannot be employed here, and every reason to further +explore the interplay of these techniques when applied to problems in computer vision. +References +[1] I.E. Lagaris, A. Likas, and D.I. Fotiadis. Artificial neural networks for solving ordinary and partial +differential equations. IEEE Transactions on Neural Networks, 9(5):987–1000, 1998. +[2] Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part I): +Data-driven solutions of nonlinear partial differential equations. arXiv:1711.10561, 2017. +[3] Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part II): +Data-driven discovery of nonlinear partial differential equations. arXiv:1711.10566, 2017. +[4] Justin Sirignano and Konstantinos Spiliopoulos. DGM: A deep learning algorithm for solving partial +differential equations. Journal of Computational Physics, 375:1339–1364, 2018. +[5] Anil Ananthaswamy. Latest neural nets solve world’s hardest equations faster than ever before. Quanta +Magazine, 2021. +[6] Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2):251– +257, 1991. +[7] Andrew R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE +Transactions on Information Theory, 39(3):930–945, 1993. +[8] Vincent Sitzmann, Julien N.P. Martel, Alexander W. Bergman, David B. Lindell, and Gordon Wetzstein. +Implicit neural representations with periodic activation functions. In Proceedings of the 34th Conference +on Neural Information Processing System, 2020. +[9] Matthew Tancik, Pratul P. Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh +Singhal, Ravi Ramamoorthi, Jonathan T. Barron, and Ren Ng. Fourier features let networks learn +high frequency functions in low dimensional domains. In Proceedings of the 34th Conference on Neural +Information Processing Systems, 2020. +17 + +[10] Akshunna S. Dogra. Error estimation and correction from within neural network differential equation +solvers. arXiv:2007.04433, 2020. +[11] Akshunna S. Dogra et. al. Neural network differential equation solvers allow unsupervised error analysis +and correction. under review, 2023. +[12] Marios Mattheakis, David Sondak, Akshunna S. Dogra, and Pavlos Protopapas. Hamiltonian neural +networks for solving equations of motion. Physical Review E, 105(6), 2022. +[13] Akshunna S. Dogra and William T. Redman. Local error quantification for neural network differential +equation solvers. arXiv:2008.12190, 2021. +[14] Akshunna S. Dogra. Dynamical systems and neural networks. arXiv:2004.11826, 2020. +[15] Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal +approximators. Neural Networks, 2(5):359–366, 1989. +[16] G. H. Hardy, John E. Littlewood, and George P´olya. Inequalities. Cambridge University Press, Cam- +bridge, second edition, 1988. +[17] Simon S. Du, Jason D. Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai. Gradient descent finds global +minima of deep neural networks. +In Proceedings of the 36th International Conference on Machine +Learning, 2019. +[18] Rudrajit Das. Recent Advances in Non-Convex Optimization for Deep Learning. https://rudrajit15. +github.io/posts/2018/09/blog-post-2/. Accessed: 2 May 2022. +[19] Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse rectifier neural networks. In Proceedings +of the 14th International Conference on Artificial Intelligence and Statistics, 2011. +[20] Ronen Basri, Meirav Galun, Amnon Geifman, David Jacobs, Yoni Kasten, and Shira Kritchman. Fre- +quency bias in neural networks for input of non-uniform density. In Proceedings of the 37th International +Conference on Machine Learning, 2020. +[21] Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred A. Hamprecht, Yoshua +Bengio, and Aaron Courville. +On the spectral bias of neural networks. +In Proceedings of the 36th +International Conference on Machine Learning, 2019. +[22] Yuanzhi Li and Yingyu Liang. +Learning overparameterized neural networks via stochastic gradient +descent on structured data. In Proceedings of the 32nd Conference on Neural Information Processing +Systems, 2018. +[23] Ronen Basri, David Jacobs, Yoni Kasten, and Shira Kritchman. The convergence rate of neural net- +works for learned functions of different frequencies. In Proceedings of the 33rd Conference on Neural +Information Processing Systems, 2019. +[24] Arthur Jacot, Franck Gabriel, and Clement Hongler. Neural tangent kernel: Convergence and gener- +alization in neural networks. In Proceedings of the 32nd Conference on Neural Information Processing +Systems, 2018. +[25] Akshunna S. Dogra and William T. Redman. Optimizing neural networks via koopman operator theory. +Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020. +[26] Johannes Nicaise. Lecture notes in real analysis, 2020. +[27] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of +the 3rd International Conference for Learning Representations, 2015. +18 + diff --git a/W9FPT4oBgHgl3EQfrzVr/content/tmp_files/load_file.txt b/W9FPT4oBgHgl3EQfrzVr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..398c7151488ff2db60858eead6afc51c029f4795 --- /dev/null +++ b/W9FPT4oBgHgl3EQfrzVr/content/tmp_files/load_file.txt @@ -0,0 +1,441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf,len=440 +page_content='Enhancing Neural Network Differential Equation Solvers Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Wright December 2022 Abstract We motivate the use of neural networks for the construction of numerical solutions to differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We prove that there exists a feed-forward neural network that can arbitrarily minimise an objective function that is zero at the solution of Poisson’s equation, allowing us to guarantee that neural network solution estimates can get arbitrarily close to the exact solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We also show how these estimates can be appreciably enhanced through various strategies, in particular through the construction of error correction networks, for which we propose a general method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We conclude by providing numerical experiments that attest to the validity of all such strategies for variants of Poisson’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The source code for this project can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='com/mjhwright/error-correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 1 Introduction Differential equations are among the most ubiquitous problems in contemporary mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In recent years, developments in artificial neural networks have prompted new research into their capacity to be excellent differential equation solvers [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' They are universal approximators [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' they can circumvent the curse of dimensionality [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' and they are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, practically, their construction and optimisation costs are enough to deter the discerning user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In this paper, we explain a method by which neural networks can numerically solve differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We further this by providing three strategies that can be targeted to improve the efficacy of the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The first two – sinusoidal representation networks [8] and random Fourier features [9] – are well-established in the field of artificial neural networks and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The third is a novel technique called error correction [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We explain how error correction can be implemented recursively, with little modification to the original solver, to give enhanced numerical solutions to differential equations, and we present results that demonstrate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This paper is designed to give a flavour of the competence of artificial neural networks in this field, while also highlighting their certain limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 2 Background Throughout this paper, we consider differential equations with solution φ : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Consequently, our neural network approximation is a function N : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='1 Universal approximation theorems The realisation of neural networks’ capabilities to learn seemingly any function has brought about numerous universal approximation theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' These state that, under certain conditions, neural networks are able to approximate any function to arbitrary closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We recall one of these theorems by Hornik [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' First, define the set of all functions represented by a neural network with a single hidden layer of width n and identity activation on the output layer as 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='13146v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='NA] 28 Dec 2022 A n(σ) = � N : Rd → R, N(x) = W(1) � σ � W(0)x + b(0)�� + b(1)� where x ∈ Rd, W(0) ∈ Rn×d, W(1) ∈ R1×n, b(0) ∈ Rn, b(1) ∈ R, and σ : R → R is applied element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, A (σ) = ∞ � n=1 A n(σ) (1) is the set of all such functions with any number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Define also Cm(Rd) as the space of all functions that, together with their partial derivatives of order |α| ≤ m, are continuous on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [6] If σ ∈ Cm(Rd) is nonconstant and bounded, then A (σ) is uniformly m-dense on all compact sets of Cm(Rd), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' for all φ ∈ Cm(Rd), for all compact sets Ω ⊂ Rd and for all ϵ > 0, there exists N ∈ A (σ) such that max |α|≤m sup x∈Ω |∂(α) x N(x) − ∂(α) x φ(x)| < ϵ Theorem 1 illustrates the universal approximation quality for single-layer networks of arbitrary width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Ap- plying results from an earlier paper by Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [15], this can be extended to multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Crucially, these theorems tell us that neural networks are dense on certain function spaces, but they do not tell us how to train a network to realise this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='2 Neural network differential equation solvers Using neural networks to solve differential equations was introduced in the late 1990s [1], but experienced a modern resurgence through the publication of two papers [2,3] on physics-informed neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The deep Galerkin method [4] which we describe below is very similar to the method described in [2] only, instead of using experimental data, we train a network on points randomly sampled across the domain of the differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Consider Poisson’s equation with Dirichlet boundary conditions: � ∇2φ = f in Ω φ = g on ∂Ω (2) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Let Ω ⊂ Rd be a smooth, compact domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then there exists at most one solution φ to (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Suppose φ and ϕ both satisfy the conditions of (2) and let ω = φ − ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then ω is harmonic in Ω and zero on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, � Ω ω(x)∇2ω(x) dx = � ∂Ω ω(x)δnω(x) dx − � Ω ||∇ω(x)||2 dx = − � Ω ||∇ω(x)||2 dx = 0 and ∇ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, ω = 0 and φ = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We now seek an approximation N to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Define the objective function J (N) = � Ω |∇2N(x) − f(x)|2ν1(x) dx + � ∂Ω |N(x) − g(x)|2ν2(x) dx 2 for probability distributions ν1 on Ω and ν2 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' By uniqueness of φ, J (N) = 0 =⇒ N = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, minimising the objective function directly is impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' First, we transform the problem into a machine learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Our approximation N = N(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θ) becomes a neural network with parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Deep Galerkin method We demonstrate the algorithm for the deep Galerkin method [4] when applied to Poisson’s equation (2): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Randomly sample points {xi}M i=1 from Ω and {yj}N j=1 from ∂Ω according to respective probability distributions ν1 and ν2, and propagate them through a feed-forward neural network N(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Calculate the loss: L(θ) = 1 M M � i=1 � ∇2N(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θ) − f(xi) �2 + 1 N N � j=1 (N(yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θ) − g(yj))2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Update parameters θt+1 = θt − η∇θL(θt) with learning rate η > 0 and t ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Repeat until ∇θL(θt) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is a minibatch gradient descent implementation, where M and N are the size of the minibatches and M > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' E[∇θL(θt)|θt] = ∇θJ (N(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt)) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Assume L sufficiently smooth and bounded to interchange derivatives and integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, E[∇θL(θt)|θt] = ∇θ � � 1 M M � i=1 E � (∇2N(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt) − f(xi))2� + 1 N N � j=1 E � (N(yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt) − g(yj))2� � � = ∇θ � � 1 M M � i=1 � Ω (∇2N(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt) − f(x))2ν1(x) dx + 1 N N � j=1 � ∂Ω (N(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt) − g(y))2ν2(y) dy � � = ∇θ �� Ω (∇2N(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt) − f(x))2ν1(x) dx + � ∂Ω (N(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt) − g(y))2ν2(y) dy � = ∇θJ (N(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt)) Therefore, the ∇θL(θt) are unbiased estimates of ∇θJ (N(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θt)), and we can assume a step in the descent direction of L is also one in J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, any minimisation of L should translate to a local minimisation of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Minimisation of J (N) We prove the following theorem, adapted from the original deep Galerkin method paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Let A (σ) be given by (1), for nonconstant, bounded σ, and let Ω ∈ Rd be a compact domain and consider measures ν1, ν2 whose supports are contained in Ω, ∂Ω respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Assume further that ∇2φ is locally Lipschitz with Lipschitz constant that can have at most polynomial growth on ∇φ, uniformly with respect to x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' |∇2N − ∇2φ| ≤ � ||∇N|| a 2 + ||∇φ|| b 2 � ||∇N − ∇φ|| (3) for some constants 0 ≤ a, b < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, for all ϵ > 0, there exists a constant κ > 0 such that there exists a function N ∈ A (σ) with J (N) ≤ κϵ 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The condition given by (3) implies that |∇2N − ∇2φ|2 ≤ � ||∇N|| a 2 + ||∇φ|| b 2 �2 ||∇N − ∇φ||2 ≤ � ||∇N||a + ||∇φ||b + 2||∇N|| a 2 ||∇φ|| b 2 � ||∇N − ∇φ||2 ≤ 2 � ||∇N||a + ||∇φ||b� ||∇N − ∇φ||2 with the last line following from Young’s inequality [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, � Ω |∇2N(x) − ∇2φ(x)|2 dν1(x) ≤ 2 � Ω � ||∇N(x)||a + ||∇φ(x)||b� ||∇N(x) − ∇φ(x)||2 dν1(x) ≤ 2 �� Ω � ||∇N(x)||a + ||∇φ(x)||b�p dν1(x) � 1 p �� Ω ||∇N(x) − ∇φ(x)||2q dν1(x) � 1 q if we apply H¨older’s inequality [16] for exponents p, q satisfying 1 p + 1 q = 1 and 1 ≤ p, q ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Furthermore, � Ω |∇2N(x) − ∇2φ(x)|2 dν1(x) ≤ K �� Ω � ||∇N(x) − ∇φ(x)||a + ||∇φ(x)||max{a,b}�p dν1(x) � 1 p �� Ω ||∇N(x) − ∇φ(x)||2q dν1(x) � 1 q ≤ K(ϵa + sup x∈Ω ||∇φ(x)||max{a,b})ϵ2 for some constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The last line follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Applying this result and Theorem 1 again to the objective function J , we obtain: J (N) = � Ω |∇2N(x) − f(x)|2 dν1(x) + � ∂Ω |N(x) − g(x)|2 dν2(x) = � Ω |∇2N(x) − ∇2φ(x)|2 dν1(x) + � ∂Ω |N(x) − φ(x)|2 dν2(x) ≤ K(ϵa + sup x∈Ω ||∇φ(x)||max{a,b})ϵ2 + ϵ2 Finally, a rescaling of ϵ > 0 yields J (N) ≤ κϵ for some constant κ > 0 which may depend on sup x∈Ω ||∇φ(x)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Theorem 4 guarantees the existence of a feed-forward neural network N that, under relatively relaxed conditions, makes the objective function J (N) for Poisson’s equation arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, neural network objective functions are highly non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This means they have numerous minima and, while gradient descent algorithms like the deep Galerkin method are extremely effective at reaching said minima [17], there is no guarantee of achieving the global minimum i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=', in our case, finding the unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Many authors research such ideas in non-convex optimisation [18], but we do not touch on them here, and present only empirical evidence of our solver finding/not finding global minima in the Results section (see 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 4 3 Methods We now present three highly accessible methods to enhance the performance of a neural network trained to solve differential equations via the deep Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='1 Sinusoidal representation networks Consider a neural network that is trained to approximate a function directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We need only the first- order derivatives of the activation functions to backpropagate, and thus ReLU seems a natural choice [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, our framework requires a network to learn a function via its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' ReLU networks cannot do this without significant loss of information since they have second derivative zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' They are incapable of accurately modelling a signal’s higher-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' A recent paper [8] highlighting these limitations proposes something the authors call a sinusoidal repre- sentation network or SIREN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is a neural network that implicitly defines a function, in our case N, with sinusoidal activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, while regular feed-forward networks with, say, ReLU activation may be excellent function approximators, a SIREN can further accurately fit derivatives of functions φ through its own derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' ReLU networks typically cannot, due to their piecewise linear nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This idea is hidden in Theorem 1 since ReLU is continuous but not differentiable, and so a network N with ReLU activation could only achieve sup x |∂(α) x N(x) − ∂(α) x φ(x)| < ϵ for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' By contrast, sin ∈ C∞, so the equivalent statement is true for any |α| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Evaluating the gradient of a SIREN scales quadratically in the number of layers of the SIREN [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' So, fitting higher-order derivatives is no easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, for simple differential equations like Poisson’s equation, it is computationally feasible, and the authors of [8] provide experimental results that show SIRENs are excellent at modelling first and second-order derivatives of complicated signals, as well as the high-frequency signals themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='2 Random Fourier features Recent works [20, 21] have described a spectral bias inherent to neural networks learning functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' They prioritise learning the low-frequency modes of the functions and thus, high frequencies are captured much later in the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In many ways, this is a key reason behind the immense success of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Often, they are over- parameterised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' the number of parameters far exceeds the number of training samples yet, counter- intuitively, they still show remarkable capacity to generalise well [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Spectral bias may explain part of this phenomenon because it suggests, if there is a way to fit data effectively with only low frequencies, then a neural network will do just this, without needing to resort to high frequencies that overfit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, this also means that neural networks struggle to learn high frequency functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Theoretical results in [23] show that a one-dimensional function of pure frequency ω, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' cos(ωx), is learned in time that scales with ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is ratified experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' A 2020 paper [9] publishes results on the use of a Fourier feature mapping to effectively overcome this spectral bias, and allow multilayer perceptrons (MLPs) to learn high frequency functions in low-dimensional domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The authors motivate such work with neural tangent kernel (NTK) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' NTKs have been shown to model the behaviour of MLPs in the infinite-width limit during training [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We do not describe them in detail here, but give a summary of the main idea behind Fourier feature mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' For two different inputs x, x′ to the MLP, the corresponding NTK can be given by 5 NTK(x, x′) = h(xT x′) where h is some scalar function [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The mapping γ(x) = [cos(2πBx), sin(2πBx)]T (4) is a Gaussian random Fourier feature mapping for x ∈ Rd, where each entry in B ∈ Rn×d is sampled from a normal distribution with mean zero and variance Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Therefore, NTK(γ(x), γ(x′)) = h(γ(x)T γ(x′)) = h (cos(2πBx) cos(2πBx′) + sin(2πBx) sin(2πBx′)) = h(cos(2πB(x − x′))) Crucially, this defines a kernel function with width controlled by the random matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Kernel functions are used to fit data, and their width directly influences whether they overfit (with high frequencies) or underfit (with low frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' So, given that this function characterises the evolution of the MLP during training, we can tune the network towards learning particular frequencies by simply changing Σ: A small Σ gives a wide kernel that will underfit a high-frequency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' A large Σ gives a narrow kernel that will overfit a low-frequency function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In our framework, Σ is now just another hyperparameter, and we can find the optimal Σ through a simple sweep of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We choose the value that gives the fastest convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The authors of [9] also advise that n, the number of Fourier features, improves performance with size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Of course, there is a computational cost associated with increasing n, so it is best taken ‘as small as gives good enough results.’ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='3 Error correction We introduce the main work of this paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' the novel technique error correction [10–14] is designed to increase the efficacy of any neural network differential equation solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This method is general and can be applied to all differential equations, in combination with any such similar strategies, such as Koopman boosting [25] or those presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Much of the work here was proposed in [10] and formalised in [11], which the reader should refer to as supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' When dealing with neural networks, we bank on the idea that a ‘small enough’ loss implies a ‘good enough’ accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Now, in many scenarios, this ideology fails because zero loss would represent drastic overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Conveniently, this does not concern us as we want our network to fit the (training) data as accurately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Still, the original problem remains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' how can we know how close we are to the true solution φ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' It turns out analysis and estimation of the unknown error between φ and N is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Indeed, in [12], the author shows how you can obtain specific bounds on this error, without knowledge of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In this section, we provide a correction method (based on this error) to enhance neural network differential equation solvers, by overcoming performance saturation when the network settles around a local minimum of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Here, we also make use of differential equation operators which send true solutions to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Consider this for Poisson’s equation (2): F[·] = ∇2[·] − f 6 Define φϵ = φ − N as the error between the unknown solution φ and a fixed approximation N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Clearly, F[N] = ∇2N − f = ∇2[φ − φϵ] − f = ∇2φ − f − ∇2φϵ = −∇2φϵ since F[φ] = ∇2φ − f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, F[N] + ∇2φϵ = 0 and, given that F[N] is completely independent to φϵ, we have defined a new Poisson’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Our general strategy now will be to train a neural network Nϵ to approximate φϵ through the conditions of this new differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, N + Nϵ ≈ N + φϵ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Before we formalise and evaluate this method, note that it applies also to differential equations with non- linear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Consider the Poisson-Boltzmann equation with Dirichlet boundary conditions: � ∇2φ + sinh φ = f in Ω φ = g on ∂Ω Define the operator G[·] = ∇2[·] + sinh[·] − f and, once again, have φϵ = φ − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, G[N] = ∇2N + sinh N − f = ∇2[φ − φϵ] + sinh N + sinh φ − sinh φ − f = ∇2φ + sinh φ − f − ∇2φϵ + sinh N − sinh φ = −∇2φϵ + sinh N − sinh(N + φϵ) since G[φ] = ∇2φ + sinh φ − f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' A clever trick of adding and subtracting sinh φ allows the G[φ] term to be removed from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In the last line, we simply seek to keep the equation explicit in N and φϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Theoretical results Now, we formalise this idea of error correction, adapting the approach from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Consider a differential equation over Ω in operator form: F0[φ] = A[φ] + B[φ] + C = 0 (5) where A represents the terms that depend linearly on φ, B represents those that depend non-linearly on φ, and C is independent of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The solution φ may also admit some constraints on the boundary ∂Ω but, for now, these are not of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Assume also that φ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We first prove a result that follows from the inverse function theorem [26]: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' (Inverse function theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Suppose that F : Rn → Rn is continuously differentiable in some open set containing x∗, and suppose moreover that the Jacobian DF(x∗) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then there exists open sets U, V ⊂ Rn with x∗ ∈ U and F(x∗) ∈ V such that F : U → V is a bijection, and F −1 : V → U is continuously differentiable for all y ∈ V with DF −1(y) = � DF(F −1(y)) �−1 7 Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Suppose that F0 : R → R in (5) is continuously differentiable in some open set containing φ∗, that DF0[φ∗] is invertible, and F0[φ∗] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, there is a neighbourhood of 0 small enough such that F0[N] → 0 =⇒ N → φ∗ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' By Theorem 5, choose neighbourhoods U, V ⊂ R with φ∗ ∈ U, 0 ∈ V such that F0 : U → V is a bijection and F0 −1 : V → U is continuous differentiable for all y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' For N ∈ U, the continuity of F0 −1 implies that F0[N] → 0 =⇒ N → φ∗ Thus, assuming we can minimise the loss function for some neural network N such that F0[N] → 0 at all points, then N → φ at all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' So, let us train such a network N0 to approximate φ via (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Define also φ1 = φ − N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' F0[N0] = A[N0] + B[N0] + C = A[φ − φ1] + B[N0] + B[φ] − B[φ] + C = A[φ] + B[φ] + C − A[φ1] + B[N0] − B[φ] = −A[φ1] + B[N0] − B[N0 + φ1] since F0[φ] = A[φ] + B[φ] + C = 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We have defined a new differential equation in operator form: F1[φ1] = F0[N0] + A[φ1] − B[N0] + B[N0 + φ1] = 0 φ1 solves the above equation exactly and, given the uniqueness of φ, is also unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Now, train some other neural network N1 to approximate φ1, and define φ2 = φ1 − N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Once again, F1[N1] = F0[N0] + A[N1] − B[N0] + B[N0 + N1] = F0[N0] + A[φ1 − φ2] − B[N0] + B[N0 + N1] + B[N0 + φ1] − B[N0 + φ1] = F0[N0] + A[φ1] − B[N0] + B[N0 + φ1] − A[φ2] + B[N0 + N1] − B[N + φ1] = −A[φ2] + B[N0 + N1] − B[N0 + φ1] = −A[φ2] + B[N0 + N1] − B[N0 + N1 + φ2] since F1[φ1] = F0[N0] + A[φ1] − B[N0] + B[N0 + φ1] = 0, and φ1 = N1 + φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We define a further differential equation in operator form: F2[φ2] = F1[N1] + A[φ2] − B[N0 + N1] + B[N0 + N1 + φ2] = 0 Now, repeat the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This algorithm can continue indefinitely, and we summarise the steps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The idea is that our error-corrected approximation N0 +N1 +N2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' will be more accurate than the once-trained approximation N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This strategy is not unseen in the field of numerical methods to differential equations, we just apply it here to neural network solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Let us define a recursive differential equation for the kth error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' At this point, we have trained the initial network N0, and also a further k − 1 residual networks N1, N2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=', Nk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Our current error-corrected 8 approximation is N (k−1) = N0 +N1 +N2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='+Nk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Define φk = φk−1 − Nk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Now, train a new network Nk to approximate φk through the following differential equation: Fk[φk] = Fk−1[Nk−1] + A[φk] − B[N (k−1)] + B[N (k−1) + φk] = 0 (6) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Fk[Nk] ≡ F0[N (k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Suppose that Fk : R → R in (6) is continuously differentiable in some open set containing φ∗ k, that DFk[φ∗ k] is invertible, and Fk[φ∗ k] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Then, there is a neighbourhood of 0 small enough such that Fk[Nk] → 0 =⇒ Nk → φ∗ k Furthermore, |N (k) − φ| = O (|Fk[Nk]|) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The first result follows analogously from the inverse function theorem as in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' By Theorem 5, F0 −1 is continuously differentiable on some open set around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, it is also locally Lipschitz continuous around 0, meaning there exists some constant α ≥ 0 such that |N (k) − φ| = |F0 −1[F0[N (k)] − F0 −1[F0[φ]]| ≤ α|F0[N (k)] − F0[φ]| ≤ α|F0[N (k)]| ≤ α|Fk[Nk]| and therefore, |N (k) − φ| = O (|Fk[Nk]|) Finally, given Dirichlet boundary conditions φ = g on ∂Ω, any φk is known exactly over ∂Ω since φk = φ − N (k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, the loss function for the kth error correction can be defined as Lk(θ(k)) = 1 M M � i=1 � Fk � Nk � xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θ(k)���2 + 1 N N � j=1 � Nk � yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' θ(k)� − φk(yj) �2 (7) for some randomly sampled points {xi}M i=1 from Ω and {yj}N j=1 from ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Algorithm The error correction algorithm to order K proceeds as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Train a neural network N0 to satisfy the conditions of a differential equation given by (5) and constraint conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Once the loss has converged, stop training and freeze the parameters of N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Initiate and train new neural networks {Nk}K k=1 in sequence to satisfy differential equations given by (6), via loss functions (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Once the loss has converged, stop training, freeze the parameters of Nk, and proceed with Nk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The solution to (5) is approximated by N := N (K) = K � k=0 Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is given above for Dirichlet boundary conditions, but works generally if you incorporate the constraint conditions into all loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Poisson’s equation For Poisson’s equation (2), B ≡ 0 since the Laplacian is linear, so we can write the kth differential equation as Fk[φk] = Fk−1[Nk−1] + ∇2[φk] = 0 which is a Poisson’s equation with our usual f = −Fk−1[Nk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, we can apply Theorem 4 to guarantee that there are neural networks out there that can get very, very close to the φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In the next section, we provide evidence that shows, if we can train just two or three of these networks to reasonably approximate their true solutions, our error-corrected approximation will be a more accurate numerical solution to the original differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 4 Results We present results for a variety of different Poisson’s equations (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Our choice of Poisson’s equation is motivated by its immense application in many areas of theoretical physics, including electrostatics and fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' It is also the simplest second-order, linear PDE, making for a concise yet insightful demonstration of the power of error correction in neural network differential equation solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' To achieve this, we choose the function f on the RHS to force a particular solution φ that we want to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' For example, f(x) = 1 would force the solution φ(x) = 1 2x2 + c1x + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, in general, f can be anything, particularly something which does not admit a closed-form solution to (2), and we do this for ease of visualising φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Knowing the ground truth solution φ in closed form also allows us to compute the relative error � x∈S (φ(x) − N(x))2 � x∈S φ(x)2 at each epoch (iteration) of the training procedure, so we have an understanding of the success of our solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' It is important to note that, while we know φ and the relative error associated with our approximation, the neural network does not, and is solely trained via the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' All neural networks used are SIRENs with 5 hidden layers and 128 hidden units per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' They are trained on batches of 256, using the stochastic gradient descent variant Adam [27], and learning rates are manually tuned for each case of Poisson’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' All experiments are run on a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='8 GHz Dual-Core Intel Core i5 CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='1 3D Poisson’s Equation Figure 1: φ(x, y, z) = sin(5x) sin(5y) sin(5z) at z = π 10 Figure 1 shows the solution to Poisson’s equation: � ∇2φ = −75 sin(5x) sin(5y) sin(5z) in Ω = [−π, π] × [−π, π] × [−π, π] φ = 0 on ∂Ω (8) Figure 2 shows our numerical solutions, with N (0) = N0 on the left, N (1) = N0 + N1 in the centre, and N (2) = N0 + N1 + N2 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' We refer to these as Error Correction 0, 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Visually, all error corrections seem to capture the solution well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Furthermore, each correction decreases the relative error (printed at the bottom of of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Error Correction 1 does so significantly, while the improvement in accuracy from Error Correction 2 is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is further captured in Figure 3, which is a plot of the loss and relative error per epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' After finding a local minimum in Error Correction 0, the loss fluctuates erratically until we initialise Error Correction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The improvement is truly appreciable, and felt across the trends in relative error too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 11 Error Correction 0 Error Correction 1 Error Correction 2 Ground Truth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 3： 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0 2 2 0 2 2 0 2 2 0 2 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='156 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='028 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='020Figure 2: Numerical solutions N (0), N (1) and N (2) to (8) Figure 3: Per-epoch loss and relative errors for numerical solutions to (8) 12 ORDER 0 ORDER 1 ORDER 2 102 LosS 101 0 10000 20000 30000 40000 50000 60000 Epochs ORDER0 ORDER1 ORDER 2 101 Relative error 100 10-1 0 10000 20000 30000 40000 50000 60000 EpochsError Correction 0 Error Correction 1 Error Correction 2 Ground Truth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 3： 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0 2 2 0 2 2 0 2 2 0 2 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='156 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='028 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='0204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='2 2D Poisson’s Equation (i) Figure 4: φ(x, y) = sin(20x) sin(20y) Figure 4 shows the solution to Poisson’s equation: � ∇2φ = −800 sin(5x) sin(5y) in Ω = [−π, π] × [−π, π] φ = 0 on ∂Ω (9) Due to the highly oscillatory nature of the solution, a neural network will struggle to accurately capture its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is demonstrated in Figure 5, where the approximation cannot account for so many peaks and troughs in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Figure 5: Naive attempt at a numerical solution to (9) To obtain a realistic solution, we apply a Gaussian random Fourier feature mapping to the input, before passing it through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' After a simple sweep of values, we take Σ = 1 and n = 256, as defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Figures 6 and 7 show similar trends to those in the previous experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 13 3 2 1 0 1 2 3 3 2 1 0 1 2 3ErrorCorrection o ErrorCorrection1 ErrorCorrection2 Ground Truth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 2 0 2 2 0 2 0 2 2 0 2 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='063 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='040 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='033Figure 6: Numerical solutions N (0), N (1) and N (2) to (9), trained using random Fourier features with Σ = 1 and n = 256 Figure 7: Per-epoch loss and relative errors for numerical solutions to (9) 14 ORDER 0 ORDER 1 ORDER 2 105 104 LosS 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 102 h 0 10000 20000 30000 40000 50000 60000 Epochs ORDER0 ORDER1 ORDER 2 Relative error 100 10-1 10000 20000 30000 40000 50000 60000 EpochsErrorCorrection o ErrorCorrection1 ErrorCorrection2 Ground Truth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='00 2 0 2 2 0 2 0 2 2 0 2 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='063 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='040 Relative error: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='033(ii) Figure 8: φ(x, y) = (π2 − y2) sin(10x) Figure 8 shows the solution to Poisson’s equation: � ∇2φ = (100y2 − 100π2 − 2) sin(10x) in Ω = [−π, π] × [−π, π] φ = 0 on ∂Ω (10) In Figure 9, we train a neural network N (0) to approximate the solution to (10) for 211 epochs, but we save its parameter states after 210 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' These define a new network which we call N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The fully-trained N (0) achieves a reasonable relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Roughness is clearly visible in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Figure 10, we plot the half-trained N0 on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' As expected, it has not yet reached the accuracy of N (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' However, we also initiate an error correction N1, of N0, that trains for another 210 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Thus, we produce an approximation N (1) = N0+N1 that has also trained for a total of 211 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This is significantly more accurate than N (0), and the plot is visibly smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Figures 9 and 10 provide a clear exemplification of the immediate fruitfulness of a single error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 15 Groundtruth 0 5 m 2 L 3 0 2 1 0 2 3 3Figure 9: Numerical solution N (0) to (10), trained for 211 epochs Figure 10: Numerical solutions N0 and N (1) to (10), trained for a total of 211 epochs 16 Error Correction O 10 5 0 5 10 15 2 3 1 32-1 0 Relative Error = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='244 1 0 1 2 3 3ErrorCorrection ErrorCorrection1 10 10 5 5 0 0 5 10 5 15 10 3 3 2 2 1 1 0 3 0 Relative Error = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='423 1 Relative Error = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='035 1 1 0 1 2 1 ~2 2 3 2 3 3 35 Discussion Our results do not endorse error correction as a tool to marginally reduce error across tens of corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Instead, they suggest training a network for half the allotted time, and devoting the other half to a single error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This can yield significantly more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Error correction is not without cost however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In our implementation, we train correction networks on newly sampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' This means that to obtain Fk[Nk], we must first make k forward passes of the new data through N0, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=', Nk−1 and differentiate these to compute Fk−1[Nk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The time complexity of producing a kth order approximation N (k), assuming the number of epochs E and batch size B per correction, and optimisation costs, are kept constant across all corrections, is O � EB(k + 1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' If we instead pass identical batches through each correction network, storing the Fk−1[Nk−1] in memory, we can have a time complexity of O(EB(k + 1)), however the space complexity would be substantially increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 6 Further Work Over time, this study of neural network differential equation solvers naturally lent itself to hot-off-the-press topics in machine learning like sinusoidal representation networks [8] and random Fourier features [9], for the simple reason that such concepts are inextricably linked through their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Outside of differential equations, neural networks as continuous parameterisations of discrete signals have immense potential in 3D shape representation, but also in image, video and audio representation and reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' These problems may utilise neural networks as function approximators or, as we did, derivative approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' There is no reason to suggest why the ideas of error correction cannot be employed here, and every reason to further explore 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+page_content=' Accessed: 2 May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [19] Xavier Glorot, Antoine Bordes, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Deep sparse rectifier neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 14th International Conference on Artificial Intelligence and Statistics, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [20] Ronen Basri, Meirav Galun, Amnon Geifman, David Jacobs, Yoni Kasten, and Shira Kritchman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Fre- quency bias in neural networks for input of non-uniform density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 37th International Conference on Machine Learning, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [21] Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Hamprecht, Yoshua Bengio, and Aaron Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' On the spectral bias of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [22] Yuanzhi Li and Yingyu Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Learning overparameterized neural networks via stochastic gradient descent on structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 32nd Conference on Neural Information Processing Systems, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [23] Ronen Basri, David Jacobs, Yoni Kasten, and Shira Kritchman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' The convergence rate of neural net- works for learned functions of different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 33rd Conference on Neural Information Processing Systems, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [24] Arthur Jacot, Franck Gabriel, and Clement Hongler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Neural tangent kernel: Convergence and gener- alization in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 32nd Conference on Neural Information Processing Systems, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [25] Akshunna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Dogra and William T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Redman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Optimizing neural networks via koopman operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [26] Johannes Nicaise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Lecture notes in real analysis, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' [27] Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' In Proceedings of the 3rd International Conference for Learning Representations, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FPT4oBgHgl3EQfrzVr/content/2301.13146v1.pdf'} diff --git a/X9E0T4oBgHgl3EQfWQAZ/content/2301.02274v1.pdf b/X9E0T4oBgHgl3EQfWQAZ/content/2301.02274v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6f2662a49ee713a967c8df9dfa642f13e584e97f --- /dev/null +++ b/X9E0T4oBgHgl3EQfWQAZ/content/2301.02274v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a2580e7dc33ff5002220a4e4fcf7374066e7560658db3665699d09b26584d023 +size 857397 diff --git a/X9FJT4oBgHgl3EQf6C07/content/tmp_files/2301.11672v1.pdf.txt b/X9FJT4oBgHgl3EQf6C07/content/tmp_files/2301.11672v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..45c6fe59168de6aabd26c8add8498edb45ff7afa --- /dev/null +++ b/X9FJT4oBgHgl3EQf6C07/content/tmp_files/2301.11672v1.pdf.txt @@ -0,0 +1,1199 @@ +Excitons and trions with negative effective masses in two-dimensional semiconductors +M.A. Semina +Ioffe Institute, 26 Polytechnicheskaya, 194021, St.-Petersburg, Russia +J.V. Mamedov +National Research University, Higher School of Economics, +3A Kantemirovskaya, 194100, St.-Petersburg, Russia +M.M. Glazov +Ioffe Institute, 26 Polytechnicheskaya, 194021, St.-Petersburg, Russia and +National Research University, Higher School of Economics, +3A Kantemirovskaya, 194100, St.-Petersburg, Russia +We study theoretically fundamental Coulomb-correlated complexes: neutral and charged exci- +tons, also known as trions, in transition metal dichalogenides monolayers. We focus on the situation +where one of the electrons occupies excited, high-lying, conduction band characterized by a negative +effective mass. We develop the theory of such high-lying excitons and trions with negative effective +mass and demonstrate the key role of the non-parabolicity of the high-lying conduction band disper- +sion in formation of the bound exciton and trion states. We present simple, accurate and physically +justified trial wavefunctions for calculating the binding energies of Coulomb-bound complexes and +compare the results of variational calculations with those of a fully numerical approach. Within +the developed model we discuss recent experimental results on observation of high-lying negative +effective mass trions [K.-Q. Lin et al., Nat. Commun. 13, 6980 (2022)]. +Keywords: transition metal dichalcogenides, exciton, trion, negative effective mass, non-parabolic dispersion +I. +INTRODUCTION +Atomically +thin +transition-metal +dichalcogenides +(TMDC) provide a versatile platform for two-dimensional +(2D) materials with tailored functionalities and fasci- +nating physical properties [1]. +These semiconducting +materials demonstrate outstanding optical properties – +absorption, reflection, emission – due to excitons and +trions, the Coulomb-correlated states of electrons and +holes [2–4], see Refs. [5–7] for review. +Controllable +light-matter interaction [8–10] and ability to form van +der Waals heterostructures [11] make these materials +prime candidates for nanophotonics applications [12–14]. +Usually, excitons, bound electron-hole pairs, and tri- +ons, three particle complexes formed of the electron and +two holes or two electrons and a hole, involve charge car- +riers from the bottom conduction and topmost valence +band [15–17]. +In specific cases, like bulk cuprous ox- +ide, several excitonic series are observed that originate +from closely-lying bands [18, 19]. In this respect, TMDC +monolayers (MLs) show unique properties. In recent ex- +periments, the high-lying excitons and trions were ob- +served [20, 21] that originate from the topmost valence +band holes and electrons in the excited conduction band. +Corresponding optical transitions lie in the ultraviolet +spectral range and can be advantageous for various ap- +plications. +Interestingly, the effective mass of the electron in this +excited conduction band is negative. +It makes energy +spectrum and structure of the Coulomb-correlated com- +plexes different from that in conventional situation where +the effective masses of the involved charge carriers are +positive. Such situation calls for special investigation. +Here, motivated by recent experiments [20, 21], we +study the excitons and trions where one of the charge +carriers, namely, the electron, has a negative effective +mass. We demonstrate the importance of non-parabolic +k4 terms in the high-lying electron dispersion and present +numerical and analytical results of the binding energies +and wavefunctions of excitons and trions with negative- +mass electrons. +The paper is organized as follows: After brief intro- +duction (Sec. I) we formulate the model in Sec. II and +present the results for the excitons in Sec. III and tri- +ons in Sec. IV. Main results are summarized and a brief +outlook is given in Sec. V. +II. +MODEL +We consider a simplified band structure of the TMDC +monolayer that includes the topmost valence band vb, +bottom conduction band cb and the high-lying conduc- +tion band cb+2 in notations of Refs. [6, 20–22]. Figure 1 +shows schematics of the band structure in the vicinity +of the K± points of the Brillouin zone where the direct +band gap of TMDC monolayers is realized. The disper- +sion of the bands nearest conduction and valence bands +(vb and cb) is taken in the isotropic parabolic form: +Evb +k = −Eg − ℏ2k2 +2mh +, +Ecb +k = ℏ2k2 +2m1 +, +(1) +while in the dispersion of the high-lying cb+2 we take into +account also a non-parabolic contribution in the form of +arXiv:2301.11672v1 [cond-mat.mes-hall] 27 Jan 2023 + +2 +the k4 term: +Ecb+2 +k += E′ +g + ℏ2k2 +2m2 ++ Bk4. +(2) +Here k is the electron wavevector, Eg > 0 and E′ +g > 0 +are the band gaps between cb ↔ vb and cb + 2 ↔ cb, +respectively, m1 > 0 and m2 < 0 are, respectively, the +electron effective masses in the bottom conduction band +and high-lying band, and mh > 0 is the effective mass +of the valence band hole; the electron effective mass in +vb mvb = −mh < 0. The coefficient B > 0 describes +the non-parabolic contribution to the dispersion of the +high-lying electron. +Note that in absence of k4 terms the energy Ecb+2 +k +can become lower than Ecb +k making the band notations +meaningless, while Bk4 renders the problem well-defined. +Thus, the dispersions (1) and (2) with B > 0 repre- +sent a minimum model that allows us to have a con- +sistent picture of the high-lying excitons and trions. As +a result of the interplay of the k2 and k4 the disper- +sion in the cb + 2 band has a loop (ring) of exterma at +k∗ = +� +−ℏ2/(4m2B). In real TMDC monolayers charac- +terized by the three-fold rotational symmetry, the disper- +sion of the charge carriers is anisotropic in the plane and, +instead of the extrema loop, three minima can be formed. +We briefly discuss the effects of anisotropy in the end of +the paper. Note that a non-parabolicity in the nearest cb +and vb is related to the interband k · p-mixing [23, 24], +we disregard such effects for simplicity. +To describe the excitons and trions we need to intro- +duce the Coulomb interaction. We use it in the Rytova- +Keldysh form [25, 26] +Vij(ρ) = πqiqj +2r0κ +� +H0 +� ρ +r0 +� +− Y0 +� ρ +r0 +�� +. +(3) +Figure 1. Schematic illustration (not to scale) of the band +structure of TMDC monolayer in the vicinity of the K± points +of the Brillouin zone. The topmost valence, bottom and high- +lying conduction bands are denoted as vb, cb, and cb + 2, re- +spectively. Arrows denote the electron spin orientation; states +with the opposite spin orientations are not shown for clarity. +Here qi,j are the charges of the corresponding carriers +(qe = e < 0 is the electron charge, qh = −e > 0 is +the hole charge), κ is the effective dielectric constant +of the environment, ρ is the interparticle distance, r0 +is the dielectric screening radius, H0(x) and Y0(x) are +the Struve and Neumann functions. At large distances +and/or small screening radius ρ/r0 ≫ 1 the potential en- +ergy takes the Coulomb form ∝ 1/ρ, while at small dis- +tances and/or large screening radius the potential is loga- +rithmic function of the distance ∝ ln ρ/r0. The potential +energy in the form of Eq. (3) is adequate for describing +the Coulomb interaction in atomically thin semiconduc- +tors, see Refs. [27–32] for details. +III. +EXCITONS +We start with the theory of the two-particle bound +states – high-lying excitons (HX) – formed from the va- +lence band hole and high-lying electron. +The effective +Hamiltonian describing the relative motion of the elec- +tron and hole in the HX reads +H = − ℏ2 +2µ2 +∆ + B∆2 + Veh(ρ), +(4) +where µ2 is the high-lying electron and hole reduced +mass, +µ1 = +m1mh +m1 + mh +, +µ2 = +m2mh +m2 + mh +, +(5) +and ∆ is the Laplace operator acting on a wavefunction +ψ(ρ) with the relative electron-hole coordinate ρ. Since +the contribution Eg + Eg′ is excluded from the Hamil- +tonian (4) the total energy of the high-lying exciton is +Eg + Eg′ − Eb,HX where Eb,HX is the binding energy. +We recall that in the parabolic approximation, B = 0, +the HX can be bound only if µ2 > 0 for attractive +Veh(ρ) < 0: Indeed, the inversion of the sign of the mass +can be formally considered as an inversion of the inter- +action potential energy sign [33]. Hence, for µ2 < 0 and +Veh < 0 a bound HX state is absent. By contrast for +positive µ2 > 0, the binding energy is given by +Eb,HX = 2µ2e4 +κ2ℏ2 ζ +�r0µ2e2 +κℏ2 +� +, +µ2 > 0, +(6) +where the function 0 ⩽ ζ(x) ⩽ 1 takes into account the +dielectric screenig effect: At x → 0 the function ζ(x) → 1 +recovering the two-dimensional hydrogen model and at +x → ∞ we have ζ(x) ∼ ln(x)/x [6, 26]. Interestingly, for +the negative reduced mass a two-electron state can be +bound despite the Coulomb repulsion between them [33], +see also Ref. [34] where the electron pairing due to the +spin-orbit interaction is discussed. Note that if µ2 > 0, +but m2 < 0 the HX translational mass mHX = m2+mh < +0, cf. Ref. [35]. +The presence of non-parabolic contribution to the dis- +persion B > 0 makes HX bound for any sign and value + +3 +of the reduced mass µ2, and, hence, for any value of +the high-lying electron effective mass m2, both positive +and negative. To illustrate it we consider, instead of a +Coulomb potential, a shallow short-range potential V0(ρ) +Veh(ρ) < V0(ρ) < 0. +(7) +The presence of the bound state for V0 naturally implies +the bound state for a deeper (Rytova-Keldysh) potential. +For a shallow short-range interaction potential we trans- +form the Sch¨odinger equation Hψ = Eψ to the k-space +and approximate the potential energy as +� +k′ +V0;k−k′ψk′ ≈ V0;0 +� +k′ +ψk′, +where +V0,q += +� +dρV0(ρ) exp (iqρ), +ψk += +� +dρψ(ρ) exp (ikρ) +are +the +Fourier-components +of +the potential energy and wavefunction, respectively, and +the normalization area is set to unity; V0,0 = V0,q=0 < 0. +Thus, +ψk ∝ +1 +E − Ek +, +(8) +and the Schr¨odinger equation reduces to an algebraic +equation; the bound state energy is found from the self- +consistency requirement (see Supplementary Materials +for Ref. [21]): +V0;0 +� +k +1 +E − Ek += 1, +Ek = Ak2/2 + Bk4, +(9) +with A = ℏ2/µ2. In the case of A > 0 we obtain the +bound-state energy in the form +E = − A2 +4B +1 +1 − exp (−A/V0;0) ≈ − A2 +4B eA/V0;0, +(10) +where the approximate equality holds for V0;0 → 0. The +binding energy is Eb = −E. In this situation we recover +exponentially shallow bound state as expected for two- +dimensional system with parabolic dispersion [36]. The +non-parabolicity terms play a role of the high-momentum +cut-off and determine the prefactor in the exponent in +Eq. (10). +At A < 0 (negative reduced mass) Eq. (9) can be trans- +formed to the following form +arctan +A +√ +−16BE − A2 = π +2 + +√ +−16BE − A2 +2V +. +(11) +The minimum of the relative motion dispersion is in this +case E∗ = −A2/(16B) corresponding to +k∗ = +� +− A +4B . +(12) +Thus the binding energy is Eb = E∗ − E. One can check +that Eq. (11) has solutions with E < 0 for any relation +(a) +(b) +Figure 2. (a) Relative motion dispersion, Eq. (9) (dark red), +and the wavefunction absolute value squared, Eq. (8) (dark +blue), in the k-space. (b) Absolute value squared of the rela- +tive motion wavefunction in the real space shown in the log- +linear scale to make oscillations more pronounced. For illus- +trative purposes we use arb. units. +between A and B in the reduced motion dispersion. In +the important limits, +Eb = +� (πV0,0)2 +4B +, +|V0,0| ≪ |A|, +(πV0,0)2 +16B ++ AV0,0 +4B , +|V0,0| ≫ |A|. +(13) +For the negative reduced mass case the bound state is +formed in the vicinity of the minima loop in the k-space +with the relevant wavevectors k ≈ k∗, Fig. 2(a). Thus, +as shown in Fig. 2(b), the relative motion wavefunction +oscillates in the real space. Another specific feature of +the wavefunctions is their behavior at ρ → 0: ψ(ρ) = +const + ρ2 ln ρ owing to the presence of k4 terms in the +dispersion. This function is sufficiently smooth at ρ → 0 +in contrast to the of the parabolic dispersion where the +wavefunction for the shallow short-range potential well +diverges as ln ρ. +The analysis performed above forms a basis for calcu- +lating the excitonic states in the case of the Coulomb, +−e2/(κρ), and Rytova-Keldysh potential (3) and allows +us to formulate convenient trial functions to calculate the +high-lying exciton binding energy. Namely, the ground +state wavefunctions both for µ2 > 0 and µ2 < 0 should +behave as const + ρ2 at ρ → 0, otherwise divergence oc- +curs due to k4 terms and, for µ2 < 0, the wavefunc- +tion should oscillate in space. Naturally, the bound state +wavefunctions should decay at ρ → ∞. We use the fol- + +4 +lowing trial functions for the HX +ψ± +HX(ρ; a, b) ∝ +� +exp (−a +� +b2 + ρ2), +µ2 > 0, +J0(aρ) exp (−bρ2), +µ2 < 0, +(14) +with a and b being the variational parameters and su- +perscript ± corresponds to the sign of µ2; hereafter the +normalization factors are omitted. +Both functions are +smooth at ρ → 0, the wavefunction for µ2 < 0 oscillates +as a function of ρ. We used the Bessel function J0(ρ) +as it is convenient oscillating function with decaying am- +plitude with increase in ρ, which reasonably matches the +oscillating behavior of the exact solution (8) in the short- +range interaction model with variational parameter b con- +trolling the period of oscillations. We have checked ac- +curacy of these trial functions by comparing the exciton +energy found by minimizing the expectation value of the +Hamiltonian (4) with the results of numerical diagonal- +ization of Hamiltonian matrix using the non-orthogonal +basis of Gaussian functions φi(ρ) = exp(−αiρ2). Here +the parameters αi were taken as geometric progression. +The total number N of basic functions and specific val- +ues of αi were chosen to optimize both the numerical +convergence and computational costs [37, 38], typically, +N ≈ 50 − 100 was sufficient for excitons, further increase +of N did not affect the result. Note, that with the cho- +sen basis we can obtain only exciton ground state and +axially-symmetric (s-shell) excited states. In Fig. 3 the +dotted and solid lines show Eb,HX as a function of µ2 for +different values of B calculated variationally (dots) and +numerically. Here and in what follows we use +E = µ1e4/(κ2ℏ2), +a = κℏ2/(µ1e2), +(15) +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 + B*=0.1 + B*=0.25 + B*=1 + B*=5 + B*=10 +Reduced mass ratio µ2/µ1 +HX binding energy Eb,HX/E +Figure 3. Exciton binding energy as a function of the high- +lying electron-hole reduced mass calculated for the Coulomb +potential (r0 = 0 in Eq. (3)) using the variational approach +with the trial functions (14) (dots) and numerical diagonal- +ization (solid lines). +0.001 +0.01 +0.1 +1 +10 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 + µ2/µ1 = - 3 + µ2/µ1 = -1 + µ2/µ1 = -0.5 + µ2/µ1 = 0.5 + µ2/µ1 = 1 + µ2/µ1 = 3 +HX binding energy Eb,HX/E +B* +r0/a = 1 +Figure 4. Exciton binding energy as function of the parameter +B∗ = Be4µ3 +1/(κ2ℏ6) characterizing the non-parabolicity of +the dispersion. +as units of the energy and length. Accordingly, the non- +parabolic term in the dispersion is given by the dimen- +sionless value B∗ = Be4µ3 +1/(κ2ℏ6). Overall, very good +agreement between the two approaches is seen. The ex- +citon state is bound for any µ2 (positive or negative) in +agreement with the analysis above. +Figure 4 shows the HX binding energy as a function +the non-parabolicity parameter B for several values of +µ2: solid lines correspond to µ2 > 0 and dashed lines to +µ2 < 0. For large B∗ the HX binding energy approaches +the asymptotic behavior +Eb,HX = C +E +(B∗)1/3 , +(16) +with the numerical coefficient C ≈ 0.8. The B−1/3 power +law dependence follows from the dimensional arguments +taking into account that for a bound state the mean +values of kinetic and potential energies of the exciton +should be of the same order of magnitude and the co- +efficient C has been found by variational approach with +the Gaussian trial function. At small B the HX bind- +ing energy saturates: for µ2 > 0 it reaches the value +for the parabolic dispersion, Eq. (6) with a correction in +the form ∼ (µ2/µ1)B∗ ln B∗E. The ln B∗ factor arises +because, strictly speaking, the first-order perturbation +theory contribution related to the quantum mechanical +average of Bk4 term logarithmically diverges for hydro- +genic wavefunction. Interestingly, for µ2 < 0 the Eb,HX +also approaches a constant value that depends on µ2. +The detailed analysis of this limit is an interesting task +for future. Here we just note that for sufficiently small +B the radial motion takes place in the vicinity of the +minimum in the dispersion with k ≈ k∗. In the vicinity +of the minimum the dispersion is parabolic and does not +depend on B and, hence, Eb,HX ∼ |µ2/µ1|E. + +5 +Finally, Fig. 5 shows the binding energies of HX ground +and excited states for two values of µ2/µ1 = ±1 and +three values of B∗. +The figure shows the energies of +axially-symmetric (s-shell) HX states with the princi- +pal quantum numbers up to n = 10. +The effect of +non-parabolicity is clearly seen. Deviations from the 2D +hydrogenic model in the case of the Coulomb potential +[black squares in Fig. 5(a)] are clearly visible. Particu- +larly, for positive µ2 and B ̸= 0 the binding energies of +excitonic states are smaller than for the parabolic dis- +persion: It is because the dispersion is steeper and hence +the kinetic energy contribution which reduces the bind- +ing energy is larger. For negative µ2 the exciton energies +are higher than for the parabolic case, this is because the +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0.01 +0.1 +1 + µ2/µ1=1, � ∗=0 + µ2/µ1=1, � ∗=0.1 + µ2/µ1=1, � ∗=10 + µ2/µ1=−1, � ∗=0.1 + µ2/µ1=−1, � ∗=10 +HX binding energy, Eb,HX/E +Principal quantum number, n +(a) +r0/a = 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0.01 +0.1 +1 + µ2/µ1=1, � ∗=0 + µ2/µ1=1, � ∗=0.1 + µ2/µ1=1, � ∗=10 + µ2/µ1=−1, � ∗=0.1 + µ2/µ1=−1, � ∗=10 +HX binding energy, Eb,HX/E +Principal quantum number, n +(b) +r0/a = 1 +Figure 5. Excitonic series for Coulomb potential (r0 = 0 in +Eq. (3)), panel (a), and for the screened potential (r0/a = 1), +panel (b). s-shell exciton binding energies as a function of the +principal quantum number n are shown. +dispersion for small k ≲ k∗ is smoother. +IV. +TRIONS +Now we study the high-lying trions, the three particle +complexes consisting of two holes occupying the topmost +valence bands and one electron in the high-lying cb + 2 +band (HX+ trion) or a hole in vb and two electrons one of +those occupying the conduction band cb and another one +occupying the high-lying band cb + 2 (HX− trion). We +consider here only symmetric trions where the envelope +function is symmetric with respect to the permutations +of identical particles while the correspondig two-particle +Bloch function is antisymmetric with respect to the per- +mutations [30]; these states are optically active at low +carrier densities. Note that antisymmetric trions can also +manifest themselves in the optical response but their os- +cillator strength is proportional to the second power of +the free carrier density [39]. Similarly to the band edge +trions where are two HX− states: intravalley (or so-called +singlet) and intervalley (or triplet) ones where two elec- +trons are, respectively, in the same valley, or in the dif- +ferent valleys [30, 40–42] resulting in the fine structure of +the HX−. Since the fine structure of high-lying negative +trion is related to the short-range part of the electron- +electron interaction [cf. Ref. [30]] and, consequently, the +splitting between the intra- and intervalley states is by +far smaller than the trion binding energy (note that this +splitting has not been observed in Ref. [21]), we disregard +the difference between the intra- and intervalley trions in +what follows. +A. +Parabolic dispersion +It is instructive to start with the parabolic dispersion +model neglecting Bk4 terms in the cb + 2 dispersion. Let +us consider first the HX+ state. The relative motion of +the holes with respect to an electron is governed by the +Hamiltonian +HHX+ = − ℏ2 +2µ2 +� +∆1 + ∆2 + +2σ2 +σ2 + 1∇1∇2 +� ++ Vhh(ρ1 − ρ2) + Veh(ρ1) + Veh(ρ2), +(17) +where ρi are the relative coordinates of two holes (i = +1, 2) with respect to the electron, ∇i and ∆i are the +gradient and Laplace operators acting on functions of ρi, +µ2 is the reduced mass of the high-lying electron and a +hole, Eq. (5), and σ2 = mh/m2 is the hole-to-electron +mass ratio, cf. Refs. [30, 43, 44]. We recall that for the +neutral HX to be bound µ2 should be positive in the +parabolic approximation, see Eq. (6). In this case HX+ +is bound as well, since Eq. (17) describes the positive- +mass situation, see [30] for details. Its binding energy is +a fraction of the high-lying exciton binding energy +Eb,HX+ = χEb,HX, +(18) + +6 +where the coefficient χ ∼ 0.1 depends on the screening +radius r0 and effective masses via µ2 and σ2. +The situation with HX− is more involved. The relative +motion Hamiltonian within a parabolic approximation +takes the form +HHX− = − ℏ2 +2µ1 +∆1 − ℏ2 +2µ2 +∆2 − ℏ2 +mh +∇1∇2 ++ Vee(ρ1 − ρ2) + Veh(ρ1) + Veh(ρ2). +(19) +In this case ρi are the relative coordinates of two elec- +trons with respect to a hole. Taking into account that +the HX− envelope function is symmetric with respect to +permutation of electrons +ψHX−(ρ1, ρ2) = ψHX−(ρ2, ρ1), +the Hamiltonian can be mapped to the symmetrized one +(cf. Eq. (17) in supplement to Ref. [21]) +H = − ℏ2 +2¯µ +� +∆1 + ∆2 + +2¯σ +¯σ + 1∇1∇2 +� ++ Vee(ρ1 − ρ2) + Veh(ρ1) + Veh(ρ2), +(20) +with the renormalized values of the parameters +1 +¯µ = 1 +2 +� 1 +µ1 ++ 1 +µ2 +� +, +¯σ = +¯µ +mh − ¯µ = +2m1m2 +mh(m1 + m2). +(21) +Similarly to the case of the HX+ one can find square- +integrable eigenfunction of Hamiltonian (20). However, +it does not automatically mean that the corresponding +negative high-lying trion is bound, since its energy can be +above the energy of a neutral HX energy. Formally this is +because such a trion is bound with respect to the exciton +with the reduced mass ¯µ [with corresponding “effective” +binding energy ¯Eb,HX− = 2χ¯µe4/(ℏκ)2, cf. +Eq. (18)] +rather than HX with the reduced mass µ2. +Following +Suppementary Materials to Ref. [21] we obtain for the +HX− binding energy +Eb,HX− = 2µ2e4 +ℏ2κ2 +� ¯µ +µ2 +(1 + χ) − 1 +� +. +(22) +The binding energy should be positive, thus, in addition +to µ2 > 0, the following conditions should hold +� +0 < m2 < m∗ ≡ m1mh(1+2χ) +mh−2χm1 , +if +m∗ > 0, +0 < m2 +or +m2 < m∗, +if +m∗ < 0. +(23) +Thus, for negative m2 the condition for HX− to be bound +requires |m2| to be sufficiently large. This condition can +be understood from the following qualitative arguments: +to form a bound trion state the HX considered as a rigid +particle should bind with the cb-electron. The interaction +between HX and electron is typically attractive due to +both the exchange and polarization contributions [45–47]. +Hence, corresponding reduced mass of HX and electron +should be positive yielding m2 < −m1 − mh < 0 where +we made use of the fact that the translational mass of the +HX is mHX = m2 + mh < 0 (for µ2 > 0) and µe−HX = +m1mHX/(m1 + mHX) > 0. +B. +HX− with non-parabolic dispersion +Next we address the effects of cb+2 band nonparabol- +icity on trions. We focus here mainly on the negatively +charged high-lying trion, because this situation is par- +ticularly interesting due to an interplay of the exciton +and trion binding for µ2 < 0. We perform two types of +calculations of the HX− ground state. +The first type of calculations is variational. +In the +variational calculation we use symmetrized combinations +of HX trial functions, Eq. (14) in the form +ψHX−(ρ1, ρ2; a1, a2, b1, b2) ∝ +ψα +HX(ρ1; a1, b1)ψβ +HX(ρ2; a2, b2) + {1 ↔ 2}, +(24) +where ai, bi (i = 1, 2) are the variational parameters, +and α, β = ± determine the particular form of the high- +lying exciton wavefunction in Eq. (14): For µ2 > 0 we +use α = β = +, while for µ2 < 0 we use α = + and +β = −. In the latter case such sign convention allows +us to take into account that one of the electrons in the +HX− (from cb) has a positive effective mass and the other +one (from cb + 2) has a negative mass such that reduced +mass µ2 < 0. We have also checked a trial function with +both α = β = − for trions with the negative reduced +mass, µ2 < 0, and the resulting energies were very close +to obtained with function with α = + and β = −. +The second type of calculations is used to test the vari- +ational approach and provide more accurate numerical +framework for determining the high-lying trion states. +In this calculation the HX− wavefunction is decomposed +as +ψHX−(ρ1, ρ2) += +� +i,j,k +Cijk +� +e−αiρ2 +1−βjρ2 +2 + e−αiρ2 +2−βjρ2 +1 +� +e−δk|ρ1−ρ2|2, +(25) +where αi, βj, δk are parameters whose values were taken +as geometric progression. Similarly to calculation of the +high-lying excitons presented above, the total number of +basic functions and specific values of αi, βj, δk were cho- +sen for the best combination of convergence and compu- +tational costs. The coefficients Cijk were determined by +minimizing the total energy. The wavefunction (25) pro- +vides rather accurate form of the radial wavefunctions for +relative motion of electrons and, importantly, takes into +account, via the factor exp (−δk|ρ1 − ρ2|2), correlation +between the electron motion. +Figure 6 shows the dependence of the HX− binding +energy on the high-lying electron to hole reduced mass +µ2 calculated for several values of the non-parabolicity +parameter B. +Solid lines show the results of the full +numerical calculation, while dotted lines in Fig. 6(a) +demonstrate the results of the variational approach with +the trial functions (24). +The variational calculation +gives reasonable estimate of the binding energy being by + +7 +-3 +-2 +-1 +0 +1 +2 +3 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 + B*=0.1 + B*=0.25 + B*=1 + B*=5 + B*=10 +HX− binding energy Eb,HX−/E +r0/a = 0 +(a) +Reduced mass ratio µ2/µ1 +-3 +-2 +-1 +0 +1 +2 +3 +-0.3 +-0.2 +-0.1 +0.0 +0.1 + B*=0.1 + B*=0.25 + B*=1 + B*=5 + B*=10 +HX− binding energy Eb,HX−/E +r0/a = 1 +(b) +Reduced mass ratio µ2/µ1 +Figure 6. HX− binding energy as a function of the high-lying +electron-hole reduced mass calculated for the Coulomb poten- +tial (a) and for screened potential (b). Dotted lines in (a) show +the results of variational calculation and solid lines [in panels +(a) and (b)] show the results of full numerical approach. +10% . . . 30% lower than the “exact” value found using the +wavefunction (25). We have also performed the numeri- +cal calculation with the function in the form of Eq. (25) +but without correlation factors, i.e., setting δk ≡ 0. +These results turn out to be almost indistinguishable +from the results of variational calculation, which justifies +the choice of the trial functions (24) for the variational +calculation. +For negative µ2 the larger is |µ2|, the larger is the high- +lying trion binding energy. For positive µ2 a maximum +in the dependence of Eb,HX− on µ2 is seen for small B. +Qualitatively, this maximum can be understood within +the framework of the analytical expression for the HX− +binding energy in the parabolic approximation, Eq. (22). +It appears as a result of an interplay of two terms: the +first, positive term, weakly increases with increase in µ2, +while the absolute value of the second, negative term, +increases linearly with µ2. This maximum becomes more +pronounced in the case of the screened Rytova-Keldysh +potential, Fig. 6(b). +The dependence of the high-lying trion binding energy +on the non-parabolic contribution to the dispersion char- +acterized by the parameter B is shown in Fig. 7. For +small B (B∗ ≪ 1) the HX− binding energy increases with +increase in B and strongly depends on µ2. Hence, the +presence of k4 terms in the high-lying electron dispersion +makes trions more stable. For large non-parabolic term +(B∗ ≫ 1) the Eb,HX− decreases with increasing B regard- +less the value of µ2 following the same B−1/3 power law +as for the HX, Eq. (16) with a different coefficient yield- +ing relatively large high-lying trion-to high-lying exciton +binding energy ratio +Eb,HX− +Eb,HX +≈ 0.3. +(26) +To summarize, the presence of k4 terms in the high- +lying electron dispersion significantly expand the range +µ2/µ1 where the high-lying HX− trion is bound. As for +positive trion, HX+ it is bound already in the parabolic +approximation and our estimates show that it remains +bound in the presence of non-parabolic contributions to +the dispersion. +C. +Discussion of the results +Let us now briefly discuss the obtained results in view +of experimental data reported in Ref. [21]. For rough es- +timates we note that in TMDC monolayers the valence +band hole and the conduction band (cb) electron effec- +tive masses are about the same, mh ≈ m1. The electron +0.001 +0.01 +0.1 +1 +10 +100 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 + µ2/µ1 = - 3 + µ2/µ1 = - 1 + µ2/µ1 = - 0.5 + µ2/µ1 = 0.5 + µ2/µ1 = 1 + µ2/µ1 = 3 +HX− binding energy Eb,HX−/E +r0/a = 1 +B* +Figure 7. +High-lying trion binding energy as a function of +non-parabolicity parameter B∗. + +8 +effective mass in cb+2 conduction band has a similar ab- +solute value, but it is negative. Estimates based on the +DFT approach presented in Ref. [20] show that for WSe2 +monolayers |m2| ≈ 0.46m0 > mh ≈ 0.36m0 with m0 +being the free-electron mass, making the reduced mass +µ2 ≈ 1.66m0 > 0 in Eq. (5). +Additional evidence for +|m2| > mh follows from the strong phonon progression of +HX observed in Ref. [20] indicating that the translational +mass of the high-lyign exciton is negative. Thus, neutral +high-lying exciton, HX, is bound even if Bk4 terms are +neglected in the cb + 2 dispersion. For such parameters, +however, m∗ > 0 in Eq. (23) (for m1 ≈ mh and reason- +able χ ≈ 0.2 the m∗ ≈ 2.3). Hence, according to Eq. (23) +the HX− is not bound in the parabolic approximation. +Thus, we come to the conclusion that Bk4 contribution +should be sizeable to make HX− bound. +Note that experimentally observed HX− binding ener- +gies are ≈ 43 meV for WSe2 monolayer and ≈ 21 meV for +MoSe2 monolayer [21]. In the former case it is slightly +larger than the band-edge trion, X−, binding energy, +while in the latter case it is slightly smaller than that +of X−. Depending on µ2/µ1 and B the HX− binding en- +ergy can be on the order of 10% . . . 25% of the HX bind- +ing energy, thus, somewhat increased values of Eb,HX− +compared to Eb,X− can be related to (i) enhancement +of the Eb,HX due to rather large µ2/µ1 ≈ 6 . . . 10 for the +estimated parameters and (ii) to large B∗ where, as men- +tionned above, the trion-to-exciton binding energy ratio +turns out to be quite large, Eq. (26). +In this theoretical paper we abstain from further anal- +ysis of the experimental data and more detailed com- +parison of the calculations with experiment. The main +reason is that the dispersion in cb + 2 band is quite com- +plicated [20, 22] and contains, in addition to simplified +Eq. (2), anisotropic terms. Also, experimental data on +high-lying excitons binding energy are not available to +the best of our knowledge. Still our theoretical results +in combination with experimental data [21] indicate im- +portance of the non-parabolic terms in the high-lying +electron dispersion for formation of the three-particle +Coulomb complexes. +V. +CONCLUSION AND OUTLOOK +To conclude, we have developed the theory of high- +lying excitons and trions in two-dimensional semiconduc- +tors. Such Coulomb-bound complexes involve one elec- +tron in the excited conduction band with the negative ef- +fective mass and a non-parabolic dispersion. We have de- +veloped (i) variational method for calculating such com- +plexes with simple and physically justified trial functions +and (ii) the efficient and accurate numerical approach +based on decomposition of the wavefunctions of Gaus- +sians. We have demonstrated the importance of the band +non-parabolicity for formation of the high-lying excitons +and trions. In particular, for negative reduced mass the +presence of k4 terms in the high-lying electron dispersion +makes exciton bound and strongly enhances the range of +stability of the negatively charged high-lying trions. Our +estimates show that the high-lying trion binding energies +can be in the range of 10% . . . 30% of the high-lying ex- +citon binding energy, i.e., on the order of several tens of +meV for transition-metal dichalcogenide monolayers. +The developed theory is not limited to the mono- +layer transition-metal dichalcogenides. 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Pikhtin, Indi- +rect exciton dispersion in III-V semiconductors: ‘Camel’s +back’ in GaP, Solid State Communications 30, 631 +(1979). + diff --git a/X9FJT4oBgHgl3EQf6C07/content/tmp_files/load_file.txt b/X9FJT4oBgHgl3EQf6C07/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0bbfecb6fa53e6d4a12e47fc85e4501ee6e75cb8 --- /dev/null +++ b/X9FJT4oBgHgl3EQf6C07/content/tmp_files/load_file.txt @@ -0,0 +1,842 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf,len=841 +page_content='Excitons and trions with negative effective masses in two-dimensional semiconductors M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Semina Ioffe Institute, 26 Polytechnicheskaya, 194021, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='-Petersburg, Russia J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Mamedov National Research University, Higher School of Economics, 3A Kantemirovskaya, 194100, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='-Petersburg, Russia M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Glazov Ioffe Institute, 26 Polytechnicheskaya, 194021, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='-Petersburg, Russia and National Research University, Higher School of Economics, 3A Kantemirovskaya, 194100, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='-Petersburg, Russia We study theoretically fundamental Coulomb-correlated complexes: neutral and charged exci- tons, also known as trions, in transition metal dichalogenides monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We focus on the situation where one of the electrons occupies excited, high-lying, conduction band characterized by a negative effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We develop the theory of such high-lying excitons and trions with negative effective mass and demonstrate the key role of the non-parabolicity of the high-lying conduction band disper- sion in formation of the bound exciton and trion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We present simple, accurate and physically justified trial wavefunctions for calculating the binding energies of Coulomb-bound complexes and compare the results of variational calculations with those of a fully numerical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Within the developed model we discuss recent experimental results on observation of high-lying negative effective mass trions [K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=', Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 13, 6980 (2022)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Keywords: transition metal dichalcogenides, exciton, trion, negative effective mass, non-parabolic dispersion I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' INTRODUCTION Atomically thin transition-metal dichalcogenides (TMDC) provide a versatile platform for two-dimensional (2D) materials with tailored functionalities and fasci- nating physical properties [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' These semiconducting materials demonstrate outstanding optical properties – absorption, reflection, emission – due to excitons and trions, the Coulomb-correlated states of electrons and holes [2–4], see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [5–7] for review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Controllable light-matter interaction [8–10] and ability to form van der Waals heterostructures [11] make these materials prime candidates for nanophotonics applications [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Usually, excitons, bound electron-hole pairs, and tri- ons, three particle complexes formed of the electron and two holes or two electrons and a hole, involve charge car- riers from the bottom conduction and topmost valence band [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In specific cases, like bulk cuprous ox- ide, several excitonic series are observed that originate from closely-lying bands [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this respect, TMDC monolayers (MLs) show unique properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In recent ex- periments, the high-lying excitons and trions were ob- served [20, 21] that originate from the topmost valence band holes and electrons in the excited conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Corresponding optical transitions lie in the ultraviolet spectral range and can be advantageous for various ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Interestingly, the effective mass of the electron in this excited conduction band is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' It makes energy spectrum and structure of the Coulomb-correlated com- plexes different from that in conventional situation where the effective masses of the involved charge carriers are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Such situation calls for special investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Here, motivated by recent experiments [20, 21], we study the excitons and trions where one of the charge carriers, namely, the electron, has a negative effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We demonstrate the importance of non-parabolic k4 terms in the high-lying electron dispersion and present numerical and analytical results of the binding energies and wavefunctions of excitons and trions with negative- mass electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The paper is organized as follows: After brief intro- duction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' I) we formulate the model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' II and present the results for the excitons in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' III and tri- ons in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Main results are summarized and a brief outlook is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' MODEL We consider a simplified band structure of the TMDC monolayer that includes the topmost valence band vb, bottom conduction band cb and the high-lying conduc- tion band cb+2 in notations of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [6, 20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Figure 1 shows schematics of the band structure in the vicinity of the K± points of the Brillouin zone where the direct band gap of TMDC monolayers is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The disper- sion of the bands nearest conduction and valence bands (vb and cb) is taken in the isotropic parabolic form: Evb k = −Eg − ℏ2k2 2mh , Ecb k = ℏ2k2 2m1 , (1) while in the dispersion of the high-lying cb+2 we take into account also a non-parabolic contribution in the form of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='11672v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='mes-hall] 27 Jan 2023 2 the k4 term: Ecb+2 k = E′ g + ℏ2k2 2m2 + Bk4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (2) Here k is the electron wavevector, Eg > 0 and E′ g > 0 are the band gaps between cb ↔ vb and cb + 2 ↔ cb, respectively, m1 > 0 and m2 < 0 are, respectively, the electron effective masses in the bottom conduction band and high-lying band, and mh > 0 is the effective mass of the valence band hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' the electron effective mass in vb mvb = −mh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The coefficient B > 0 describes the non-parabolic contribution to the dispersion of the high-lying electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Note that in absence of k4 terms the energy Ecb+2 k can become lower than Ecb k making the band notations meaningless, while Bk4 renders the problem well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Thus, the dispersions (1) and (2) with B > 0 repre- sent a minimum model that allows us to have a con- sistent picture of the high-lying excitons and trions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' As a result of the interplay of the k2 and k4 the disper- sion in the cb + 2 band has a loop (ring) of exterma at k∗ = � −ℏ2/(4m2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In real TMDC monolayers charac- terized by the three-fold rotational symmetry, the disper- sion of the charge carriers is anisotropic in the plane and, instead of the extrema loop, three minima can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We briefly discuss the effects of anisotropy in the end of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Note that a non-parabolicity in the nearest cb and vb is related to the interband k · p-mixing [23, 24], we disregard such effects for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' To describe the excitons and trions we need to intro- duce the Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We use it in the Rytova- Keldysh form [25, 26] Vij(ρ) = πqiqj 2r0κ � H0 � ρ r0 � − Y0 � ρ r0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (3) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Schematic illustration (not to scale) of the band structure of TMDC monolayer in the vicinity of the K± points of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The topmost valence, bottom and high- lying conduction bands are denoted as vb, cb, and cb + 2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Arrows denote the electron spin orientation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' states with the opposite spin orientations are not shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Here qi,j are the charges of the corresponding carriers (qe = e < 0 is the electron charge, qh = −e > 0 is the hole charge), κ is the effective dielectric constant of the environment, ρ is the interparticle distance, r0 is the dielectric screening radius, H0(x) and Y0(x) are the Struve and Neumann functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' At large distances and/or small screening radius ρ/r0 ≫ 1 the potential en- ergy takes the Coulomb form ∝ 1/ρ, while at small dis- tances and/or large screening radius the potential is loga- rithmic function of the distance ∝ ln ρ/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The potential energy in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (3) is adequate for describing the Coulomb interaction in atomically thin semiconduc- tors, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [27–32] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' EXCITONS We start with the theory of the two-particle bound states – high-lying excitons (HX) – formed from the va- lence band hole and high-lying electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The effective Hamiltonian describing the relative motion of the elec- tron and hole in the HX reads H = − ℏ2 2µ2 ∆ + B∆2 + Veh(ρ), (4) where µ2 is the high-lying electron and hole reduced mass, µ1 = m1mh m1 + mh , µ2 = m2mh m2 + mh , (5) and ∆ is the Laplace operator acting on a wavefunction ψ(ρ) with the relative electron-hole coordinate ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Since the contribution Eg + Eg′ is excluded from the Hamil- tonian (4) the total energy of the high-lying exciton is Eg + Eg′ − Eb,HX where Eb,HX is the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We recall that in the parabolic approximation, B = 0, the HX can be bound only if µ2 > 0 for attractive Veh(ρ) < 0: Indeed, the inversion of the sign of the mass can be formally considered as an inversion of the inter- action potential energy sign [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Hence, for µ2 < 0 and Veh < 0 a bound HX state is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' By contrast for positive µ2 > 0, the binding energy is given by Eb,HX = 2µ2e4 κ2ℏ2 ζ �r0µ2e2 κℏ2 � , µ2 > 0, (6) where the function 0 ⩽ ζ(x) ⩽ 1 takes into account the dielectric screenig effect: At x → 0 the function ζ(x) → 1 recovering the two-dimensional hydrogen model and at x → ∞ we have ζ(x) ∼ ln(x)/x [6, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Interestingly, for the negative reduced mass a two-electron state can be bound despite the Coulomb repulsion between them [33], see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [34] where the electron pairing due to the spin-orbit interaction is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Note that if µ2 > 0, but m2 < 0 the HX translational mass mHX = m2+mh < 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The presence of non-parabolic contribution to the dis- persion B > 0 makes HX bound for any sign and value 3 of the reduced mass µ2, and, hence, for any value of the high-lying electron effective mass m2, both positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' To illustrate it we consider, instead of a Coulomb potential, a shallow short-range potential V0(ρ) Veh(ρ) < V0(ρ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (7) The presence of the bound state for V0 naturally implies the bound state for a deeper (Rytova-Keldysh) potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For a shallow short-range interaction potential we trans- form the Sch¨odinger equation Hψ = Eψ to the k-space and approximate the potential energy as � k′ V0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='k−k′ψk′ ≈ V0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 � k′ ψk′, where V0,q = � dρV0(ρ) exp (iqρ), ψk = � dρψ(ρ) exp (ikρ) are the Fourier-components of the potential energy and wavefunction, respectively, and the normalization area is set to unity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' V0,0 = V0,q=0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Thus, ψk ∝ 1 E − Ek , (8) and the Schr¨odinger equation reduces to an algebraic equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' the bound state energy is found from the self- consistency requirement (see Supplementary Materials for Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [21]): V0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 � k 1 E − Ek = 1, Ek = Ak2/2 + Bk4, (9) with A = ℏ2/µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In the case of A > 0 we obtain the bound-state energy in the form E = − A2 4B 1 1 − exp (−A/V0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0) ≈ − A2 4B eA/V0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0, (10) where the approximate equality holds for V0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The binding energy is Eb = −E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this situation we recover exponentially shallow bound state as expected for two- dimensional system with parabolic dispersion [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The non-parabolicity terms play a role of the high-momentum cut-off and determine the prefactor in the exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' At A < 0 (negative reduced mass) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (9) can be trans- formed to the following form arctan A √ −16BE − A2 = π 2 + √ −16BE − A2 2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (11) The minimum of the relative motion dispersion is in this case E∗ = −A2/(16B) corresponding to k∗ = � − A 4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (12) Thus the binding energy is Eb = E∗ − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' One can check that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (11) has solutions with E < 0 for any relation (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (a) Relative motion dispersion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (9) (dark red), and the wavefunction absolute value squared, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (8) (dark blue), in the k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (b) Absolute value squared of the rela- tive motion wavefunction in the real space shown in the log- linear scale to make oscillations more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For illus- trative purposes we use arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' between A and B in the reduced motion dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In the important limits, Eb = � (πV0,0)2 4B , |V0,0| ≪ |A|, (πV0,0)2 16B + AV0,0 4B , |V0,0| ≫ |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (13) For the negative reduced mass case the bound state is formed in the vicinity of the minima loop in the k-space with the relevant wavevectors k ≈ k∗, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Thus, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 2(b), the relative motion wavefunction oscillates in the real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Another specific feature of the wavefunctions is their behavior at ρ → 0: ψ(ρ) = const + ρ2 ln ρ owing to the presence of k4 terms in the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' This function is sufficiently smooth at ρ → 0 in contrast to the of the parabolic dispersion where the wavefunction for the shallow short-range potential well diverges as ln ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The analysis performed above forms a basis for calcu- lating the excitonic states in the case of the Coulomb, −e2/(κρ), and Rytova-Keldysh potential (3) and allows us to formulate convenient trial functions to calculate the high-lying exciton binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Namely, the ground state wavefunctions both for µ2 > 0 and µ2 < 0 should behave as const + ρ2 at ρ → 0, otherwise divergence oc- curs due to k4 terms and, for µ2 < 0, the wavefunc- tion should oscillate in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Naturally, the bound state wavefunctions should decay at ρ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We use the fol- 4 lowing trial functions for the HX ψ± HX(ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' a, b) ∝ � exp (−a � b2 + ρ2), µ2 > 0, J0(aρ) exp (−bρ2), µ2 < 0, (14) with a and b being the variational parameters and su- perscript ± corresponds to the sign of µ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' hereafter the normalization factors are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Both functions are smooth at ρ → 0, the wavefunction for µ2 < 0 oscillates as a function of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We used the Bessel function J0(ρ) as it is convenient oscillating function with decaying am- plitude with increase in ρ, which reasonably matches the oscillating behavior of the exact solution (8) in the short- range interaction model with variational parameter b con- trolling the period of oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We have checked ac- curacy of these trial functions by comparing the exciton energy found by minimizing the expectation value of the Hamiltonian (4) with the results of numerical diagonal- ization of Hamiltonian matrix using the non-orthogonal basis of Gaussian functions φi(ρ) = exp(−αiρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Here the parameters αi were taken as geometric progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The total number N of basic functions and specific val- ues of αi were chosen to optimize both the numerical convergence and computational costs [37, 38], typically, N ≈ 50 − 100 was sufficient for excitons, further increase of N did not affect the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Note, that with the cho- sen basis we can obtain only exciton ground state and axially-symmetric (s-shell) excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 3 the dotted and solid lines show Eb,HX as a function of µ2 for different values of B calculated variationally (dots) and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Here and in what follows we use E = µ1e4/(κ2ℏ2), a = κℏ2/(µ1e2), (15) 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 B*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 B*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='25 B*=1 B*=5 B*=10 Reduced mass ratio µ2/µ1 HX binding energy Eb,HX/E Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Exciton binding energy as a function of the high- lying electron-hole reduced mass calculated for the Coulomb potential (r0 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (3)) using the variational approach with the trial functions (14) (dots) and numerical diagonal- ization (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='4 µ2/µ1 = - 3 µ2/µ1 = -1 µ2/µ1 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 µ2/µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 µ2/µ1 = 1 µ2/µ1 = 3 HX binding energy Eb,HX/E B* r0/a = 1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Exciton binding energy as function of the parameter B∗ = Be4µ3 1/(κ2ℏ6) characterizing the non-parabolicity of the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' as units of the energy and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Accordingly, the non- parabolic term in the dispersion is given by the dimen- sionless value B∗ = Be4µ3 1/(κ2ℏ6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Overall, very good agreement between the two approaches is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The ex- citon state is bound for any µ2 (positive or negative) in agreement with the analysis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Figure 4 shows the HX binding energy as a function the non-parabolicity parameter B for several values of µ2: solid lines correspond to µ2 > 0 and dashed lines to µ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For large B∗ the HX binding energy approaches the asymptotic behavior Eb,HX = C E (B∗)1/3 , (16) with the numerical coefficient C ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The B−1/3 power law dependence follows from the dimensional arguments taking into account that for a bound state the mean values of kinetic and potential energies of the exciton should be of the same order of magnitude and the co- efficient C has been found by variational approach with the Gaussian trial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' At small B the HX bind- ing energy saturates: for µ2 > 0 it reaches the value for the parabolic dispersion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (6) with a correction in the form ∼ (µ2/µ1)B∗ ln B∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The ln B∗ factor arises because, strictly speaking, the first-order perturbation theory contribution related to the quantum mechanical average of Bk4 term logarithmically diverges for hydro- genic wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Interestingly, for µ2 < 0 the Eb,HX also approaches a constant value that depends on µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The detailed analysis of this limit is an interesting task for future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Here we just note that for sufficiently small B the radial motion takes place in the vicinity of the minimum in the dispersion with k ≈ k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In the vicinity of the minimum the dispersion is parabolic and does not depend on B and, hence, Eb,HX ∼ |µ2/µ1|E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 5 Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 5 shows the binding energies of HX ground and excited states for two values of µ2/µ1 = ±1 and three values of B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The figure shows the energies of axially-symmetric (s-shell) HX states with the princi- pal quantum numbers up to n = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The effect of non-parabolicity is clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Deviations from the 2D hydrogenic model in the case of the Coulomb potential [black squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 5(a)] are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Particu- larly, for positive µ2 and B ̸= 0 the binding energies of excitonic states are smaller than for the parabolic dis- persion: It is because the dispersion is steeper and hence the kinetic energy contribution which reduces the bind- ing energy is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For negative µ2 the exciton energies are higher than for the parabolic case, this is because the 1 2 3 4 5 6 7 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 1 µ2/µ1=1, � ∗=0 µ2/µ1=1, � ∗=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 µ2/µ1=1, � ∗=10 µ2/µ1=−1, � ∗=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 µ2/µ1=−1, � ∗=10 HX binding energy, Eb,HX/E Principal quantum number, n (a) r0/a = 0 1 2 3 4 5 6 7 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 1 µ2/µ1=1, � ∗=0 µ2/µ1=1, � ∗=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 µ2/µ1=1, � ∗=10 µ2/µ1=−1, � ∗=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 µ2/µ1=−1, � ∗=10 HX binding energy, Eb,HX/E Principal quantum number, n (b) r0/a = 1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Excitonic series for Coulomb potential (r0 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (3)), panel (a), and for the screened potential (r0/a = 1), panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' s-shell exciton binding energies as a function of the principal quantum number n are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' dispersion for small k ≲ k∗ is smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' TRIONS Now we study the high-lying trions, the three particle complexes consisting of two holes occupying the topmost valence bands and one electron in the high-lying cb + 2 band (HX+ trion) or a hole in vb and two electrons one of those occupying the conduction band cb and another one occupying the high-lying band cb + 2 (HX− trion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We consider here only symmetric trions where the envelope function is symmetric with respect to the permutations of identical particles while the correspondig two-particle Bloch function is antisymmetric with respect to the per- mutations [30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' these states are optically active at low carrier densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Note that antisymmetric trions can also manifest themselves in the optical response but their os- cillator strength is proportional to the second power of the free carrier density [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Similarly to the band edge trions where are two HX− states: intravalley (or so-called singlet) and intervalley (or triplet) ones where two elec- trons are, respectively, in the same valley, or in the dif- ferent valleys [30, 40–42] resulting in the fine structure of the HX−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Since the fine structure of high-lying negative trion is related to the short-range part of the electron- electron interaction [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [30]] and, consequently, the splitting between the intra- and intervalley states is by far smaller than the trion binding energy (note that this splitting has not been observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [21]), we disregard the difference between the intra- and intervalley trions in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Parabolic dispersion It is instructive to start with the parabolic dispersion model neglecting Bk4 terms in the cb + 2 dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Let us consider first the HX+ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The relative motion of the holes with respect to an electron is governed by the Hamiltonian HHX+ = − ℏ2 2µ2 � ∆1 + ∆2 + 2σ2 σ2 + 1∇1∇2 � + Vhh(ρ1 − ρ2) + Veh(ρ1) + Veh(ρ2), (17) where ρi are the relative coordinates of two holes (i = 1, 2) with respect to the electron, ∇i and ∆i are the gradient and Laplace operators acting on functions of ρi, µ2 is the reduced mass of the high-lying electron and a hole, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (5), and σ2 = mh/m2 is the hole-to-electron mass ratio, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [30, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We recall that for the neutral HX to be bound µ2 should be positive in the parabolic approximation, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this case HX+ is bound as well, since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (17) describes the positive- mass situation, see [30] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Its binding energy is a fraction of the high-lying exciton binding energy Eb,HX+ = χEb,HX, (18) 6 where the coefficient χ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 depends on the screening radius r0 and effective masses via µ2 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The situation with HX− is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The relative motion Hamiltonian within a parabolic approximation takes the form HHX− = − ℏ2 2µ1 ∆1 − ℏ2 2µ2 ∆2 − ℏ2 mh ∇1∇2 + Vee(ρ1 − ρ2) + Veh(ρ1) + Veh(ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (19) In this case ρi are the relative coordinates of two elec- trons with respect to a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Taking into account that the HX− envelope function is symmetric with respect to permutation of electrons ψHX−(ρ1, ρ2) = ψHX−(ρ2, ρ1), the Hamiltonian can be mapped to the symmetrized one (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (17) in supplement to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [21]) H = − ℏ2 2¯µ � ∆1 + ∆2 + 2¯σ ¯σ + 1∇1∇2 � + Vee(ρ1 − ρ2) + Veh(ρ1) + Veh(ρ2), (20) with the renormalized values of the parameters 1 ¯µ = 1 2 � 1 µ1 + 1 µ2 � , ¯σ = ¯µ mh − ¯µ = 2m1m2 mh(m1 + m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (21) Similarly to the case of the HX+ one can find square- integrable eigenfunction of Hamiltonian (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' However, it does not automatically mean that the corresponding negative high-lying trion is bound, since its energy can be above the energy of a neutral HX energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Formally this is because such a trion is bound with respect to the exciton with the reduced mass ¯µ [with corresponding “effective” binding energy ¯Eb,HX− = 2χ¯µe4/(ℏκ)2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (18)] rather than HX with the reduced mass µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Following Suppementary Materials to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [21] we obtain for the HX− binding energy Eb,HX− = 2µ2e4 ℏ2κ2 � ¯µ µ2 (1 + χ) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (22) The binding energy should be positive, thus, in addition to µ2 > 0, the following conditions should hold � 0 < m2 < m∗ ≡ m1mh(1+2χ) mh−2χm1 , if m∗ > 0, 0 < m2 or m2 < m∗, if m∗ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (23) Thus, for negative m2 the condition for HX− to be bound requires |m2| to be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' This condition can be understood from the following qualitative arguments: to form a bound trion state the HX considered as a rigid particle should bind with the cb-electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The interaction between HX and electron is typically attractive due to both the exchange and polarization contributions [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Hence, corresponding reduced mass of HX and electron should be positive yielding m2 < −m1 − mh < 0 where we made use of the fact that the translational mass of the HX is mHX = m2 + mh < 0 (for µ2 > 0) and µe−HX = m1mHX/(m1 + mHX) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' HX− with non-parabolic dispersion Next we address the effects of cb+2 band nonparabol- icity on trions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We focus here mainly on the negatively charged high-lying trion, because this situation is par- ticularly interesting due to an interplay of the exciton and trion binding for µ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We perform two types of calculations of the HX− ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The first type of calculations is variational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In the variational calculation we use symmetrized combinations of HX trial functions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (14) in the form ψHX−(ρ1, ρ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' a1, a2, b1, b2) ∝ ψα HX(ρ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' a1, b1)ψβ HX(ρ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' a2, b2) + {1 ↔ 2}, (24) where ai, bi (i = 1, 2) are the variational parameters, and α, β = ± determine the particular form of the high- lying exciton wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (14): For µ2 > 0 we use α = β = +, while for µ2 < 0 we use α = + and β = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In the latter case such sign convention allows us to take into account that one of the electrons in the HX− (from cb) has a positive effective mass and the other one (from cb + 2) has a negative mass such that reduced mass µ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We have also checked a trial function with both α = β = − for trions with the negative reduced mass, µ2 < 0, and the resulting energies were very close to obtained with function with α = + and β = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The second type of calculations is used to test the vari- ational approach and provide more accurate numerical framework for determining the high-lying trion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this calculation the HX− wavefunction is decomposed as ψHX−(ρ1, ρ2) = � i,j,k Cijk � e−αiρ2 1−βjρ2 2 + e−αiρ2 2−βjρ2 1 � e−δk|ρ1−ρ2|2, (25) where αi, βj, δk are parameters whose values were taken as geometric progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Similarly to calculation of the high-lying excitons presented above, the total number of basic functions and specific values of αi, βj, δk were cho- sen for the best combination of convergence and compu- tational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The coefficients Cijk were determined by minimizing the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The wavefunction (25) pro- vides rather accurate form of the radial wavefunctions for relative motion of electrons and, importantly, takes into account, via the factor exp (−δk|ρ1 − ρ2|2), correlation between the electron motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Figure 6 shows the dependence of the HX− binding energy on the high-lying electron to hole reduced mass µ2 calculated for several values of the non-parabolicity parameter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Solid lines show the results of the full numerical calculation, while dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 6(a) demonstrate the results of the variational approach with the trial functions (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The variational calculation gives reasonable estimate of the binding energy being by 7 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='4 B*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 B*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='25 B*=1 B*=5 B*=10 HX− binding energy Eb,HX−/E r0/a = 0 (a) Reduced mass ratio µ2/µ1 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 B*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 B*=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='25 B*=1 B*=5 B*=10 HX− binding energy Eb,HX−/E r0/a = 1 (b) Reduced mass ratio µ2/µ1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' HX− binding energy as a function of the high-lying electron-hole reduced mass calculated for the Coulomb poten- tial (a) and for screened potential (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Dotted lines in (a) show the results of variational calculation and solid lines [in panels (a) and (b)] show the results of full numerical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 10% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 30% lower than the “exact” value found using the wavefunction (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We have also performed the numeri- cal calculation with the function in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (25) but without correlation factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=', setting δk ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' These results turn out to be almost indistinguishable from the results of variational calculation, which justifies the choice of the trial functions (24) for the variational calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For negative µ2 the larger is |µ2|, the larger is the high- lying trion binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For positive µ2 a maximum in the dependence of Eb,HX− on µ2 is seen for small B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Qualitatively, this maximum can be understood within the framework of the analytical expression for the HX− binding energy in the parabolic approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' It appears as a result of an interplay of two terms: the first, positive term, weakly increases with increase in µ2, while the absolute value of the second, negative term, increases linearly with µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' This maximum becomes more pronounced in the case of the screened Rytova-Keldysh potential, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The dependence of the high-lying trion binding energy on the non-parabolic contribution to the dispersion char- acterized by the parameter B is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For small B (B∗ ≪ 1) the HX− binding energy increases with increase in B and strongly depends on µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Hence, the presence of k4 terms in the high-lying electron dispersion makes trions more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For large non-parabolic term (B∗ ≫ 1) the Eb,HX− decreases with increasing B regard- less the value of µ2 following the same B−1/3 power law as for the HX, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (16) with a different coefficient yield- ing relatively large high-lying trion-to high-lying exciton binding energy ratio Eb,HX− Eb,HX ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (26) To summarize, the presence of k4 terms in the high- lying electron dispersion significantly expand the range µ2/µ1 where the high-lying HX− trion is bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' As for positive trion, HX+ it is bound already in the parabolic approximation and our estimates show that it remains bound in the presence of non-parabolic contributions to the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Discussion of the results Let us now briefly discuss the obtained results in view of experimental data reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For rough es- timates we note that in TMDC monolayers the valence band hole and the conduction band (cb) electron effec- tive masses are about the same, mh ≈ m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='10 µ2/µ1 = - 3 µ2/µ1 = - 1 µ2/µ1 = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 µ2/µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='5 µ2/µ1 = 1 µ2/µ1 = 3 HX− binding energy Eb,HX−/E r0/a = 1 B* Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' High-lying trion binding energy as a function of non-parabolicity parameter B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 8 effective mass in cb+2 conduction band has a similar ab- solute value, but it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Estimates based on the DFT approach presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [20] show that for WSe2 monolayers |m2| ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='46m0 > mh ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='36m0 with m0 being the free-electron mass, making the reduced mass µ2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='66m0 > 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Additional evidence for |m2| > mh follows from the strong phonon progression of HX observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [20] indicating that the translational mass of the high-lyign exciton is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Thus, neutral high-lying exciton, HX, is bound even if Bk4 terms are neglected in the cb + 2 dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' For such parameters, however, m∗ > 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (23) (for m1 ≈ mh and reason- able χ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='2 the m∗ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Hence, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (23) the HX− is not bound in the parabolic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Thus, we come to the conclusion that Bk4 contribution should be sizeable to make HX− bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Note that experimentally observed HX− binding ener- gies are ≈ 43 meV for WSe2 monolayer and ≈ 21 meV for MoSe2 monolayer [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In the former case it is slightly larger than the band-edge trion, X−, binding energy, while in the latter case it is slightly smaller than that of X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Depending on µ2/µ1 and B the HX− binding en- ergy can be on the order of 10% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 25% of the HX bind- ing energy, thus, somewhat increased values of Eb,HX− compared to Eb,X− can be related to (i) enhancement of the Eb,HX due to rather large µ2/µ1 ≈ 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 10 for the estimated parameters and (ii) to large B∗ where, as men- tionned above, the trion-to-exciton binding energy ratio turns out to be quite large, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this theoretical paper we abstain from further anal- ysis of the experimental data and more detailed com- parison of the calculations with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The main reason is that the dispersion in cb + 2 band is quite com- plicated [20, 22] and contains, in addition to simplified Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' (2), anisotropic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Also, experimental data on high-lying excitons binding energy are not available to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Still our theoretical results in combination with experimental data [21] indicate im- portance of the non-parabolic terms in the high-lying electron dispersion for formation of the three-particle Coulomb complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK To conclude, we have developed the theory of high- lying excitons and trions in two-dimensional semiconduc- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Such Coulomb-bound complexes involve one elec- tron in the excited conduction band with the negative ef- fective mass and a non-parabolic dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We have de- veloped (i) variational method for calculating such com- plexes with simple and physically justified trial functions and (ii) the efficient and accurate numerical approach based on decomposition of the wavefunctions of Gaus- sians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' We have demonstrated the importance of the band non-parabolicity for formation of the high-lying excitons and trions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In particular, for negative reduced mass the presence of k4 terms in the high-lying electron dispersion makes exciton bound and strongly enhances the range of stability of the negatively charged high-lying trions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Our estimates show that the high-lying trion binding energies can be in the range of 10% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 30% of the high-lying ex- citon binding energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=', on the order of several tens of meV for transition-metal dichalcogenide monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' The developed theory is not limited to the mono- layer transition-metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In several other material platforms, including few-layer Ga- and In- monoseledines and monosulfides [48] the dispersion with a ring of extrema in the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Similar situa- tion is probably realized in two-dimensional hexagonal BN [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this respect, we can also mention bulk GaP with the camel’s back dispersion [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Impor- tantly, dispersion engineering, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=', in moire lattices can be used to realize the non-parabolic dispersion with an extremum ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' In this regard, developed theoretical ap- proaches will be helpful for studying the fundamental quasiparticles, excitons and trions, in a wide range of material systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors are grateful to Kai-Qiang Lin, Jonas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Ziegler, and Alexey Chernikov for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Kolobov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} +page_content=' Pikhtin, Indi- rect exciton dispersion in III-V semiconductors: ‘Camel’s back’ in GaP, Solid State Communications 30, 631 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FJT4oBgHgl3EQf6C07/content/2301.11672v1.pdf'} diff --git a/X9FOT4oBgHgl3EQf9TQp/content/tmp_files/2301.12969v1.pdf.txt b/X9FOT4oBgHgl3EQf9TQp/content/tmp_files/2301.12969v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..096890fdb45484828d0b55c7b5bf1f6a0164f3fd --- /dev/null +++ b/X9FOT4oBgHgl3EQf9TQp/content/tmp_files/2301.12969v1.pdf.txt @@ -0,0 +1,2495 @@ +Using n-akṣaras to model Sanskrit +& Sanskrit-adjacent texts +Presented at Perspectives of Digital Humanities in the Field of +Buddhist Studies, Universität Hamburg, 13 January 2023. +Revised 25 January 2023. +Charles Li + +Centre nationale de la recherche scientifique +Paris, France +Abstract +Despite — or perhaps because of — their simplicity, n-grams, or contiguous +sequences of tokens, have been used with great success in computational linguistics +since their introduction in the late 20 +th century. Recast as k-mers, or contiguous +sequences of monomers, they have also found applications in computational biology. +When applied to the analysis of texts, n-grams usually take the form of sequences +of words. But if we try to apply this model to the analysis of Sanskrit texts, we are +faced with the arduous task of, firstly, resolving sandhi to split a phrase into words, +and, secondly, splitting long compounds into their components. This paper presents +a simpler method of tokenizing a Sanskrit text for n-grams, by using n-akṣaras, or +contiguous sequences of akṣaras. This model reduces the need for sandhi resolution, +making it much easier to use on raw text. It is also possible to use this model on +Sanskrit-adjacent texts, e.g., a Tamil commentary on a Sanskrit text. As a test +case, the commentaries on Amarakoṣa 1.0.1 have been modelled as n-akṣaras, +showing patterns of text reuse across ten centuries and nine languages. Some initial +observations are made concerning Buddhist commentarial practices. + +This paper is an outcome of the Texts Surrounding Texts project of the CNRS, in collaboration with the +Bibliothèque nationale de France, and jointly funded by the ANR & DFG (FRAL 2018). + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI Introduction +N-grams were introduced in 1997, as a strategy for efficiently clustering documents on the In- +ternet. Using the AltaVista search engine, researchers were able to perform syntactic cluster- +ing on “every document on the World Wide Web” at the time. In that paper, n-grams are +called “shingles”: +We view each document as a sequence of words, and start by lexically analyzing it +into a canonical sequence of tokens…. A contiguous subsequence contained in [a +document] is called a shingle. +1 +The example given in the paper is the sentence “a rose is a rose is a rose.” Split into 4-grams, +or contiguous sequences of 4 words, this sentence yields the set “a rose is a,” “rose is a rose,” “is +a rose is.” This set can then be compared to other sentences, and the resemblance between +them can be quantified by the difference between the sets. +This simple method of modelling a text — as subsequences of words — has been applied +widely, not only on texts, but also, for example, on DNA. In fact, any phenomenon that has +some extension in space and/or time can be modelled in this way. +2 The crucial question in +each case is how to tokenize — how do you take a continuous reality and cast it into discrete +units that will form the basis for n-grams? +Where does meaning lie? +When we think of a text, we usually think of it as consisting of words, phrases, paragraphs, +verses, etc. But it is important not to forget that every text also has a material basis — a text +equally consists of incisions in a palm leaf, ink on a page, or sound waves propagating through +air. These aspects, too, can be tokenized, and they can yield meaningful insights about a text +and how it is transmitted — for example, in the case of palaeographic analysis or speech tone +analysis. +1 +Broder et al. 1997, 1158. +2 +For a further example, see Huang et al., 2012, on n-grams used to cluster heartbeat signals. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI We can consider a text as a continuum — spanning its material basis, on the one end, and its +linguistic or conceptual idealization, on the other. Focusing on any single point in the con- +tinuum implies some trade-off, some meaning lost and gained. For example, if we analyze a +written text as words, with a normalized spelling, we ignore different orthographies that are +used in different regions. On the other hand, if we analyze a text as a sequence of marks, per- +haps for the purpose of handwriting analysis, we largely ignore its linguistic content. For +Sanskrit texts, akṣaras are a good compromise between materiality and linguistics — they are +a good representation of the sequence of marks on a written page while also giving some sense +of semantics and syntax. +Akṣaras vs. words +Akṣaras are well-defined and easy to tokenize. An akṣara is a single syllable ending in a vowel, +anusvāra, or visarga. For example, the phrase akṣaraḥ kartā is tokenized as +a kṣa raḥ ka rtā +There is no need for splitting words and compounds, and there is no need for stemming. This +is especially important in cases where word splitting is undesirable, such as when the text is +(intentionally) ambiguous, damaged or difficult to decipher. For example, take this manuscript +fragment, where the commentary, written in the top and bottom margins, has been occluded +by damage. +Characters, ligatures +Marks, shapes, sounds +Akṣaras +Word stems +Phrases +Materials +Words + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI In a case such as this, where much of the text is lost, it can be difficult to determine where a +phrase or even a word begins and ends. In order to split the text as words, it would be neces- +sary to speculatively emend the text; if the emendation is wrong, then it will become im- +possible to match this text with similar passages in other manuscripts. Wrong emendations +can also lead to compounding errors, bringing into being phantom words that may never have +existed previously. +3 But if we tokenize this text as akṣaras, then we don’t need to emend at +all: +[...]śritvabhāvaka[…] → śri tva bhā va ka +Akṣaras vs. characters +It is possible to analyze akṣaras further into characters, i.e., as consonants and vowels. But +n-grams composed of characters are less meaningful than n-grams composed of akṣaras. For +example, compare these 2-grams of the previous example phrase, akṣaraḥ kartā, first as char- +acters and then as akṣaras. +3 +See Li 2022 for an example in Buddhist Hybrid Sanskrit. +Figure 1: Endangered Archives Project, Santipur Bangiya Puran Parishad. EAP781/1/1/1061, +folio 1r. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI a k ṣ a r a ḥ k a r t ā → ak, kṣ, ṣa, ar, ra, aḥ, ḥk, ka, rt, tā + a kṣa raḥ ka rtā → akṣa, kṣaraḥ, raḥka, kartā +With characters, most of our n-grams don’t convey much information; they are extremely +common. In order to get meaningful results, we would need much longer sequences. But with +akṣaras, even sequences composed of two tokens already capture single words or recognizable +parts of words. +Moreover, akṣaras more closely match the written sequence of a text than characters do, even +across the many different scripts used to write Sanskrit. For example, if we split the akṣara kē +into (Romanized) characters, we get the sequence k ē. But in Indic writing systems, each con- +sonant has an inherent vowel that needs to be suppressed with a virāma sign in order to ex- +press the consonant alone. And in many scripts, the vowel sign ē is written before the conson- +ant k. +Malayalam: kē → + +കേ� +k ē → + + +�് ഏ +Kannada: kē → + +ಕೇ� +k ē → + +ಕ್ ಏ +By tokenizing a Sanskrit text as characters, we may end up misrepresenting the actual written +sequence of the text, depending on the script used to express it. +Commentaries on Amarakoṣa 1.0.1 +As a test case, a corpus composed of commentaries on the first verse of the Amarakoṣa was +modelled as n-akṣaras. The Amarakoṣa, or Nāmaliṅgānuśāsana, is a well-known Sanskrit lex- +icon by Amarasiṃha of uncertain date, but it is certainly the most widespread Sanskrit lex- +icon and perhaps the most widespread Sanskrit text in existence. It is also likely the most +commented-upon Sanskrit text; it counts at least 80 known commentaries, +4 spanning from the +10 +th to the 21 +st centuries, not including anonymous and marginal commentaries. Perhaps one +4 +Vogel 2015, 25. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI reason why the text attracted so much commentary is the ambiguity of its first, benedictory +verse: +yasya jñānadayāsindhor agādhasyānaghā guṇāḥ | +sevyatām akṣayo dhīrāḥ sa śriye cāmṛtāya ca || 1.0.1 || +Hey wise guys! For glory and for immortality, you should worship the one who is an +unfathomable ocean of knowledge and compassion, whose qualities are faultless. +Amarasiṃha was almost certainly a Buddhist, and here, the one who should be worshipped al- +most certainly refers to the Buddha. But, because no deity is explicitly named, this verse has +led to over a thousand years of commentarial speculation, interpretation, and hermeneutics: +ihānukto 'pi buddho viśeṣaṇena spaṣṭaṃ pratīyate iti +Here, even though unsaid, the Buddha is obviously understood through his +qualities. +Raghunātha Cakravartin (17th c.) +yady api śrīmadamarasiṃho buddhamatānuyāyī… tathāpi… śivasambandhivyākhyā- +naṃ naḥ sutarāṃ rocate +Even if Amarasiṃha was a follower of Buddhism, still an explanation relating to +Śiva would please us very much. +Brahmānanda Tripāṭhin (20th c.) +svāmī tu jinam anusmṛtyeti… āha | tan na, jinavācakapadasyātrādarśanāt +But [Kṣīra]svāmin said, “having memorialized the Jina.” Not so, because a word +expressing “Jina” does not appear in the verse. +Bhānuji Dīkṣita (17th c.) quoting Kṣīrasvāmin (11th c.) +he dhīra saḥ aḥ viṣṇuḥ sevyatāṃ | akāro viṣṇuḥ +Oh wise one! He, aḥ, or Viṣṇu, should be worshipped. a is Viṣṇu. +anonymous marginal commentary, Shantipur Bangiya Puran Parishad MS A482 + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI These commentaries are highly intertextual — they reuse and rephrase passages from one an- +ther, quote one another, and debate different interpretations of the verse. Many of them are +also very inventive — the last excerpt, for example, divides the word dhīrāḥ into dhīra and aḥ. +In fact, many commentators split the verse in different ways in order to obtain different mean- +ings; they view the verse as a sequence of akṣaras that can be freely partitioned in different +places. If we take the verse and split it into a canonical sequence of words, then that would +only capture one out of its many possible meanings — we would be missing other interpreta- +tions of it as given by commentators over the centuries. +Document similarity using n-akṣaras +By modelling the commentaries as n-akṣaras, we can quantify their similarity. For example, +take these two phrases from two different commentaries: +ihānukto 'pi buddho viśeṣaṇena spaṣṭaṃ pratīyate +atrānukto pi budho viśeṣeṇaiḥ sūcayati +After normalizing the orthography — for features such as consonant gemination, e.g., dho vs. +ddho +5 — we can compare these two phrases as sets of 4-akṣaras: +ihānukto, hānukto'pi, nukto'pibu, kto'pibuddho, pibuddhovi, buddhoviśe… +atrānukto, trānuktopi, nuktopibu, ktopibudho, pibudhovi, budhoviśe… +There are a number of similarity metrics that can then be used with these results; the Jaccard +index, +6 for example, would quantify the similarity between these two sets as 0.5 (4 items in +common / 8 unique items total). +5 +On strategies for normalizing Sanskrit, see Li 2017. +6 +Jaccard 1912. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI n-akṣaras across languages +Although the majority of commentaries on the Amarakoṣa are written in Sanskrit, many are +written in other languages, such as Hindi, Newar, Tamil, etc. But since they often take inspir- +ation from Sanskrit commentaries, or even quote them outright, there is a high degree of liter- +al intertextuality between them that can be detected using n-akṣaras. For example, take these +two passages from two commentaries in different languages: +Marathi: yasya jyā parameśvarāce → ya sya jyā pa ra me śva rā ce +Newar: amo parameśvara → a mo pa ra me śva ra +Here, we are able to find a 4-akṣara match since the same Sanskrit word has been used in +both commentaries, even if it is inflected differently. But we can do better by implementing +some fuzzy matching: +Sanskrit: śriye saṃpataye → śri ye saṃ pa ta ye → śri ye saṃ pa ta ye + normalization ignore n-2 vowels +Malayalam: śrī = sanpattŭ → śrī saṃ pa tŭ → śrī saṃ pa tŭ +Another strategy that we could employ is skip-grams, +7 i.e., we can skip akṣaras. This is espe- +cially useful because the languages we are working with feature inflectional suffixes; if we skip +these suffixes, we can match sequences of word stems that are common across languages, +without having to do any formal stemming: +Hindi: satya, śauca, dayā, kṣāṃti, tyāga ādi + skip 1 akṣara +Sanskrit: satyaṃ śaucaṃ dayā kṣāṃtiḥ tyāgaḥ +7 +Guthrie et al. 2006. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI Text reuse in Amarakoṣa commentaries +So far, 105 commentaries +8 on Amarakoṣa 1.0.1, in 9 languages, have been collected and mod- +elled as sets of 2-, 3-, 4-, and 5-akṣaras. Using the Dice coefficient +9 as a measure of similarity, a +minimum spanning tree was created, revealing patterns of text reuse. +From this graph, some obvious clusters emerge — manuscripts of the same text are grouped +very closely together. But the boundaries of these clusters are not always obvious. Especially +8 +This count does not distinguish between “texts” and “witnesses”; even different versions of the same “text” are +considered distinct texts themselves. See below. +9 +Dice 1945. +Figure 2: Minimum spanning tree, using the Dice coefficient for edge weights. +An interactive version of this figure can be found at chchch.github.io/amarakosa + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI in the case of anonymous commentaries, it is not always clear whether the intent was to copy +an existing commentary or to create a new commentary based on older material. Scholars of- +ten distinguish between a text and witnesses of that text, as if there is one model which is +simply copied imperfectly; but in the case of these commentaries, it can be very difficult to +make that distinction, to know where one text ends and another begins. Perhaps a more ac- +curate way to describe these clusters — rather than as witnesses of a distinct text — is as +overlapping families of texts, without any particular centre considered as the urtext. +Geographic regions and languages also create their own clusters — unsurprisingly, Tamil com- +mentaries form a branch, connected to commentaries in other southern languages, while Hindi +commentaries form a different branch along with the Nepali commentary, which seems to be a +translation of a Hindi commentary. However, what seems at first to be a distinction between +Sanskrit and non-Sanskrit commentaries may rather be a distinction between long, erudite +commentaries and short, concise commentaries. While many of the Sanskrit commentaries are +full of grammatical derivations and multiple interpretations, most of the commentaries in oth- +er languages are aimed at children, bearing titles such as Bālapriyā or Bālabodhinī, and they +feature simple, common glosses. +But there are also many interesting outliers. RE22704, a Tamil and Telugu commentary pre- +served in a palm-leaf manuscript from the French Institute of Pondicherry, is conspicuously +not connected to other Tamil commentaries but to PpEd, the Padapārijāta, a Sanskrit com- +mentary. RE22704 is an unusually long and detailed commentary, and, in fact, it quotes the +Padapārijāta — this is something that was already pointed out in a study of this manuscript +by Giovanni Ciotti and R. Sathyanarayan, +10 and our quantitative analysis nicely mirrors their +scholarly conclusions. The Padapārijāta itself is also difficult to place — although its author, +Mallinātha, hails from south India, he quotes widely from both northern and southern Indian +texts, brahmanical and Buddhist alike. +11 +10 Ciotti & Sathyanarayan 2020, 455-457. +11 Ramanathan 1971, xlvi-xlvii. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI Buddhisms +What is not apparent in the graph are clusters based on religion. Commentators did not limit +themselves to only reading and quoting from authors of their same religious affiliation. +Moreover, two commentators from what seems like the same religion may have radically differ- +ent interpretations of the text. This is perhaps most evident when comparing two Buddhist +commentaries: JEd, the 10 +th-century Sanskrit commentary of Jātarūpa, and KA2b, a Newar +commentary preserved in a 16 +th-century manuscript in Kathmandu. +12 +Both of these commentaries interpret the person who should be worshipped in Amarakoṣa +1.0.1 to be the Buddha, but the way in which the Buddha is described is completely different. +Jātarūpa uses fairly technical language, describing the Buddha as paramakāruṇika and vima- +labuddhi. On the other hand, KA2b uses the unusual phrase parameśvara tathāgatatvaṃ. +These two commentaries have very little in common; at the level of 3-akṣaras, there are only a +handful of matches, reflecting glosses that are common to many other commentaries. In fact, +KA2b is less similar to JEd (Dice coefficient: 0.018) than it is to a Tamil Vaiṣṇava comment- +12 On JEd, see Pant 2000. On KA2b, see the description of A2b at newari.net/source.html. +Figure 3: Comparison of JEd and KA2b, with 3-akṣara matches highlighted. The online +version of this figure can be reached at chchch.github.io/amarakosa: click the node JEd and +then the node KA2b. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI ary that use parameśvara as an epithet of Viṣṇu (RE37121, Dice coefficient: 0.029). JEd and +KA2b are each well-connected within their own geographical and cultural milieu — Jātarūpa, +who may have been from Bengal, is quoted by other Bengali commentators, and KA2b is very +similar to the other Newar commentaries — but they are practically unrelated to one another. +Going further +As has been suggested by other scholars at the conference, this study on text reuse in Amara- +koṣa commentaries is confined to a small corpus, and it is a corpus in which a great deal of +similarity between texts would already be expected. Further studies on a larger corpus may re- +veal surprising and unexpected connections between texts. In addition, a more formal compar- +ison between character-, akṣara-, and word-level tokenization would be welcome, with per- +formance metrics for different tasks and situations. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI Bibliography +British Library. 2014. “Nāmaliṅgānuśāsana or Amarakośa (with gloss).” Collection of +manuscripts digitized from Shantipur Bangiya Puran Parishad, Endangered Archives +Programme. doi:10.15130/EAP781 eap.bl.uk/archive-file/EAP781-1-1-1061 +Broder, Andrei Z. et al. 1997. “Syntactic clustering of the Web.” Computer Networks and +ISDN Systems 29: 1157-1166. +Ciotti, Giovanni & Sathyanarayan, R. 2020. “A multilingual commentary of the first verse of +the Nāmaliṅgānuśāsana as found in ms. IFP RE22704.” In Anandakichenin, S. & +D’Avella, V., eds., The Commentary Idioms of the Tamil Learned Traditions: 443-489. +Pondichéry: École française d’Extrême-Orient. +Dice, Lee R. 1945. “Measures of the Amount of Ecologic Association Between Species.” Ecology +26(3): 297–302. doi:10.2307/1932409 + +Guthrie, David et al. 2006. “A Closer Look at Skip-gram Modelling.” Proceedings of the Fifth +International Conference on Language Resources and Evaluation (LREC’06), Genoa, +Italy: 1222-1225. +Huang, Yu-Chen et al. 2012. “Using n-gram analysis to cluster heartbeat signals.” BMC Medic- +al Informatics and Decision Making 12(64). doi:10.1186/1472-6947-12-64 +Jaccard, Paul. 1912. “The Distribution of the Flora in the Alpine Zone.” New Phytologist +11(2): 37–50. doi:10.1111/j.1469-8137.1912.tb05611.x + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI Li, Charles. 2017. “Critical diplomatic editing: Applying text-critical principles as algorithms.” +Advances in Digital Scholarly Editing, ed. P. Boot et al. Leiden: Sidestone Press. +Li, Charles. 2022. “Akālaka: a lexical phantom in Buddhist Hybrid Sanskrit.” The World of the +Orient 4: 203-210. doi:10.15407/orientw2022.04.203 +Nepal Bhasha Dictionary Committee. “Description of Source manuscripts of Amarakośas.” Ne- +wari Lexicon based on the Amarakosa. newari.net/source.html +Pant, Mahes Raj. 2000. Jātarūpa’s Commentary on the Amarakoṣa. Delhi: Motilal Banarsi- +dass. +Ramanathan, A. A. 1971. Amarakośa [1] with the unpublished south Indian commentaries. +Madras: Adyar Library and Research Centre. +Vogel, Claus. 2015. Indian Lexicography. Revised and Enlarged edition. München: P. Kirch- +heim. + +：●Sanskrit +●Hindi +•Kannada +●Nepali +●Newar +·Malayalam +•Marathi +• Tamil +.TeluguD3351 +RPEd +D3347 +BhoEd +GKH2 +A122-218 +RDEd +VEd +BKEd +MPEd +. +DhaEd +A126-307 +A122-223 +B626 +VSEd +ABEdM +ABEdR +. +KK933 +B623 +SA145 +GKHI +. +D1205 +S242 +S591 +TEd +A131-466 +S427 +ABEdT +S344 +S161 +EAP-886-1-21 +. +P743 +SAg8: +B1379 +TEd +D3377 +B612m +. +KA324-10 +R483 +JEd +PcEd +B612r +W155 +. +KA328-19 +. +B612 +. +B619 +ABEdPc +B549 +KA320-13 +KA324-1 +. +R824 +PpEd +B1379m +KkEd +KA322-4b +. +. +KB2fh-2 +RE22704 +RLEd +EAP584-1-88 +P1481 +KA322-4a +KAg24-9 +NcEd +KA2b +KA7 +KA322-5 +BPEd +MEd +KA5 +SAg82 +MsEd +PEd +KA6 +PvEdC +EAP248-1-81 +LBS322 +GBPEd +PvG +KAaa +PvEd +CAddi6g8 +EO1272 +AEd +A125-266 +RE37121 +. +EAP584-5-300 +B614 +RE45807 +Ad70820 +KEd +Dn83 +B620 +. +RE32661 +Ad71010 +ORI3317 +. +RE33660 +JM278 +Ad6g312 +KPEd +Ad72614 +C43 +RSEd +EAPs84-2-29 +.2-aksaras +3-aksaras +4-aksaras +5-aksaras +JEd +KA2b +Jatarupa'sCommentaryontheAmarakosa(ed.Pant2ooo) +Asha Archives (Newari.net) +yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh +samudrasya +gonasa atmasa jnana nom daya nom mahasamudratvam them +paripiurnna +yana +agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa +conanana nihyapa jusyam vanamgva (guna?)thvala amo paramesvara tathagatatvam +madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad +aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka +anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah +sakalasyam sevarapa gvana +samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari +po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- +tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- +jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- +dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- +sah trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki si- +ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda +gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva +ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate jnanadayakrta +ca sa tasyaparipirnateti trtiya tat krtarthena...iti samasah |trtiya ca kartrkaranayoh... +ity anenaiva lI \ No newline at end of file diff --git a/X9FOT4oBgHgl3EQf9TQp/content/tmp_files/load_file.txt b/X9FOT4oBgHgl3EQf9TQp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b5486a4a59317051e432bfa8aaf478dfc21c30e --- /dev/null +++ b/X9FOT4oBgHgl3EQf9TQp/content/tmp_files/load_file.txt @@ -0,0 +1,795 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf,len=794 +page_content='Using n-akṣaras to model Sanskrit & Sanskrit-adjacent texts Presented at Perspectives of Digital Humanities in the Field of Buddhist Studies, Universität Hamburg, 13 January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Revised 25 January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Charles Li Centre nationale de la recherche scientifique Paris, France Abstract Despite — or perhaps because of — their simplicity, n-grams, or contiguous sequences of tokens, have been used with great success in computational linguistics since their introduction in the late 20 th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Recast as k-mers, or contiguous sequences of monomers, they have also found applications in computational biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' When applied to the analysis of texts, n-grams usually take the form of sequences of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But if we try to apply this model to the analysis of Sanskrit texts, we are faced with the arduous task of, firstly, resolving sandhi to split a phrase into words, and, secondly, splitting long compounds into their components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This paper presents a simpler method of tokenizing a Sanskrit text for n-grams, by using n-akṣaras, or contiguous sequences of akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This model reduces the need for sandhi resolution, making it much easier to use on raw text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' It is also possible to use this model on Sanskrit-adjacent texts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=', a Tamil commentary on a Sanskrit text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' As a test case, the commentaries on Amarakoṣa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1 have been modelled as n-akṣaras, showing patterns of text reuse across ten centuries and nine languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Some initial observations are made concerning Buddhist commentarial practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This paper is an outcome of the Texts Surrounding Texts project of the CNRS, in collaboration with the Bibliothèque nationale de France, and jointly funded by the ANR & DFG (FRAL 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': 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4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI Introduction N-grams were introduced in 1997, as a strategy for efficiently clustering documents on the In- ternet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Using the AltaVista search engine, researchers were able to perform syntactic cluster- ing on “every document on the World Wide Web” at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In that paper, n-grams are called “shingles”: We view each document as a sequence of words, and start by lexically analyzing it into a canonical sequence of tokens….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' A contiguous subsequence contained in [a document] is called a shingle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 1 The example given in the paper is the sentence “a rose is a rose is a rose.” Split into 4-grams, or contiguous sequences of 4 words, this sentence yields the set “a rose is a,” “rose is a rose,” “is a rose is.” This set can then be compared to other sentences, and the resemblance between them can be quantified by the difference between the sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This simple method of modelling a text — as subsequences of words — has been applied widely, not only on texts, but also, for example, on DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In fact, any phenomenon that has some extension in space and/or time can be modelled in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2 The crucial question in each case is how to tokenize — how do you take a continuous reality and cast it into discrete units that will form the basis for n-grams?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Where does meaning lie?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' When we think of a text, we usually think of it as consisting of words, phrases, paragraphs, verses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But it is important not to forget that every text also has a material basis — a text equally consists of incisions in a palm leaf, ink on a page, or sound waves propagating through air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' These aspects, too, can be tokenized, and they can yield meaningful insights about a text and how it is transmitted — for example, in the case of palaeographic analysis or speech tone analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 1 Broder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 1997, 1158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2 For a further example, see Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=', 2012, on n-grams used to cluster heartbeat signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': 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LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI We can consider a text as a continuum — spanning its material basis, on the one end, and its linguistic or conceptual idealization, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Focusing on any single point in the con- tinuum implies some trade-off, some meaning lost and gained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For example, if we analyze a written text as words, with a normalized spelling, we ignore different orthographies that are used in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' On the other hand, if we analyze a text as a sequence of marks, per- haps for the purpose of handwriting analysis, we largely ignore its linguistic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For Sanskrit texts, akṣaras are a good compromise between materiality and linguistics — they are a good representation of the sequence of marks on a written page while also giving some sense of semantics and syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Akṣaras vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' words Akṣaras are well-defined and easy to tokenize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' An akṣara is a single syllable ending in a vowel, anusvāra, or visarga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For example, the phrase akṣaraḥ kartā is tokenized as a kṣa raḥ ka rtā There is no need for splitting words and compounds, and there is no need for stemming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This is especially important in cases where word splitting is undesirable, such as when the text is (intentionally) ambiguous, damaged or difficult to decipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For example, take this manuscript fragment, where the commentary, written in the top and bottom margins, has been occluded by damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Characters, ligatures Marks, shapes, sounds Akṣaras Word stems Phrases Materials Words ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI In a case such as this, where much of the text is lost, it can be difficult to determine where a phrase or even a word begins and ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In order to split the text as words, it would be neces- sary to speculatively emend the text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' if the emendation is wrong, then it will become im- possible to match this text with similar passages in other manuscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Wrong emendations can also lead to compounding errors, bringing into being phantom words that may never have existed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 3 But if we tokenize this text as akṣaras, then we don’t need to emend at all: [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=']śritvabhāvaka[…] → śri tva bhā va ka Akṣaras vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' characters It is possible to analyze akṣaras further into characters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=', as consonants and vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But n-grams composed of characters are less meaningful than n-grams composed of akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For example, compare these 2-grams of the previous example phrase, akṣaraḥ kartā, first as char- acters and then as akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 3 See Li 2022 for an example in Buddhist Hybrid Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Figure 1: Endangered Archives Project, Santipur Bangiya Puran Parishad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP781/1/1/1061, folio 1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': 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LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI a k ṣ a r a ḥ k a r t ā → ak, kṣ, ṣa, ar, ra, aḥ, ḥk, ka, rt, tā a kṣa raḥ ka rtā → akṣa, kṣaraḥ, raḥka, kartā With characters, most of our n-grams don’t convey much information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' they are extremely common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In order to get meaningful results, we would need much longer sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But with akṣaras, even sequences composed of two tokens already capture single words or recognizable parts of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Moreover, akṣaras more closely match the written sequence of a text than characters do, even across the many different scripts used to write Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For example, if we split the akṣara kē into (Romanized) characters, we get the sequence k ē.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But in Indic writing systems, each con- sonant has an inherent vowel that needs to be suppressed with a virāma sign in order to ex- press the consonant alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' And in many scripts, the vowel sign ē is written before the conson- ant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Malayalam: kē → കേ� k ē → �് ഏ Kannada: kē → ಕೇ� k ē → ಕ್ ಏ By tokenizing a Sanskrit text as characters, we may end up misrepresenting the actual written sequence of the text, depending on the script used to express it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Commentaries on Amarakoṣa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1 As a test case, a corpus composed of commentaries on the first verse of the Amarakoṣa was modelled as n-akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' The Amarakoṣa, or Nāmaliṅgānuśāsana, is a well-known Sanskrit lex- icon by Amarasiṃha of uncertain date, but it is certainly the most widespread Sanskrit lex- icon and perhaps the most widespread Sanskrit text in existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' It is also likely the most commented-upon Sanskrit text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' it counts at least 80 known commentaries, 4 spanning from the 10 th to the 21 st centuries, not including anonymous and marginal commentaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Perhaps one 4 Vogel 2015, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI reason why the text attracted so much commentary is the ambiguity of its first, benedictory verse: yasya jñānadayāsindhor agādhasyānaghā guṇāḥ | sevyatām akṣayo dhīrāḥ sa śriye cāmṛtāya ca || 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1 || Hey wise guys!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For glory and for immortality, you should worship the one who is an unfathomable ocean of knowledge and compassion, whose qualities are faultless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Amarasiṃha was almost certainly a Buddhist, and here, the one who should be worshipped al- most certainly refers to the Buddha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=" But, because no deity is explicitly named, this verse has led to over a thousand years of commentarial speculation, interpretation, and hermeneutics: ihānukto 'pi buddho viśeṣaṇena spaṣṭaṃ pratīyate iti Here, even though unsaid, the Buddha is obviously understood through his qualities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Raghunātha Cakravartin (17th c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=') yady api śrīmadamarasiṃho buddhamatānuyāyī… tathāpi… śivasambandhivyākhyā- naṃ naḥ sutarāṃ rocate Even if Amarasiṃha was a follower of Buddhism, still an explanation relating to Śiva would please us very much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Brahmānanda Tripāṭhin (20th c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=') svāmī tu jinam anusmṛtyeti… āha | tan na, jinavācakapadasyātrādarśanāt But [Kṣīra]svāmin said, “having memorialized the Jina.” Not so, because a word expressing “Jina” does not appear in the verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Bhānuji Dīkṣita (17th c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=') quoting Kṣīrasvāmin (11th c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=') he dhīra saḥ aḥ viṣṇuḥ sevyatāṃ | akāro viṣṇuḥ Oh wise one!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' He, aḥ, or Viṣṇu, should be worshipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' a is Viṣṇu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' anonymous marginal commentary, Shantipur Bangiya Puran Parishad MS A482 ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI These commentaries are highly intertextual — they reuse and rephrase passages from one an- ther, quote one another, and debate different interpretations of the verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Many of them are also very inventive — the last excerpt, for example, divides the word dhīrāḥ into dhīra and aḥ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In fact, many commentators split the verse in different ways in order to obtain different mean- ings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' they view the verse as a sequence of akṣaras that can be freely partitioned in different places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' If we take the verse and split it into a canonical sequence of words, then that would only capture one out of its many possible meanings — we would be missing other interpreta- tions of it as given by commentators over the centuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Document similarity using n-akṣaras By modelling the commentaries as n-akṣaras, we can quantify their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=" For example, take these two phrases from two different commentaries: ihānukto 'pi buddho viśeṣaṇena spaṣṭaṃ pratīyate atrānukto pi budho viśeṣeṇaiḥ sūcayati After normalizing the orthography — for features such as consonant gemination, e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=', dho vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=" ddho 5 — we can compare these two phrases as sets of 4-akṣaras: ihānukto, hānukto'pi, nukto'pibu, kto'pibuddho, pibuddhovi, buddhoviśe… atrānukto, trānuktopi, nuktopibu, ktopibudho, pibudhovi, budhoviśe… There are a number of similarity metrics that can then be used with these results;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' the Jaccard index, 6 for example, would quantify the similarity between these two sets as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='5 (4 items in common / 8 unique items total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 5 On strategies for normalizing Sanskrit, see Li 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 6 Jaccard 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI n-akṣaras across languages Although the majority of commentaries on the Amarakoṣa are written in Sanskrit, many are written in other languages, such as Hindi, Newar, Tamil, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But since they often take inspir- ation from Sanskrit commentaries, or even quote them outright, there is a high degree of liter- al intertextuality between them that can be detected using n-akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' For example, take these two passages from two commentaries in different languages: Marathi: yasya jyā parameśvarāce → ya sya jyā pa ra me śva rā ce Newar: amo parameśvara → a mo pa ra me śva ra Here, we are able to find a 4-akṣara match since the same Sanskrit word has been used in both commentaries, even if it is inflected differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But we can do better by implementing some fuzzy matching: Sanskrit: śriye saṃpataye → śri ye saṃ pa ta ye → śri ye saṃ pa ta ye normalization ignore n-2 vowels Malayalam: śrī = sanpattŭ → śrī saṃ pa tŭ → śrī saṃ pa tŭ Another strategy that we could employ is skip-grams, 7 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=', we can skip akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This is espe- cially useful because the languages we are working with feature inflectional suffixes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' if we skip these suffixes, we can match sequences of word stems that are common across languages, without having to do any formal stemming: Hindi: satya, śauca, dayā, kṣāṃti, tyāga ādi skip 1 akṣara Sanskrit: satyaṃ śaucaṃ dayā kṣāṃtiḥ tyāgaḥ 7 Guthrie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI Text reuse in Amarakoṣa commentaries So far, 105 commentaries 8 on Amarakoṣa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1, in 9 languages, have been collected and mod- elled as sets of 2-, 3-, 4-, and 5-akṣaras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Using the Dice coefficient 9 as a measure of similarity, a minimum spanning tree was created, revealing patterns of text reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' From this graph, some obvious clusters emerge — manuscripts of the same text are grouped very closely together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But the boundaries of these clusters are not always obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Especially 8 This count does not distinguish between “texts” and “witnesses”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' even different versions of the same “text” are considered distinct texts themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' See below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 9 Dice 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Figure 2: Minimum spanning tree, using the Dice coefficient for edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' An interactive version of this figure can be found at chchch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='io/amarakosa ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI in the case of anonymous commentaries, it is not always clear whether the intent was to copy an existing commentary or to create a new commentary based on older material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Scholars of- ten distinguish between a text and witnesses of that text, as if there is one model which is simply copied imperfectly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' but in the case of these commentaries, it can be very difficult to make that distinction, to know where one text ends and another begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Perhaps a more ac- curate way to describe these clusters — rather than as witnesses of a distinct text — is as overlapping families of texts, without any particular centre considered as the urtext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Geographic regions and languages also create their own clusters — unsurprisingly, Tamil com- mentaries form a branch, connected to commentaries in other southern languages, while Hindi commentaries form a different branch along with the Nepali commentary, which seems to be a translation of a Hindi commentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' However, what seems at first to be a distinction between Sanskrit and non-Sanskrit commentaries may rather be a distinction between long, erudite commentaries and short, concise commentaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' While many of the Sanskrit commentaries are full of grammatical derivations and multiple interpretations, most of the commentaries in oth- er languages are aimed at children, bearing titles such as Bālapriyā or Bālabodhinī, and they feature simple, common glosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' But there are also many interesting outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE22704, a Tamil and Telugu commentary pre- served in a palm-leaf manuscript from the French Institute of Pondicherry, is conspicuously not connected to other Tamil commentaries but to PpEd, the Padapārijāta, a Sanskrit com- mentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE22704 is an unusually long and detailed commentary, and, in fact, it quotes the Padapārijāta — this is something that was already pointed out in a study of this manuscript by Giovanni Ciotti and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Sathyanarayan, 10 and our quantitative analysis nicely mirrors their scholarly conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' The Padapārijāta itself is also difficult to place — although its author, Mallinātha, hails from south India, he quotes widely from both northern and southern Indian texts, brahmanical and Buddhist alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 11 10 Ciotti & Sathyanarayan 2020, 455-457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 11 Ramanathan 1971, xlvi-xlvii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI Buddhisms What is not apparent in the graph are clusters based on religion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Commentators did not limit themselves to only reading and quoting from authors of their same religious affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Moreover, two commentators from what seems like the same religion may have radically differ- ent interpretations of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' This is perhaps most evident when comparing two Buddhist commentaries: JEd, the 10 th-century Sanskrit commentary of Jātarūpa, and KA2b, a Newar commentary preserved in a 16 th-century manuscript in Kathmandu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 12 Both of these commentaries interpret the person who should be worshipped in Amarakoṣa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1 to be the Buddha, but the way in which the Buddha is described is completely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Jātarūpa uses fairly technical language, describing the Buddha as paramakāruṇika and vima- labuddhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' On the other hand, KA2b uses the unusual phrase parameśvara tathāgatatvaṃ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' These two commentaries have very little in common;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' at the level of 3-akṣaras, there are only a handful of matches, reflecting glosses that are common to many other commentaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In fact, KA2b is less similar to JEd (Dice coefficient: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='018) than it is to a Tamil Vaiṣṇava comment- 12 On JEd, see Pant 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' On KA2b, see the description of A2b at newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net/source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Figure 3: Comparison of JEd and KA2b, with 3-akṣara matches highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' The online version of this figure can be reached at chchch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='io/amarakosa: click the node JEd and then the node KA2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI ary that use parameśvara as an epithet of Viṣṇu (RE37121, Dice coefficient: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='029).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' JEd and KA2b are each well-connected within their own geographical and cultural milieu — Jātarūpa, who may have been from Bengal, is quoted by other Bengali commentators, and KA2b is very similar to the other Newar commentaries — but they are practically unrelated to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Going further As has been suggested by other scholars at the conference, this study on text reuse in Amara- koṣa commentaries is confined to a small corpus, and it is a corpus in which a great deal of similarity between texts would already be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Further studies on a larger corpus may re- veal surprising and unexpected connections between texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' In addition, a more formal compar- ison between character-, akṣara-, and word-level tokenization would be welcome, with per- formance metrics for different tasks and situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA324-10 R483 JEd PcEd B612r W155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KA328-19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B612 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' B619 ABEdPc B549 KA320-13 KA324-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' R824 PpEd B1379m KkEd KA322-4b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KB2fh-2 RE22704 RLEd EAP584-1-88 P1481 KA322-4a KAg24-9 NcEd KA2b KA7 KA322-5 BPEd MEd KA5 SAg82 MsEd PEd KA6 PvEdC EAP248-1-81 LBS322 GBPEd PvG KAaa PvEd CAddi6g8 EO1272 AEd A125-266 RE37121 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' EAP584-5-300 B614 RE45807 Ad70820 KEd Dn83 B620 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE32661 Ad71010 ORI3317 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' RE33660 JM278 Ad6g312 KPEd Ad72614 C43 RSEd EAPs84-2-29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI Bibliography British Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' “Nāmaliṅgānuśāsana or Amarakośa (with gloss).” Collection of manuscripts digitized from Shantipur Bangiya Puran Parishad, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Huang, Yu-Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' “Using n-gram analysis to cluster heartbeat signals.” BMC Medic- al Informatics and Decision Making 12(64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1186/1472-6947-12-64 Jaccard, Paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' “The Distribution of the Flora in the Alpine Zone.” New Phytologist 11(2): 37–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1469-8137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='tb05611.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="2-aksaras 3-aksaras 4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI Li, Charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' “Critical diplomatic editing: Applying text-critical principles as algorithms.” Advances in Digital Scholarly Editing, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Boot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Leiden: Sidestone Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Li, Charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' “Akālaka: a lexical phantom in Buddhist Hybrid Sanskrit.” The World of the Orient 4: 203-210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='15407/orientw2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='203 Nepal Bhasha Dictionary Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' “Description of Source manuscripts of Amarakośas.” Ne- wari Lexicon based on the Amarakosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net/source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='html Pant, Mahes Raj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Jātarūpa’s Commentary on the Amarakoṣa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Delhi: Motilal Banarsi- dass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Ramanathan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Amarakośa [1] with the unpublished south Indian commentaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Madras: Adyar Library and Research Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Vogel, Claus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Indian Lexicography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Revised and Enlarged edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' München: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' Kirch- heim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ：●Sanskrit Hindi Kannada Nepali Newar Malayalam Marathi Tamil .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='TeluguD3351 RPEd D3347 BhoEd GKH2 A122-218 RDEd VEd BKEd MPEd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' DhaEd A126-307 A122-223 B626 VSEd ABEdM ABEdR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' KK933 B623 SA145 GKHI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' D1205 S242 S591 TEd A131-466 S427 ABEdT S344 S161 EAP-886-1-21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' P743 SAg8: B1379 TEd D3377 B612m .' 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4-aksaras 5-aksaras JEd KA2b Jatarupa'sCommentaryontheAmarakosa(ed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='Pant2ooo) Asha Archives (Newari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='net) yasyeti I yasyakhilavastuvisayena mahata jnanena dayaya ca sindhoh samudrasya gonasa atmasa jnana nom daya nom mahasamudratvam them paripiurnna yana agadhasyanyair anadhigatajnanadayaparatvad gambhirasya I guna maitriksamopasa conanana nihyapa jusyam vanamgva (guna?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' )thvala amo paramesvara tathagatatvam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="madayah I anaghah kamakrodhaprabhavadosarahitah I so'ksayo nityaparipirnatvad " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='aksayaavinasi jusyam vanamgva ihalokasa laksmih paralokasa moksa gava jnaniloka ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='anapacayo dayasindhutvat paramakaruniko bhogalalasena sriye sampattaye ca punah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sakalasyam sevarapa gvana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='samsaraklesabhayad amrtaya moksaya ca I jnanasindhutvad vimalabuddhir bodhari ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='po bhagavan bho dhirah sevyatamIyasya sindhor agadhasya gunah srijanmabhiumi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="tvapiyisaprabhavatvadayah I anagha avinasSinah I so 'ksayo nityaparipurnatvan maha- " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jalanidhih|sriye camrtaya piyusaya catadarthinam sevitum yogya evananucitavi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='dhanam etat | atra jnanadayabhyam sindhur iti vigrhya trtiya iti yogavibhagat sama- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='sah ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='trtiya ca trtiyavidhane prakrtyadibhya upasankhyanam ity aupasankhyaniki ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='si- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ndhusabdas ca sindhur iva pirnah I sindhur iti gauno 'yam I athava sarve sabda " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="gunasamudaye vartante kvacid ekadese'pi itiparipirnatam gunam abhidhaya tad eva " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content="ca nimittam upadaya sindhusabdo'yam tadvati devatavisese vartate " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='jnanadayakrta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='ca sa tasyaparipirnateti trtiya tat krtarthena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='iti samasah |trtiya ca kartrkaranayoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} +page_content=' ity anenaiva lI' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FOT4oBgHgl3EQf9TQp/content/2301.12969v1.pdf'} diff --git a/XdE2T4oBgHgl3EQfDwbQ/content/tmp_files/2301.03629v1.pdf.txt b/XdE2T4oBgHgl3EQfDwbQ/content/tmp_files/2301.03629v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..59626e43acba34d0420c1b38228abb506dd2bb08 --- /dev/null +++ b/XdE2T4oBgHgl3EQfDwbQ/content/tmp_files/2301.03629v1.pdf.txt @@ -0,0 +1,1511 @@ +Astronomy & Astrophysics manuscript no. SMACS0723 +©ESO 2023 +January 11, 2023 +On the correlation between dark matter, intracluster light and +globular cluster distribution in SMACS0723. +J.M. Diego1,⋆, M. Pascale2, B. Frye3, A. Zitrin4, T. Broadhurst5, 6, 7, G. Mahler8, 9, G.B. Caminha10, M. Jauzac8, 9, 11, 12, +Myung Gyoon Lee13, Jang Ho Bae13, In Sung Jang14, and Mireia Montes15, 16 +1 Instituto de Física de Cantabria (CSIC-UC). Avda. Los Castros s/n. 39005 Santander, Spain +2 Department of Astronomy, University of California, 501 Campbell Hall #3411, Berkeley, CA 94720, USA +3 Department of Astronomy/Steward Observatory, University of Arizona, 933 N Cherry Ave., Tucson, AZ 85721, USA +4 Physics Department, Ben-Gurion University of the Negev, P. O. Box 653, Be’er-Sheva, 8410501, Israel +5 Department of Physics, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain +6 DIPC, Basque Country UPV/EHU, E-48080 San Sebastian, Spain +7 Ikerbasque, Basque Foundation for Science, E-48011 Bilbao, Spain +8 Centre for Extragalactic Astronomy, Durham University, South Road, Durham DH1 3LE, UK +9 Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK +10 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany +11 Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa +12 School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Durban 4041, South +Africa +13 Astronomy Program, Department of Physics and Astronomy, SNUARC, Seoul National University, 1 Gwanak-ro, Gwanak-gu, +Seoul 08826, Republic of Korea +14 Department of Astronomy & Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA +15 Instituto de Astrofísica de Canarias, c/ Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain +16 Departamento de Astrofísica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, +January 11, 2023 +ABSTRACT +We present a free-form model of SMACS0723, the first cluster observed with JWST. This model makes no strong assumptions about +the distribution of mass (mostly dark matter) in the cluster and we use it to study the possible correlation between dark matter with +the intracluster light and distribution of globular clusters. To explore the uncertainty in mass modelling, we derive three lens models +based on spectroscopically confirmed systems and new candidate systems with redshifts predicted by the lens model derived from the +spectroscipic systems. We find that beyond the radius of influence of the BCG, the total mass does not trace the ICL, implying the need +for a dark component (dark matter). Two loop-like structures observed in the intracluster light do not have an obvious correspondence +with the total mass (mostly dark matter) distribution. The radial profiles of the ICL and the distribution of globular clusters are similar +to each other, but steeper than the profile of the lens model. More specifically, we find that the total mass is shallower by 1 dex in log +scale than both ICL and globular cluster profiles. This is in excellent agreement with N-body simulations of cold dark matter. +Key words. gravitational lensing – dark matter – cosmology +1. Introduction +After its launch on December 25th 2021, on July 11st 2022, the +first color image from James Webb Space Telescope (JWST) was +presented to the world. The image showed a view of the distant +infrared universe with a detail and depth never seen before at +these wavelengths. This image was centered on a massive galaxy +cluster at z = 0.39, SMACS J0723.3-7327 (or SMACS0723 +hereafter) acting as a powerful gravitational lens. This natural +lens magnifies the galaxies in the background. Some of these +background galaxies appear repeated several times in the image, +since they take different paths which are later refocused by the +gravitational lens into the JWST telescope. +In anticipation for these first JWST data, a lens model based +on HST data and listing the first few sets of multiple images and +some spectroscopic redshifts for them was posted on the arXiv +⋆ jdiego@ifca.unican.es +by Golubchik et al. (2022) on the same date (July 11th). The +JWST data became itself public on July 13th 2022, and just a +day after the data release, two papers presenting new candidates +to multiply lensed galaxies and new lens models were submit- +ted simultaneously to arXiv (Mahler et al. 2022; Pascale et al. +2022).1. A day after these two papers appeared on arXiv, a third +one was submitted presenting an additional lens model and new +lensed system candidates (Caminha et al. 2022). Other papers fo- +cusing on the high-redshift galaxies lensed by SMACS0723 and +their properties quickly followed (Ferreira et al. 2022; Cheng +et al. 2022; Laporte et al. 2022; Adams et al. 2023; Carnall et al. +2023). This frenzy over the new data reflects the excitement and +anticipation of the community for the new JWST data. JWST is +revolutionizing the field of astronomy in a similar fashion as it +1 The difference between the two submission times was just 13 sec- +onds! +Article number, page 1 of 12 +arXiv:2301.03629v1 [astro-ph.CO] 9 Jan 2023 + +A&A proofs: manuscript no. SMACS0723 +was done by its predecessor, the Hubble Space Telescope (HST), +at the end of the 20th and beginning of the 21st centuries. +The first image of JWST reveals approximately two dozen +lensed system candidates, five of which have spectroscopic +redshift estimations from MUSE and JWST data (Golubchik +et al. 2022; Sharon et al. 2022; Pascale et al. 2022; Mahler +et al. 2022; Caminha et al. 2022)2. One of the surprises in +the new data is the presence of hundreds of point like sources +near the large member galaxies in the cluster (possibly stripped +galactic nuclei or compact globular clusters) (Lee et al. 2022; +Faisst et al. 2022). Some of the lensed galaxies also show small +unresolved structures which could be compact star forming +regions, globular clusters, groups of stars or even individual +stars in cases of extreme magnification (Mowla et al. 2022). +These can prove very valuable in upcoming works, which will +look for flux anomalies between pairs of counterimages (Pooley +et al. 2012; Chan et al. 2020). The unresolved nature of these +substructures, together with the large magnification of some +of them can be used to study models of dark matter (DM) +which predict anomalous flux ratios between these pairs of +images. An additional surprise in the new data is the unusual +distribution of the intracluster light (or ICL hereafter), already +noted in Pascale et al. (2022) and Mahler et al. (2022) and +studied in more detail in Montes & Trujillo (2022). The ICL is +formed by stars not bound to any galaxy of the cluster, but to +the gravitational potential of the cluster as a whole (see Montes +2022, for a review). ICL is observed to be older near the centre +of the clusters (Montes & Trujillo 2018), suggesting an earlier +accretion of the central ICL region. The ICL is particularly +interesting in the context of DM. Similarly to DM particles, the +stars responsible for the ICL (as well as the globular clusters and +galactic core remnants) can be considered as non-interacting +particles that respond only to gravity. Hence, one would expect +a tight correlation between the distribution of ICL and DM +(Montes & Trujillo 2019; Alonso Asensio et al. 2020). In the +case of SMACS0723, the ICL departs from the expected smooth +distribution predicted for DM from N-body simulations (Alonso +Asensio et al. 2020) and shows two loop-like structures at ≈ 200 +kpc from the central BCG, one to the east and one to the west +of the cluster core. These loop-like structures may be the result +of a relatively recent merger but given the expected connection +between the stars in the ICL and the DM it is interesting to +study if similar structures can be found in the distribution of +dark matter from the lens model. +The previous lens models rely on some parameterization of +the mass distribution, usually by placing ellipsoids at the posi- +tions of galaxies, and/or large elliptical halos near the center of +the cluster to account for the contribution from DM, or by as- +suming it follows the cluster galaxy distribution. Hence, they are +less than ideal to study the possible correlation between the DM +and ICL distributions. In this paper, we present an additional lens +model based on a free-form technique that makes no assump- +tions about the underlying distribution of DM. A comparison +between the DM and ICL (or globular cluster) distributions can +then be done without being subject to assumptions made about +the distribution of DM. +The paper is organized as follows. In section 2, we discuss +the lensing constraints used to derive the lens model. Section +3 gives a brief introduction to the free-form algorithm used to +derive the lens model, making no assumptions about the dis- +2 As this paper was being finished, a new spectroscopic redshift for +system 4 (z=2.211) is provided in Noirot et al. (2022) +tribution of DM. In section 4, we present the driver model, +or Model-1, which is derived using only lensed systems with +known redshifts (spectroscopic). Section 5 uses the driver model +to make predictions for the redshifts of the candidate lensed sys- +tems without spectroscopic redshifts. These redshifts are an in- +teresting alternative to (and some times are more precise than) +the more common photometric redshifts. In section 6, we use the +lens model predicted redshifts and present two additional mod- +els (Model-2 and Model-3), which use all the additional systems +with constrained redshifts and, in the case of Model-3, increases +also the spatial resolution of the DM component. Model-2 and +Model-3 are useful to explore the uncertainty in the lens model +due to i) the lens system definition (spectroscopic sample vs full +sample), and ii) the spatial resolution in the lens model. In sec- +tion 7, we study the correlation between the ICL, globular clus- +ter distribution and DM. We discuss our results and present our +conclusions in section 8. We adopt a standard flat cosmological +model with Ωm = 0.3 and h = 0.7. At the redshift of the lens +(z=0.39), and for this cosmology, one arcsecond corresponds to +5.29 kpc. +2. Lensing constraints +The lensing constraints used in this work are compiled from +the three most recent works discussed previously (Pascale et al. +2022; Mahler et al. 2022; Caminha et al. 2022). Positions, and +IDs of these systems are presented in Table A.1 in appendix +A. When possible, we maintain the original ID of earlier work. +Candidates 6.3 and 16.3 in Pascale et al. (2022) are updated +with the nearby candidates 7.3 and 11.3 respectively from +Mahler et al. (2022). For convenience, Table A.1 also includes +the IDs used in earlier works. Among the systems in this table, +five of them have spectroscopic redshifts. All systems are +marked in Figure 1 with circles and their corresponding ID. The +five systems with spectroscopic redshifts are highlighted in bold. +In addition to the classic lensing constraints, we add the po- +sition of critical curves which can be determined from the radial +arc in system 5 (at z=1.425), and the merging pair of images +in system 7 (at z=5.17). The positions of these critical points +are added at the end of table A.1 and labeled CP5 and CP7 re- +spectively. Each critical point contributes with two constraints as +detailed in Diego et al. (2022b). Since at a critical point the mag- +nification diverges, this can be easily incorporated by applying a +rotation to the data by the angle determined by the elongation of +the arc. After this rotation one can simply impose that the inverse +of the tangential magnification equals zero, or similarly 1 = κ−γ. +The second constraint is simply γ2 = 0, which is satisfied when +the rotation is applied. +3. WSLAP+ +To optimize the lens model we use the code WSLAP+ (Diego +et al. 2005a, 2007; Sendra et al. 2014; Diego et al. 2016). A +lens model derived using WSLAP+ is considered a hybrid type +of model as it combines a free-form decomposition of the lens +plane for the smooth large-scale component with a small-scale +contribution from the member galaxies. Details can be found in +references above. Here we give a brief description of the method. +We start with the classic definition of the lens equation +β = θ − α(θ, Σ) , +(1) +Article number, page 2 of 12 + +Diego et al.: WSLAPping SMACS0723 + + +5” +E +N +1.1 +1.2 +1.3 +2.1 +2.2 +2.3 +3.1 +3.2 +3.3 +3.4 +4.2 +4.3 +4.1 +5.1 +5.2 +5.3 +6.1 +6.2 +6.3 +7.1 +7.2 +7.3 +8.2 +8.1 +8.3 +9.3 +9.2 +9.1 +10.1 +10.2 +10.3 +11.2 +11.1 +12.1 +12.2 +12.3 +13.3 +13.2 +13.4 +13.1 +14.1 +14.2 +14.3 +15.3 +15.2 +15.1 +16.1,2 +16.3 +17.1 +17.2 +17.3 +18.1 +18.2 +18.3 +19.3 +19.2 +19.1 +20.1 +20.2 +21.1 +21.2 +21.3 +22.2 +22.1 +23.1 +23.2 +23.3 +24.1 +24.2 +24.3 +25.1 +25.2 +25.3 +26.1 +26.2 +27.3 +27.2 +27.1 +Fig. 1. Central ≈ 1 arcminute region of SMACS0723 with systems of lensed galaxies. Systems in white have spectroscopic redshifts and are +the ones used to build the driver model or Model-1. Systems in yellow do not have spectroscopic redshifts but are used in combination with the +spectroscopic systems to build the lens Model-2 and Model-3. Note the hundreds of unresolved sources surrounding the BCG. These are mostly +globular clusters and galactic core remnants. Unless otehrwise noted, all figures in this plot are in the same orientation as this one. +where θ is the observed position of the source, α is the deflection +angle, Σ(θ) is the unknown surface mass-density of the cluster at +the position θ, and β is the unknown position of the background +source. The optimization of the WSLAP+ solution takes advan- +tage of the fact that the lens equation can be expressed as a linear +function of the surface mass density, Σ. WSLAP+ parameterizes +Σ as a linear superposition of functions, which translates into +α(θ, Σ) being also linear in Σ. +In WSLAP+, the surface mass density, Σ, is described by +the combination of two components; i) a smooth component +(usually parameterized as superposition of Gaussians) corre- +sponding to the free-form part of the model, or large scale +cluster potential; and ii) a compact component that accounts for +the mass associated with the individual galaxies in the cluster. +For the smooth component we use Gaussian functions defined +over a grid of points. A Gaussian function is simple and enables +fast computation of the deflection field, but also provides a good +compromise between the desired compactness and smoothness +of the basis function. The grid configuration can be defined as +regular (all grid points have the same size) or irregular (grid +points near the centre are in general smaller). Adopting a regular +grid is similar to a flat prior in the mass distribution while an +Article number, page 3 of 12 + +A&A proofs: manuscript no. SMACS0723 +irregular grid can be interpreted as a model with a prior on the +mass distribution with higher mass density assigned to smaller +cells. Since one of the goals of this paper is to study the possible +correlation between the DM distribution and the ICL, we adopt +a regular grid, since this makes minimal assumptions about the +mass distribution. +For the compact component, we directly adopt the light +distribution in the JWST band F277W around the brightest +member elliptical galaxies in the cluster. For each galaxy, we +assign a mass proportional to its surface brightness. This is the +only free parameter. This mass is later re-adjusted as part of +the optimization process. The number of parameters connected +with the compact component depends on the number of adopted +layers. Each layer contains a number of member galaxies. The +minimum number of layers is 1, corresponding to the case +where all galaxies are placed in the same layer, that is, they +are all assumed to have the same mass-to-light ratio. In this +case, the single layer is proportional to the light distribution +of all member galaxies, and is assigned a fiducial mass for the +entire mass of the member galaxies. For each layer there is one +extra parameter which accounts for the renormalization constant +multiplying the map of the mass distribution, that is optimized +by WSLAP+. For the particular case of SMACS0723, we use +4 layers. The first layer contains the main BCG, the second +layer contains a large elliptical galaxy ≈ 9” west of the main +BCG. The third layer contains two large elliptical galaxies near +the Beret galaxy discussed in Mahler et al. (2022). Finally, the +fourth layer contains all remaining member galaxies. Member +galaxies are selected from the standard red-sequence, and we +also make sure that spectroscopic members identified in Mahler +et al. (2022) are included in this set. +As shown by Diego et al. (2005a, 2007), the strong and weak +lensing problem can be expressed as a system of linear equations +that can be represented in a compact form, +Θ = ΓX, +(2) +where the measured strong lensing observables (and weak lens- +ing if available) are contained in the array Θ of dimension +NΘ = 2Nsl (plus 2Nwl if weak lensing data is available), the un- +known surface mass density and source positions are in the array +X of dimension +NX = Nc + Nl + 2Ns, +(3) +and the matrix Γ is known (for a given grid configuration and +fiducial galaxy deflection field) and has dimension NΘ × NX. +Nsl is the number of strong lensing observables (each one con- +tributing with two constraints, x, and y), Nc is the number of grid +points (or cells) that we use to divide the field of view, Nl is the +number of layers (Nl = 4 in our case as mentioned above), and +Ns is the number of background sources being strongly lensed +(each source represent two unknowns in X, βx, and βy). +The solution, X, of the system of equations 2 is found after +minimizing a quadratic function of X (derived from the system of +equations 2 as described in Diego et al. 2005a). The minimiza- +tion of the quadratic function is done with the constraint that +the solution, X, has to be positive. Since the vector X contains +the grid masses, the renormalization factors for the galaxy de- +flection field and the background source positions, and all these +quantities are always positive (the zero of the source positions +is defined in the bottom left corner of the field of view). Impos- +ing X > 0 helps constrain the space of meaningful solutions, +and to regularise the solution, as it avoids unwanted large nega- +tive and positive contiguous fluctuations. A detailed discussion +of the quadratic algorithm can be found in Diego et al. (2005a). +For a discussion of its convergence and performance (based on +simulated data), see Sendra et al. (2014). +As discussed in Diego et al. (2022b), critical points can also +be added as extra constraints. We identify two such constraints +in systems 5 (at z=1.425) and system 7 (at z=5.1727) with spec- +troscopic redshifts, and include them in our set of lensing con- +straints. The addition of these two points act as anchors for the +lens model, enforcing the critical curve to pass through the de- +sired point at the given redshift. +4. Driver lens model +Using the constraints listed in Table A.1, we first derive the +driver model, or Model-1. This model is only based on systems +with spectroscopic redshifts. For the case of SMACS0723, and +at the time of writing these paper, five systems are known to have +spectroscopic redshifts3. These are marked in bold in Table A.1. +For the grid, we use a regular distribution of 20 × 20 = 400 grid +points. Given the relatively small number of lensing constraints, +a significantly larger number of grid points results in nonphysi- +cal solutions with large mass fluctuations. +Based on the 5 lensed systems with spectroscopic redshift, +the driver model can be used to predict the redshift of the other +system candidates in Table A.1. We do so in the next section. +5. Redshifts predicted by the lens model +Using the driver model, we derive redshifts for all systems listed +in table A.1. The probability of a system to be at redshift z is +computed by; +P(z) = exp(−V(z)/(2σ2)), +(4) +where V(z) is the variance between the arc positions of a given +system projected on the source plane at redshift z. The projection +is done with the deflection field of the driver model (computed +at redshift z = 3) which is re-scaled to the desired redshift. The +dispersion, σ, in the expression above is fixed to three pixels, or +≈ 0.18”. This is a reasonable choice for well constrained sys- +tems, resulting in relatively narrow distributions for the redshift, +and with uncertainty in the error prediction consistent with the +observed error (see for instance Diego et al. 2022a, for a more +in depth analysis of the errors expected with this technique and +for WSLAp+). Systems that are well reproduced by the driver +model result in a small variance V(z) near the optimal redshift, +which in turn result in maximum values of P(z) close to 1. Sys- +tems that are poorly reproduced by the driver model have larger +values of V(z), which reduce the maximum value of P(z). A low +maximum probability for P(z) does not necessarily mean that the +system is a bad candidate. This can simply be the result of the +driver model not being well constrained in that part of the lens +system. Systems at high redshift tend to have broader probabil- +ities, since for source redshift z > 2 the deflection field varies +slowly with redshift. +3 As noted earlier, a new spectroscopic redshift was recently made +available for system 4 in Noirot et al. (2022) at the time of finishing +this paper. This new redshift (z=2.211) was not used in our analysis +where we adopted our geometric redshift estimate (z=2). The difference +in redshift is small and is not expected to have any significant impact in +our results. +Article number, page 4 of 12 + +Diego et al.: WSLAPping SMACS0723 +Fig. 2. Redshifts predicted by the driver model for the case of well con- +strained systems. +The derived probabilities P(z) can be divided in two groups. +In the first group we find systems with well defined and rela- +tively narrow probabilities. The probabilities for these systems +are shown in Figure 2. Among these we find the systems with +spectroscopic redshifts that were used to derive the driver model. +Naturally, the maximum of P(z) for these systems falls very close +to the spectroscopic value. System 4 had its spectroscopic red- +shift estimated recently in Noirot et al. (2022) where they find +z = 2.211. As shown in Figure 2, the P(z) for this system con- +tains the correct redshift within the 95% confidence interval. +In a different group we find systems for which the redshift is +not so well constrained. The probabilities for these systems is +shown in Figure 3. Two systems (14 and 26) have no constrain on +their redshift (z>13). The bad performance of these systems can +be easily understood since they correspond to cases of galaxy- +galaxy lensing, where the member galaxy acting as a lens is not +optimized individually (these galaxies are part of layer 4 dis- +cussed in Section 3). +System 8 has a very low probability of P(z). This probability +is shown as a dashed line in Figure 3, and the probability has +been multiplied by a factor 100, to make it visible in the figure. +The low probability of system 8 can be interpreted as being a bad +system or being in a region with poor constraints in the driver +model. This is the case on the western part of the cluster, where +system 8 lies, since only one system has a spectroscopic redshift +in this region of the lens. +Redshifts predicted by gravitational lenses are an interesting +alternative to photometric redshifts, specially for high-redshift +candidates for which photometry may be poor or nonexistent in +more than one band (such as in the case of dropouts). A similar +technique has been used in the past in other lenses with positive +results and was recently applied in Diego et al. (2022a) to predict +the redshift of the new systems identified in the JWST data of El +Gordo cluster. +6. Full-sample lens models +Taking advantage of the redshifts predicted by the lens model +discussed in the previous section, we expand the number of +constraints and update the lens model. As discussed above, the +redshift for systems 14 and 26 cannot be constrained by the +lens model so we exclude these two systems from our list of +constraints. The remaining number of systems totals 25, and the +Fig. 3. Redshifts predicted by the driver model for the case of poorly +constrained systems. The probability of system 8, shown as a dashed +line at z ≈ 8, has been multiplied by a factor 100. +number of constraints exceeds 150 (x and y positions of each +arc plus the two critical point positions, each contributing also +with two constraints). Using these constraints we derive two +models. One that we refer to as Model-2, which is derived with +the full set of constraints (excluding systems 14 and 26), and a +regular grid of 20 × 20 = 400 points. We increase the number of +grid points to 25 × 25 = 625 in a third model that we refer to +as Model-3. Increasing the number of grid points even further +can result in unstable solutions. For instance, with a grid of +30 × 30 = 900 points we obtain a solution that places too much +mass in the edges of the field of view and introduces relatively +large mass fluctuations across the entire field so we do not +consider solutions with more than 625 grid points. The critical +curves for Model-2 and Model-3 are shown in Figure 4 as green +and blue curves respectively. For convenience we include again +in this figure the critical curve for the driver model (or Model-1) +in red. All three curves are again computed at the redshift of +system 7 (z = 5.1727). The three models produce consistent +results in the eastern part of the lens, which is the portion of +the cluster where the number density of spectroscopic redshifts +is the highest. In contrast, the critical curves in the west differ +significantly from one another, indicating that the western part +of the cluster is more poorly constrained. The addition of new +spectroscopic systems in this part of the lens will reduce the +uncertainty in the lens model. +In terms of mass, we can compare with previous published +results based on parametric models. Both Mahler et al. (2022) +and Caminha et al. (2022) quote the total projected mass within +a cylinder of radius 128 kpc centered in the BCG. This radius +corresponds approximately to the Einstein radius for a source at +z > 2, and it is the radius within which the lens model can be +properly constrained with strong lensing data. They find masses +of 8.26 ± 0.04 × 1013M⊙ and 8.7 ± 0.2 × 1013M⊙ respectively, +and within the aforementioned 128 kpc radius. For our three lens +models we find 7.28×1013M⊙, 7.31×1013M⊙, and 7.15×1013M⊙ +for Model-1, Model-2 and Model-3 respectively, and within the +same radius. This is approximately 10% less than in the para- +metric models. +Article number, page 5 of 12 + +A&A proofs: manuscript no. SMACS0723 +Fig. 4. Critical curves of alternative lens models. All critical curves are +computed at the redshift of system 7 (z=5.1727). The red curve corre- +sponds to the driver model derived with the five spectroscopic systems +and a grid of 20x20 points. The green curve uses the same grid configu- +ration but is derived from the 25 systems with constrained redshifts. The +blue curve uses the same 25 systems but is based on a higher resolution +grid of 25x25 points. +7. Correlation between the dark matter, intracluster +light, and globular cluster distributions +In all three models discussed in the previous sections we find +that the critical curves for the three models are consistent among +them, with the largest differences concentrating in the west +portion of the cluster. Hence the lens model is relatively well +constrained for different choices of lensed systems and grid +configurations. In this section, we pay special attention to the +distribution of light in the ICL, and the distribution of globular +clusters. We are interested in the possible correlation between +the ICL, globular cluster and the DM distributions. Figure 5 +shows how the ICL presents two loop-like structures to the east +and west of the cluster. At ≈ 200 kpc from the central BCG and +towards west, a cavity-like structure can be appreciated in the +ICL (Giant West Loop in Montes & Trujillo 2022). Although +not as clear, a similar cavity can be also observed towards the +east at approximately the same distance from the central BCG. +This is a surprising feature in the ICL where one expects to find +more uniform distributions. Recent merger activity can result in +trails of stars being stripped away from their host galaxy by tidal +forces. These tails are however much smaller than the observed +loops in the ICL of SMACS0723, and connect with the host +galaxy. In the case of the loops of SMACS0723, one cannot +establish any correspondence between the loops and a member +galaxy. On the other hand, as noted by Mahler et al. (2022), +the offset between the radial velocity of the central BCG and +the mean redshift of the cluster suggests a recent past merger +(a relaxed cluster would have no offset), offering a possible +explanation for the odd distribution of the ICL. +Whatever the cause for the morphology of the ICL, it is +interesting to compare its two-dimensional distribution with the +distribution of mass from our lens models. Since approximately +85% of the projected mass of the cluster is expected to be dark +matter, if dark matter and the ICL are related, we should expect +a correlation between the two. In Figure 5 we show as yellow +contours the DM distribution from our Model-2 while in blue +we show the contours for the DM distribution from our Model-3 +(Model-1 is not shown but it is very similar to Model-2). The +DM component is obtained after subtracting the mass associated +to the galaxies from the total mass. In general we find good +correspondence between the distribution of the ICL and the two +DM models. +A more quantitative comparison is shown in Figure 6 where +we compute the average of the ICL or the DM along a straight +line. This line is shown in Figure 5 and it intersects the ICL from +west to east, passing through the central BCG in the middle. The +average is computed at each position as the mean over a box of +size 0.18” × 0.18” and centered in the line. The black curve in +Figure 6 corresponds to the light distribution. The colored lines +are for the driver model or Model-1 (red), Model-2 (green), and +Model-3 (blue). The curves for the DM models have been re- +scaled by an arbitrary number to match the black curve. +In the east part of the cluster we find good correspondence +between all three models and the ICL. This is not true in the +west part of the cluster, where the cavity clearly seen in the ICL +at ≈ −200 kpc in Figure 6 is not observed in any of the DM +models. +In addition to the ICL, another possible tracer of the potential +are globular clusters, whose distribution could correlate with the +distribution of dark matter, since as the stars in the ICL, globular +clusters respond to gravitational forces. The superior sensitivity +and spatial resolution of JWST allows to detect these clusters +with unprecedented detail. Preliminary results based on JWST +data in SMACS0723 are presented in Lee et al. (2022); Faisst +et al. (2022). It is interesting to compare our results with those +from earlier work. Figure 7 compares the observed ICL in the +F356W filter with the distribution of dark matter (yellow con- +tour) and the distribution of globular clusters (blue contours) +from Lee et al. (2022). To compute the blue contours we have +smoothed the distribution of globular clusters with a Gaussian of +FWHM=1.5". To first order, there is a good spatial correspon- +dence between the DM, ICL and globular cluster distribution, +with all three components centered in the same point (BCG) and +having similar alignments in the east-west direction. As in the +case of the ICL, the distribution of globular clusters appears to +show a similar deficit in number density at the position of the +cavity on the west side of the cluster. This cavity has no corre- +spondence in the distribution of DM. +In terms of radial profiles, we show a comparison of our lens +model with the ICL profiles from Montes & Trujillo (2022) and +the globular cluster profile from Lee et al. (2022) in Figure 8. +For the globular clusters, we have re-scaled the surface number +density (expressed as number per kpc2) by a factor 2 × 109 in +order for the resulting profile to overlap with the ICL profile. +For comparison we plot a power law R−1.3 as a dashed line. +This power law reproduces well the profile of the ICL and the +globular cluster number density. +The mass profiles from the three lens models are shown as +a solid lines. Within the inner 20 kpc region, the total mass and +the ICL have similar profiles. This is expected in our lens model +since the compact component of the lens model takes directly +the light distribution of member galaxies, including the central +BCG. Since near the centre of the BCG, the bulk of the mass +is expected to come from stars (or the baryonic component in +general), by comparing our lens model with the ICL profile from +Montes & Trujillo (2022), we find that either i) there is ≈ 10 +times more dark matter than stellar mass within the central 20 +Article number, page 6 of 12 + +CDiego et al.: WSLAPping SMACS0723 +Fig. 5. Projected total mass vs ICL. The contours represent the smooth +component of the lens model obtained with the 25 constrained systems. +The yellow contour is obtained with a regular grid of 20x20 cells while +the blue contour is obtained with a higher resolution grid of 25x25 cells. +The image is a masked version of the F277W band, where the ICL light +can be better appreciated. The contours correspond to values of the con- +vergence, κ, computed at a fiducial source redshift of zs = 3. Space be- +tween contours correspond to δκ = 0.1, with values starting at κ = 0.5. +The last contour is for κ = 1.15. The white straight line marks the di- +rection over which we construct the one-dimensional scan of the light +profile and DM models. +kpc or ii) the stellar mass from the ICL is underestimated by +some factor. +Beyond ≈ 20 kpc, the total mass profile is clearly shallower +than the profile of the ICL and the number density of globular +clusters. This departure is interesting and needs to be studied in +other clusters with more constraints. Increasing the number of +lensing constraints will allow to improve the spatial resolution +of the lens model. +8. Discussion and conclusions +The new data from JWST reveals a wealth of new candidate +lensed galaxies. Future observations of these candidates will se- +cure their redshifts, which can then be compared with the ge- +ometric redshift estimate based on our driver model. If spec- +troscopic confirmation validates the method of estimating dis- +tances through geometric redshifts, future observations by the +JWST can take advantage of a similar technique, where a hand- +ful of spectroscopic lensed galaxies may suffice to calibrate a +lens model for distance estimation. Recent work has shown how +photometric redshifts can predict erroneous redshifts for the case +of dropout galaxies in the JWST bands (Harikane et al. 2022; +Naidu et al. 2022; Zavala et al. 2022). An independent estimation +of the distance to these galaxies can help reduce the uncertainty +in the estimation of the redshift, and identify those galaxies that +have large photometric redshifts (z>10) yet they are predicted by +the lens model to be at much lower redshift. +Lens models like the one presented in this work are +also needed to interpret sources near caustics. In the case of +SMACS0723, Pascale et al. (2022) discuss a small pair of knots +in the middle of the merging pair of images of system 5 (see their +Fig. 6. One dimensional scan of the light distribution vs DM. The x- +axis is the distance to the BCG. The solid black line shows the mean +of the light emission in the F277W band along the straight line shown +in Figure 5. The mean is computed over a box of 10x10 pixels at each +position. The colored lines are the corresponding mean of the DM com- +ponent for the three lens models discussed in this work. The red color +is for the driver model, the green line is for Model-2, or low-resolution +(20x20 grid points) with 25 systems, and the blue model is for the high- +resolution (25x25 grid points) Model-3 with 25 systems. The DM pro- +files are re-scaled by arbitrary units to visually match the profile of the +light emission. See Figure 8 for a direct comparison of the profiles with- +out the re-scaling. +Figure 2). Since the lens model has a resolution comparable to +the separation between the knots in the pair, the magnification in +these knots is better estimated by interpolating the magnification. +Based on symmetry arguments, the critical curve must pass be- +tween these two points, so they are equidistant to it (d ≈ 0.08”). +Since the magnification near a fold caustic scales as µ = A/d +(Schneider et al. 1992), we can estimate A from our lens model. +We find A ≈ 58”, which results in µ ≈ 725 for each one of the +images in the pair. This estimate matches very well the value +quoted in Pascale et al. (2022) of µ ≈ 750. +The greater sensitivity of JWST to the ICL offers new op- +portunities to study the correlation between the DM and ICL. In +addition, the improved spatial resolution in the infrared bands +allows for detection of small clumps of old stellar populations in +the cluster stripped from their hosts galaxies. The first image of +JWST on this cluster reveals hundreds of unresolved clumps that +are interpreted as globular clusters (Lee et al. 2022; Faisst et al. +2022), but could be also the surviving remnants (after a close en- +counter with a larger galaxy in the cluster, such as the BCG) of +compact galactic cores. Both the stars in the ICL and the globu- +lar clusters are expected to interact with the rest of the matter in +the cluster mostly through gravitational forces, and hence behave +similar to dark matter. +In this work we use a free-form modelling technique which +makes minimal assumptions about the distribution of dark mat- +ter, and find that in general the DM traces well the ICL and glob- +ular cluster distribution but we also find that the small loop-like +structures (and associated cavities) to the east and west of the +central region of SMACS0723 have no obvious correspondence +in the DM distribution. At distances from the centre comparable +to the Einstein radius (∼ 100 kpc, and hence well constrained by +the available data) we find that the dark matter profile is signifi- +cantly shallower than the ICL and globular cluster distributions. +This is also found in simulation of galaxy clusters. In Alonso +Article number, page 7 of 12 + +A&A proofs: manuscript no. SMACS0723 +Fig. 7. Comparison of the dark matter and globular cluster distribution +(number density). The image corresponds to the F356W JWST filter. +Yellow contours are the smooth component of the dark matter distribu- +tion (Model-2), while blue contours are for a Gaussian filtered version +(FWHM=1.5") of the distribution of globular clusters from Lee et al. +(2022). +Asensio et al. (2020), the authors analyze the EAGLE simula- +tions and find that the ICL profile is steeper than the total mass +profile. In particular, they find that the ratio between the ICL and +total mass profiles is a power law with slope −1. Interestingly, +in the range between ≈ 20 kpc and ≈ 200 kpc we find a similar +ratio between the total mass and ICL (and globular cluster) pro- +files, with the ICL and globular cluster profiles falling as ∼ R−1.3 +while the total mass falls as ∼ R−0.3 (see Figure 8). Similar con- +clusions are found in Pillepich et al. (2018) where, based on the +IllustrisTNG simulations, the 3D profile of the ICL in massive +clusters is found to fall faster (by approximately 1 dex from their +Figure 6 at around 100 kpc distance) than the canonical NFW +profile commonly used to describe dark matter profiles. Earlier +work based on the EAGLE simulations shows a similar trend +(Schaller et al. 2015). Hence, our results on SMACS0723 are in +agreement with the ones derived from N-body simulations. +We observe differences in the range ≈ 20–200 kpc between +the ICL (and globular cluster) profiles and the total mass (mostly +dominated by dark matter in this distance range). We specu- +late that this may be related to the different formation times +of the cluster dark matter halo and ICL. Since dark matter is +more loosely bounded to their host halos (as it mostly resides on +the outskirts of the galaxies, with the central region being more +baryon dominated), it can be stripped more easily during the first +encounters with the cluster and hence retaining the initial (rela- +tively large) angular momentum. The baryonic component (stars +in our case) is more concentrated around the centre of the satel- +lite galaxies and can survive more encounters with the cluster, +and without being stripped away. In each encounter, the satellite +galaxy looses angular momentum due to dynamical friction and +can get closer to the BCG (Contini et al. 2018; Chun et al. 2022). +Stars that are stripped at a later time lose part of their bulk kinetic +energy this way, and when stripped from their hosts can remain +at shorter radii, resulting in profiles that are steeper (more con- +Fig. 8. Comparison of the total mass profile from the three lens models +(solid lines) with the ICL profile from Montes & Trujillo (2022) (shaded +orange and blue regions for the East and West sectors respectively) and +the globular cluster number profile from Lee et al. (2022) (blue dots). +For the later, we have re-scaled the number density by an arbitrary num- +ber of 2×109 in order to overlap with the ICL profile. The black dashed +line is a power law that scales with distance as R−1.3. The red dashed +line is a power law that scales as R−0.3. +centrated) than the dark matter profiles. Globular clusters and +galactic core remnants are subject also to dynamical friction and +hence expected to orbit closer to the BCG, resulting in more con- +centrated profiles. N-body simulations also show how the radial +distribution of subhalos is steeper than the distribution of dark +matter (Gao et al. 2004). +The presence of cavities in the distribution of the ICL, and +not detected in the total mass distribution, is another interesting +difference. The formation of cavities in the ICL but not in the +DM distribution could be due to the different distribution of stars +and dark matter inside the satellite galaxies before they enter the +galaxy cluster, and the striping mechanism starts to take place. +The dark matter, forming an extended halo around the satellite +galaxy, is easily tidally stripped from its host galaxy as it enters +the cluster and starts orbiting around the BCG. The better ability +of the baryonic matter to cool down more efficiently and form +more concentrated structures like disks or bulges facilitates the +survival of the bulge (or disk) as they orbit the minimum of the +potential. During a close encounter with the BCG, parts of the +bulge or disc of a satellite galaxy can be tidally stripped, creating +the loop-like structures and associated cavities. Tidal stripping +of satellite galaxies has been claimed as responsible for filamen- +tary structures seen in the ICL of the nearby Virgo cluster (Mihos +et al. 2005). Structures that resemble the loop-cavity system are +also observed in nearby galaxies that had recent encounters with +satellite galaxies (Martinez-Delgado et al. 2022). In simulations, +faint structures in the ICL, that resemble the loop-cavity struc- +tures can be appreciated in Figures 3 and 4 in Pillepich et al. +(2018). +Perhaps one of the most interesting findings is the connec- +tion between the ICL and globular cluster distribution, both hav- +ing a similar profile. This connection could be easily explained if +the ICL corresponds to the outer envelopes of the alleged glob- +ular clusters. In this case, the globular clusters should be re- +interpreted as the surviving galactic cores of the infalling satel- +lite galaxies. +More examples like the one studied in this work are needed +in order to extract a firmer conclusion regarding the connection +between the ICL, globular cluster and DM distributions. In par- +Article number, page 8 of 12 + +N +10″Diego et al.: WSLAPping SMACS0723 +ticular, the addition of new constraints (with confirmed spectro- +scopic redshift) will allow us to increase the resolution of the +lens model, revealing perhaps finer details in the distribution of +DM that can not be unveiled with the current set of constraints. +For the particular case of SMACS0723, the number of lensing +constraints around the west cavity is very small (≈ 4 lensed +galaxies in this region). Future analysis based on JWST data, +especially of low redshift clusters for which both ICL and glob- +ular clusters are more easily detected, and with abundant lens- +ing constraints (such as the Hubble Frontier Fields Clusters) will +enable more precise conclusions on the correlation between the +ICL, globular cluster and DM distributions. +Acknowledgements. J.M.D. acknowledges the support of project PGC2018- +101814-B-100 (MCIU/AEI/MINECO/FEDER, UE) Ministerio de Ciencia, In- +vestigación y Universidades. This project was funded by the Agencia Estatal +de Investigación, Unidad de Excelencia María de Maeztu, ref. MDM-2017- +0765. M.P. was funded through the NSF Graduate Research Fellowship grant +No. DGE 1752814. A.Z. acknowledges support by Grant No. 2020750 from +the United States-Israel Binational Science Foundation (BSF) and Grant No. +2109066 from the United States National Science Foundation (NSF), and by +the Ministry of Science & Technology, Israel. G.M. acknowledges funding +from the European Union’s Horizon 2020 research and innovation programme +under the Marie Sklodowska-Curie grant agreement No MARACHAS-DLV- +896778. M.J. is supported by the United Kingdom Research and Innovation +(UKRI) Future Leaders Fellowship ‘Using Cosmic Beasts to uncover the Na- +ture of Dark Matter’ (grant number MR/S017216/1). 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Column four lists the +spectroscopic redshifts when available. Spectroscopic redshifts +are marked in bold face. Column five lists the redshifts predicted +by the driver model. In this case, errors correspond to the 68% +interval of the PDF. For all systems, only the first arc is given +with its redshift. Colums six, seven, and eight give the original +ID in Pascale et al. (2022), Caminha et al. (2022), and Mahler et +al. (2022) respectively. +‡While finishing this paper, Noirot et al. (2022) published spec- +troscopic redshifts of some galaxies in the field of SMACS0723 +including a redshift for our system 4, for which they find zspec = +2.211, in good agreement (2σ) with our geometric redshift esti- +mate (zgeo = 2 ± 0.1). No other redshifts are published for the +remaining arcs without spectroscopic redshifts. +Article number, page 10 of 12 + +Diego et al.: WSLAPping SMACS0723 +Table A.1. Arc positions and redshifts. +ID +RA +DEC +zs +zm +IDP +IDC +IDM +1.1 +110.8407240 +-73.4510787 +1.449 +1.45±0.07 +1.1 +2a +1.1 +1.2 +110.8429489 +-73.4548399 +– +– +1.2 +2b +1.2 +1.3 +110.8389887 +-73.4587844 +– +– +1.3 +2c +1.3 +2.1 +110.8387288 +-73.4510508 +1.3779 +1.37±0.06 +2.1 +3a +2.1 +2.2 +110.8407771 +-73.4552122 +– +– +2.2 +3b +2.2 +2.3 +110.8364983 +-73.4588136 +– +– +2.3 +3c +2.3 +3.1 +110.8304431 +-73.4485622 +1.9914 +1.97±0.11 +3.1 +1a +3.3 +3.2 +110.8318194 +-73.4552311 +– +– +3.2 +1b +3.2 +3.3 +110.8252159 +-73.4596604 +– +– +3.3 +1d +3.1 +3.4 +110.8232656 +-73.4548634 +– +– +3.4 +1c +3.4 +4.1 +110.8069982 +-73.4584308 +‡ +2.00±0.10 +4.2 +4c +4.2 +4.2 +110.8052367 +-73.4546325 +– +– +4.1 +4b +4.1 +4.3 +110.8132881 +-73.4487869 +– +– +4.3 +4a +4.3 +5.1 +110.8238908 +-73.4518820 +1.425 +1.45±0.06 +5.1 +6a +5.1 +5.2 +110.8223529 +-73.4527831 +– +– +5.2 +6b +5.2 +5.3 +110.8209254 +-73.4602058 +– +– +5.3 +6c +5.3 +6.1 +110.8358540 +-73.4518199 +– +1.67±0.07 +6.1 +9a +7.1 +6.2 +110.8367611 +-73.4530868 +– +– +6.2 +9b +7.2 +6.3 +110.8303933 +-73.4608436 +– +– +9c +7.3 +7.1 +110.7947604 +-73.4490975 +5.1727 +5.21+5.1 +−0.6 +7.2 +17.1 +7.2 +110.7954442 +-73.4487211 +– +– +7.1 +17.2 +7.3 +110.7996039 +-73.4470866 +– +– +7.3 +17.3 +8.1 +110.8023784 +-73.4602055 +– +8.06+1.7 +−0.8 +8.1 +7a +12.1 +8.2 +110.7995598 +-73.4553501 +– +– +8.2 +7b +12.2 +8.3 +110.8130564 +-73.4466651 +– +– +8.3 +7c +9.1 +110.8050637 +-73.4589656 +– +2.62±0.14 +9.2 +10c +13.3 +9.2 +110.8028896 +-73.4549564 +– +– +9.1 +10b +13.2 +9.3 +110.8127004 +-73.4481250 +– +– +9.3 +13.1 +10.1 +110.8235289 +-73.4517392 +– +1.45±0.06 +10.1 +10.2 +110.8216192 +-73.4528243 +– +– +10.2 +10.3 +110.8205119 +-73.4601152 +– +– +10.3 +11.1 +110.8107306 +-73.4569574 +– +1.47+8.1 +−0.2 +11.2 +19.1 +11.2 +110.8101464 +-73.4561599 +– +– +11.1 +19.2 +12.1 +110.8221364 +-73.4491504 +– +1.66±0.06 +12.1 +13a +14.1 +12.2 +110.8146179 +-73.4544119 +– +– +12.2 +13b +14.2 +12.3 +110.8173093 +-73.4593170 +– +– +12.3 +13c +14.3 +13.1 +110.8297224 +-73.4489907 +– +3.02+0.26 +−0.17 +13.1 +12a +6.3 +13.2 +110.8219150 +-73.4542067 +– +– +13.2 +12c +6.2 +13.3 +110.8231150 +-73.4617081 +– +– +13.3 +12d +6.4 +13.4 +110.8324286 +-73.4544642 +– +– +13.4 +12b +6.1 +14.1 +110.8015568 +-73.4583546 +– +– +14.1 +14.2 +110.8018148 +-73.4589480 +– +– +14.2 +14.3 +110.8022270 +-73.4590843 +– +– +14.3 +15.1 +110.8193895 +-73.4487436 +– +1.82±0.08 +15.1 +11a +15.2 +110.8113813 +-73.4546235 +– +– +15.2 +11b +15.3 +110.8139705 +-73.4590522 +– +– +15.3 +11c +16.1 +110.8206200 +-73.4527181 +– +1.26±0.05 +16.1 +11.1 +16.2 +110.8205250 +-73.4528156 +– +– +16.2 +11.2 +16.3 +110.8207626 +-73.4597746 +– +– +11.3 +17.1 +110.8239479 +-73.4575528 +– +2.33±0.11 +18.2 +8c +8.2 +17.2 +110.8231354 +-73.4558083 +– +– +18.1 +8b +8.1 +17.3 +110.8297769 +-73.4474619 +– +– +18.3 +8a +8.3 +18.1 +110.8216711 +-73.4506362 +– +1.36±0.05 +19.1 +9.1 +18.2 +110.8167450 +-73.4537968 +– +– +19.2 +9.2 +18.3 +110.8179340 +-73.4590101 +– +– +19.3 +9.3 +19.1 +110.8208804 +-73.4507461 +1.3825 +1.37±0.05 +5a +10.1 +19.2 +110.8164058 +-73.4535733 +– +– +5b +10.2 +19.3 +110.8173046 +-73.4589942 +– +– +54 +10.3 +20.1 +110.8165814 +-73.4519445 +– +1.20+3.4 +−0.07 +14a +20.2 +110.8159392 +-73.4523932 +– +– +14b +Article number, page 11 of 12 + +A&A proofs: manuscript no. SMACS0723 +Table A.1 – continued from previous page +ID +RA +DEC +zs +zm +IDP +IDC +IDM +21.1 +110.8168354 +-73.4485770 +– +2.19±0.11 +15a +21.2 +110.8086654 +-73.4541442 +– +– +15b +21.3 +110.8115827 +-73.4596446 +– +– +15c +22.1 +110.8293400 +-73.4561204 +– +2.27+0.73 +−0.22 +16a +22.2 +110.8268630 +-73.4578161 +– +– +16b +23.1 +110.8258363 +-73.4502839 +– +1.59±0.06 +15.1 +23.2 +110.8201612 +-73.4539789 +– +– +15.2 +23.3 +110.8213975 +-73.4602314 +– +– +15.3 +24.1 +110.8085708 +-73.4494083 +– +2.10±0.12 +16.1 +24.2 +110.8019579 +-73.4526322 +– +– +16.2 +24.3 +110.8058921 +-73.4595997 +– +– +16.3 +25.1 +110.7927038 +-73.4484814 +– +2.16+0.87 +−0.23 +18.1 +25.2 +110.7936842 +-73.4482439 +– +– +18.2 +25.2 +110.7964129 +-73.4469406 +– +– +18.3 +26.1 +110.7917089 +-73.4566332 +– +– +20.1 +26.2 +110.7914913 +-73.4558973 +– +– +20.2 +27.1 +110.8032246 +-73.4582886 +– +2.81+0.22 +−0.15 +21.1 +27.2 +110.8041292 +-73.4531883 +– +– +21.2 +27.3 +110.8136692 +-73.4495378 +– +– +21.3 +CP5 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 39005 Santander, Spain 2 Department of Astronomy, University of California, 501 Campbell Hall #3411, Berkeley, CA 94720, USA 3 Department of Astronomy/Steward Observatory, University of Arizona, 933 N Cherry Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=', Tucson, AZ 85721, USA 4 Physics Department, Ben-Gurion University of the Negev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Box 653,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Be’er-Sheva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 8410501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Israel 5 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' University of the Basque Country UPV/EHU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' E-48080 Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Spain 6 DIPC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Basque Country UPV/EHU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' E-48080 San Sebastian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Spain 7 Ikerbasque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Basque Foundation for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' E-48011 Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Spain 8 Centre for Extragalactic Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Durham University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' South Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Durham DH1 3LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' UK 9 Institute for Computational Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Durham University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' South Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Durham DH1 3LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' UK 10 Max-Planck-Institut für Astrophysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Karl-Schwarzschild-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Germany 11 Astrophysics Research Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' University of KwaZulu-Natal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Westville Campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Durban 4041,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' South Africa 12 School of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Statistics & Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' University of KwaZulu-Natal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Westville Campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Durban 4041,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' South Africa 13 Astronomy Program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' SNUARC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Seoul National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 1 Gwanak-ro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Gwanak-gu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Seoul 08826,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Republic of Korea 14 Department of Astronomy & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 5640 South Ellis Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' IL 60637,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' USA 15 Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' c/ Vía Láctea s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' E-38205 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Spain 16 Departamento de Astrofísica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Universidad de La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' E-38205 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2023 ABSTRACT We present a free-form model of SMACS0723,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' the first cluster observed with JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This model makes no strong assumptions about the distribution of mass (mostly dark matter) in the cluster and we use it to study the possible correlation between dark matter with the intracluster light and distribution of globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' To explore the uncertainty in mass modelling, we derive three lens models based on spectroscopically confirmed systems and new candidate systems with redshifts predicted by the lens model derived from the spectroscipic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We find that beyond the radius of influence of the BCG, the total mass does not trace the ICL, implying the need for a dark component (dark matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Two loop-like structures observed in the intracluster light do not have an obvious correspondence with the total mass (mostly dark matter) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The radial profiles of the ICL and the distribution of globular clusters are similar to each other, but steeper than the profile of the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' More specifically, we find that the total mass is shallower by 1 dex in log scale than both ICL and globular cluster profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is in excellent agreement with N-body simulations of cold dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' gravitational lensing – dark matter – cosmology 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Introduction After its launch on December 25th 2021, on July 11st 2022, the first color image from James Webb Space Telescope (JWST) was presented to the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The image showed a view of the distant infrared universe with a detail and depth never seen before at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This image was centered on a massive galaxy cluster at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='39, SMACS J0723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3-7327 (or SMACS0723 hereafter) acting as a powerful gravitational lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This natural lens magnifies the galaxies in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Some of these background galaxies appear repeated several times in the image, since they take different paths which are later refocused by the gravitational lens into the JWST telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In anticipation for these first JWST data, a lens model based on HST data and listing the first few sets of multiple images and some spectroscopic redshifts for them was posted on the arXiv ⋆ jdiego@ifca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='unican.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='es by Golubchik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) on the same date (July 11th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The JWST data became itself public on July 13th 2022, and just a day after the data release, two papers presenting new candidates to multiply lensed galaxies and new lens models were submit- ted simultaneously to arXiv (Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A day after these two papers appeared on arXiv, a third one was submitted presenting an additional lens model and new lensed system candidates (Caminha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Other papers fo- cusing on the high-redshift galaxies lensed by SMACS0723 and their properties quickly followed (Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This frenzy over the new data reflects the excitement and anticipation of the community for the new JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' JWST is revolutionizing the field of astronomy in a similar fashion as it 1 The difference between the two submission times was just 13 sec- onds!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Article number, page 1 of 12 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='03629v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='CO] 9 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' SMACS0723 was done by its predecessor, the Hubble Space Telescope (HST), at the end of the 20th and beginning of the 21st centuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The first image of JWST reveals approximately two dozen lensed system candidates, five of which have spectroscopic redshift estimations from MUSE and JWST data (Golubchik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Sharon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Caminha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' One of the surprises in the new data is the presence of hundreds of point like sources near the large member galaxies in the cluster (possibly stripped galactic nuclei or compact globular clusters) (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Faisst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Some of the lensed galaxies also show small unresolved structures which could be compact star forming regions, globular clusters, groups of stars or even individual stars in cases of extreme magnification (Mowla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' These can prove very valuable in upcoming works, which will look for flux anomalies between pairs of counterimages (Pooley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The unresolved nature of these substructures, together with the large magnification of some of them can be used to study models of dark matter (DM) which predict anomalous flux ratios between these pairs of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' An additional surprise in the new data is the unusual distribution of the intracluster light (or ICL hereafter), already noted in Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) and Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) and studied in more detail in Montes & Trujillo (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The ICL is formed by stars not bound to any galaxy of the cluster, but to the gravitational potential of the cluster as a whole (see Montes 2022, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' ICL is observed to be older near the centre of the clusters (Montes & Trujillo 2018), suggesting an earlier accretion of the central ICL region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The ICL is particularly interesting in the context of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Similarly to DM particles, the stars responsible for the ICL (as well as the globular clusters and galactic core remnants) can be considered as non-interacting particles that respond only to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Hence, one would expect a tight correlation between the distribution of ICL and DM (Montes & Trujillo 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Alonso Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In the case of SMACS0723, the ICL departs from the expected smooth distribution predicted for DM from N-body simulations (Alonso Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2020) and shows two loop-like structures at ≈ 200 kpc from the central BCG, one to the east and one to the west of the cluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' These loop-like structures may be the result of a relatively recent merger but given the expected connection between the stars in the ICL and the DM it is interesting to study if similar structures can be found in the distribution of dark matter from the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The previous lens models rely on some parameterization of the mass distribution, usually by placing ellipsoids at the posi- tions of galaxies, and/or large elliptical halos near the center of the cluster to account for the contribution from DM, or by as- suming it follows the cluster galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Hence, they are less than ideal to study the possible correlation between the DM and ICL distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In this paper, we present an additional lens model based on a free-form technique that makes no assump- tions about the underlying distribution of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A comparison between the DM and ICL (or globular cluster) distributions can then be done without being subject to assumptions made about the distribution of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In section 2, we discuss the lensing constraints used to derive the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Section 3 gives a brief introduction to the free-form algorithm used to derive the lens model, making no assumptions about the dis- 2 As this paper was being finished, a new spectroscopic redshift for system 4 (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='211) is provided in Noirot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) tribution of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In section 4, we present the driver model, or Model-1, which is derived using only lensed systems with known redshifts (spectroscopic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Section 5 uses the driver model to make predictions for the redshifts of the candidate lensed sys- tems without spectroscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' These redshifts are an in- teresting alternative to (and some times are more precise than) the more common photometric redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In section 6, we use the lens model predicted redshifts and present two additional mod- els (Model-2 and Model-3), which use all the additional systems with constrained redshifts and, in the case of Model-3, increases also the spatial resolution of the DM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Model-2 and Model-3 are useful to explore the uncertainty in the lens model due to i) the lens system definition (spectroscopic sample vs full sample), and ii) the spatial resolution in the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In sec- tion 7, we study the correlation between the ICL, globular clus- ter distribution and DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We discuss our results and present our conclusions in section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We adopt a standard flat cosmological model with Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' At the redshift of the lens (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='39), and for this cosmology, one arcsecond corresponds to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='29 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Lensing constraints The lensing constraints used in this work are compiled from the three most recent works discussed previously (Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Caminha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Positions, and IDs of these systems are presented in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1 in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' When possible, we maintain the original ID of earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Candidates 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 in Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) are updated with the nearby candidates 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 respectively from Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For convenience, Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1 also includes the IDs used in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Among the systems in this table, five of them have spectroscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' All systems are marked in Figure 1 with circles and their corresponding ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The five systems with spectroscopic redshifts are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In addition to the classic lensing constraints, we add the po- sition of critical curves which can be determined from the radial arc in system 5 (at z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='425), and the merging pair of images in system 7 (at z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The positions of these critical points are added at the end of table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1 and labeled CP5 and CP7 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Each critical point contributes with two constraints as detailed in Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since at a critical point the mag- nification diverges, this can be easily incorporated by applying a rotation to the data by the angle determined by the elongation of the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' After this rotation one can simply impose that the inverse of the tangential magnification equals zero, or similarly 1 = κ−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The second constraint is simply γ2 = 0, which is satisfied when the rotation is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' WSLAP+ To optimize the lens model we use the code WSLAP+ (Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2005a, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Sendra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A lens model derived using WSLAP+ is considered a hybrid type of model as it combines a free-form decomposition of the lens plane for the smooth large-scale component with a small-scale contribution from the member galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Details can be found in references above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Here we give a brief description of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We start with the classic definition of the lens equation β = θ − α(θ, Σ) , (1) Article number, page 2 of 12 Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' : WSLAPping SMACS0723 5” E N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='2 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Central ≈ 1 arcminute region of SMACS0723 with systems of lensed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Systems in white have spectroscopic redshifts and are the ones used to build the driver model or Model-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Systems in yellow do not have spectroscopic redshifts but are used in combination with the spectroscopic systems to build the lens Model-2 and Model-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Note the hundreds of unresolved sources surrounding the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' These are mostly globular clusters and galactic core remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Unless otehrwise noted, all figures in this plot are in the same orientation as this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' where θ is the observed position of the source, α is the deflection angle, Σ(θ) is the unknown surface mass-density of the cluster at the position θ, and β is the unknown position of the background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The optimization of the WSLAP+ solution takes advan- tage of the fact that the lens equation can be expressed as a linear function of the surface mass density, Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' WSLAP+ parameterizes Σ as a linear superposition of functions, which translates into α(θ, Σ) being also linear in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In WSLAP+, the surface mass density, Σ, is described by the combination of two components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' i) a smooth component (usually parameterized as superposition of Gaussians) corre- sponding to the free-form part of the model, or large scale cluster potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' and ii) a compact component that accounts for the mass associated with the individual galaxies in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the smooth component we use Gaussian functions defined over a grid of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A Gaussian function is simple and enables fast computation of the deflection field, but also provides a good compromise between the desired compactness and smoothness of the basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The grid configuration can be defined as regular (all grid points have the same size) or irregular (grid points near the centre are in general smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Adopting a regular grid is similar to a flat prior in the mass distribution while an Article number, page 3 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' SMACS0723 irregular grid can be interpreted as a model with a prior on the mass distribution with higher mass density assigned to smaller cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since one of the goals of this paper is to study the possible correlation between the DM distribution and the ICL, we adopt a regular grid, since this makes minimal assumptions about the mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the compact component, we directly adopt the light distribution in the JWST band F277W around the brightest member elliptical galaxies in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For each galaxy, we assign a mass proportional to its surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is the only free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This mass is later re-adjusted as part of the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The number of parameters connected with the compact component depends on the number of adopted layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Each layer contains a number of member galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The minimum number of layers is 1, corresponding to the case where all galaxies are placed in the same layer, that is, they are all assumed to have the same mass-to-light ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In this case, the single layer is proportional to the light distribution of all member galaxies, and is assigned a fiducial mass for the entire mass of the member galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For each layer there is one extra parameter which accounts for the renormalization constant multiplying the map of the mass distribution, that is optimized by WSLAP+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the particular case of SMACS0723, we use 4 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The first layer contains the main BCG, the second layer contains a large elliptical galaxy ≈ 9” west of the main BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The third layer contains two large elliptical galaxies near the Beret galaxy discussed in Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Finally, the fourth layer contains all remaining member galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Member galaxies are selected from the standard red-sequence, and we also make sure that spectroscopic members identified in Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) are included in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' As shown by Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2005a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2007),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' the strong and weak lensing problem can be expressed as a system of linear equations that can be represented in a compact form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Θ = ΓX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2) where the measured strong lensing observables (and weak lens- ing if available) are contained in the array Θ of dimension NΘ = 2Nsl (plus 2Nwl if weak lensing data is available),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' the un- known surface mass density and source positions are in the array X of dimension NX = Nc + Nl + 2Ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (3) and the matrix Γ is known (for a given grid configuration and fiducial galaxy deflection field) and has dimension NΘ × NX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Nsl is the number of strong lensing observables (each one con- tributing with two constraints, x, and y), Nc is the number of grid points (or cells) that we use to divide the field of view, Nl is the number of layers (Nl = 4 in our case as mentioned above), and Ns is the number of background sources being strongly lensed (each source represent two unknowns in X, βx, and βy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The solution, X, of the system of equations 2 is found after minimizing a quadratic function of X (derived from the system of equations 2 as described in Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2005a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The minimiza- tion of the quadratic function is done with the constraint that the solution, X, has to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since the vector X contains the grid masses, the renormalization factors for the galaxy de- flection field and the background source positions, and all these quantities are always positive (the zero of the source positions is defined in the bottom left corner of the field of view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Impos- ing X > 0 helps constrain the space of meaningful solutions, and to regularise the solution, as it avoids unwanted large nega- tive and positive contiguous fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A detailed discussion of the quadratic algorithm can be found in Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2005a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For a discussion of its convergence and performance (based on simulated data), see Sendra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' As discussed in Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022b), critical points can also be added as extra constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We identify two such constraints in systems 5 (at z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='425) and system 7 (at z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1727) with spec- troscopic redshifts, and include them in our set of lensing con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The addition of these two points act as anchors for the lens model, enforcing the critical curve to pass through the de- sired point at the given redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Driver lens model Using the constraints listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1, we first derive the driver model, or Model-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This model is only based on systems with spectroscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the case of SMACS0723, and at the time of writing these paper, five systems are known to have spectroscopic redshifts3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' These are marked in bold in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the grid, we use a regular distribution of 20 × 20 = 400 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Given the relatively small number of lensing constraints, a significantly larger number of grid points results in nonphysi- cal solutions with large mass fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Based on the 5 lensed systems with spectroscopic redshift, the driver model can be used to predict the redshift of the other system candidates in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We do so in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Redshifts predicted by the lens model Using the driver model, we derive redshifts for all systems listed in table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The probability of a system to be at redshift z is computed by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' P(z) = exp(−V(z)/(2σ2)), (4) where V(z) is the variance between the arc positions of a given system projected on the source plane at redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The projection is done with the deflection field of the driver model (computed at redshift z = 3) which is re-scaled to the desired redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The dispersion, σ, in the expression above is fixed to three pixels, or ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='18”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is a reasonable choice for well constrained sys- tems, resulting in relatively narrow distributions for the redshift, and with uncertainty in the error prediction consistent with the observed error (see for instance Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022a, for a more in depth analysis of the errors expected with this technique and for WSLAp+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Systems that are well reproduced by the driver model result in a small variance V(z) near the optimal redshift, which in turn result in maximum values of P(z) close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Sys- tems that are poorly reproduced by the driver model have larger values of V(z), which reduce the maximum value of P(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A low maximum probability for P(z) does not necessarily mean that the system is a bad candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This can simply be the result of the driver model not being well constrained in that part of the lens system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Systems at high redshift tend to have broader probabil- ities, since for source redshift z > 2 the deflection field varies slowly with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 3 As noted earlier, a new spectroscopic redshift was recently made available for system 4 in Noirot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) at the time of finishing this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This new redshift (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='211) was not used in our analysis where we adopted our geometric redshift estimate (z=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The difference in redshift is small and is not expected to have any significant impact in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Article number, page 4 of 12 Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' : WSLAPping SMACS0723 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Redshifts predicted by the driver model for the case of well con- strained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The derived probabilities P(z) can be divided in two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In the first group we find systems with well defined and rela- tively narrow probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The probabilities for these systems are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Among these we find the systems with spectroscopic redshifts that were used to derive the driver model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Naturally, the maximum of P(z) for these systems falls very close to the spectroscopic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' System 4 had its spectroscopic red- shift estimated recently in Noirot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) where they find z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' As shown in Figure 2, the P(z) for this system con- tains the correct redshift within the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In a different group we find systems for which the redshift is not so well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The probabilities for these systems is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Two systems (14 and 26) have no constrain on their redshift (z>13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The bad performance of these systems can be easily understood since they correspond to cases of galaxy- galaxy lensing, where the member galaxy acting as a lens is not optimized individually (these galaxies are part of layer 4 dis- cussed in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' System 8 has a very low probability of P(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This probability is shown as a dashed line in Figure 3, and the probability has been multiplied by a factor 100, to make it visible in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The low probability of system 8 can be interpreted as being a bad system or being in a region with poor constraints in the driver model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is the case on the western part of the cluster, where system 8 lies, since only one system has a spectroscopic redshift in this region of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Redshifts predicted by gravitational lenses are an interesting alternative to photometric redshifts, specially for high-redshift candidates for which photometry may be poor or nonexistent in more than one band (such as in the case of dropouts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A similar technique has been used in the past in other lenses with positive results and was recently applied in Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022a) to predict the redshift of the new systems identified in the JWST data of El Gordo cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Full-sample lens models Taking advantage of the redshifts predicted by the lens model discussed in the previous section, we expand the number of constraints and update the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' As discussed above, the redshift for systems 14 and 26 cannot be constrained by the lens model so we exclude these two systems from our list of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The remaining number of systems totals 25, and the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Redshifts predicted by the driver model for the case of poorly constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The probability of system 8, shown as a dashed line at z ≈ 8, has been multiplied by a factor 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' number of constraints exceeds 150 (x and y positions of each arc plus the two critical point positions, each contributing also with two constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Using these constraints we derive two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' One that we refer to as Model-2, which is derived with the full set of constraints (excluding systems 14 and 26), and a regular grid of 20 × 20 = 400 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We increase the number of grid points to 25 × 25 = 625 in a third model that we refer to as Model-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Increasing the number of grid points even further can result in unstable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For instance, with a grid of 30 × 30 = 900 points we obtain a solution that places too much mass in the edges of the field of view and introduces relatively large mass fluctuations across the entire field so we do not consider solutions with more than 625 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The critical curves for Model-2 and Model-3 are shown in Figure 4 as green and blue curves respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For convenience we include again in this figure the critical curve for the driver model (or Model-1) in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' All three curves are again computed at the redshift of system 7 (z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1727).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The three models produce consistent results in the eastern part of the lens, which is the portion of the cluster where the number density of spectroscopic redshifts is the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In contrast, the critical curves in the west differ significantly from one another, indicating that the western part of the cluster is more poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The addition of new spectroscopic systems in this part of the lens will reduce the uncertainty in the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In terms of mass, we can compare with previous published results based on parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Both Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) and Caminha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) quote the total projected mass within a cylinder of radius 128 kpc centered in the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This radius corresponds approximately to the Einstein radius for a source at z > 2, and it is the radius within which the lens model can be properly constrained with strong lensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' They find masses of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='04 × 1013M⊙ and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='2 × 1013M⊙ respectively, and within the aforementioned 128 kpc radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For our three lens models we find 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='28×1013M⊙, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='31×1013M⊙, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='15×1013M⊙ for Model-1, Model-2 and Model-3 respectively, and within the same radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is approximately 10% less than in the para- metric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Article number, page 5 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' SMACS0723 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Critical curves of alternative lens models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' All critical curves are computed at the redshift of system 7 (z=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1727).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The red curve corre- sponds to the driver model derived with the five spectroscopic systems and a grid of 20x20 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The green curve uses the same grid configu- ration but is derived from the 25 systems with constrained redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The blue curve uses the same 25 systems but is based on a higher resolution grid of 25x25 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Correlation between the dark matter, intracluster light, and globular cluster distributions In all three models discussed in the previous sections we find that the critical curves for the three models are consistent among them, with the largest differences concentrating in the west portion of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Hence the lens model is relatively well constrained for different choices of lensed systems and grid configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In this section, we pay special attention to the distribution of light in the ICL, and the distribution of globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We are interested in the possible correlation between the ICL, globular cluster and the DM distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Figure 5 shows how the ICL presents two loop-like structures to the east and west of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' At ≈ 200 kpc from the central BCG and towards west, a cavity-like structure can be appreciated in the ICL (Giant West Loop in Montes & Trujillo 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Although not as clear, a similar cavity can be also observed towards the east at approximately the same distance from the central BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is a surprising feature in the ICL where one expects to find more uniform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Recent merger activity can result in trails of stars being stripped away from their host galaxy by tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' These tails are however much smaller than the observed loops in the ICL of SMACS0723, and connect with the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In the case of the loops of SMACS0723, one cannot establish any correspondence between the loops and a member galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' On the other hand, as noted by Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022), the offset between the radial velocity of the central BCG and the mean redshift of the cluster suggests a recent past merger (a relaxed cluster would have no offset), offering a possible explanation for the odd distribution of the ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Whatever the cause for the morphology of the ICL, it is interesting to compare its two-dimensional distribution with the distribution of mass from our lens models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since approximately 85% of the projected mass of the cluster is expected to be dark matter, if dark matter and the ICL are related, we should expect a correlation between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In Figure 5 we show as yellow contours the DM distribution from our Model-2 while in blue we show the contours for the DM distribution from our Model-3 (Model-1 is not shown but it is very similar to Model-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The DM component is obtained after subtracting the mass associated to the galaxies from the total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In general we find good correspondence between the distribution of the ICL and the two DM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A more quantitative comparison is shown in Figure 6 where we compute the average of the ICL or the DM along a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This line is shown in Figure 5 and it intersects the ICL from west to east, passing through the central BCG in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The average is computed at each position as the mean over a box of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='18” × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='18” and centered in the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The black curve in Figure 6 corresponds to the light distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The colored lines are for the driver model or Model-1 (red), Model-2 (green), and Model-3 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The curves for the DM models have been re- scaled by an arbitrary number to match the black curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In the east part of the cluster we find good correspondence between all three models and the ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is not true in the west part of the cluster, where the cavity clearly seen in the ICL at ≈ −200 kpc in Figure 6 is not observed in any of the DM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In addition to the ICL, another possible tracer of the potential are globular clusters, whose distribution could correlate with the distribution of dark matter, since as the stars in the ICL, globular clusters respond to gravitational forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The superior sensitivity and spatial resolution of JWST allows to detect these clusters with unprecedented detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Preliminary results based on JWST data in SMACS0723 are presented in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Faisst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' It is interesting to compare our results with those from earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Figure 7 compares the observed ICL in the F356W filter with the distribution of dark matter (yellow con- tour) and the distribution of globular clusters (blue contours) from Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' To compute the blue contours we have smoothed the distribution of globular clusters with a Gaussian of FWHM=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='5".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' To first order, there is a good spatial correspon- dence between the DM, ICL and globular cluster distribution, with all three components centered in the same point (BCG) and having similar alignments in the east-west direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' As in the case of the ICL, the distribution of globular clusters appears to show a similar deficit in number density at the position of the cavity on the west side of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This cavity has no corre- spondence in the distribution of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In terms of radial profiles, we show a comparison of our lens model with the ICL profiles from Montes & Trujillo (2022) and the globular cluster profile from Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the globular clusters, we have re-scaled the surface number density (expressed as number per kpc2) by a factor 2 × 109 in order for the resulting profile to overlap with the ICL profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For comparison we plot a power law R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 as a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This power law reproduces well the profile of the ICL and the globular cluster number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The mass profiles from the three lens models are shown as a solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Within the inner 20 kpc region, the total mass and the ICL have similar profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is expected in our lens model since the compact component of the lens model takes directly the light distribution of member galaxies, including the central BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since near the centre of the BCG, the bulk of the mass is expected to come from stars (or the baryonic component in general), by comparing our lens model with the ICL profile from Montes & Trujillo (2022), we find that either i) there is ≈ 10 times more dark matter than stellar mass within the central 20 Article number, page 6 of 12 CDiego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' : WSLAPping SMACS0723 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Projected total mass vs ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The contours represent the smooth component of the lens model obtained with the 25 constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The yellow contour is obtained with a regular grid of 20x20 cells while the blue contour is obtained with a higher resolution grid of 25x25 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The image is a masked version of the F277W band, where the ICL light can be better appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The contours correspond to values of the con- vergence, κ, computed at a fiducial source redshift of zs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Space be- tween contours correspond to δκ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1, with values starting at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The last contour is for κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The white straight line marks the di- rection over which we construct the one-dimensional scan of the light profile and DM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' kpc or ii) the stellar mass from the ICL is underestimated by some factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Beyond ≈ 20 kpc, the total mass profile is clearly shallower than the profile of the ICL and the number density of globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This departure is interesting and needs to be studied in other clusters with more constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Increasing the number of lensing constraints will allow to improve the spatial resolution of the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Discussion and conclusions The new data from JWST reveals a wealth of new candidate lensed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Future observations of these candidates will se- cure their redshifts, which can then be compared with the ge- ometric redshift estimate based on our driver model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' If spec- troscopic confirmation validates the method of estimating dis- tances through geometric redshifts, future observations by the JWST can take advantage of a similar technique, where a hand- ful of spectroscopic lensed galaxies may suffice to calibrate a lens model for distance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Recent work has shown how photometric redshifts can predict erroneous redshifts for the case of dropout galaxies in the JWST bands (Harikane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Zavala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' An independent estimation of the distance to these galaxies can help reduce the uncertainty in the estimation of the redshift, and identify those galaxies that have large photometric redshifts (z>10) yet they are predicted by the lens model to be at much lower redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Lens models like the one presented in this work are also needed to interpret sources near caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In the case of SMACS0723, Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) discuss a small pair of knots in the middle of the merging pair of images of system 5 (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' One dimensional scan of the light distribution vs DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The x- axis is the distance to the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The solid black line shows the mean of the light emission in the F277W band along the straight line shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The mean is computed over a box of 10x10 pixels at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The colored lines are the corresponding mean of the DM com- ponent for the three lens models discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The red color is for the driver model, the green line is for Model-2, or low-resolution (20x20 grid points) with 25 systems, and the blue model is for the high- resolution (25x25 grid points) Model-3 with 25 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The DM pro- files are re-scaled by arbitrary units to visually match the profile of the light emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' See Figure 8 for a direct comparison of the profiles with- out the re-scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since the lens model has a resolution comparable to the separation between the knots in the pair, the magnification in these knots is better estimated by interpolating the magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Based on symmetry arguments, the critical curve must pass be- tween these two points, so they are equidistant to it (d ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='08”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since the magnification near a fold caustic scales as µ = A/d (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 1992), we can estimate A from our lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We find A ≈ 58”, which results in µ ≈ 725 for each one of the images in the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This estimate matches very well the value quoted in Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) of µ ≈ 750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The greater sensitivity of JWST to the ICL offers new op- portunities to study the correlation between the DM and ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In addition, the improved spatial resolution in the infrared bands allows for detection of small clumps of old stellar populations in the cluster stripped from their hosts galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The first image of JWST on this cluster reveals hundreds of unresolved clumps that are interpreted as globular clusters (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Faisst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022), but could be also the surviving remnants (after a close en- counter with a larger galaxy in the cluster, such as the BCG) of compact galactic cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Both the stars in the ICL and the globu- lar clusters are expected to interact with the rest of the matter in the cluster mostly through gravitational forces, and hence behave similar to dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In this work we use a free-form modelling technique which makes minimal assumptions about the distribution of dark mat- ter, and find that in general the DM traces well the ICL and glob- ular cluster distribution but we also find that the small loop-like structures (and associated cavities) to the east and west of the central region of SMACS0723 have no obvious correspondence in the DM distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' At distances from the centre comparable to the Einstein radius (∼ 100 kpc, and hence well constrained by the available data) we find that the dark matter profile is signifi- cantly shallower than the ICL and globular cluster distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This is also found in simulation of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In Alonso Article number, page 7 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' SMACS0723 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Comparison of the dark matter and globular cluster distribution (number density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The image corresponds to the F356W JWST filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Yellow contours are the smooth component of the dark matter distribu- tion (Model-2), while blue contours are for a Gaussian filtered version (FWHM=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='5") of the distribution of globular clusters from Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Asensio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2020), the authors analyze the EAGLE simula- tions and find that the ICL profile is steeper than the total mass profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In particular, they find that the ratio between the ICL and total mass profiles is a power law with slope −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Interestingly, in the range between ≈ 20 kpc and ≈ 200 kpc we find a similar ratio between the total mass and ICL (and globular cluster) pro- files, with the ICL and globular cluster profiles falling as ∼ R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 while the total mass falls as ∼ R−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3 (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Similar con- clusions are found in Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2018) where, based on the IllustrisTNG simulations, the 3D profile of the ICL in massive clusters is found to fall faster (by approximately 1 dex from their Figure 6 at around 100 kpc distance) than the canonical NFW profile commonly used to describe dark matter profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Earlier work based on the EAGLE simulations shows a similar trend (Schaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Hence, our results on SMACS0723 are in agreement with the ones derived from N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We observe differences in the range ≈ 20–200 kpc between the ICL (and globular cluster) profiles and the total mass (mostly dominated by dark matter in this distance range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' We specu- late that this may be related to the different formation times of the cluster dark matter halo and ICL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Since dark matter is more loosely bounded to their host halos (as it mostly resides on the outskirts of the galaxies, with the central region being more baryon dominated), it can be stripped more easily during the first encounters with the cluster and hence retaining the initial (rela- tively large) angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The baryonic component (stars in our case) is more concentrated around the centre of the satel- lite galaxies and can survive more encounters with the cluster, and without being stripped away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In each encounter, the satellite galaxy looses angular momentum due to dynamical friction and can get closer to the BCG (Contini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Chun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Stars that are stripped at a later time lose part of their bulk kinetic energy this way, and when stripped from their hosts can remain at shorter radii, resulting in profiles that are steeper (more con- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Comparison of the total mass profile from the three lens models (solid lines) with the ICL profile from Montes & Trujillo (2022) (shaded orange and blue regions for the East and West sectors respectively) and the globular cluster number profile from Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the later, we have re-scaled the number density by an arbitrary num- ber of 2×109 in order to overlap with the ICL profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The black dashed line is a power law that scales with distance as R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The red dashed line is a power law that scales as R−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' centrated) than the dark matter profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Globular clusters and galactic core remnants are subject also to dynamical friction and hence expected to orbit closer to the BCG, resulting in more con- centrated profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' N-body simulations also show how the radial distribution of subhalos is steeper than the distribution of dark matter (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The presence of cavities in the distribution of the ICL, and not detected in the total mass distribution, is another interesting difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The formation of cavities in the ICL but not in the DM distribution could be due to the different distribution of stars and dark matter inside the satellite galaxies before they enter the galaxy cluster, and the striping mechanism starts to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The dark matter, forming an extended halo around the satellite galaxy, is easily tidally stripped from its host galaxy as it enters the cluster and starts orbiting around the BCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The better ability of the baryonic matter to cool down more efficiently and form more concentrated structures like disks or bulges facilitates the survival of the bulge (or disk) as they orbit the minimum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' During a close encounter with the BCG, parts of the bulge or disc of a satellite galaxy can be tidally stripped, creating the loop-like structures and associated cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Tidal stripping of satellite galaxies has been claimed as responsible for filamen- tary structures seen in the ICL of the nearby Virgo cluster (Mihos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Structures that resemble the loop-cavity system are also observed in nearby galaxies that had recent encounters with satellite galaxies (Martinez-Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In simulations, faint structures in the ICL, that resemble the loop-cavity struc- tures can be appreciated in Figures 3 and 4 in Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Perhaps one of the most interesting findings is the connec- tion between the ICL and globular cluster distribution, both hav- ing a similar profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This connection could be easily explained if the ICL corresponds to the outer envelopes of the alleged glob- ular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In this case, the globular clusters should be re- interpreted as the surviving galactic cores of the infalling satel- lite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' More examples like the one studied in this work are needed in order to extract a firmer conclusion regarding the connection between the ICL, globular cluster and DM distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In par- Article number, page 8 of 12 N 10″Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' : WSLAPping SMACS0723 ticular, the addition of new constraints (with confirmed spectro- scopic redshift) will allow us to increase the resolution of the lens model, revealing perhaps finer details in the distribution of DM that can not be unveiled with the current set of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For the particular case of SMACS0723, the number of lensing constraints around the west cavity is very small (≈ 4 lensed galaxies in this region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Future analysis based on JWST data, especially of low redshift clusters for which both ICL and glob- ular clusters are more easily detected, and with abundant lens- ing constraints (such as the Hubble Frontier Fields Clusters) will enable more precise conclusions on the correlation between the ICL, globular cluster and DM distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' acknowledges the support of project PGC2018- 101814-B-100 (MCIU/AEI/MINECO/FEDER, UE) Ministerio de Ciencia, In- vestigación y Universidades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' This project was funded by the Agencia Estatal de Investigación, Unidad de Excelencia María de Maeztu, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' MDM-2017- 0765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' was funded through the NSF Graduate Research Fellowship grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' DGE 1752814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' acknowledges support by Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2020750 from the United States-Israel Binational Science Foundation (BSF) and Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2109066 from the United States National Science Foundation (NSF), and by the Ministry of Science & Technology, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No MARACHAS-DLV- 896778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' is supported by the United Kingdom Research and Innovation (UKRI) Future Leaders Fellowship ‘Using Cosmic Beasts to uncover the Na- ture of Dark Matter’ (grant number MR/S017216/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' was supported by the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='01816 Article number, page 9 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' SMACS0723 Appendix A: Arc positions and redshifts This appendix presents all arc system candidates used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The table is a compilation of systems presented in Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Caminha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' The last two rows are the positions of the two critical points used as extra constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' IDs of systems is shown in column one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Columns two and three give RA, DEC positions in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Column four lists the spectroscopic redshifts when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Spectroscopic redshifts are marked in bold face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Column five lists the redshifts predicted by the driver model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' In this case, errors correspond to the 68% interval of the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' For all systems, only the first arc is given with its redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Colums six, seven, and eight give the original ID in Pascale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022), Caminha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022), and Mahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' ‡While finishing this paper, Noirot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' (2022) published spec- troscopic redshifts of some galaxies in the field of SMACS0723 including a redshift for our system 4, for which they find zspec = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='211, in good agreement (2σ) with our geometric redshift esti- mate (zgeo = 2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' No other redshifts are published for the remaining arcs without spectroscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Article number, page 10 of 12 Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' : WSLAPping SMACS0723 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' Arc positions and redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfDwbQ/content/2301.03629v1.pdf'} +page_content=' ID RA DEC zs zm IDP IDC 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b/XtFQT4oBgHgl3EQfdTZh/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ccdc58ff1ae6883ce33e38bbcc8cefe10095083bd2e961d8f2799fff4409f95b +size 4128813 diff --git a/ZNE4T4oBgHgl3EQfOQxg/content/tmp_files/2301.04963v1.pdf.txt b/ZNE4T4oBgHgl3EQfOQxg/content/tmp_files/2301.04963v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ffbd67914ce36140e01147c817f787a36adfd3be --- /dev/null +++ b/ZNE4T4oBgHgl3EQfOQxg/content/tmp_files/2301.04963v1.pdf.txt @@ -0,0 +1,3210 @@ +arXiv:2301.04963v1 [math.RT] 12 Jan 2023 +Normal subgroups and support τ-tilting modules ∗† +Ryotaro KOSHIO +Yuta KOZAKAI +January 13, 2023 +Abstract +Let ˜G be a finite group, G a normal subgroup of ˜G and k an algebraically closed field of character- +istic p > 0. The first main result in this paper is to show that support τ-tilting k ˜G-modules satisfying +some properties are support τ-tilting modules as kG-modules too. As the second main result, we give +equivalent conditions for support τ-tilting k ˜G-modules to satisfy the above properties, and show that +the set of the support τ-tilting k ˜G-modules with the properties is isomorphic to the set of ˜G-invariant +support τ-tilting kG-modules as partially ordered sets. As an application, we show that the set of +˜G-invariant support τ-tilting kG-modules is isomorphic to the set of support τ-tilting k ˜G-modules in +the case that the index G in ˜G is a p-power. As a further application, we give a feature of vertices of +indecomposable τ-rigid k ˜G-modules. Finally, we give the block versions of the above results. +1 +Introduction +Since 2014 when τ-tilting theory was introduced by T. Adachi, O. Iyama, and I. Reiten [1], the theory +continues to develop rapidly. The main theme of the theory is the study of support τ-tilting modules, and +many researchers have given the work on these. In fact, the support τ-tilting modules over finite dimen- +sional algebras are under the one-to-one correspondences with the various representation-theoretically +important objects including two-term silting complexes [1], functorially finite torsion classes [1], left- +finite semibricks [3], two-term simple-minded collections [3, 15], and so on. In particular, the theory is +expected to be helpful in solving Brou´e’s abelian defect group conjecture because the theory is useful +for the classification of two-term tilting complexes over group algebras or block algebras of finite groups. +Even though the studies on the τ-tilting theory related to the modular representation theory of finite +groups are very important for these reasons, there are few such studies. Therefore, the authors have given +the studies combining the above two theories [16, 17, 18]. All of them show that the induction functors +from kG-modules to k ˜G-modules give the poset homomorphisms from the support τ-tilting modules over +kG to those over k ˜G under appropriate assumptions, where G is a normal subgroup of a finite group ˜G +and k an algebraically closed field of characteristic p > 0. Naturally, we are interested in the following +question. +Question 1.1. When the restriction functor from k ˜G-modules to kG-modules give the maps from the +support τ-tilting modules over k ˜G to those over kG? +In regarding this question, in [5], S. Breaz, A. Marcus, and G. C. Modoi gave a positive answer in +case that the quotient group ˜G/G is a p-prime group (i.e. the prime number p does not divide the order +of the factor group ˜G/G). Therefore, we consider the case that ˜G/G is not necessarily a p-prime group, +and get the following positive answer for the question. +Theorem 1.2 (Theorem 3.4 and Corollary 3.5). Let ˜G be a finite group, G a normal subgroup of ˜G, ˜ +M a +relatively G-projective support τ-tilting k ˜G-module, and ( ˜ +M, ˜P) a corresponding support τ-tilting pair. If +it holds that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M, then Res +˜ +G +G ˜ +M is a support τ-tilting kG-module, and (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) +a corresponding support τ-tilting pair. Moreover, for relatively G-projective support τ-tilting k ˜G-modules +˜ +M1 and ˜ +M2 with the property that Ind +˜ +G +GRes +˜ +G +G ˜ +Mi ∈ add ˜ +Mi for i = 1, 2, if ˜ +M1 ≥ ˜ +M2 in sτ-tilt k ˜G, then +Res +˜ +G +G ˜ +M1 ≥ Res +˜ +G +G ˜ +M2 in sτ-tilt kG. +∗Mathematics Subject Classification (2020). 20C20, 16G10. +†Keywords. Support τ-tilting modules, blocks of finite groups, induction functors, restriction functors +1 + +Let M be a ˜G-invariant support τ-tilting kG-module. +The first author showed that Ind +˜ +G +GM is a +support τ-tilting k ˜G-module [16, Theorem 3.2]. We are interested in what is the image of the set of ˜G- +invariant support τ-tilting kG-modules under the map induced by the induction functor Ind +˜ +G +G. Therefore, +we give equivalent conditions to the assumption of Theorem 1.2, and finally we clarify what the image of +the map induced by Ind +˜ +G +G is in the following theorem. +Theorem 1.3 (Theorem 3.8 and Corollary 3.9). Let ˜ +M be a support τ-tilting k ˜G-module. Then the +following conditions are equivalent: +(1) +˜ +M =add Ind +˜ +G +GM for some ˜G-invariant support τ-tilting kG-module M. +(2) Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and ˜ +M is relatively G-projective. +(3) S ⊗k ˜ +M ∈ add ˜ +M for each simple k( ˜G/G)-module S. +Moreover, denoting by (sτ-tilt kG) ˜ +G the subset of sτ-tilt kG consisting of ˜G-invariant support τ-tilting kG- +modules and by (sτ-tilt k ˜G)⋆ the subset of sτ-tilt k ˜G consisting of support τ-tilting k ˜G-modules satisfying +the above equivalent conditions, the induction functor Ind +˜ +G +G induces a poset isomorphism +(sτ-tilt kG) ˜ +G +(sτ-tilt k ˜G)⋆ +M +Ind +˜ +G +GM. +∼ +The studies on the vertices of indecomposable modules over group algebras have been done by many +researchers for a long time, for example, see [6, 7, 8, 11, 14, 22]. On the other hand, τ-rigid modules over +finite dimensional algebras are important classes and there are many studies on the modules. They have +nice properties and are essential objects for the representation theory, for example, see [1, 4, 9, 10, 13]. +Therefore, one of our interests is to give a feature of the vertices of indecomposable τ-rigid modules over +group algebras. As a further application of Theorem 1.3, we give a feature of vertices of indecomposable +τ-rigid modules. +Theorem 1.4 (See Theorem 3.14). Let ˜G be a finite group. Then any indecomposable τ-rigid k ˜G-module +has a vertex contained in a Sylow p-subgroup of ˜G properly if and only if ˜G has a normal subgroup of +p-power index in ˜G. +As a natural question, we wonder if we get the block version of our theorems for group algebras. In +particular, we are interested in how we give the block versions of Theorems 1.2 and 1.3. As the results, +we get the block versions of the theorems. Let G be a normal subgroup of a finite group ˜G and B a block +of kG. We denote by I ˜ +G(B) := +� +˜g ∈ ˜G +��� ˜gB˜g−1 = B +� +the inertial group of B in ˜G. +Theorem 1.5 (See Theorem 5.6). Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B +a block of k ˜G covering B, β the block of kI ˜ +G(B) satisfying +� +x∈[ ˜ +G/I ˜ +G(B)] +x1βx−1 = 1 ˜ +B +and ˜ +M a support τ-tilting ˜B-module. If it holds that βInd +I ˜ +G(B) +G +Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M ∈ add βRes +˜ +G +I ˜ +G(B) ˜ +M +and βRes +˜ +G +I ˜ +G(B) ˜ +M is relatively G-projective, then we have that Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M is a support τ-tilting +B-module. Moreover, if ( ˜ +M, ˜P) is a support τ-tilting pair for ˜B corresponding to ˜ +M, then the pair +(Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M, Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜P) +is a support τ-tilting pair for B corresponding to Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M. +Theorem 1.6 (See Theorem 5.7 and Corollary 5.8). Let G be a normal subgroup of a finite group ˜G, B +a block of kG, ˜B a block of k ˜G covering B, β the block of kI ˜ +G(B) satisfying +� +x∈[ ˜ +G/I ˜ +G(B)] +x1βx−1 = 1 ˜ +B +and ˜ +M a support τ-tilting ˜B-module. Then the following conditions are equivalent: +2 + +(1) +˜ +M =add ˜BInd +˜ +G +GM for some I ˜ +G(B)-invariant support τ-tilting B-module M. +(2) βInd +I ˜ +G(B) +G +Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M ∈ add βRes +˜ +G +I ˜ +G(B) ˜ +M and βRes +˜ +G +I ˜ +G(B) ˜ +M is relatively G-projective. +(3) β(S ⊗k βRes +˜ +G +I ˜ +G(B) ˜ +M) ∈ add βRes +˜ +G +I ˜ +G(B) ˜ +M for each simple k(I ˜ +G(B)/G)-module S. +Moreover, denoting by (sτ-tilt B)I ˜ +G(B) the subset of sτ-tilt B consisting of I ˜ +G(B)-invariant support τ- +tilting B-modules and by (sτ-tilt ˜B)⋆⋆⋆ the subset of sτ-tilt ˜B consisting of support τ-tilting ˜B-modules +satisfying the above equivalent conditions, the functor ˜BInd +˜ +G +G induces a poset isomorphism +(sτ-tilt B)I ˜ +G(B) +(sτ-tilt ˜B)⋆⋆⋆ +M +˜BInd +˜ +G +GM. +∼ +Our particular interest is the case that the index of G in ˜G is a p-power. +In fact, under some +assumptions, it is expected that tilting complexes over the block B of kG give those over the unique +block ˜B of k ˜G covering B (for example, see [12, 19, 25]). In this regard, the authors showed the following +result in [18]. +Theorem 1.7 ([18, Theorem 1.2]). Let G be a normal subgroup of a finite group ˜G of p-power index in +˜G, B a block of kG, and ˜B the unique block of k ˜G covering B. Assume that the following two conditions +are satisfied: +(1) Any indecomposable B-module is I ˜ +G(B)-invariant. +(2) The set of isomorphism classes of basic support τ-tilting B-modules is a finite set. +Then the induction functor Ind +˜ +G +G induces an isomorphism from sτ-tilt B to sτ-tilt ˜B of partially ordered +sets. +This theorem can be applied to the case that the block B has a cyclic defect group, but the two +conditions limit the scope of its use. For example, the theorem cannot be applied to the case that p = 2, +G is the alternating group A4 of degree 4 and that ˜G is the symmetric group S4 of degree 4, because the +nontrivial simple kA4-modules are not S4-invariant. Indeed, sτ-tilt kA4 is not isomorphic to sτ-tilt kS4 +because the number of isomorphism classes of simple kA4-modules is three and that of kS4 is two. +However, we wonder if the induction functor might give some kinds of good relation between the special +subsets of the two, and finally, as an application of Theorem 1.6, we could get the following theorem +which can be applied to the case of kA4 and kS4. The following theorem is a significant generalization of +Theorem 1.7 and enables us to explain the phenomenon occurred in [16, Example 3.9] (see Example 3.11). +Theorem 1.8 (See Theorem 5.9). Let G be a normal subgroup of a finite group ˜G, B a block of kG and +˜B a block of k ˜G covering B. If the quotient group I ˜ +G(B)/G is a p-group, then the functor Ind +˜ +G +G induces +an isomorphism as partially ordered sets between (sτ-tilt B)I ˜ +G(B) and sτ-tilt ˜B, where (sτ-tilt B)I ˜ +G(B) is +the subset of sτ-tilt B consisting of I ˜ +G(B)-invariant support τ-tilting B-modules. +Throughout this paper, we fix the following notation: +Let k be an algebraically closed field of characteristic p > 0. An algebra means a k-algebra. For a +finite dimensional algebra Λ, a Λ-module means a finite dimensional left Λ-module. For a Λ-module M, +we denote the Auslander-Reiten translate of M by τM. In case that Λ is a symmetric algebra, τM is +isomorphic to Ω2M. We denote the category of all direct summands of finite direct sums of copies of +M by add M. For Λ-modules M and N, we write M =add N if add M = add N. This relation is an +equivalence relation. We denote by sτ-tilt Λ the set of equivalence classes of support τ-tilting Λ-modules +under the equivalence relation =add. +Let G be a finite group and H a subgroup of G. We denote the restriction functor from kG-modules +to kH-modules by ResG +H and the induction functor kG ⊗kH − from kH-modules to kG-modules by IndG +H. +We denote the trivial kG-module by kG. +Let ˜G be a finite group, G a normal subgroup of ˜G. We denote a set of coset representatives of G in +˜G by [ ˜G/G]. For a kG-moduleM and ˜g ∈ ˜G, we define a kG-module ˜gM consisting of symbols ˜gm as a +set, where m ∈ M, and its kG-module structure is given by ˜gm+ ˜gm′ := ˜g(m+ m′), g(˜gm) := ˜g(˜g−1g˜gm) +and λ(˜gm) = ˜g(λm) for m, m′ ∈ M, g ∈ G and λ ∈ k. For a kG-module M, we say that M is ˜G-invariant +if M is isomorphic to ˜gM for any ˜g ∈ ˜G. +3 + +2 +Preliminaries +In this section, we give elementary facts on the modular representation theory which are helpful to prove +our results. +Proposition 2.1 (See [2, Lemma 8.5, Lemma 8.6]). Let G be a finite group, K a subgroup of G, H a +subgroup of K. For any kG-module U and kH-module V , the following hold: +(1) ResG +HU ∼= ResK +HResG +KU. +(2) IndG +HV ∼= IndG +KIndK +HV . +(3) IndG +H(V ⊗k ResG +HU) ∼= (IndG +HV ) ⊗k U. +(4) HomkG(U, IndG +HV ) ∼= HomkH(ResG +HU, V ). +(5) HomkG(IndG +HV, U) ∼= HomkH(V, ResG +HU). +(6) The functors ResG +H and IndG +H send free modules (projective modules) to free modules (projective +modules, respectively). +In the modular representation theory of finite groups, Mackey’s decomposition formula is well-known +and important. We recall Mackey’s decomposition formula for normal subgroups. +Proposition 2.2 (See [2, Lemma 8.7]). Let G be a normal subgroup of a finite group ˜G and M a +kG-module. Then the following isomorphism as kG-modules holds: +Res +˜ +G +GInd +˜ +G +GM ∼= +� +x∈[ ˜ +G/G] +xM. +The following is known as Eckmann-Shapiro Lemma. +Lemma 2.3 (See [21, Proposition 2.20.7]). Let H be a finite group of a finite group G, M a kH-module +and N a kG-module. Then for all n ∈ N, there exists an isomorphism of k-vector spaces: +Extn +kH(M, ResG +HN) ∼= Extn +kG(IndG +HM, N) +The following lemma is a refinement of [16, Lemma 3.1] which requires the ˜G-invariance for the +kG-module. +Lemma 2.4. Let G be a normal subgroup of ˜G and M a kG-module. Then the following hold: +(1) Ind +˜ +G +G(ΩM) ∼= Ω(Ind +˜ +G +GM). +(2) Ind +˜ +G +G(τM) ∼= τ(Ind +˜ +G +GM). +Proof. We enough to show that the statement (1) holds since τ ∼= Ω2 for symmetric algebras. There exists +a projective kG-module Q such that Ind +˜ +G +G(ΩM) ∼= Ω(Ind +˜ +G +GM)⊕Q and that Ind +˜ +G +GP(M) ∼= P(Ind +˜ +G +GM)⊕Q. +Hence, we have that +Res +˜ +G +GInd +˜ +G +G(ΩM) ∼= Res +˜ +G +GΩ(Ind +˜ +G +GM) ⊕ Res +˜ +G +GQ, +and the left-hand side is isomorphic to � +x∈[ ˜ +G/G] xΩM by Proposition 2.2. However, each xΩM has no +projective summands and the restricted module Res +˜ +G +GQ is a projective kG-module by Proposition 2.1 (6), +which implies that Q = 0. Therefore, we conclude that Ind +˜ +G +G(ΩM) ∼= Ω(Ind +˜ +G +GM). +Lemma 2.5. Let G be a normal subgroup of a finite group ˜G and ˜ +M be a k ˜G-module. Then Res +˜ +G +G ˜ +M is +a ˜G-invariant kG-module. +4 + +Proof. Take ˜g ∈ ˜G arbitrarily. We consider the map +f : Res +˜ +G +G ˜ +M +˜gRes +˜ +G +G ˜ +M +m +˜gm. +Clearly, this map is linear and bijective. We only show that the map is kG-homomorphism, but for any +g ∈ G and m ∈ Res +˜ +G +G ˜ +M, it holds that +f(gm) = ˜ggm = ˜gg˜g−1˜gm = g · ˜gm = g · f(m). +3 +Main Theorems +In this section, we give theorems stated in Section 1 and their proofs. Throughout this section, ˜G means +a finite group and G a normal subgroup of ˜G. +First, we start with a consideration on restricted modules of rigid modules and τ-rigid modules. Let Λ +be a finite dimensional algebra. We recall that a Λ-module M is rigid (resp. τ-rigid) if Ext1 +Λ(M, M) = 0 +(resp. HomΛ(M, τM) = 0). We remark that τ-rigid modules are rigid modules by Auslander-Reiten +duality HomΛ(X, Y ) ∼= D Ext1 +Λ(Y, τX). +Lemma 3.1. Let ˜ +M be a k ˜G-module with the property that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M. Then the following +hold: +(1) If ˜ +M is a rigid k ˜G-module, then the restricted module Res +˜ +G +G ˜ +M is a rigid kG-module. +(2) If ˜ +M is a τ-rigid k ˜G-module, then the restricted module Res +˜ +G +G ˜ +M is a τ-rigid kG-module. +Proof. (1) Let ˜ +M be a rigid k ˜G-module. Then, by Lemma 2.3, we have that +Ext1 +kG(Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜ +M) ∼= Ext1 +k ˜ +G( ˜ +M, Ind +˜ +G +GRes +˜ +G +G ˜ +M). +By the assumption that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and the rigidity of ˜ +M, we have that the right-hand side is +0. Hence, Res +˜ +G +G ˜ +M is a rigid kG-module. +(2) Let ˜ +M be a τ-rigid k ˜G-module. Then we have that +HomkG(Res +˜ +G +G ˜ +M, τRes +˜ +G +G ˜ +M) ∼= Homk ˜ +G( ˜ +M, Ind +˜ +G +GτRes +˜ +G +G ˜ +M) ∼= Homk ˜ +G( ˜ +M, τInd +˜ +G +GRes +˜ +G +G ˜ +M), +where the last isomorphism comes from Lemma 2.4. By the assumption that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and +the τ-rigidity of ˜ +M, we have that Homk ˜ +G( ˜ +M, τInd +˜ +G +GRes +˜ +G +G ˜ +M) = 0, which implies that Res +˜ +G +G ˜ +M is a τ-rigid +kG-module. +For a finite group H and a subgroup K of H, we recall that a kH-module M is relatively K-projective +if M is a direct summand of IndH +KResH +KM. +Lemma 3.2. Let +˜ +M be a relatively G-projective k ˜G-module. +Then Res +˜ +G +G(Ω ˜ +M) ∼= Ω(Res +˜ +G +G ˜ +M). +In +particular, it holds that τ(Res +˜ +G +G ˜ +M) ∼= Res +˜ +G +G(τ ˜ +M). +Proof. There exists a projective kG-module P such that Res +˜ +G +G(Ω ˜ +M) ∼= Ω(Res +˜ +G +G ˜ +M)⊕P. Hence, we enough +to show that P = 0. It is clear in the case that ˜ +M is a projective k ˜G-module. +We may assume that +˜ +M has no projective summands. Since +˜ +M is relatively G-projective, Ω ˜ +M is +relatively G-projective too (for example see [2, Proposition 20.7]). Hence, Ω ˜ +M is a direct summand +of Ind +˜ +G +GRes +˜ +G +GΩ ˜ +M. On the other hand, by the isomorphism Res +˜ +G +GΩ ˜ +M ∼= Ω(Res +˜ +G +G ˜ +M) ⊕ P, we have that +5 + +Ind +˜ +G +GRes +˜ +G +GΩ ˜ +M ∼= Ind +˜ +G +G(Ω(Res +˜ +G +G ˜ +M)) ⊕ Ind +˜ +G +GP. Here, since Ind +˜ +G +GP is a projective k ˜G-module by Proposi- +tion 2.1 (6) and Ω ˜ +M has no projective summands by the self-injectivity of k ˜G, we have that Ω ˜ +M is a +direct summand of Ind +˜ +G +G(Ω(Res +˜ +G +G ˜ +M)). Therefore, Res +˜ +G +G(Ω ˜ +M) is a direct summand of +Res +˜ +G +GInd +˜ +G +G(Ω(Res +˜ +G +G ˜ +M)) ∼= +� +˜g∈[ ˜ +G/G] +˜gΩ(Res +˜ +G +G ˜ +M) +by Proposition 2.2, which implies that Res +˜ +G +G(Ω ˜ +M) is has no projective summands because each ˜gΩ(Res +˜ +G +G ˜ +M) +has no projective summands by the self-injectivity of kG. Thus, we conclude that P = 0 and Res +˜ +G +G(Ω ˜ +M) ∼= +Ω(Res +˜ +G +G ˜ +M). +The later assertion follows from the fact that τ ∼= Ω2 and the relative G-projectivity of Ω ˜ +M. +The following is important for the proof of Theorem 3.4. +Proposition 3.3 ([1, Corollary 2.13]). Let Λ be a finite dimensional algebra. For a τ-rigid pair (M, P) +for Λ the following are equivalent: +(1) (M, P) is a support τ-tilting pair for Λ. +(2) If HomΛ(M, τX) = 0, HomΛ(X, τM) = 0 and HomΛ(P, X) = 0, then X ∈ add M. +Theorem 3.4. Let ˜ +M be a support τ-tilting k ˜G-module. If it holds that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and ˜ +M is +relatively G-projective, then we have that Res +˜ +G +G ˜ +M is a support τ-tilting kG-module. Moreover, if ( ˜ +M, ˜P) +is a support τ-tilting pair for k ˜G corresponding to ˜ +M, then (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a support τ-tilting pair +for kG corresponding to Res +˜ +G +G ˜ +M. +Proof. Let ( ˜ +M, ˜P) be a support τ-tilting pair for k ˜G corresponding to the support τ-tilting k ˜G-module +˜ +M. +First, we show that (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a τ-rigid pair for kG. Since the k ˜G-module ˜ +M is a support τ- +tilting module, it is a τ-rigid module. Hence, we have that Res +˜ +G +G ˜ +M is a τ-rigid k ˜G-module by Lemma 3.1. +On the other hand, by Proposition 2.1 we have that +HomkG(Res +˜ +G +G ˜P, Res +˜ +G +G ˜ +M) ∼= Homk ˜ +G( ˜P, Ind +˜ +G +GRes +˜ +G +G ˜ +M). +Now, by the assumption that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M, we have that HomkG(Res +˜ +G +G ˜P, Res +˜ +G +G ˜ +M) = 0 because +( ˜ +M, ˜P) is a support τ-tilting pair for k ˜G. Therefore, we conclude that (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a τ-rigid pair +for kG. +Next, we show that the τ-rigid pair (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a support τ-tilting pair for kG. We show that +X ∈ add Res +˜ +G +G ˜ +M under the assumption that +HomkG(X, τ(Res +˜ +G +G ˜ +M)) = HomkG(Res +˜ +G +G ˜ +M, τX) = HomkG(Res +˜ +G +G ˜P, X) = 0, +which implies that the pair (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a support τ-tilting pair for kG by Proposition 3.3. Under +these assumptions, we have the following: +Homk ˜ +G(Ind +˜ +G +GX, τ ˜ +M) ∼= HomkG(X, Res +˜ +G +G(τ ˜ +M)) +(Proposition 2.1) +∼= HomkG(X, τ(Res +˜ +G +G ˜ +M)) +(Lemma 3.2) += 0. +Homk ˜ +G( ˜ +M, τ(Ind +˜ +G +GX)) ∼= Homk ˜ +G( ˜ +M, Ind +˜ +G +G(τX)) +(Lemma 2.4) +∼= HomkG(Res +˜ +G +G ˜ +M, τX) +(Proposition 2.1) += 0. +6 + +Homk ˜ +G( ˜P, Ind +˜ +G +GX) ∼= Homk ˜ +G(Res +˜ +G +G ˜P, X) +(Proposition 2.1) += 0. +By these three isomorphisms and the fact that ( ˜ +M, ˜P) is a support τ-tilting pair for k ˜G, applying +Proposition 3.3, we have that Ind +˜ +G +GX ∈ add ˜ +M. Also, X is a direct summand of Res +˜ +G +GInd +˜ +G +GX by Proposi- +tion 2.2. Therefore, we have that X ∈ add Res +˜ +G +G ˜ +M. +Corollary 3.5. Let +˜ +M1 and +˜ +M2 be relatively G-projective support τ-tilting k ˜G-modules such that +Ind +˜ +G +GRes +˜ +G +G ˜ +Mi ∈ add ˜ +Mi for i = 1, 2. +Then +˜ +M1 ≥ +˜ +M2 in sτ-tilt k ˜G means that Res +˜ +G +G ˜ +M1 ≥ Res +˜ +G +G ˜ +M2 +in sτ-tilt kG. +Proof. The consequence immediately follows from Theorem 3.4 and the exactness of the functor Res +˜ +G +G. +We consider equivalent conditions to the assumption of Theorem 3.4. First, we give the lemmas which +can be applied in case of rigid k ˜G-modules not only support τ-tilting k ˜G-modules. +Lemma 3.6. Let ˜ +M be a rigid k ˜G-module and L a k ˜G-module. If it holds that S ⊗k ˜ +M ∈ add ˜ +M for +any composition factor S of L, then the following isomorphism as k ˜G-modules holds: +L ⊗k ˜ +M ∼= +� +S +S ⊗k ˜ +M, +where S runs over all composition factors of L. +Proof. Let L be an arbitrary k ˜G-module and ˜ +M a rigid k ˜G-module satisfying that +S ⊗k ˜ +M ∈ add ˜ +M for any composition factor of S of L. +(3.1) +We use induction on the composition length ℓ(L) of L. If ℓ(L) = 1, there is nothing to prove. Hence, we +assume that ℓ(L) ≥ 2 and that the statement for any k ˜G-module L′ satisfying ℓ(L′) < ℓ(L) is true. Let +T be a simple submodule of L. We get the exact sequence +0 +T ⊗k ˜ +M +L ⊗k ˜ +M +L/T ⊗k ˜ +M +0 +(3.2) +obtained by applying the exact functor − ⊗k ˜ +M to the exact sequence +0 +T +L +L/T +0. +By the rigidity of ˜ +M, the assumption (3.1) and the assumption of this induction, the sequence (3.2) splits, +and we get that +L ⊗k ˜ +M ∼= T ⊗k ˜ +M ⊕ L/T ⊗k ˜ +M ∼= T ⊗k ˜ +M ⊕ +� +S′ +S′ ⊗k ˜ +M ∼= +� +S +S ⊗k ˜ +M, +where S′ and S run over all composition factors of L/T and L, respectively. +Lemma 3.7. Let ˜ +M be a rigid k ˜G-module. Then the following conditions are equivalent: +(1) Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and ˜ +M is relatively G-projective. +(2) S ⊗k ˜ +M ∈ add ˜ +M for each simple k( ˜G/G)-module S. +Proof. By Proposition 2.1, we have that +Ind +˜ +G +GRes +˜ +G +G ˜ +M ∼= Ind +˜ +G +G(kG ⊗k Res +˜ +G +G ˜ +M) ∼= (Ind +˜ +G +GkG) ⊗k ˜ +M ∼= k( ˜G/G) ⊗k ˜ +M. +7 + +(1) ⇒ (2). By the assumptions, we have that Ind +˜ +G +GRes +˜ +G +G ˜ +M =add ˜ +M. Hence, by Proposition 2.1 we get +that +S ⊗k ˜ +M =add S ⊗k Ind +˜ +G +GRes +˜ +G +G ˜ +M +∼= Ind +˜ +G +G(Res +˜ +G +GS ⊗k Res +˜ +G +G ˜ +M) +∼= Ind +˜ +G +G(k⊕ dimk S +G +⊗k Res +˜ +G +G ˜ +M) +=add Ind +˜ +G +GRes +˜ +G +G ˜ +M +=add ˜ +M, +for any simple k( ˜G/G)-module S, which implies that S ⊗k ˜ +M ∈ add ˜ +M. +(2) ⇒ (1). By Lemma 3.6, we have that +Ind +˜ +G +GRes +˜ +G +G ˜ +M ∼= k( ˜G/G) ⊗k ˜ +M ∼= +� +S +S ⊗k ˜ +M, +where S runs over all composition factors of the k ˜G-module k( ˜G/G). Therefore, the assumption implies +that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M. Moreover, since the trivial k ˜G-module k ˜ +G appears as a composition factor +of k( ˜G/G), we have that the module ˜ +M appears as a direct summand of Ind +˜ +G +GRes +˜ +G +G ˜ +M, that is ˜ +M is a +relatively G-projective k ˜G-module. +We give the equivalent conditions to the assumption of Theorem 3.4. +Theorem 3.8. Let ˜ +M be a support τ-tilting k ˜G-module. Then the following conditions are equivalent: +(1) +˜ +M =add Ind +˜ +G +GM for some ˜G-invariant support τ-tilting kG-module M. +(2) Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and ˜ +M is relatively G-projective. +(3) S ⊗k ˜ +M ∈ add ˜ +M for each simple k( ˜G/G)-module S. +Proof. (1) ⇒ (2). Assume that ˜ +M =add Ind +˜ +G +GM for some ˜G-invariant support τ-tilting kG-module M. +Then clearly ˜ +M is a relatively G-projective k ˜G-module (see [2, 3.9.1]), and by Proposition 2.2, we have +that +Ind +˜ +G +GRes +˜ +G +G ˜ +M =add Ind +˜ +G +GRes +˜ +G +GInd +˜ +G +GM ∼= Ind +˜ +G +G( +� +˜g∈[ ˜ +G/G] +˜gM) ∼= +� +˜g∈[ ˜ +G/G] +Ind +˜ +G +GM ∈ add ˜ +M. +(2) ⇒ (1). Assume that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and that ˜ +M is relatively G-projective. Put M := Res +˜ +G +G ˜ +M. +Then by Lemma 2.5 and Theorem 3.4, M is a ˜G-invariant support τ-tilting kG-module. We show that +Ind +˜ +G +GM =add ˜ +M, that is add(Ind +˜ +G +GM) = add ˜ +M. By the assumption that Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M, we have +add(Ind +˜ +G +GM) ⊂ add ˜ +M. On the other hand, since ˜ +M is relatively G-projective, ˜ +M is a direct summand +of Ind +˜ +G +GRes +˜ +G +G ˜ +M = Ind +˜ +G +GM. Hence, we have add ˜ +M ⊂ add(Ind +˜ +G +GM). +(2) ⇔ (3). +Since support τ-tilting k ˜G-modules are rigid k ˜G-modules, the equivalence follows from +Lemma 3.7. +Corollary 3.9. Let (sτ-tilt kG) ˜ +G be the subset of sτ-tilt kG consisting of ˜G-invariant support τ-tilting +kG-modules and (sτ-tilt k ˜G)⋆ the subset of sτ-tilt k ˜G consisting of support τ-tilting k ˜G-modules satisfying +the equivalent conditions of Theorem 3.8. Then the induction functor Ind +˜ +G +G induces a poset isomorphism +(sτ-tilt kG) ˜ +G +(sτ-tilt k ˜G)⋆ +M +Ind +˜ +G +GM. +∼ +(3.3) +In particular, the induction functor Ind +˜ +G +G induces the poset monomorphism +(sτ-tilt kG) ˜ +G +sτ-tilt k ˜G +M +Ind +˜ +G +GM. +(3.4) +8 + +Proof. By [16, Theorem 3.2], the map (3.4) is well-defined. Moreover, by the exactness of the functor +Ind +˜ +G +G, if N ≤ M in sτ-tilt kG then Ind +˜ +G +GN ≤ Ind +˜ +G +GM in sτ-tilt k ˜G for any support τ-tilting kG-modules +N and M. Therefore, the map (3.4) is a poset homomorphism. +We show that the map (3.4) restricts to a bijection (3.3). By the definition of (sτ-tilt k ˜G)⋆ and the +above argument, the map (3.3) is well-defined and a poset homomorphism. For any relatively G-projective +support τ-tilting k ˜G-module +˜ +M with Ind +˜ +G +GRes +˜ +G +G ˜ +M ∈ +˜ +M, by Theorem 3.8, we can take a ˜G-invariant +support τ-tilting kG-module M satisfying Ind +˜ +G +GM =add ˜ +M. Hence, the map is surjective. Also, assume +that two ˜G-invariant support τ-tilting kG-modules M and N satisfy that Ind +˜ +G +GM =add Ind +˜ +G +GN. Then we +have that +� +˜g∈[ ˜ +G/G] +˜gM ∼= Res +˜ +G +GInd +˜ +G +GM =add Res +˜ +G +GInd +˜ +G +GN ∼= +� +˜g∈[ ˜ +G/G] +˜gN +by Proposition 2.2, which means that M =add N by the ˜G-invariances of M and N. Hence, the map is +injective. This completes the proof of the first assertion. +The latter assertion immediately follows from the fact that the map (3.4) is the composition of the +poset isomorphism (3.3) and the inclusion map (sτ-tilt k ˜G)⋆ +sτ-tilt k ˜G. +As an application of Theorem 3.8, we consider the case that ˜G/G is a p-group. +Theorem 3.10. Let ˜G be a finite group and G a normal subgroup of ˜G of p-power index in ˜G. Then +the induction functor Ind +˜ +G +G induces an isomorphism as partially ordered sets between (sτ-tilt kG) ˜ +G and +sτ-tilt k ˜G, where (sτ-tilt kG) ˜ +G is the subset of sτ-tilt kG consisting of ˜G-invariant support τ-tilting kG- +module. +Proof. By Corollary 3.9, the map (3.3) is a poset isomorphism. We enough to show that (sτ-tilt k ˜G)⋆ = +sτ-tilt k ˜G. It is clear that (sτ-tilt k ˜G)⋆ ⊂ sτ-tilt k ˜G. To prove the reverse inclusion, take an arbitrary +support τ-tilting k ˜G-module ˜ +M. Since ˜G/G is a p-group, the only simple k( ˜G/G)-module is the trivial +k( ˜G/G)-module. Hence, the condition (3) of Theorem 3.8 is satisfied in our situation because k ˜ +G/G ⊗k ˜ +M +is isomorphic to ˜ +M. Therefore, we conclude that sτ-tilt k ˜G ⊂ (sτ-tilt k ˜G)⋆. +The following example can be seen in [16]. +Example 3.11. Let k be an algebraically closed field of characteristic p = 2. We consider that the case +that G is the alternating group A4 of degree 4 and ˜G is the symmetric group S4 of degree 4. The algebras +kA4 and kS4 are Brauer graph algebras associated to the Brauer graphs in Figure 1(a) and Figure 1(b), +respectively: +kA4 = 1 +2 +3 +(a) The Brauer graph of kA4 +multiplicity: 2 +2′ +1′ = kS4 +(b) The Brauer graph of kS4 +Figure 1: Brauer graphs +Now we draw the Hasse diagram H(sτ-tilt kA4) of the partially ordered set sτ-tilt kA4 as follows: +9 + +H(sτ-tilt kA4) : +P(1) ⊕ P(2) ⊕ P(3) +P1 ⊕ +3 +1 +1 +2 +3 ⊕ P3 +3 +1 ⊕ +2 +3 +3 +3 +2 ⊕ 1 +3 +3 +1 ⊕ 3 +2 ⊕ P3 +3 +1 ⊕ 3 +2 ⊕ 3 +3 +1 ⊕ 3 +3 +2 ⊕ 3 +3 +3 +1 ⊕ 1 +3 +3 +1 ⊕ +3 +1 +1 +2 +3 ⊕ P3 +P1 ⊕ +3 +1 +1 +2 +3 ⊕ 1 +3 +2 +3 +3 +1 +2 ⊕ P2 ⊕ P3 +P1 ⊕ P2 ⊕ +1 +2 +2 +3 +1 +2 +3 +3 +1 +2 ⊕ P2 ⊕ 2 +3 +2 +3 +3 +1 +2 ⊕ 3 +2 ⊕ 2 +3 +3 +2 ⊕ 2 +3 +2 ⊕ 2 +3 +2 +2 +1 ⊕ P2 ⊕ 2 +3 +2 +1 ⊕ 2 ⊕ 2 +3 +2 +3 +3 +1 +2 ⊕ 3 +2 ⊕ P3 +P1 ⊕ 1 +2 ⊕ +1 +2 +2 +3 +1 +P1 ⊕ 1 +2 ⊕ 1 +3 +1 ⊕ 1 +2 ⊕ 1 +3 +2 +1 ⊕ 1 +2 ⊕ +1 +2 +2 +3 +1 +2 +1 ⊕ 1 +2 +2 +1 ⊕ 2 +1 ⊕ 1 +2 +1 ⊕ 1 +3 +1 +2 +1 ⊕ P2 ⊕ +1 +2 +2 +3 +1 +0 +Figure 2: The Hasse diagram of sτ-tilt kA4 +10 + +The enclosed support τ-tilting modules in Figure 2 are all the invariant support τ-tilting modules under +the action of S4. Next, we draw the Hasse diagram H(sτ-tilt kS4) of partially ordered set sτ-tilt kS4 as +follows: +H(sτ-tilt(kS4)) : +P1′ ⊕ P2′ +2′ +2′ +1′ +2′ +1′ +2′ ⊕ P2′ +2′ +2′ +1′ +2′ +1′ +2′ ⊕ 2′ +2′ +2′ +2′ +0 +P1′ ⊕ +1′ +1′ +2′ +1′ +1′ ⊕ +1′ +1′ +2′ +1′ +1′ +Figure 3: The Hasse diagram of sτ-tilt kS4 +The functor IndS4 +A4 takes each enclosed S4-invariant support τ-tilting kA4-module in Figure 2 to that in +Figure 3 with the same square. +Remark 3.12. Let (Ind +˜ +G +G)−1(sτ-tilt k ˜G) := {M ∈ sτ-tilt kG | Ind +˜ +G +GM ∈ sτ-tilt k ˜G}. Then (sτ-tilt kG) ˜ +G +is contained in (Ind +˜ +G +G)−1(sτ-tilt k ˜G) by [16, Theorem 3.2]. On the other hand, they do not coincide in +general. Moreover, though the poset homomorphism +(sτ-tilt kG) ˜ +G +sτ-tilt k ˜G +M +Ind +˜ +G +GM +is a monomorphism by Corollary 3.9, the one +(Ind +˜ +G +G)−1(sτ-tilt k ˜G) +sτ-tilt k ˜G +M +Ind +˜ +G +GM +is not a monomorphism in general. +For example, for p = 2, the alternating group A4 of degree 4 and the symmetric group S4 of degree +4, a kA4-module M := 1 ⊕ 1 +2 is a support τ-tilting kA4-module, where 1 means the trivial kA4-module +and 2 a non-trivial kA4-module. +Also, it holds that σM ∼= 1 ⊕ 1 +3 for σ ∈ S4 \ A4, where 3 means +the non-trivial simple kA4-module not isomorphic to 2. Therefore, we have that M ̸∈ (sτ-tilt kA4)S4. +However, IndS4 +A4M ∼= 1′ +1′ ⊕ +1′ +1′ +2′ is a support τ-tilting kS4-module, where 1′ means the trivial kS4-module +and 2′ the simple kS4-module of dimension 2. This implies that M ∈ (IndS4 +A4)−1(sτ-tilt kS4). Moreover, +for N := 1 ⊕ 1 +2 ⊕ 1 +3 ∈ (sτ-tilt kA4)S4, it holds that IndS4 +A4N ∼= 1′ +1′ ⊕ +1′ +1′ +2′ ⊕ +1′ +1′ +2′ =add 1′ +1′ ⊕ +1′ +1′ +2′ (=add IndS4 +A4M). +Therefore, the map +(IndS4 +A4)−1(sτ-tilt kS4) +sτ-tilt kS4 +M +IndS4 +A4M +is not a monomorphism. +11 + +At the end of this section, we discuss a feature of vertices of indecomposable τ-rigid k ˜G-modules. +Lemma 3.13. Let ˜G be a finite group. Then the trivial k ˜G-module k ˜ +G is a τ-rigid if and only if ˜G has +no normal subgroup of index p. +Proof. By [20, Chap. I, Corollary 10.13], there exists a normal subgroup of ˜G of index p if and only if +Ext1 +k ˜ +G(k ˜ +G, k ˜ +G) ̸= 0. Also, by the simplicity of the trivial k ˜G-module and Auslander-Reiten duality, we +have that +Homk ˜ +G(k ˜ +G, τk ˜ +G) ∼= Homk ˜ +G(k ˜ +G, τk ˜ +G) ∼= D Ext1 +k ˜ +G(k ˜ +G, k ˜ +G). +Therefore, we get the result. +Theorem 3.14. Let ˜G be a finite group. Then any indecomposable τ-rigid k ˜G-module has a vertex +contained in a Sylow p-subgroup of ˜G properly if and only if ˜G has a proper normal subgroup of p-power +index. +Proof. Assume that ˜G has no proper normal subgroup of p-power index. Then by Lemma 3.13, the trivial +k ˜G-module, whose vertex is a Sylow p-subgroup of ˜G, is a τ-rigid module. +Conversely, assume that ˜G has normal subgroup of p-power index. In this case, there exists a normal +subgroup G of ˜G of index p. Let ˜X be an arbitrary τ-rigid k ˜G-module. Then, ˜X is a direct summand +of a support τ-tilting k ˜G-module ˜ +M by [1, Theorem 2.10], that is, ˜X is relatively G-projective. Also, +there exists a ˜G-invariant support τ-tilting kG-module M such that ˜ +M =add Ind +˜ +G +GM by Theorem 3.10. +Hence, ˜X is a direct summand of Ind +˜ +G +GM. Therefore, ˜X has a vertex contained in a Sylow p-subgroup of +˜G properly. +4 +Preliminaries for the block version of the main results +We recall the definition of blocks of group algebras. Let G be a finite group. The group algebra kG has +a unique decomposition +kG = B0 × · · · × Bl +into the direct product of indecomposable algebras. We call each indecomposable direct product compo- +nent Bi a block of kG and the decomposition above the block decomposition. We remark that any block +Bi is a two-sided ideal of kG. +For any indecomposable kG-module U, there exists a unique block Bi of kG such that U = BiU and +BjU = 0 for all j ̸= i. Then we say that U lies in the block Bi or simply U is a Bi-module. We denote +by B0(kG) the principal block of kG, in which the trivial kG-module kG lies. +Let G be a normal subgroup of a finite group ˜G, B a block of kG and ˜B a block of k ˜G. We say that ˜B +covers B (or that B is covered by ˜B) if 1B1 ˜ +B ̸= 0, where 1B and 1 ˜ +B mean the respective identity element +of B and ˜B. +Proposition 4.1 (See [2, Theorem 15.1, Lemma 15.3]). With the notation above, the following are +equivalent: +(1) The block ˜B covers B. +(2) There exists a non-zero ˜B-module ˜U such that Res +˜ +G +G ˜U has a non-zero direct summand lying in B. +(3) For any non-zero ˜B-module ˜U, there exists a non-zero direct summand of Res +˜ +G +G ˜U lying in B. +(4) For any non-zero ˜B-module ˜U and indecomposable direct summand V of Res +˜ +G +G ˜U, there exists ˜g ∈ ˜G +such that V lies in the block ˜gB˜g−1. +(5) The block B is a direct summand of ˜B as a (kG, kG)-bimodule. +(6) The block ˜B is a direct summand of k ˜GB ˜G as a (k ˜G, k ˜G)-bimodule. +We denote by I ˜ +G(B) the inertial group of B in ˜G, that is +I ˜ +G(B) := +� +˜g ∈ ˜G +��� ˜gB˜g−1 = B +� +. +12 + +Remark 4.2. For a block ˜B of k ˜G and a block B of kG, the block ˜B covers only B if and only if +I ˜ +G(B) = ˜G by [2, Theorem 15.1 (1)]. Since Res +˜ +G +Gk ˜ +G ∼= kG, the principal block B0(kG) of kG is the only +block of kG covered by the principal block B0(k ˜G) of k ˜G by the equivalence of Proposition 4.1 (1), (3). +Therefore, we have that I ˜ +G(B0(kG)) = ˜G. +Proposition 4.3. Let G be a normal subgroup of a finite group ˜G, B a block of kG and U an indecom- +posable B-module. Then the following hold: +(1) For a block ˜B of k ˜G covering the block B, the module Res +˜ +G +G ˜BInd +˜ +G +GU has a direct summand isomor- +phic to U. In particular, the ˜B-module ˜BInd +˜ +G +GU is non-zero. +(2) Any indecomposable direct summand ˜V of Ind +˜ +G +GU lies in a block of k ˜G covering B. +Proof. Let U be an indecomposable B-module. By the equivalence of Proposition 4.1 (1), (5), the block +˜B has a direct summand B as a (kG, kG)-bimodule. Hence, there exists a (kG, kG)-bimodule B′ such +that ˜B ∼= B ⊕ B′ as a (kG, kG)-bimodule. Therefore, we have that +Res +˜ +G +G ˜BInd +˜ +G +GU ∼= Res +˜ +G +G ˜B(k ˜G ⊗kG U) ∼= Res +˜ +G +G ˜B ⊗kG U ∼= (B ⊕ B′) ⊗kG U ∼= U ⊕ (B′ ⊗kG U), +which prove (1). +Let ˜V be an indecomposable direct summand of Ind +˜ +G +GU lying in a block ˜A of k ˜G. Since the restricted +module Res +˜ +G +G ˜V is a direct summand of the kG-module Res +˜ +G +GInd +˜ +G +GU, we have that the block ˜A covers B +by Proposition 2.2 and the equivalences of Proposition 4.1 (1), (2), (4). Hence, we get that (2). +The following is a generalization of [23, Corollary 5.5.6] (or [24, Corollary 9.9.6]). +Proposition 4.4. Let G be a normal subgroup of a finite group ˜G and B a block of kG. If there exists +an indecomposable B-module X such that Ind +˜ +G +GX is an indecomposable k ˜G-module, then there exists +only one block of k ˜G covering B. +Proof. Let ˜A and ˜B be a block of k ˜G covering B. The modules ˜AInd +˜ +G +GX and ˜BInd +˜ +G +GX are non-zero +direct summands of the indecomposable k ˜G-module Ind +˜ +G +GX by Proposition 4.3 (1). Hence, we get that +˜BInd +˜ +G +GX ∼= ˜AInd +˜ +G +GX ∼= Ind +˜ +G +GX by the indecomposability of Ind +˜ +G +GX. +Since the non-zero k ˜G-module +Ind +˜ +G +GX lies in the blocks ˜A and ˜B, we get that ˜A = ˜B. +Corollary 4.5 (See [23, Corollary 5.5.6] or [24, Corollary 9.9.6]). If ˜G/G is a p-group, then there exists +only one block of k ˜G covering B. +Proof. It immediately follows from Proposition 4.4 and Green’s indecomposability theorem (for example, +see [2, 11, 23]). +Proposition 4.6 (See [21, Theorem 6.8.3] or [23, Theorem 5.5.10, Theorem 5.5.12]). Let G be a normal +subgroup of a finite group ˜G and B a block of kG. Then the following hold: +(1) For any block β of kI ˜ +G(B) covering B, there exists a block ˜B of k ˜G such that +� +x∈[ ˜G/I ˜ +G(B)] +x1βx−1 = 1 ˜ +B, +and then ˜B covers B. Moreover, the correspondence sending β to ˜B induces a bijection between +the set of blocks of kI ˜ +G(B) covering B and those of k ˜G covering B. +(2) If ˜B corresponds to β under the bijection of (1), then the induction functor +Ind +˜ +G +I ˜ +G(B) : kI ˜ +G(B)-mod +k ˜G-mod +restricts to a Morita equivalence +Ind +˜ +G +I ˜ +G(B) : β-mod +˜B-mod +13 + +and its inverse functor is given by +βRes +˜ +G +I ˜ +G(B) : ˜B-mod +β-mod . +Proposition 4.7. Let G be a normal subgroup of a finite group ˜G, B a block of kG, U a B-module, β +a block of kI ˜ +G(B) covering B and ˜B a block of k ˜G covering B such that +� +x∈[ ˜ +G/I ˜ +G(B)] +x1βx−1 = 1 ˜ +B. +Then ˜BInd +˜ +G +GU ∼= Ind +˜ +G +I ˜ +G(B)βInd +I ˜ +G(B) +G +U. +Proof. Let ˜ +B1 = ˜B, . . . , ˜ +Be be the all blocks of k ˜G covering B. By Proposition 4.6, we can take β1 = +β, . . . , βe the blocks of kI ˜ +G(B) satisfying the induction functor Ind +˜ +G +I ˜ +G(B) restricts to a Morita equivalence +Ind +˜ +G +I ˜ +G(B) : βi-mod +˜Bi-mod +for any i = 1, . . . , e. By Proposition 4.3 (2), we get the following isomorphism: +Ind +I ˜ +G(B) +G +U ∼= β1Ind +I ˜ +G(B) +G +U ⊕ · · · ⊕ βeInd +I ˜ +G(B) +G +U. +Moreover, by Proposition 2.1 (2), we have that +Ind +˜ +G +GU ∼= Ind +˜ +G +I ˜ +G(B)Ind +I ˜ +G(B) +G +U ∼= Ind +˜ +G +I ˜ +G(B)β1Ind +I ˜ +G(B) +G +U ⊕ · · · ⊕ Ind +˜ +G +I ˜ +G(B)βeInd +I ˜ +G(B) +G +U. +Since the k ˜G-module Ind +˜ +G +I ˜ +G(B)βiInd +I ˜ +G(B) +G +U lies in the block ˜Bi for any i = 1, . . . , e, we get that +˜BiInd +˜ +G +GU ∼= Ind +˜ +G +I ˜ +G(B)βiInd +I ˜ +G(B) +G +U. +Therefore, we complete the proof. +5 +Block version of main results +In this section, we give the block versions of our theorem stated in Section 3. Let Λ be a finite dimensional +algebra. For Λ-modules M and N, we write M ≤add N if add M ⊂ add N. This relation is clearly reflexive +and transitive. Moreover, if M ≤add N and N ≤add M then M =add N for any Λ-modules M and N. +The following is the special case of the block version of Theorem 3.4. +Theorem 5.1. Let G be a normal subgroup of a finite group ˜G, B a block of kG satisfying I ˜ +G(B) = ˜G, +˜B a block of k ˜G covering B and ˜ +M a support τ-tilting ˜B-module. If it holds that ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M +and ˜ +M is relatively G-projective, then we have that Res +˜ +G +G ˜ +M is a support τ-tilting B-module. Moreover, if +( ˜ +M, ˜P) is a support τ-tilting pair for ˜B corresponding to ˜ +M, then (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a support τ-tilting +pair for B corresponding to Res +˜ +G +G ˜ +M. +Proof. Let ( ˜ +M, ˜P) be a support τ-tilting pair for ˜B corresponding to the support τ-tilting ˜B-module ˜ +M. +Our assumption that I ˜ +G(B) = ˜G means the block B is the only block of kG covered by ˜B by Remark 4.2. +Hence, we have that the restricted modules Res +˜ +G +G ˜ +M and Res +˜ +G +G ˜P are B-modules by Proposition 4.1 (4). +First, we show that (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a τ-rigid pair for B. Since the ˜B-module ˜ +M is a support +τ-tilting ˜B-module, it is a τ-rigid ˜B-module. Hence, we have that Res +˜ +G +G ˜ +M is a τ-rigid B-module by +Lemma 3.1. On the other hand, by Proposition 2.1 (4) we have that +HomB(Res +˜ +G +G ˜P, Res +˜ +G +G ˜ +M) ∼= HomkG(Res +˜ +G +G ˜P, Res +˜ +G +G ˜ +M) +∼= Homk ˜ +G( ˜P, Ind +˜ +G +GRes +˜ +G +G ˜ +M) +∼= Hom ˜ +B( ˜P, ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M). +14 + +Now, by the assumption that ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M, we have that Hom ˜ +B( ˜P, ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M) = 0 +because ( ˜ +M, ˜P) is a support τ-tilting pair for ˜B. Therefore, we conclude that (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a +τ-rigid pair for B. +Next, we show that the τ-rigid pair (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a support τ-tilting pair for B. We show that +X ∈ add Res +˜ +G +G ˜ +M under the assumption that +HomB(X, τ(Res +˜ +G +G ˜ +M)) = HomB(Res +˜ +G +G ˜ +M, τX) = HomB(Res +˜ +G +G ˜P, X) = 0, +which implies that the pair (Res +˜ +G +G ˜ +M, Res +˜ +G +G ˜P) is a support τ-tilting pair for B by Proposition 3.3. Under +these assumptions, we have the following: +Hom ˜ +B( ˜BInd +˜ +G +GX, τ ˜ +M) ∼= Homk ˜ +G(Ind +˜ +G +GX, τ ˜ +M) +(τ ˜ +M is a ˜B-module) +∼= HomkG(X, Res +˜ +G +G(τ ˜ +M)) +(Proposition 2.1 (5)) +∼= HomkG(X, τ(Res +˜ +G +G ˜ +M)) +(Lemma 3.2) +∼= HomB(X, τ(Res +˜ +G +G ˜ +M)) +(X and τ(Res +˜ +G +G ˜ +M) are B-modules) += 0. +Hom ˜ +B( ˜ +M, τ( ˜BInd +˜ +G +GX)) ∼= Hom ˜ +B( ˜ +M, ˜BInd +˜ +G +G(τX)) +(Lemma 2.4) +∼= Homk ˜ +G( ˜ +M, Ind +˜ +G +G(τX)) +( ˜ +M is the ˜B-module) +∼= HomkG(Res +˜ +G +G ˜ +M, τX) +(Proposition 2.1 (4)) +∼= HomB(Res +˜ +G +G ˜ +M, τX) +(Res +˜ +G +G ˜ +M and τX are B-modules) += 0. +Hom ˜ +B( ˜P, ˜BInd +˜ +G +GX) ∼= Homk ˜ +G( ˜P, Ind +˜ +G +GX) +( ˜P is a ˜B-module) +∼= HomkG(Res +˜ +G +G ˜P, X) +(Proposition 2.1 (4)) +∼= HomB(Res +˜ +G +G ˜P, X) +(Res +˜ +G +G ˜P and X are B-modules) += 0. +By these three isomorphisms and the fact that ( ˜ +M, ˜P) is a support τ-tilting pair for ˜B, applying +Proposition 3.3, we have that ˜BInd +˜ +G +GX ∈ add ˜ +M. Also, since the block ˜B covers B, the B-module X is a +direct summand of Res +˜ +G +G ˜BInd +˜ +G +GX by Proposition 4.3 (1). Therefore, we have that X ∈ add Res +˜ +G +G ˜ +M. +We consider equivalent conditions to the assumption of Theorem 5.1. First, we give the lemmas which +can be applied in case of rigid ˜B-modules not only support τ-tilting ˜B-modules. The following lemma is +the block version of Lemma 3.6, which is helpful to prove Theorem 5.4. +Lemma 5.2. Let ˜G be a finite group, ˜B a block of k ˜G, ˜ +M a rigid ˜B-module and L a k ˜G-module. If it +holds that ˜B(S ⊗k ˜ +M) ∈ add ˜ +M for any composition factor S of L, then the following isomorphism as +˜B-modules holds: +˜B(L ⊗k ˜ +M) ∼= +� +S +˜B(S ⊗k ˜ +M), +where S runs over all composition factors of L. +Proof. A similar proof of Lemma 3.6 works in this setting. Let L be an arbitrarily k ˜G-module and ˜ +M a +rigid ˜B-module satisfying that +˜B(S ⊗k ˜ +M) ∈ add ˜ +M for any composition factors S of L. +(5.1) +15 + +We use induction on the composition length ℓ(L) of L. If ℓ(L) = 1, there is nothing to prove. Hence, we +assume that ℓ(L) ≥ 2 and that the statement for any k ˜G-module L′ satisfying ℓ(L′) < ℓ(L) is true. Let +T be a simple submodule of L. We get the exact sequence +0 +˜B(T ⊗k ˜ +M) +˜B(L ⊗k ˜ +M) +˜B((L/T ) ⊗k ˜ +M) +0 +(5.2) +obtained by applying the exact functor ˜B(− ⊗k ˜ +M) to the exact sequence +0 +T +L +L/T +0. +By the rigidity of ˜ +M, the assumption (5.1) and the assumption of the induction, the sequence (5.2) splits, +and we get that +˜B(L ⊗k ˜ +M) ∼= ˜B(T ⊗k ˜ +M) ⊕ ˜B((L/T ) ⊗k ˜ +M) ∼= ˜B(T ⊗k ˜ +M) ⊕ +� +S′ +˜B(S′ ⊗k ˜ +M) ∼= +� +S +˜B(S ⊗k ˜ +M), +where S′ and S run over all composition factors of L/T and L, respectively. +Lemma 5.3. Let G be a normal subgroup of a finite group ˜G, ˜B a block of k ˜G and ˜ +M a rigid ˜B-module. +Then the following conditions are equivalent: +(1) ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and ˜ +M is relatively G-projective. +(2) ˜B(S ⊗k ˜ +M) ∈ add ˜ +M for each simple k( ˜G/G)-module S. +Proof. By Proposition 2.1, we have that +˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∼= ˜BInd +˜ +G +G(kG ⊗k Res +˜ +G +G ˜ +M) ∼= ˜B((Ind +˜ +G +GkG) ⊗k ˜ +M) ∼= ˜B(k( ˜G/G) ⊗k ˜ +M). +(1) ⇒ (2). By the assumptions, we have that ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M =add ˜ +M. Hence, by Proposition 2.1 we get +that +˜B(S ⊗k ˜ +M) =add ˜B(S ⊗k ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M) +≤add ˜B(S ⊗k Ind +˜ +G +GRes +˜ +G +G ˜ +M) +∼= ˜BInd +˜ +G +G(Res +˜ +G +GS ⊗k Res +˜ +G +G ˜ +M) +∼= ˜BInd +˜ +G +G(k⊕ dimk S +G +⊗k Res +˜ +G +G ˜ +M) +=add ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M +=add ˜ +M, +for any simple k( ˜G/G)-module S, which implies that ˜B(S ⊗k ˜ +M) ∈ add ˜ +M. +(2) ⇒ (1). By Lemma 5.2, we have that +˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∼= ˜B(k( ˜G/G) ⊗k ˜ +M) ∼= +� +S +˜B(S ⊗k ˜ +M), +where S runs over all composition factors of the k ˜G-module k( ˜G/G). Therefore, the assumption implies +that ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M. Moreover, since the trivial k ˜G-module k ˜ +G appears as a composition factor +of k( ˜G/G), we have that the ˜B-module ˜ +M appears as a direct summand of ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M, that is ˜ +M is +a relatively G-projective k ˜G-module. +We give the equivalent conditions to the assumption of Theorem 5.1. +Theorem 5.4. Let G be a normal subgroup of a finite group, B a block of kG satisfying I ˜ +G(B) = ˜G and +˜B a block of k ˜G covering B. Let ˜ +M be a support τ-tilting ˜B-module. Then the following conditions are +equivalent: +16 + +(1) +˜ +M =add ˜BInd +˜ +G +GM for some ˜G-invariant support τ-tilting B-module M. +(2) ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and ˜ +M is relatively G-projective. +(3) ˜B(S ⊗k ˜ +M) ∈ add ˜ +M for each simple k( ˜G/G)-module S. +Proof. (1) ⇒ (2). Assume that ˜ +M =add ˜BInd +˜ +G +GM for some ˜G-invariant support τ-tilting B-module M. +Then clearly ˜ +M is a relatively G-projective ˜B-module, and we get that +˜BInd +˜ +G +GRes +˜ +G +G ˜ +M =add ˜BInd +˜ +G +GRes +˜ +G +G ˜BInd +˜ +G +GM +≤add ˜BInd +˜ +G +GRes +˜ +G +GInd +˜ +G +GM +∼= ˜BInd +˜ +G +G( +� +˜g∈[ ˜ +G/G] +˜gM) +∼= +� +˜g∈[ ˜ +G/G] +˜BInd +˜ +G +GM +=add ˜ +M. +Hence, we get ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M. +(2) ⇒ (1). Assume that ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M and that ˜ +M is relatively G-projective. Put M := Res +˜ +G +G ˜ +M. +Then by Lemma 2.5, Proposition 4.1 (4), Remark 4.2 and Theorem 5.1, M is a ˜G-invariant support τ- +tilting B-module. We show that ˜BInd +˜ +G +GM =add ˜ +M, that is add( ˜BInd +˜ +G +GM) = add ˜ +M. By the assumption +that ˜BInd +˜ +G +GM = ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ add ˜ +M, we have that add( ˜BInd +˜ +G +GM) ⊂ add ˜ +M. On the other hand, +since ˜ +M is relatively G-projective, ˜ +M is a direct summand of Ind +˜ +G +GRes +˜ +G +G ˜ +M = Ind +˜ +G +GM. Moreover, since +˜ +M lies in ˜B, ˜ +M is a direct summand of ˜BInd +˜ +G +GM. Hence, we have add ˜ +M ⊂ add( ˜BInd +˜ +G +GM). +(2) ⇔ (3) Since support τ-tilting ˜B-modules are rigid ˜B-modules, the equivalence follows from Lemma 5.3. +Corollary 5.5. Let G be a normal subgroup of a finite group ˜G, B a block of kG satisfying I ˜ +G(B) = ˜G +and ˜B a block of k ˜G covering B. +We denote by (sτ-tilt B) ˜ +G the subset of sτ-tilt B consisting of ˜G- +invariant support τ-tilting B-modules and by (sτ-tilt ˜B)⋆⋆ the subset of sτ-tilt ˜B consisting of support +τ-tilting ˜B-modules satisfying the equivalent conditions of Theorem 5.4. Then the functor ˜BInd +˜ +G +G induces +a poset isomorphism +(sτ-tilt B) ˜ +G +(sτ-tilt ˜B)⋆⋆ +M +˜BInd +˜ +G +GM. +∼ +(5.3) +In particular, the functor ˜BInd +˜ +G +G induces the poset monomorphism +(sτ-tilt B) ˜ +G +sτ-tilt ˜B +M +˜BInd +˜ +G +GM. +(5.4) +Proof. By [16, Theorem 3.3], the map (5.4) is well-defined. Moreover, by the exactness of the functor +˜BInd +˜ +G +G, if N ≤ M in sτ-tilt B then ˜BInd +˜ +G +GN ≤ ˜BInd +˜ +G +GM in sτ-tilt ˜B. Therefore, the map (5.4) is a poset +homomorphism. +It remains to show that the map (5.4) restricts to a poset isomorphism (5.3). By the definition of +(sτ-tilt ˜B)⋆⋆ and the above argument, the map (5.3) is well-defined and a poset homomorphism. Also, +for any relatively G-projective support τ-tilting ˜B-module ˜ +M with ˜BInd +˜ +G +GRes +˜ +G +G ˜ +M ∈ ˜ +M, by Theorem 5.4, +we can take a ˜G-invariant support τ-tilting B-module M satisfying ˜BInd +˜ +G +GM =add ˜ +M. Hence, the map +(5.3) is surjective. Also, assume that two ˜G-invariant support τ-tilting B-modules M and N satisfy that +˜BInd +˜ +G +GM =add ˜BInd +˜ +G +GN. We have that +M ≤add Res +˜ +G +G ˜BInd +˜ +G +GM ≤add Res +˜ +G +GInd +˜ +G +GM ∼= +� +˜g∈[ ˜ +G/G] +˜gM =add M +17 + +by Proposition 4.3 (1), Proposition 2.2 and the ˜G-invariance of M. Therefore, we have that M =add +Res +˜ +G +G ˜BInd +˜ +G +GM and that N =add Res +˜ +G +G ˜BInd +˜ +G +GN similarly. Hence, we get that M =add Res +˜ +G +G ˜BInd +˜ +G +GM =add +Res +˜ +G +G ˜BInd +˜ +G +GN =add N, which implies that the map (5.3) is injective. This completes the proof of the +first assertion. +The latter assertion immediately follows from the fact that the map (5.4) is the composition of the +poset isomorphism (5.3) and the inclusion map (sτ-tilt ˜B)⋆⋆ +sτ-tilt ˜B. +The following is the block version of Theorem 3.4. +Theorem 5.6. Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B a block of k ˜G +covering B, β the block of kI ˜ +G(B) satisfying +� +x∈[ ˜ +G/I ˜ +G(B)] +x1βx−1 = 1 ˜ +B +and ˜ +M a support τ-tilting ˜B-module. If it holds that βInd +I ˜ +G(B) +G +Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M ∈ add βRes +˜ +G +I ˜ +G(B) ˜ +M +and βRes +˜ +G +I ˜ +G(B) ˜ +M is relatively G-projective, then we have that Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M is a support τ-tilting +B-module. Moreover, if ( ˜ +M, ˜P) is a support τ-tilting pair for ˜B corresponding to ˜ +M, then the pair +(Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M, Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜P) +is a support τ-tilting pair for B corresponding to Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M. +Proof. Since the functor +βRes +˜ +G +I ˜ +G(B) : ˜B-mod +β-mod +is a Morita equivalence by Proposition 4.6, the module βRes +˜ +G +I ˜ +G(B) ˜ +M is a support τ-tilting β-module and +(βRes +˜ +G +I ˜ +G(B) ˜ +M, βRes +˜ +G +I ˜ +G(B) ˜P) +is a corresponding support τ-tilting pair for β. Hence, by Theorem 5.1 it immediately follows the conse- +quence. +The following is the block version of Theorem 3.8. +Theorem 5.7. Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B a block of k ˜G +covering B, β the block of kI ˜ +G(B) satisfying +� +x∈[ ˜ +G/I ˜ +G(B)] +x1βx−1 = 1 ˜ +B +and ˜ +M a support τ-tilting ˜B-module. Then the following conditions are equivalent: +(1) +˜ +M =add ˜BInd +˜ +G +GM for some I ˜ +G(B)-invariant support τ-tilting B-module M. +(2) βInd +I ˜ +G(B) +G +Res +I ˜ +G(B) +G +βRes +˜ +G +I ˜ +G(B) ˜ +M ∈ add βRes +˜ +G +I ˜ +G(B) ˜ +M and βRes +˜ +G +I ˜ +G(B) ˜ +M is relatively G-projective. +(3) β(S ⊗k βRes +˜ +G +I ˜ +G(B) ˜ +M) ∈ add βRes +˜ +G +I ˜ +G(B) ˜ +M for each simple k(I ˜ +G(B)/G)-module S. +Proof. We remark that the module βRes +˜ +G +I ˜ +G(B) ˜ +M is a support τ-tilting β-module since the functor +βRes +˜ +G +I ˜ +G(B) : ˜B-mod +β-mod +(5.5) +is a Morita equivalence by Proposition 4.6. +18 + +(1) ⇒ (2). Assume that ˜ +M =add ˜BInd +˜ +G +GM for some I ˜ +G(B)-invariant support τ-tilting B-module M. By +[16, Theorem 3.3], the module βInd +I ˜ +G(B) +G +M is a support τ-tilting β-module. Since the functor +Ind +˜ +G +I ˜ +G(B) : β-mod +˜B-mod +(5.6) +is a Morita equivalence with the inverse functor (5.5). we have Ind +˜ +G +I ˜ +G(B)βRes +˜ +G +I ˜ +G(B) ˜ +M ∼= +˜ +M. Also, by +the assumption and Proposition 4.7, we get that ˜ +M =add ˜BInd +˜ +G +GM ∼= Ind +˜ +G +I ˜ +G(B)βInd +I ˜ +G(B) +G +M. Therefore, +we have that Ind +˜ +G +I ˜ +G(B)βRes +˜ +G +I ˜ +G(B) ˜ +M =add Ind +˜ +G +I ˜ +G(B)βInd +I ˜ +G(B) +G +M. Hence, by the fact that the functor (5.6) +is a Morita equivalence again, we have that βRes +˜ +G +I ˜ +G(B) ˜ +M =add βInd +I ˜ +G(B) +G +M. +Therefore, we get the +consequence (2) by the equivalence of Theorem 5.4. (1) and (2). +(2) ⇒ (1). Since βRes +˜ +G +I ˜ +G(B) ˜ +M is a support τ-tilting β-module, there exists an I ˜ +G(B)-invariant support +τ-tilting B-module M such that βRes +˜ +G +I ˜ +G(B) ˜ +M =add βInd +I ˜ +G(B) +G +M by the assumptions and Theorem 5.4. +Therefore, by Proposition 4.7, we get that +˜ +M ∼= Ind +˜ +G +I ˜ +G(B)βRes +˜ +G +I ˜ +G(B) ˜ +M =add Ind +˜ +G +I ˜ +G(B)βInd +I ˜ +G(B) +G +M ∼= ˜BInd +˜ +G +GM. +(2) ⇔ (3). +Since the support τ-tilting β-module βRes +˜ +G +I ˜ +G(B) ˜ +M is the rigid β-module, the equivalence +follows from Lemma 5.3. +Corollary 5.8. Let (sτ-tilt B)I ˜ +G(B) be the subset of sτ-tilt B consisting of I ˜ +G(B)-invariant support +τ-tilting B-modules and (sτ-tilt ˜B)⋆⋆⋆ the subset of sτ-tilt ˜B consisting of support τ-tilting ˜B-modules +satisfying the equivalent conditions of Theorem 5.7. Then the functor ˜BInd +˜ +G +G induces a poset isomorphism +(sτ-tilt B)I ˜ +G(B) +(sτ-tilt ˜B)⋆⋆⋆ +M +˜BInd +˜ +G +GM. +(5.7) +In particular, the functor ˜BInd +˜ +G +G induces the poset monomorphism +(sτ-tilt B)I ˜ +G(B) +sτ-tilt ˜B +M +˜BInd +˜ +G +GM. +Proof. Let (sτ-tilt β)⋆⋆ be the subset of sτ-tilt β consisting of support τ-tilting β-modules satisfying the +equivalent conditions of Theorem 5.4. Since the functor +Ind +˜ +G +I ˜ +G(B) : β-mod +˜B-mod +is a Morita equivalence, we have poset isomorphisms +sτ-tilt β +sτ-tilt ˜B +M +Ind +˜ +G +I ˜ +G(B)M +and +(sτ-tilt β)⋆⋆ +(sτ-tilt ˜B)⋆⋆⋆ +M +Ind +˜ +G +I ˜ +G(B)M. +(5.8) +By Theorem 5.4, we get the poset isomorphism +(sτ-tilt B)I ˜ +G(B) +(sτ-tilt β)⋆⋆ +M +βInd +I ˜ +G(B) +G +M. +(5.9) +By Proposition 4.7, the map (5.7) is the composition of the poset isomorphisms (5.9) and (5.8). Hence, +we complete the proof. +19 + +As an application of Corollary 5.8, we consider the case that I ˜ +G(B)/G is a p-group. The following +theorem is a significant generalization of [18, Theorem 1.2] and [10, Theorem 15]. +Theorem 5.9. Let G be a normal subgroup of a finite group ˜G, B a block of kG and ˜B a block of k ˜G +covering B. If the quotient group I ˜ +G(B)/G is a p-group, then the functor Ind +˜ +G +G induces an isomorphism +as partially ordered sets between (sτ-tilt B)I ˜ +G(B) and sτ-tilt ˜B, where (sτ-tilt B)I ˜ +G(B) is the subset of +sτ-tilt B consisting of I ˜ +G(B)-invariant support τ-tilting B-modules. +Proof. It immediately follows from Corollary 4.5, Theorem 5.7 (3), Corollary 5.8 and the fact that the +only simple k(I ˜ +G(B)/G)-module is the trivial k(I ˜ +G(B)/G)-module. +References +[1] T. Adachi, O. Iyama, and I. Reiten. τ-tilting theory. Compos. Math., 150(3) pages 415–452, 2014. +DOI 10.1112/S0010437X13007422. +[2] J. L. Alperin. Local representation theory, volume 11 of Cambridge Studies in Advanced Mathematics. +Cambridge University Press, Cambridge, 1986. 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In Modular representation theory of +finite groups (Charlottesville, VA, 1998), pages 101–146. de Gruyter, Berlin, 2001. +Ryotaro KOSHIO +Department of Mathematics, Tokyo University of Science +1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan +E-mail: 1120702@ed.tus.ac.jp +Yuta KOZAKAI +Department of Mathematics, Tokyo University of Science +1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan +E-mail: kozakai@rs.tus.ac.jp +21 + diff --git a/ZNE4T4oBgHgl3EQfOQxg/content/tmp_files/load_file.txt b/ZNE4T4oBgHgl3EQfOQxg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0e7ba6411591bc79f108d1bf4b7c1000dee1f03 --- /dev/null +++ b/ZNE4T4oBgHgl3EQfOQxg/content/tmp_files/load_file.txt @@ -0,0 +1,1133 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf,len=1132 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='04963v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='RT] 12 Jan 2023 Normal subgroups and support τ-tilting modules ∗† Ryotaro KOSHIO Yuta KOZAKAI January 13, 2023 Abstract Let ˜G be a finite group, G a normal subgroup of ˜G and k an algebraically closed field of character- istic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The first main result in this paper is to show that support τ-tilting k ˜G-modules satisfying some properties are support τ-tilting modules as kG-modules too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As the second main result, we give equivalent conditions for support τ-tilting k ˜G-modules to satisfy the above properties, and show that the set of the support τ-tilting k ˜G-modules with the properties is isomorphic to the set of ˜G-invariant support τ-tilting kG-modules as partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As an application, we show that the set of ˜G-invariant support τ-tilting kG-modules is isomorphic to the set of support τ-tilting k ˜G-modules in the case that the index G in ˜G is a p-power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As a further application, we give a feature of vertices of indecomposable τ-rigid k ˜G-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Finally, we give the block versions of the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 1 Introduction Since 2014 when τ-tilting theory was introduced by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Adachi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Iyama, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Reiten [1], the theory continues to develop rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The main theme of the theory is the study of support τ-tilting modules, and many researchers have given the work on these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In fact, the support τ-tilting modules over finite dimen- sional algebras are under the one-to-one correspondences with the various representation-theoretically important objects including two-term silting complexes [1], functorially finite torsion classes [1], left- finite semibricks [3], two-term simple-minded collections [3, 15], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In particular, the theory is expected to be helpful in solving Brou´e’s abelian defect group conjecture because the theory is useful for the classification of two-term tilting complexes over group algebras or block algebras of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Even though the studies on the τ-tilting theory related to the modular representation theory of finite groups are very important for these reasons, there are few such studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, the authors have given the studies combining the above two theories [16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' All of them show that the induction functors from kG-modules to k ˜G-modules give the poset homomorphisms from the support τ-tilting modules over kG to those over k ˜G under appropriate assumptions, where G is a normal subgroup of a finite group ˜G and k an algebraically closed field of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Naturally, we are interested in the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' When the restriction functor from k ˜G-modules to kG-modules give the maps from the support τ-tilting modules over k ˜G to those over kG?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In regarding this question, in [5], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Breaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Marcus, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Modoi gave a positive answer in case that the quotient group ˜G/G is a p-prime group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' the prime number p does not divide the order of the factor group ˜G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we consider the case that ˜G/G is not necessarily a p-prime group, and get the following positive answer for the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group, G a normal subgroup of ˜G, ˜ M a relatively G-projective support τ-tilting k ˜G-module, and ( ˜ M, ˜P) a corresponding support τ-tilting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M, then Res ˜ G G ˜ M is a support τ-tilting kG-module, and (Res ˜ G G ˜ M, Res ˜ G G ˜P) a corresponding support τ-tilting pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, for relatively G-projective support τ-tilting k ˜G-modules ˜ M1 and ˜ M2 with the property that Ind ˜ G GRes ˜ G G ˜ Mi ∈ add ˜ Mi for i = 1, 2, if ˜ M1 ≥ ˜ M2 in sτ-tilt k ˜G, then Res ˜ G G ˜ M1 ≥ Res ˜ G G ˜ M2 in sτ-tilt kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' ∗Mathematics Subject Classification (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 20C20, 16G10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' †Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Support τ-tilting modules, blocks of finite groups, induction functors, restriction functors 1 Let M be a ˜G-invariant support τ-tilting kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The first author showed that Ind ˜ G GM is a support τ-tilting k ˜G-module [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We are interested in what is the image of the set of ˜G- invariant support τ-tilting kG-modules under the map induced by the induction functor Ind ˜ G G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we give equivalent conditions to the assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2, and finally we clarify what the image of the map induced by Ind ˜ G G is in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a support τ-tilting k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) ˜ M =add Ind ˜ G GM for some ˜G-invariant support τ-tilting kG-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) S ⊗k ˜ M ∈ add ˜ M for each simple k( ˜G/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, denoting by (sτ-tilt kG) ˜ G the subset of sτ-tilt kG consisting of ˜G-invariant support τ-tilting kG- modules and by (sτ-tilt k ˜G)⋆ the subset of sτ-tilt k ˜G consisting of support τ-tilting k ˜G-modules satisfying the above equivalent conditions, the induction functor Ind ˜ G G induces a poset isomorphism (sτ-tilt kG) ˜ G (sτ-tilt k ˜G)⋆ M Ind ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' ∼ The studies on the vertices of indecomposable modules over group algebras have been done by many researchers for a long time, for example, see [6, 7, 8, 11, 14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, τ-rigid modules over finite dimensional algebras are important classes and there are many studies on the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' They have nice properties and are essential objects for the representation theory, for example, see [1, 4, 9, 10, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, one of our interests is to give a feature of the vertices of indecomposable τ-rigid modules over group algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As a further application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3, we give a feature of vertices of indecomposable τ-rigid modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4 (See Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then any indecomposable τ-rigid k ˜G-module has a vertex contained in a Sylow p-subgroup of ˜G properly if and only if ˜G has a normal subgroup of p-power index in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As a natural question, we wonder if we get the block version of our theorems for group algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In particular, we are interested in how we give the block versions of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As the results, we get the block versions of the theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G and B a block of kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote by I ˜ G(B) := � ˜g ∈ ˜G ��� ˜gB˜g−1 = B � the inertial group of B in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5 (See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B a block of k ˜G covering B, β the block of kI ˜ G(B) satisfying � x∈[ ˜ G/I ˜ G(B)] x1βx−1 = 1 ˜ B and ˜ M a support τ-tilting ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that βInd I ˜ G(B) G Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M ∈ add βRes ˜ G I ˜ G(B) ˜ M and βRes ˜ G I ˜ G(B) ˜ M is relatively G-projective, then we have that Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M is a support τ-tilting B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, if ( ˜ M, ˜P) is a support τ-tilting pair for ˜B corresponding to ˜ M, then the pair (Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M, Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜P) is a support τ-tilting pair for B corresponding to Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6 (See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B a block of k ˜G covering B, β the block of kI ˜ G(B) satisfying � x∈[ ˜ G/I ˜ G(B)] x1βx−1 = 1 ˜ B and ˜ M a support τ-tilting ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: 2 (1) ˜ M =add ˜BInd ˜ G GM for some I ˜ G(B)-invariant support τ-tilting B-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) βInd I ˜ G(B) G Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M ∈ add βRes ˜ G I ˜ G(B) ˜ M and βRes ˜ G I ˜ G(B) ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) β(S ⊗k βRes ˜ G I ˜ G(B) ˜ M) ∈ add βRes ˜ G I ˜ G(B) ˜ M for each simple k(I ˜ G(B)/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, denoting by (sτ-tilt B)I ˜ G(B) the subset of sτ-tilt B consisting of I ˜ G(B)-invariant support τ- tilting B-modules and by (sτ-tilt ˜B)⋆⋆⋆ the subset of sτ-tilt ˜B consisting of support τ-tilting ˜B-modules satisfying the above equivalent conditions, the functor ˜BInd ˜ G G induces a poset isomorphism (sτ-tilt B)I ˜ G(B) (sτ-tilt ˜B)⋆⋆⋆ M ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' ∼ Our particular interest is the case that the index of G in ˜G is a p-power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In fact, under some assumptions, it is expected that tilting complexes over the block B of kG give those over the unique block ˜B of k ˜G covering B (for example, see [12, 19, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In this regard, the authors showed the following result in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7 ([18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G of p-power index in ˜G, B a block of kG, and ˜B the unique block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that the following two conditions are satisfied: (1) Any indecomposable B-module is I ˜ G(B)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) The set of isomorphism classes of basic support τ-tilting B-modules is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the induction functor Ind ˜ G G induces an isomorphism from sτ-tilt B to sτ-tilt ˜B of partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' This theorem can be applied to the case that the block B has a cyclic defect group, but the two conditions limit the scope of its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For example, the theorem cannot be applied to the case that p = 2, G is the alternating group A4 of degree 4 and that ˜G is the symmetric group S4 of degree 4, because the nontrivial simple kA4-modules are not S4-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Indeed, sτ-tilt kA4 is not isomorphic to sτ-tilt kS4 because the number of isomorphism classes of simple kA4-modules is three and that of kS4 is two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' However, we wonder if the induction functor might give some kinds of good relation between the special subsets of the two, and finally, as an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6, we could get the following theorem which can be applied to the case of kA4 and kS4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following theorem is a significant generalization of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7 and enables us to explain the phenomenon occurred in [16, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9] (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8 (See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG and ˜B a block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If the quotient group I ˜ G(B)/G is a p-group, then the functor Ind ˜ G G induces an isomorphism as partially ordered sets between (sτ-tilt B)I ˜ G(B) and sτ-tilt ˜B, where (sτ-tilt B)I ˜ G(B) is the subset of sτ-tilt B consisting of I ˜ G(B)-invariant support τ-tilting B-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Throughout this paper, we fix the following notation: Let k be an algebraically closed field of characteristic p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' An algebra means a k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a finite dimensional algebra Λ, a Λ-module means a finite dimensional left Λ-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a Λ-module M, we denote the Auslander-Reiten translate of M by τM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In case that Λ is a symmetric algebra, τM is isomorphic to Ω2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote the category of all direct summands of finite direct sums of copies of M by add M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For Λ-modules M and N, we write M =add N if add M = add N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' This relation is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote by sτ-tilt Λ the set of equivalence classes of support τ-tilting Λ-modules under the equivalence relation =add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a finite group and H a subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote the restriction functor from kG-modules to kH-modules by ResG H and the induction functor kG ⊗kH − from kH-modules to kG-modules by IndG H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote the trivial kG-module by kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group, G a normal subgroup of ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote a set of coset representatives of G in ˜G by [ ˜G/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a kG-moduleM and ˜g ∈ ˜G, we define a kG-module ˜gM consisting of symbols ˜gm as a set, where m ∈ M, and its kG-module structure is given by ˜gm+ ˜gm′ := ˜g(m+ m′), g(˜gm) := ˜g(˜g−1g˜gm) and λ(˜gm) = ˜g(λm) for m, m′ ∈ M, g ∈ G and λ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a kG-module M, we say that M is ˜G-invariant if M is isomorphic to ˜gM for any ˜g ∈ ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 3 2 Preliminaries In this section, we give elementary facts on the modular representation theory which are helpful to prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (See [2, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a finite group, K a subgroup of G, H a subgroup of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For any kG-module U and kH-module V , the following hold: (1) ResG HU ∼= ResK HResG KU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) IndG HV ∼= IndG KIndK HV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) IndG H(V ⊗k ResG HU) ∼= (IndG HV ) ⊗k U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (4) HomkG(U, IndG HV ) ∼= HomkH(ResG HU, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5) HomkG(IndG HV, U) ∼= HomkH(V, ResG HU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (6) The functors ResG H and IndG H send free modules (projective modules) to free modules (projective modules, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In the modular representation theory of finite groups, Mackey’s decomposition formula is well-known and important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We recall Mackey’s decomposition formula for normal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 (See [2, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G and M a kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following isomorphism as kG-modules holds: Res ˜ G GInd ˜ G GM ∼= � x∈[ ˜ G/G] xM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following is known as Eckmann-Shapiro Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 (See [21, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let H be a finite group of a finite group G, M a kH-module and N a kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then for all n ∈ N, there exists an isomorphism of k-vector spaces: Extn kH(M, ResG HN) ∼= Extn kG(IndG HM, N) The following lemma is a refinement of [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1] which requires the ˜G-invariance for the kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of ˜G and M a kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following hold: (1) Ind ˜ G G(ΩM) ∼= Ω(Ind ˜ G GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) Ind ˜ G G(τM) ∼= τ(Ind ˜ G GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We enough to show that the statement (1) holds since τ ∼= Ω2 for symmetric algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' There exists a projective kG-module Q such that Ind ˜ G G(ΩM) ∼= Ω(Ind ˜ G GM)⊕Q and that Ind ˜ G GP(M) ∼= P(Ind ˜ G GM)⊕Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we have that Res ˜ G GInd ˜ G G(ΩM) ∼= Res ˜ G GΩ(Ind ˜ G GM) ⊕ Res ˜ G GQ, and the left-hand side is isomorphic to � x∈[ ˜ G/G] xΩM by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' However, each xΩM has no projective summands and the restricted module Res ˜ G GQ is a projective kG-module by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (6), which implies that Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we conclude that Ind ˜ G G(ΩM) ∼= Ω(Ind ˜ G GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G and ˜ M be a k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then Res ˜ G G ˜ M is a ˜G-invariant kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Take ˜g ∈ ˜G arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We consider the map f : Res ˜ G G ˜ M ˜gRes ˜ G G ˜ M m ˜gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Clearly, this map is linear and bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We only show that the map is kG-homomorphism, but for any g ∈ G and m ∈ Res ˜ G G ˜ M, it holds that f(gm) = ˜ggm = ˜gg˜g−1˜gm = g · ˜gm = g · f(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 3 Main Theorems In this section, we give theorems stated in Section 1 and their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Throughout this section, ˜G means a finite group and G a normal subgroup of ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' First, we start with a consideration on restricted modules of rigid modules and τ-rigid modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let Λ be a finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We recall that a Λ-module M is rigid (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' τ-rigid) if Ext1 Λ(M, M) = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' HomΛ(M, τM) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We remark that τ-rigid modules are rigid modules by Auslander-Reiten duality HomΛ(X, Y ) ∼= D Ext1 Λ(Y, τX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a k ˜G-module with the property that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following hold: (1) If ˜ M is a rigid k ˜G-module, then the restricted module Res ˜ G G ˜ M is a rigid kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) If ˜ M is a τ-rigid k ˜G-module, then the restricted module Res ˜ G G ˜ M is a τ-rigid kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (1) Let ˜ M be a rigid k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3, we have that Ext1 kG(Res ˜ G G ˜ M, Res ˜ G G ˜ M) ∼= Ext1 k ˜ G( ˜ M, Ind ˜ G GRes ˜ G G ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the assumption that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and the rigidity of ˜ M, we have that the right-hand side is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, Res ˜ G G ˜ M is a rigid kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) Let ˜ M be a τ-rigid k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then we have that HomkG(Res ˜ G G ˜ M, τRes ˜ G G ˜ M) ∼= Homk ˜ G( ˜ M, Ind ˜ G GτRes ˜ G G ˜ M) ∼= Homk ˜ G( ˜ M, τInd ˜ G GRes ˜ G G ˜ M), where the last isomorphism comes from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the assumption that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and the τ-rigidity of ˜ M, we have that Homk ˜ G( ˜ M, τInd ˜ G GRes ˜ G G ˜ M) = 0, which implies that Res ˜ G G ˜ M is a τ-rigid kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a finite group H and a subgroup K of H, we recall that a kH-module M is relatively K-projective if M is a direct summand of IndH KResH KM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a relatively G-projective k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then Res ˜ G G(Ω ˜ M) ∼= Ω(Res ˜ G G ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In particular, it holds that τ(Res ˜ G G ˜ M) ∼= Res ˜ G G(τ ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' There exists a projective kG-module P such that Res ˜ G G(Ω ˜ M) ∼= Ω(Res ˜ G G ˜ M)⊕P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we enough to show that P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' It is clear in the case that ˜ M is a projective k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We may assume that ˜ M has no projective summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since ˜ M is relatively G-projective, Ω ˜ M is relatively G-projective too (for example see [2, Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, Ω ˜ M is a direct summand of Ind ˜ G GRes ˜ G GΩ ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, by the isomorphism Res ˜ G GΩ ˜ M ∼= Ω(Res ˜ G G ˜ M) ⊕ P, we have that 5 Ind ˜ G GRes ˜ G GΩ ˜ M ∼= Ind ˜ G G(Ω(Res ˜ G G ˜ M)) ⊕ Ind ˜ G GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Here, since Ind ˜ G GP is a projective k ˜G-module by Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (6) and Ω ˜ M has no projective summands by the self-injectivity of k ˜G, we have that Ω ˜ M is a direct summand of Ind ˜ G G(Ω(Res ˜ G G ˜ M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, Res ˜ G G(Ω ˜ M) is a direct summand of Res ˜ G GInd ˜ G G(Ω(Res ˜ G G ˜ M)) ∼= � ˜g∈[ ˜ G/G] ˜gΩ(Res ˜ G G ˜ M) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2, which implies that Res ˜ G G(Ω ˜ M) is has no projective summands because each ˜gΩ(Res ˜ G G ˜ M) has no projective summands by the self-injectivity of kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Thus, we conclude that P = 0 and Res ˜ G G(Ω ˜ M) ∼= Ω(Res ˜ G G ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The later assertion follows from the fact that τ ∼= Ω2 and the relative G-projectivity of Ω ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following is important for the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ([1, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let Λ be a finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a τ-rigid pair (M, P) for Λ the following are equivalent: (1) (M, P) is a support τ-tilting pair for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) If HomΛ(M, τX) = 0, HomΛ(X, τM) = 0 and HomΛ(P, X) = 0, then X ∈ add M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a support τ-tilting k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective, then we have that Res ˜ G G ˜ M is a support τ-tilting kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, if ( ˜ M, ˜P) is a support τ-tilting pair for k ˜G corresponding to ˜ M, then (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a support τ-tilting pair for kG corresponding to Res ˜ G G ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ( ˜ M, ˜P) be a support τ-tilting pair for k ˜G corresponding to the support τ-tilting k ˜G-module ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' First, we show that (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a τ-rigid pair for kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the k ˜G-module ˜ M is a support τ- tilting module, it is a τ-rigid module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we have that Res ˜ G G ˜ M is a τ-rigid k ˜G-module by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 we have that HomkG(Res ˜ G G ˜P, Res ˜ G G ˜ M) ∼= Homk ˜ G( ˜P, Ind ˜ G GRes ˜ G G ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Now, by the assumption that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M, we have that HomkG(Res ˜ G G ˜P, Res ˜ G G ˜ M) = 0 because ( ˜ M, ˜P) is a support τ-tilting pair for k ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we conclude that (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a τ-rigid pair for kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Next, we show that the τ-rigid pair (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a support τ-tilting pair for kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We show that X ∈ add Res ˜ G G ˜ M under the assumption that HomkG(X, τ(Res ˜ G G ˜ M)) = HomkG(Res ˜ G G ˜ M, τX) = HomkG(Res ˜ G G ˜P, X) = 0, which implies that the pair (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a support τ-tilting pair for kG by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Under these assumptions, we have the following: Homk ˜ G(Ind ˜ G GX, τ ˜ M) ∼= HomkG(X, Res ˜ G G(τ ˜ M)) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) ∼= HomkG(X, τ(Res ˜ G G ˜ M)) (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Homk ˜ G( ˜ M, τ(Ind ˜ G GX)) ∼= Homk ˜ G( ˜ M, Ind ˜ G G(τX)) (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) ∼= HomkG(Res ˜ G G ˜ M, τX) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 6 Homk ˜ G( ˜P, Ind ˜ G GX) ∼= Homk ˜ G(Res ˜ G G ˜P, X) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By these three isomorphisms and the fact that ( ˜ M, ˜P) is a support τ-tilting pair for k ˜G, applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3, we have that Ind ˜ G GX ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, X is a direct summand of Res ˜ G GInd ˜ G GX by Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that X ∈ add Res ˜ G G ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M1 and ˜ M2 be relatively G-projective support τ-tilting k ˜G-modules such that Ind ˜ G GRes ˜ G G ˜ Mi ∈ add ˜ Mi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then ˜ M1 ≥ ˜ M2 in sτ-tilt k ˜G means that Res ˜ G G ˜ M1 ≥ Res ˜ G G ˜ M2 in sτ-tilt kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The consequence immediately follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4 and the exactness of the functor Res ˜ G G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We consider equivalent conditions to the assumption of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' First, we give the lemmas which can be applied in case of rigid k ˜G-modules not only support τ-tilting k ˜G-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a rigid k ˜G-module and L a k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that S ⊗k ˜ M ∈ add ˜ M for any composition factor S of L, then the following isomorphism as k ˜G-modules holds: L ⊗k ˜ M ∼= � S S ⊗k ˜ M, where S runs over all composition factors of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let L be an arbitrary k ˜G-module and ˜ M a rigid k ˜G-module satisfying that S ⊗k ˜ M ∈ add ˜ M for any composition factor of S of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) We use induction on the composition length ℓ(L) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If ℓ(L) = 1, there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we assume that ℓ(L) ≥ 2 and that the statement for any k ˜G-module L′ satisfying ℓ(L′) < ℓ(L) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let T be a simple submodule of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We get the exact sequence 0 T ⊗k ˜ M L ⊗k ˜ M L/T ⊗k ˜ M 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2) obtained by applying the exact functor − ⊗k ˜ M to the exact sequence 0 T L L/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the rigidity of ˜ M, the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) and the assumption of this induction, the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2) splits, and we get that L ⊗k ˜ M ∼= T ⊗k ˜ M ⊕ L/T ⊗k ˜ M ∼= T ⊗k ˜ M ⊕ � S′ S′ ⊗k ˜ M ∼= � S S ⊗k ˜ M, where S′ and S run over all composition factors of L/T and L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a rigid k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) S ⊗k ˜ M ∈ add ˜ M for each simple k( ˜G/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1, we have that Ind ˜ G GRes ˜ G G ˜ M ∼= Ind ˜ G G(kG ⊗k Res ˜ G G ˜ M) ∼= (Ind ˜ G GkG) ⊗k ˜ M ∼= k( ˜G/G) ⊗k ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 7 (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the assumptions, we have that Ind ˜ G GRes ˜ G G ˜ M =add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 we get that S ⊗k ˜ M =add S ⊗k Ind ˜ G GRes ˜ G G ˜ M ∼= Ind ˜ G G(Res ˜ G GS ⊗k Res ˜ G G ˜ M) ∼= Ind ˜ G G(k⊕ dimk S G ⊗k Res ˜ G G ˜ M) =add Ind ˜ G GRes ˜ G G ˜ M =add ˜ M, for any simple k( ˜G/G)-module S, which implies that S ⊗k ˜ M ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6, we have that Ind ˜ G GRes ˜ G G ˜ M ∼= k( ˜G/G) ⊗k ˜ M ∼= � S S ⊗k ˜ M, where S runs over all composition factors of the k ˜G-module k( ˜G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, the assumption implies that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, since the trivial k ˜G-module k ˜ G appears as a composition factor of k( ˜G/G), we have that the module ˜ M appears as a direct summand of Ind ˜ G GRes ˜ G G ˜ M, that is ˜ M is a relatively G-projective k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We give the equivalent conditions to the assumption of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a support τ-tilting k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) ˜ M =add Ind ˜ G GM for some ˜G-invariant support τ-tilting kG-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) S ⊗k ˜ M ∈ add ˜ M for each simple k( ˜G/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that ˜ M =add Ind ˜ G GM for some ˜G-invariant support τ-tilting kG-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then clearly ˜ M is a relatively G-projective k ˜G-module (see [2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1]), and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2, we have that Ind ˜ G GRes ˜ G G ˜ M =add Ind ˜ G GRes ˜ G GInd ˜ G GM ∼= Ind ˜ G G( � ˜g∈[ ˜ G/G] ˜gM) ∼= � ˜g∈[ ˜ G/G] Ind ˜ G GM ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and that ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Put M := Res ˜ G G ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4, M is a ˜G-invariant support τ-tilting kG-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We show that Ind ˜ G GM =add ˜ M, that is add(Ind ˜ G GM) = add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the assumption that Ind ˜ G GRes ˜ G G ˜ M ∈ add ˜ M, we have add(Ind ˜ G GM) ⊂ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, since ˜ M is relatively G-projective, ˜ M is a direct summand of Ind ˜ G GRes ˜ G G ˜ M = Ind ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we have add ˜ M ⊂ add(Ind ˜ G GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇔ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since support τ-tilting k ˜G-modules are rigid k ˜G-modules, the equivalence follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let (sτ-tilt kG) ˜ G be the subset of sτ-tilt kG consisting of ˜G-invariant support τ-tilting kG-modules and (sτ-tilt k ˜G)⋆ the subset of sτ-tilt k ˜G consisting of support τ-tilting k ˜G-modules satisfying the equivalent conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the induction functor Ind ˜ G G induces a poset isomorphism (sτ-tilt kG) ˜ G (sτ-tilt k ˜G)⋆ M Ind ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' ∼ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) In particular, the induction functor Ind ˜ G G induces the poset monomorphism (sτ-tilt kG) ˜ G sτ-tilt k ˜G M Ind ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2], the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, by the exactness of the functor Ind ˜ G G, if N ≤ M in sτ-tilt kG then Ind ˜ G GN ≤ Ind ˜ G GM in sτ-tilt k ˜G for any support τ-tilting kG-modules N and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) is a poset homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We show that the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) restricts to a bijection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the definition of (sτ-tilt k ˜G)⋆ and the above argument, the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) is well-defined and a poset homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For any relatively G-projective support τ-tilting k ˜G-module ˜ M with Ind ˜ G GRes ˜ G G ˜ M ∈ ˜ M, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8, we can take a ˜G-invariant support τ-tilting kG-module M satisfying Ind ˜ G GM =add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, the map is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, assume that two ˜G-invariant support τ-tilting kG-modules M and N satisfy that Ind ˜ G GM =add Ind ˜ G GN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then we have that � ˜g∈[ ˜ G/G] ˜gM ∼= Res ˜ G GInd ˜ G GM =add Res ˜ G GInd ˜ G GN ∼= � ˜g∈[ ˜ G/G] ˜gN by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2, which means that M =add N by the ˜G-invariances of M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, the map is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' This completes the proof of the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The latter assertion immediately follows from the fact that the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) is the composition of the poset isomorphism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) and the inclusion map (sτ-tilt k ˜G)⋆ sτ-tilt k ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' As an application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8, we consider the case that ˜G/G is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group and G a normal subgroup of ˜G of p-power index in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the induction functor Ind ˜ G G induces an isomorphism as partially ordered sets between (sτ-tilt kG) ˜ G and sτ-tilt k ˜G, where (sτ-tilt kG) ˜ G is the subset of sτ-tilt kG consisting of ˜G-invariant support τ-tilting kG- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9, the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) is a poset isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We enough to show that (sτ-tilt k ˜G)⋆ = sτ-tilt k ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' It is clear that (sτ-tilt k ˜G)⋆ ⊂ sτ-tilt k ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' To prove the reverse inclusion, take an arbitrary support τ-tilting k ˜G-module ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since ˜G/G is a p-group, the only simple k( ˜G/G)-module is the trivial k( ˜G/G)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, the condition (3) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8 is satisfied in our situation because k ˜ G/G ⊗k ˜ M is isomorphic to ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we conclude that sτ-tilt k ˜G ⊂ (sτ-tilt k ˜G)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following example can be seen in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We consider that the case that G is the alternating group A4 of degree 4 and ˜G is the symmetric group S4 of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The algebras kA4 and kS4 are Brauer graph algebras associated to the Brauer graphs in Figure 1(a) and Figure 1(b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='respectively: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='kA4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='(a) The Brauer graph of kA4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='multiplicity: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1′ = kS4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='(b) The Brauer graph of kS4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='Figure 1: Brauer graphs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='Now we draw the Hasse diagram H(sτ-tilt kA4) of the partially ordered set sτ-tilt kA4 as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='H(sτ-tilt kA4) : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='P(1) ⊕ P(2) ⊕ P(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='P1 ⊕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ⊕ P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ⊕ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ⊕ P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ⊕ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ⊕ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ⊕ P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='P1 ⊕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ⊕ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ⊕ P2 ⊕ P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='P1 ⊕ P2 ⊕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ⊕ P2 ⊕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='Figure 2: The Hasse diagram of sτ-tilt kA4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='The enclosed support τ-tilting modules in Figure 2 are all the invariant support τ-tilting modules under ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='the action of S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Next, we draw the Hasse diagram H(sτ-tilt kS4) of partially ordered set sτ-tilt kS4 as follows: H(sτ-tilt(kS4)) : P1′ ⊕ P2′ 2′ 2′ 1′ 2′ 1′ 2′ ⊕ P2′ 2′ 2′ 1′ 2′ 1′ 2′ ⊕ 2′ 2′ 2′ 2′ 0 P1′ ⊕ 1′ 1′ 2′ 1′ 1′ ⊕ 1′ 1′ 2′ 1′ 1′ Figure 3: The Hasse diagram of sτ-tilt kS4 The functor IndS4 A4 takes each enclosed S4-invariant support τ-tilting kA4-module in Figure 2 to that in Figure 3 with the same square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let (Ind ˜ G G)−1(sτ-tilt k ˜G) := {M ∈ sτ-tilt kG | Ind ˜ G GM ∈ sτ-tilt k ˜G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then (sτ-tilt kG) ˜ G is contained in (Ind ˜ G G)−1(sτ-tilt k ˜G) by [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, they do not coincide in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, though the poset homomorphism (sτ-tilt kG) ˜ G sτ-tilt k ˜G M Ind ˜ G GM is a monomorphism by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9, the one (Ind ˜ G G)−1(sτ-tilt k ˜G) sτ-tilt k ˜G M Ind ˜ G GM is not a monomorphism in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For example, for p = 2, the alternating group A4 of degree 4 and the symmetric group S4 of degree 4, a kA4-module M := 1 ⊕ 1 2 is a support τ-tilting kA4-module, where 1 means the trivial kA4-module and 2 a non-trivial kA4-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, it holds that σM ∼= 1 ⊕ 1 3 for σ ∈ S4 \\ A4, where 3 means the non-trivial simple kA4-module not isomorphic to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that M ̸∈ (sτ-tilt kA4)S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' However, IndS4 A4M ∼= 1′ 1′ ⊕ 1′ 1′ 2′ is a support τ-tilting kS4-module, where 1′ means the trivial kS4-module and 2′ the simple kS4-module of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' This implies that M ∈ (IndS4 A4)−1(sτ-tilt kS4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, for N := 1 ⊕ 1 2 ⊕ 1 3 ∈ (sτ-tilt kA4)S4, it holds that IndS4 A4N ∼= 1′ 1′ ⊕ 1′ 1′ 2′ ⊕ 1′ 1′ 2′ =add 1′ 1′ ⊕ 1′ 1′ 2′ (=add IndS4 A4M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, the map (IndS4 A4)−1(sτ-tilt kS4) sτ-tilt kS4 M IndS4 A4M is not a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 11 At the end of this section, we discuss a feature of vertices of indecomposable τ-rigid k ˜G-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the trivial k ˜G-module k ˜ G is a τ-rigid if and only if ˜G has no normal subgroup of index p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By [20, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' I, Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='13], there exists a normal subgroup of ˜G of index p if and only if Ext1 k ˜ G(k ˜ G, k ˜ G) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, by the simplicity of the trivial k ˜G-module and Auslander-Reiten duality, we have that Homk ˜ G(k ˜ G, τk ˜ G) ∼= Homk ˜ G(k ˜ G, τk ˜ G) ∼= D Ext1 k ˜ G(k ˜ G, k ˜ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then any indecomposable τ-rigid k ˜G-module has a vertex contained in a Sylow p-subgroup of ˜G properly if and only if ˜G has a proper normal subgroup of p-power index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that ˜G has no proper normal subgroup of p-power index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='13, the trivial k ˜G-module, whose vertex is a Sylow p-subgroup of ˜G, is a τ-rigid module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Conversely, assume that ˜G has normal subgroup of p-power index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In this case, there exists a normal subgroup G of ˜G of index p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜X be an arbitrary τ-rigid k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then, ˜X is a direct summand of a support τ-tilting k ˜G-module ˜ M by [1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='10], that is, ˜X is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, there exists a ˜G-invariant support τ-tilting kG-module M such that ˜ M =add Ind ˜ G GM by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, ˜X is a direct summand of Ind ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, ˜X has a vertex contained in a Sylow p-subgroup of ˜G properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 4 Preliminaries for the block version of the main results We recall the definition of blocks of group algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The group algebra kG has a unique decomposition kG = B0 × · · · × Bl into the direct product of indecomposable algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We call each indecomposable direct product compo- nent Bi a block of kG and the decomposition above the block decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We remark that any block Bi is a two-sided ideal of kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For any indecomposable kG-module U, there exists a unique block Bi of kG such that U = BiU and BjU = 0 for all j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then we say that U lies in the block Bi or simply U is a Bi-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote by B0(kG) the principal block of kG, in which the trivial kG-module kG lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG and ˜B a block of k ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We say that ˜B covers B (or that B is covered by ˜B) if 1B1 ˜ B ̸= 0, where 1B and 1 ˜ B mean the respective identity element of B and ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (See [2, Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' With the notation above, the following are equivalent: (1) The block ˜B covers B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) There exists a non-zero ˜B-module ˜U such that Res ˜ G G ˜U has a non-zero direct summand lying in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) For any non-zero ˜B-module ˜U, there exists a non-zero direct summand of Res ˜ G G ˜U lying in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (4) For any non-zero ˜B-module ˜U and indecomposable direct summand V of Res ˜ G G ˜U, there exists ˜g ∈ ˜G such that V lies in the block ˜gB˜g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5) The block B is a direct summand of ˜B as a (kG, kG)-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (6) The block ˜B is a direct summand of k ˜GB ˜G as a (k ˜G, k ˜G)-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote by I ˜ G(B) the inertial group of B in ˜G, that is I ˜ G(B) := � ˜g ∈ ˜G ��� ˜gB˜g−1 = B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 12 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For a block ˜B of k ˜G and a block B of kG, the block ˜B covers only B if and only if I ˜ G(B) = ˜G by [2, Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since Res ˜ G Gk ˜ G ∼= kG, the principal block B0(kG) of kG is the only block of kG covered by the principal block B0(k ˜G) of k ˜G by the equivalence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (1), (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that I ˜ G(B0(kG)) = ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG and U an indecom- posable B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following hold: (1) For a block ˜B of k ˜G covering the block B, the module Res ˜ G G ˜BInd ˜ G GU has a direct summand isomor- phic to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In particular, the ˜B-module ˜BInd ˜ G GU is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) Any indecomposable direct summand ˜V of Ind ˜ G GU lies in a block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let U be an indecomposable B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the equivalence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (1), (5), the block ˜B has a direct summand B as a (kG, kG)-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, there exists a (kG, kG)-bimodule B′ such that ˜B ∼= B ⊕ B′ as a (kG, kG)-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that Res ˜ G G ˜BInd ˜ G GU ∼= Res ˜ G G ˜B(k ˜G ⊗kG U) ∼= Res ˜ G G ˜B ⊗kG U ∼= (B ⊕ B′) ⊗kG U ∼= U ⊕ (B′ ⊗kG U), which prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜V be an indecomposable direct summand of Ind ˜ G GU lying in a block ˜A of k ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the restricted module Res ˜ G G ˜V is a direct summand of the kG-module Res ˜ G GInd ˜ G GU, we have that the block ˜A covers B by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 and the equivalences of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (1), (2), (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we get that (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following is a generalization of [23, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6] (or [24, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G and B a block of kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If there exists an indecomposable B-module X such that Ind ˜ G GX is an indecomposable k ˜G-module, then there exists only one block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜A and ˜B be a block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The modules ˜AInd ˜ G GX and ˜BInd ˜ G GX are non-zero direct summands of the indecomposable k ˜G-module Ind ˜ G GX by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we get that ˜BInd ˜ G GX ∼= ˜AInd ˜ G GX ∼= Ind ˜ G GX by the indecomposability of Ind ˜ G GX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the non-zero k ˜G-module Ind ˜ G GX lies in the blocks ˜A and ˜B, we get that ˜A = ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5 (See [23, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6] or [24, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If ˜G/G is a p-group, then there exists only one block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' It immediately follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4 and Green’s indecomposability theorem (for example, see [2, 11, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6 (See [21, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3] or [23, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G and B a block of kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following hold: (1) For any block β of kI ˜ G(B) covering B, there exists a block ˜B of k ˜G such that � x∈[ ˜G/I ˜ G(B)] x1βx−1 = 1 ˜ B, and then ˜B covers B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, the correspondence sending β to ˜B induces a bijection between the set of blocks of kI ˜ G(B) covering B and those of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) If ˜B corresponds to β under the bijection of (1), then the induction functor Ind ˜ G I ˜ G(B) : kI ˜ G(B)-mod k ˜G-mod restricts to a Morita equivalence Ind ˜ G I ˜ G(B) : β-mod ˜B-mod 13 and its inverse functor is given by βRes ˜ G I ˜ G(B) : ˜B-mod β-mod .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG, U a B-module, β a block of kI ˜ G(B) covering B and ˜B a block of k ˜G covering B such that � x∈[ ˜ G/I ˜ G(B)] x1βx−1 = 1 ˜ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then ˜BInd ˜ G GU ∼= Ind ˜ G I ˜ G(B)βInd I ˜ G(B) G U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ B1 = ˜B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' , ˜ Be be the all blocks of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6, we can take β1 = β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' , βe the blocks of kI ˜ G(B) satisfying the induction functor Ind ˜ G I ˜ G(B) restricts to a Morita equivalence Ind ˜ G I ˜ G(B) : βi-mod ˜Bi-mod for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 (2), we get the following isomorphism: Ind I ˜ G(B) G U ∼= β1Ind I ˜ G(B) G U ⊕ · · · ⊕ βeInd I ˜ G(B) G U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (2), we have that Ind ˜ G GU ∼= Ind ˜ G I ˜ G(B)Ind I ˜ G(B) G U ∼= Ind ˜ G I ˜ G(B)β1Ind I ˜ G(B) G U ⊕ · · · ⊕ Ind ˜ G I ˜ G(B)βeInd I ˜ G(B) G U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the k ˜G-module Ind ˜ G I ˜ G(B)βiInd I ˜ G(B) G U lies in the block ˜Bi for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' , e, we get that ˜BiInd ˜ G GU ∼= Ind ˜ G I ˜ G(B)βiInd I ˜ G(B) G U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 5 Block version of main results In this section, we give the block versions of our theorem stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let Λ be a finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' For Λ-modules M and N, we write M ≤add N if add M ⊂ add N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' This relation is clearly reflexive and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, if M ≤add N and N ≤add M then M =add N for any Λ-modules M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following is the special case of the block version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG satisfying I ˜ G(B) = ˜G, ˜B a block of k ˜G covering B and ˜ M a support τ-tilting ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective, then we have that Res ˜ G G ˜ M is a support τ-tilting B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, if ( ˜ M, ˜P) is a support τ-tilting pair for ˜B corresponding to ˜ M, then (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a support τ-tilting pair for B corresponding to Res ˜ G G ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ( ˜ M, ˜P) be a support τ-tilting pair for ˜B corresponding to the support τ-tilting ˜B-module ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Our assumption that I ˜ G(B) = ˜G means the block B is the only block of kG covered by ˜B by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we have that the restricted modules Res ˜ G G ˜ M and Res ˜ G G ˜P are B-modules by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' First, we show that (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a τ-rigid pair for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the ˜B-module ˜ M is a support τ-tilting ˜B-module, it is a τ-rigid ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we have that Res ˜ G G ˜ M is a τ-rigid B-module by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (4) we have that HomB(Res ˜ G G ˜P, Res ˜ G G ˜ M) ∼= HomkG(Res ˜ G G ˜P, Res ˜ G G ˜ M) ∼= Homk ˜ G( ˜P, Ind ˜ G GRes ˜ G G ˜ M) ∼= Hom ˜ B( ˜P, ˜BInd ˜ G GRes ˜ G G ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 14 Now, by the assumption that ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M, we have that Hom ˜ B( ˜P, ˜BInd ˜ G GRes ˜ G G ˜ M) = 0 because ( ˜ M, ˜P) is a support τ-tilting pair for ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we conclude that (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a τ-rigid pair for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Next, we show that the τ-rigid pair (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a support τ-tilting pair for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We show that X ∈ add Res ˜ G G ˜ M under the assumption that HomB(X, τ(Res ˜ G G ˜ M)) = HomB(Res ˜ G G ˜ M, τX) = HomB(Res ˜ G G ˜P, X) = 0, which implies that the pair (Res ˜ G G ˜ M, Res ˜ G G ˜P) is a support τ-tilting pair for B by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Under these assumptions, we have the following: Hom ˜ B( ˜BInd ˜ G GX, τ ˜ M) ∼= Homk ˜ G(Ind ˜ G GX, τ ˜ M) (τ ˜ M is a ˜B-module) ∼= HomkG(X, Res ˜ G G(τ ˜ M)) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (5)) ∼= HomkG(X, τ(Res ˜ G G ˜ M)) (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2) ∼= HomB(X, τ(Res ˜ G G ˜ M)) (X and τ(Res ˜ G G ˜ M) are B-modules) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hom ˜ B( ˜ M, τ( ˜BInd ˜ G GX)) ∼= Hom ˜ B( ˜ M, ˜BInd ˜ G G(τX)) (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) ∼= Homk ˜ G( ˜ M, Ind ˜ G G(τX)) ( ˜ M is the ˜B-module) ∼= HomkG(Res ˜ G G ˜ M, τX) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (4)) ∼= HomB(Res ˜ G G ˜ M, τX) (Res ˜ G G ˜ M and τX are B-modules) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hom ˜ B( ˜P, ˜BInd ˜ G GX) ∼= Homk ˜ G( ˜P, Ind ˜ G GX) ( ˜P is a ˜B-module) ∼= HomkG(Res ˜ G G ˜P, X) (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (4)) ∼= HomB(Res ˜ G G ˜P, X) (Res ˜ G G ˜P and X are B-modules) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By these three isomorphisms and the fact that ( ˜ M, ˜P) is a support τ-tilting pair for ˜B, applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3, we have that ˜BInd ˜ G GX ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, since the block ˜B covers B, the B-module X is a direct summand of Res ˜ G G ˜BInd ˜ G GX by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that X ∈ add Res ˜ G G ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We consider equivalent conditions to the assumption of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' First, we give the lemmas which can be applied in case of rigid ˜B-modules not only support τ-tilting ˜B-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following lemma is the block version of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6, which is helpful to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜G be a finite group, ˜B a block of k ˜G, ˜ M a rigid ˜B-module and L a k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that ˜B(S ⊗k ˜ M) ∈ add ˜ M for any composition factor S of L, then the following isomorphism as ˜B-modules holds: ˜B(L ⊗k ˜ M) ∼= � S ˜B(S ⊗k ˜ M), where S runs over all composition factors of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' A similar proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6 works in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let L be an arbitrarily k ˜G-module and ˜ M a rigid ˜B-module satisfying that ˜B(S ⊗k ˜ M) ∈ add ˜ M for any composition factors S of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) 15 We use induction on the composition length ℓ(L) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If ℓ(L) = 1, there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we assume that ℓ(L) ≥ 2 and that the statement for any k ˜G-module L′ satisfying ℓ(L′) < ℓ(L) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let T be a simple submodule of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We get the exact sequence 0 ˜B(T ⊗k ˜ M) ˜B(L ⊗k ˜ M) ˜B((L/T ) ⊗k ˜ M) 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2) obtained by applying the exact functor ˜B(− ⊗k ˜ M) to the exact sequence 0 T L L/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the rigidity of ˜ M, the assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1) and the assumption of the induction, the sequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2) splits, and we get that ˜B(L ⊗k ˜ M) ∼= ˜B(T ⊗k ˜ M) ⊕ ˜B((L/T ) ⊗k ˜ M) ∼= ˜B(T ⊗k ˜ M) ⊕ � S′ ˜B(S′ ⊗k ˜ M) ∼= � S ˜B(S ⊗k ˜ M), where S′ and S run over all composition factors of L/T and L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, ˜B a block of k ˜G and ˜ M a rigid ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ˜B(S ⊗k ˜ M) ∈ add ˜ M for each simple k( ˜G/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1, we have that ˜BInd ˜ G GRes ˜ G G ˜ M ∼= ˜BInd ˜ G G(kG ⊗k Res ˜ G G ˜ M) ∼= ˜B((Ind ˜ G GkG) ⊗k ˜ M) ∼= ˜B(k( ˜G/G) ⊗k ˜ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the assumptions, we have that ˜BInd ˜ G GRes ˜ G G ˜ M =add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 we get that ˜B(S ⊗k ˜ M) =add ˜B(S ⊗k ˜BInd ˜ G GRes ˜ G G ˜ M) ≤add ˜B(S ⊗k Ind ˜ G GRes ˜ G G ˜ M) ∼= ˜BInd ˜ G G(Res ˜ G GS ⊗k Res ˜ G G ˜ M) ∼= ˜BInd ˜ G G(k⊕ dimk S G ⊗k Res ˜ G G ˜ M) =add ˜BInd ˜ G GRes ˜ G G ˜ M =add ˜ M, for any simple k( ˜G/G)-module S, which implies that ˜B(S ⊗k ˜ M) ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2, we have that ˜BInd ˜ G GRes ˜ G G ˜ M ∼= ˜B(k( ˜G/G) ⊗k ˜ M) ∼= � S ˜B(S ⊗k ˜ M), where S runs over all composition factors of the k ˜G-module k( ˜G/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, the assumption implies that ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, since the trivial k ˜G-module k ˜ G appears as a composition factor of k( ˜G/G), we have that the ˜B-module ˜ M appears as a direct summand of ˜BInd ˜ G GRes ˜ G G ˜ M, that is ˜ M is a relatively G-projective k ˜G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We give the equivalent conditions to the assumption of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group, B a block of kG satisfying I ˜ G(B) = ˜G and ˜B a block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let ˜ M be a support τ-tilting ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: 16 (1) ˜ M =add ˜BInd ˜ G GM for some ˜G-invariant support τ-tilting B-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) ˜B(S ⊗k ˜ M) ∈ add ˜ M for each simple k( ˜G/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that ˜ M =add ˜BInd ˜ G GM for some ˜G-invariant support τ-tilting B-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then clearly ˜ M is a relatively G-projective ˜B-module, and we get that ˜BInd ˜ G GRes ˜ G G ˜ M =add ˜BInd ˜ G GRes ˜ G G ˜BInd ˜ G GM ≤add ˜BInd ˜ G GRes ˜ G GInd ˜ G GM ∼= ˜BInd ˜ G G( � ˜g∈[ ˜ G/G] ˜gM) ∼= � ˜g∈[ ˜ G/G] ˜BInd ˜ G GM =add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we get ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M and that ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Put M := Res ˜ G G ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 (4), Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1, M is a ˜G-invariant support τ- tilting B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We show that ˜BInd ˜ G GM =add ˜ M, that is add( ˜BInd ˜ G GM) = add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the assumption that ˜BInd ˜ G GM = ˜BInd ˜ G GRes ˜ G G ˜ M ∈ add ˜ M, we have that add( ˜BInd ˜ G GM) ⊂ add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' On the other hand, since ˜ M is relatively G-projective, ˜ M is a direct summand of Ind ˜ G GRes ˜ G G ˜ M = Ind ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, since ˜ M lies in ˜B, ˜ M is a direct summand of ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we have add ˜ M ⊂ add( ˜BInd ˜ G GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇔ (3) Since support τ-tilting ˜B-modules are rigid ˜B-modules, the equivalence follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG satisfying I ˜ G(B) = ˜G and ˜B a block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We denote by (sτ-tilt B) ˜ G the subset of sτ-tilt B consisting of ˜G- invariant support τ-tilting B-modules and by (sτ-tilt ˜B)⋆⋆ the subset of sτ-tilt ˜B consisting of support τ-tilting ˜B-modules satisfying the equivalent conditions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the functor ˜BInd ˜ G G induces a poset isomorphism (sτ-tilt B) ˜ G (sτ-tilt ˜B)⋆⋆ M ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' ∼ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) In particular, the functor ˜BInd ˜ G G induces the poset monomorphism (sτ-tilt B) ˜ G sτ-tilt ˜B M ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3], the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, by the exactness of the functor ˜BInd ˜ G G, if N ≤ M in sτ-tilt B then ˜BInd ˜ G GN ≤ ˜BInd ˜ G GM in sτ-tilt ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) is a poset homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' It remains to show that the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) restricts to a poset isomorphism (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By the definition of (sτ-tilt ˜B)⋆⋆ and the above argument, the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) is well-defined and a poset homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, for any relatively G-projective support τ-tilting ˜B-module ˜ M with ˜BInd ˜ G GRes ˜ G G ˜ M ∈ ˜ M, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4, we can take a ˜G-invariant support τ-tilting B-module M satisfying ˜BInd ˜ G GM =add ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, assume that two ˜G-invariant support τ-tilting B-modules M and N satisfy that ˜BInd ˜ G GM =add ˜BInd ˜ G GN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We have that M ≤add Res ˜ G G ˜BInd ˜ G GM ≤add Res ˜ G GInd ˜ G GM ∼= � ˜g∈[ ˜ G/G] ˜gM =add M 17 by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3 (1), Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2 and the ˜G-invariance of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that M =add Res ˜ G G ˜BInd ˜ G GM and that N =add Res ˜ G G ˜BInd ˜ G GN similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we get that M =add Res ˜ G G ˜BInd ˜ G GM =add Res ˜ G G ˜BInd ˜ G GN =add N, which implies that the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' This completes the proof of the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The latter assertion immediately follows from the fact that the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4) is the composition of the poset isomorphism (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3) and the inclusion map (sτ-tilt ˜B)⋆⋆ sτ-tilt ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following is the block version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B a block of k ˜G covering B, β the block of kI ˜ G(B) satisfying � x∈[ ˜ G/I ˜ G(B)] x1βx−1 = 1 ˜ B and ˜ M a support τ-tilting ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If it holds that βInd I ˜ G(B) G Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M ∈ add βRes ˜ G I ˜ G(B) ˜ M and βRes ˜ G I ˜ G(B) ˜ M is relatively G-projective, then we have that Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M is a support τ-tilting B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Moreover, if ( ˜ M, ˜P) is a support τ-tilting pair for ˜B corresponding to ˜ M, then the pair (Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M, Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜P) is a support τ-tilting pair for B corresponding to Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the functor βRes ˜ G I ˜ G(B) : ˜B-mod β-mod is a Morita equivalence by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6, the module βRes ˜ G I ˜ G(B) ˜ M is a support τ-tilting β-module and (βRes ˜ G I ˜ G(B) ˜ M, βRes ˜ G I ˜ G(B) ˜P) is a corresponding support τ-tilting pair for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1 it immediately follows the conse- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following is the block version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG, ˜B a block of k ˜G covering B, β the block of kI ˜ G(B) satisfying � x∈[ ˜ G/I ˜ G(B)] x1βx−1 = 1 ˜ B and ˜ M a support τ-tilting ˜B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) ˜ M =add ˜BInd ˜ G GM for some I ˜ G(B)-invariant support τ-tilting B-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) βInd I ˜ G(B) G Res I ˜ G(B) G βRes ˜ G I ˜ G(B) ˜ M ∈ add βRes ˜ G I ˜ G(B) ˜ M and βRes ˜ G I ˜ G(B) ˜ M is relatively G-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (3) β(S ⊗k βRes ˜ G I ˜ G(B) ˜ M) ∈ add βRes ˜ G I ˜ G(B) ˜ M for each simple k(I ˜ G(B)/G)-module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' We remark that the module βRes ˜ G I ˜ G(B) ˜ M is a support τ-tilting β-module since the functor βRes ˜ G I ˜ G(B) : ˜B-mod β-mod (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5) is a Morita equivalence by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 18 (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Assume that ˜ M =add ˜BInd ˜ G GM for some I ˜ G(B)-invariant support τ-tilting B-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' By [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3], the module βInd I ˜ G(B) G M is a support τ-tilting β-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the functor Ind ˜ G I ˜ G(B) : β-mod ˜B-mod (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6) is a Morita equivalence with the inverse functor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' we have Ind ˜ G I ˜ G(B)βRes ˜ G I ˜ G(B) ˜ M ∼= ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Also, by the assumption and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7, we get that ˜ M =add ˜BInd ˜ G GM ∼= Ind ˜ G I ˜ G(B)βInd I ˜ G(B) G M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we have that Ind ˜ G I ˜ G(B)βRes ˜ G I ˜ G(B) ˜ M =add Ind ˜ G I ˜ G(B)βInd I ˜ G(B) G M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, by the fact that the functor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='6) is a Morita equivalence again, we have that βRes ˜ G I ˜ G(B) ˜ M =add βInd I ˜ G(B) G M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, we get the consequence (2) by the equivalence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since βRes ˜ G I ˜ G(B) ˜ M is a support τ-tilting β-module, there exists an I ˜ G(B)-invariant support τ-tilting B-module M such that βRes ˜ G I ˜ G(B) ˜ M =add βInd I ˜ G(B) G M by the assumptions and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Therefore, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7, we get that ˜ M ∼= Ind ˜ G I ˜ G(B)βRes ˜ G I ˜ G(B) ˜ M =add Ind ˜ G I ˜ G(B)βInd I ˜ G(B) G M ∼= ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (2) ⇔ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the support τ-tilting β-module βRes ˜ G I ˜ G(B) ˜ M is the rigid β-module, the equivalence follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let (sτ-tilt B)I ˜ G(B) be the subset of sτ-tilt B consisting of I ˜ G(B)-invariant support τ-tilting B-modules and (sτ-tilt ˜B)⋆⋆⋆ the subset of sτ-tilt ˜B consisting of support τ-tilting ˜B-modules satisfying the equivalent conditions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Then the functor ˜BInd ˜ G G induces a poset isomorphism (sτ-tilt B)I ˜ G(B) (sτ-tilt ˜B)⋆⋆⋆ M ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7) In particular, the functor ˜BInd ˜ G G induces the poset monomorphism (sτ-tilt B)I ˜ G(B) sτ-tilt ˜B M ˜BInd ˜ G GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let (sτ-tilt β)⋆⋆ be the subset of sτ-tilt β consisting of support τ-tilting β-modules satisfying the equivalent conditions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Since the functor Ind ˜ G I ˜ G(B) : β-mod ˜B-mod is a Morita equivalence, we have poset isomorphisms sτ-tilt β sτ-tilt ˜B M Ind ˜ G I ˜ G(B)M and (sτ-tilt β)⋆⋆ (sτ-tilt ˜B)⋆⋆⋆ M Ind ˜ G I ˜ G(B)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8) By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='4, we get the poset isomorphism (sτ-tilt B)I ˜ G(B) (sτ-tilt β)⋆⋆ M βInd I ˜ G(B) G M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9) By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7, the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7) is the composition of the poset isomorphisms (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Hence, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' 19 As an application of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8, we consider the case that I ˜ G(B)/G is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' The following theorem is a significant generalization of [18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='2] and [10, Theorem 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Let G be a normal subgroup of a finite group ˜G, B a block of kG and ˜B a block of k ˜G covering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' If the quotient group I ˜ G(B)/G is a p-group, then the functor Ind ˜ G G induces an isomorphism as partially ordered sets between (sτ-tilt B)I ˜ G(B) and sτ-tilt ˜B, where (sτ-tilt B)I ˜ G(B) is the subset of sτ-tilt B consisting of I ˜ G(B)-invariant support τ-tilting B-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' It immediately follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='5, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='7 (3), Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='8 and the fact that the only simple k(I ˜ G(B)/G)-module is the trivial k(I ˜ G(B)/G)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Adachi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Iyama, and I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' II, volume 92 of London Mathematical Society Student Texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Linckelmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' A note on vertices of indecomposable tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Group Theory, 23(3) pages 385–391, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1515/jgth-2019-0130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Nagao and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Tsushima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Representations of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Academic Press, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=', Boston, MA, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Navarro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Characters and blocks of finite groups, volume 250 of London Mathematical Society Lec- ture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='1017/CBO9780511526015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Rouquier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Block theory via stable and Rickard equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' In Modular representation theory of finite groups (Charlottesville, VA, 1998), pages 101–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' de Gruyter, Berlin, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content=' Ryotaro KOSHIO Department of Mathematics, Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan E-mail: 1120702@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='tus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='jp Yuta KOZAKAI Department of Mathematics, Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan E-mail: kozakai@rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='tus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} +page_content='jp 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE4T4oBgHgl3EQfOQxg/content/2301.04963v1.pdf'} diff --git a/_9AyT4oBgHgl3EQf3vnw/content/2301.00776v1.pdf b/_9AyT4oBgHgl3EQf3vnw/content/2301.00776v1.pdf new file mode 100644 index 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sha256:6c5379e735644e5de5401f3f7ae76011b672aeaa4f778e7850fe3e2f5271c606 +size 4718637 diff --git a/atFAT4oBgHgl3EQf4x5l/content/tmp_files/load_file.txt b/atFAT4oBgHgl3EQf4x5l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..89681c5b07bf1ecfd5ce3586fbe19dc8e3217605 --- /dev/null +++ b/atFAT4oBgHgl3EQf4x5l/content/tmp_files/load_file.txt @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf,len=817 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='08728v1 [math-ph] 20 Jan 2023 New Mexico Tech (January 20, 2023) Spectral Asymptotics of Elliptic Operators on Manifolds Ivan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail: ivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='avramidi@nmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='edu Abstract The study of spectral properties of natural geometric elliptic partial dif- ferential operators acting on smooth sections of vector bundles over Rie- mannian manifolds is a central theme in global analysis, differential ge- ometry and mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Instead of studying the spectrum of a differential operator L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spec- tral zeta function ζ(s) = TrL−s and the heat trace Θ(t) = Tr exp(−tL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The kernel U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) of the heat semigroup exp(−tL), called the heat kernel, plays a major role in quantum field theory and quantum gravity, index the- orems, non-commutative geometry, integrable systems and financial math- ematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We review some recent progress in the study of spectral asymp- totics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We study more general spectral functions, such as Tr f(tL), that we call quantum heat traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Also, we define new invariants of differen- tial operators that depend not only on the their eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the ge- ometry of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Furthermore, we study some new invariants, such as Tr exp(−tL+)exp(−sL−), that contain relative spectral information of two differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Finally we show how the convolution of the semi- groups of two different operators can be computed by using purely algebraic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1 1 Introduction The study of spectral properties of natural geometric partial differential opera- tors is a central theme in global analysis, differential geometry and mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In particular, the basic question of spectral geometry is: “To what extent does the spectrum of an elliptic partial differential operator determine the geom- etry of the underlying manifold?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=', or as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Kac put it in his famous paper [24]: “Can one hear the shape of a drum?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In general, the answer to Kac’s question is “no” [28, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Instead of studying the spectrum of a differential operator directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function and the heat trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat trace is the trace of the heat kernel, which is the fundamental solution of the heat equation for an elliptic partial differential operator with a positive leading sym- bol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The existence of non-isometric isospectral manifolds demonstrates that the spectrum alone does not determine the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' That is why, it makes sense to study more general invariants of partial differential operators, maybe even such invariants that are not spectral invariants, that is, invariants that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Another motivation to study the heat kernel comes from quantum field theory and statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The main objects of interest are the effective action (or the partition function) and the Green functions (or the correlation functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' All these objects are expressed in terms of the functional determinants of some self- adjoint elliptic partial differential operators, their heat traces and their resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' It turns out that all of them can be expressed in terms of the heat kernels of those operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In financial mathematics one uses stochastic differential equations to model the random behavior of some financial assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then the behavior of the corre- sponding derivative securities is determined by deterministic parabolic partial dif- ferential equation (such as the diffusion or heat equation) with an elliptic partial differential operator of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The conditional probability density is then nothing else but the heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Many problems in mathematics and physics naturally lead to the presence of boundaries and to the corresponding boundary value problems for partial differen- tial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The type of the boundary conditions is not limited to the classical Dirichlet and Neumann ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In some applications, such as quantum gravity and applied mathematics, there appear mixed boundary conditions on vector bundles (that mix the Dirichlet and the Neumann one), oblique boundary conditions (that hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1 2 involve the tangential derivatives to the boundary), and even discontinuous bound- ary conditions, so called Zaremba boundary conditions, (that jump from Dirichlet to Neumann across a co-dimension 2 submanifold in the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Most natural elliptic partial differential operators are second order operators with scalar leading symbols, so called Laplace type operators, or first order op- erators whose square is a Laplace type operator, so called Dirac type operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' However, in some applications, such as in gauge field theories and quantum grav- ity, there appear second-order elliptic partial differential operators with non-scalar leading symbols, so called non-Laplace type operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Another motivation for studying such operators is non-commutative geometry where the metric tensor, that is, the inner product in the tangent bundle, is endomorphism valued in some vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In the generic situation when it is impossible to compute the heat kernel ex- actly, it becomes very important to study various asymptotic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Of special interest is the study of the short-time asymptotic expansion of the heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This expansion is closely related to the semi-classical expansion in quantum the- ory and the high-temperature expansion in statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The coefficients of this expansion, called the heat invariants, are spectral invariants associated with the asymptotic properties of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' There are also non-trivial links be- tween spectral invariants and non-linear completely integrable systems, such as the Korteweg-de Vries hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In many interesting cases such systems are, in fact, infinite-dimensional Hamiltonian systems, and the infinite set of integrals of motion of these systems is related to the spectral invariants of a linear elliptic partial differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 2 Heat Kernel 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1 Elliptic Operators Let (M,g) be a smooth compact Riemannian manifold of dimension n equipped with a positive definite Riemannian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We denote the local coordinates by xµ, with Greek indices running over 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The Riemannian volume element is defined as usual by dvol = dx g1/2, where g = detgµν and dx = dx1 ∧ ··· ∧ dxn is the standard Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let V be a smooth vector bundle over M with the typical fiber V of dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let C∞(V) be the space of smooth sections of the bundle V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The completion of the space C∞(V) defines the Hilbert space L2(V) of square integrable sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 2 3 Let ∇V : C∞(V) → C∞(T ∗M ⊗V) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) be a compatible connection on the vector bundle V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' By using the Levi-Civita connection of the metric g the connection is given its unique natural extension to bundles in the tensor algebra over V, its dual V∗ and the tangent and cotangent bundles, T M and T ∗M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' the resulting connection will usually be denoted just by ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The fiber inner product ⟨ , ⟩ defines a natural L2 inner product, ( , ), on the bundle V and the L2-trace, Tr, using the invariant Riemannian measure on the manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let ∇∗ : C∞(T M ⊗V) → C∞(V) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) be the formal adjoint of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let a, B and Q be smooth maps a : T ∗M ⊗V → T M ⊗V, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) B : T ∗M ⊗V → V, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) Q : V → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) defined by endomorphism-valued tensors satisfying aµν = aνµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) (aµν)∗ = aµν, (Bµ)∗ = −Bµ, Q∗ = Q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) Every formally self-adjoint second-order partial differential operator L :C∞(V) → C∞(V) has the form L = −∇µaµν∇ν + Bµ∇µ +∇µBµ + Q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) Natural non-Laplace type operators can be constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let Γ : T ∗M ⊗V → V (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) be a smooth map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' this defines the Dirac type operator D = Γ∇ : C∞(V) → C∞(V) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) and the operator L = D∗D : C∞(V) → C∞(V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 3 4 in this case aµν = 1 2 �Γµ ∗Γν +Γν ∗Γµ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) More generally, let Vj, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=', s, be some vector bundles and Pj : T ∗M ⊗V → Vj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) be some smooth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This defines the gradients G j = Pj∇ : C∞(V) → C∞(Vj), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) and the operator L = s � j=1 αjG∗ jG j : C∞(V) → C∞(V), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='15) where αj are some real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In this case aµν = s � j=1 αj 1 2 � Pµ ∗ j Pν j + Pν ∗ j Pµ j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='16) The leading symbol of the operator L is given by the endomorphism H(x,ξ) = aµν(x)ξµξν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='17) with x ∈ M and ξ ∈ T ∗ x M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Since this matrix is self-adjoint all its eigenvalues must be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The operator L is elliptic if all eigenvalues are positive, in other words, the matrix H(x,ξ) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The operator L is also self-adjoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' strictly speaking, it is essentially self-adjoint, that is, it has a unique self-adjoint extension (from now on, we will just say that the operator L is self-adjoint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' If all eigenvalues of the leading symbol of the operator L are equal, that is, the operator L has a scalar positive definite leading symbol defined by the Riemannian metric H(x,ξ) = |ξ|2I, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='18) with |ξ|2 = gµν(x)ξµξν, then the operator is called of Laplace type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In this case the connection can be redefined to absorb the vector Bµ, so that every Laplace type operator has the form L = −∆+ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='19) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 4 5 where ∆ = gµν∇µ∇ν = g−1/2(∂µ +Aµ)g1/2gµν(∂ν +Aν) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='20) is the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Here and below ∂µ = ∂/∂xµ denotes the partial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' A Laplace type operator is defined in terms of three pieces of local information: the Riemannian metric g, the connection one-form A and the potential Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2 Heat Kernel Let L be a self-adjoint elliptic positive partial differential operator of second order on the Hilbert space L2(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For manifolds with boundary the domain of the op- erator L has to be supplemented with some suitable elliptic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For compact manifolds the spectrum of the operator L is an increasing real se- quence of eigenvalues {λk}k∈Z+ with the corresponding orthonormal eigensections {ϕk}k∈Z+ (counted with multiplicities) defined by (L−λk)ϕk = 0, (ϕi,ϕ j)L2 = δi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='21) For noncompact manifolds the spectrum of the operator L is continuous, it goes from a positive real constant c to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In general, it is impossible to compute the spectrum exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' That is why, it becomes of special importance the study of the asymptotics of the eigenvalues (and the eigensections) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Rather than doing this directly it is more convenient to study the asymptotics of some spectral functions and special traces such as the heat trace and the zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let V ⊠V∗ be the external tensor product of the bundles V and V∗ over the product manifold M × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat kernel U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) of the operator L is a one- parameter family of smooth sections of V ⊠V∗ defined by requiring it to satisfy the heat equation (∂t + L)U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='22) for t > 0 with the initial condition U(0+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = δ(x, x′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='23) that is, U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = exp(−tL)δ(x, x′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='24) where δ(x, x′) is the covariant Dirac delta-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 5 6 The heat kernel is regular at the diagonal with a well defined diagonal value Udiag(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x) = U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='25) which is a section of the endomorphism bundle End(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat kernel diag- onal is well defined on both compact and noncompact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For compact manifolds the heat kernel can be computed in terms of the spectral data U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = ∞ � k=1 e−tλkϕk(x)ϕ∗ k(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='26) and has a well defined heat trace Θ(t) = Tr exp(−tL), Θ(t) = ∞ � k=1 e−tλk = � M dvol trUdiag(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='27) here and below tr denotes the fiber trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This also enables one to define the spectral zeta function by the Mellin-Laplace transform of the heat trace ζ(s,λ) = ∞ � k=1 1 (λk −λ)s = 1 Γ(s) ∞ � 0 ts−1etλΘ(t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='28) where λ is a complex parameter with a sufficiently large negative real part, and the regularized spectral determinant Det(L−λ) = exp � −∂sζ(s,λ) ���s=0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='29) The resolvent G(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) of the operator L is a section of V ⊠ V∗ depending on a complex parameter λ defined by (L−λ)G(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = δ(x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='30) The resolvent is related to the heat kernel by the Laplace transform G(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = ∞ � 0 dt etλU(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='31) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 6 7 For compact manifolds there is the spectral representation of the resolvent G(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = ∞ � k=1 1 λk −λϕk(x)ϕ∗ k(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='32) Off diagonal, that is, for x � x′, the resolvent is an analytic function of λ for Reλ < c with sufficiently large negative real constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' It has a diagonal singularity as x → x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' On a general manifold the heat kernel cannot be computed exactly, and that is why the short-time asymptotic expansion as t → 0 of the heat kernel and its heat trace is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For Laplace type operators on a it has a rather simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let x′ be a fixed point in the interior of the manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' we consider a sufficiently small geodesic ball centered at x′, so that each point x of the ball can be connected by a unique geodesic with the point x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This can be always done if the size of the ball is smaller than the injectivity radius of the manifold rinj(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let σ = σ(x, x′) be the Ruse-Synge function defined by σ(x, x′) = 1 2r2(x, x′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='33) where r(x, x′) is the geodesic distance between the points x and x′, and ∆(x, x′) be the Van Vleck-Morette determinant ∆(x, x′) = g−1/2(x)det � −∂2σ(x, x′) ∂xµ∂xν′ � g−1/2(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='34) Near the diagonal of M × M these two-point functions are smooth single-valued functions of the coordinates of the points x and x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In the interior of the manifold (on a finite distance from the boundary, if present) there is a local short time asymptotic expansion of the heat kernel of a Laplace type operator as t → 0+ U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) ∼ (4πt)−n/2∆1/2(x, x′)exp � −σ(x, x′) 2t � ∞ � k=0 tka2k(x, x′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='35) where ak(x, x′) are the so-called local off-diagonal heat kernel coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Notice that there are no odd-order coefficients here, that is, a2k+1(x, x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat kernel coefficients can be computed in form of a covariant Taylor series from the recurence relations obtained by substituting this ansatz in the heat equation [1] with the initial condition a0(x, x′) = P(x, x′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='36) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 7 8 where P(x, x′) is the operator of parallel transport of sections along the geodesic from the point x′ to the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This expansion immediately gives the asymptotic expansion as t → 0+ of the heat kernel diagonal Udiag(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x) ∼ (4πt)−n/2 ∞ � k=0 tkadiag 2k (x, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='37) where adiag k (x) = ak(x, x) are the diagonal local heat kernel coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For manifolds without boundary this also gives the asymptotic expansion of the heat trace Θ(t) ∼ (4πt)−n/2 ∞ � k=0 tkA2k, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='38) where A2k = � M dvoltradiag 2k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='39) are the global heat kernel coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat trace as well as the global heat kernel coefficients are obviously spectral invariants of the operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' These co- efficients were computed up to adiag 8 (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The low-order asymptotics of the heat trace of a Laplace type operator has the form Θ(t) = (4πt)−n/2 � Nvol(M)+t � M dvol �N 6 R−trQ � +O � t2�� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='40) where R is the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 3 Boundary Value Problems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1 Mixed Boundary Value Problem Now, let M be a compact Riemannian manifold with smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let N be the inward-pointing unit normal vector field to the boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' To make the operator L elliptic we need to impose some boundary conditions of the boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The classical boundary conditions are ϕ ���∂M = 0, (Dirichlet), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) ∇Nϕ ���∂M = 0, (Neumann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 8 9 The Robin boundary condition is a slight generalization of the Neumann one (∇N +S )ϕ ����∂M = 0, (Robin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) where S is a smooth self-adjoint endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' One can go further and mix these boundary conditions as follows (I −Π)ϕ ���∂M = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) Π(∇N +S )Πϕ ���∂M = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) where Π is a self-adjoint projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' A Laplace type operator L equipped with mixed boundary conditions is essentially self-adjoint and elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat kernel asymptotics as t → 0 does not depend on the boundary con- ditions in the interior of the manifold and has the same form as the heat kernel for manifolds without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Near the boundary there is a narrow strip of the width of order t1/2 where one should add an additional term, whose role is to satisfy the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In a narrow strip near the boundary this compen- sating term in the heat kernel diagonal behaves as a distribution near the boundary as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' It is precisely this feature that leads to the presence of boundary terms in the global heat kernel coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The heat trace asymptotic expansion as t → 0+ has the half-integer powers of t, Θ(t) ∼ (4πt)−n/2 ∞ � k=0 tk/2Ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) where Ak = � M dvol tr adiag k + � ∂M dvolbk , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) where the boundary heat kernel coefficients bk are local invariants constructed polynomially from the jets of the symbols of both the operator L and the boundary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The low-order asymptotics have the form Θ(t) = (4πt)−n/2 � Nvol(M)+t1/2 � ∂M dvol √π 2 (2trΠ− N) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) +t \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � M dvol �N 6 R−trQ � + � ∂M dvol �N 3 K +2trΠS �\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb+O � t3/2�� , where K is the trace of the extrinsic curvature of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 9 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2 Grubb-Gilkey-Smith Boundary Value Problem Let ˆ∇i be the tangential covariant derivative, Γi be a vector-valued anti-self-adjoint endomorphism and Λ be a first-order formally self-adjoint tangential differential operator defined by Λ = 1 2 � Γi ˆ∇i + ˆ∇iΓi� +S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) The Grubb-Gilkey-Smith boundary conditions then read (I −Π)ϕ ����∂M = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) Π(∇N +Λ)Πϕ ����∂M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) A Laplace type operator L equipped with such boundary conditions is essen- tially self-adjoint but not necessarily elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' To be elliptic the operator Λ has to satisfy the strong ellipticity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let T be a matrix defined by the leading symbol of the operator Λ, T(ξ) = Γj ˆξ j, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) where ˆξ ∈ T ∗∂M is a covector on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Since the matrices Γi are anti- self-adjoint, the matrix T 2(ξ) is self-adjoint and negative, T 2(ξ) < 0, for any ˆξ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then the oblique boundary value problem is elliptic if the matrix I|ˆξ|2 +T 2(ξ) > 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) is positive for any ˆξ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Here |ˆξ|2 = ˆgi jˆξiˆξ j is defined with the metric ˆgi j on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' When the boundary value problem is elliptic the heat trace asymptotic ex- pansion has the canonical form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Contrary to the classical boundary value problems (Dirichlet, Neumann, mixed), because of the non-commutativity of the matrices Γi, the explicit form of the coefficients bk is unknown, in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' the low-order asymptotic expansion has the form [19, 18] Θ(t) = (4πt)−n/2 \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 Nvol(M)+t1/2 � ∂M dvol √π 2 � 2trΠ−3N +2γ � +O(t) \uf8fc\uf8f4\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8f4\uf8fe , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) where γ = � Rn−1 dˆξ π(n−1)/2 tr exp � −|ˆξ|2 −T 2(ξ) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='15) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 10 11 This can be computed explicitly in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' If the matrices Γi commute then γ = tr � I +Γ2�−1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='16) where Γ2 = ˆgi jΓiΓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' One can also compute the coefficient γ explicitly in the non- commutative case when the matrices Γi form a Clifford algebra ΓiΓj +ΓjΓi = −2κΠˆgi j (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='17) with a real parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This problem is elliptic if κ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' we obtain γ = (1−κ)−(n−1)/2trΠ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='18) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3 Zaremba Boundary Value Problem Let the boundary ∂M be decomposed as a disjoint union ∂M = Σ1∪Σ2∪Σ0, where Σ1 and Σ2 are smooth compact submanifolds of ∂M of dimension (n−1), with the boundary Σ0 which is a smooth compact manifold without boundary of dimension (n − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Zaremba boundary value problem is defined by the following boundary conditions ϕ ���Σ1 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='19) (∇N +S )ϕ ���Σ2 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='20) where S is a smooth self-adjoint endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Since the boundary operator is discontinuous, this problem is a singular bound- ary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' It is well known that in this case the heat trace asymptotic expansion as t → 0+ contains, in general, logarithmic terms Θ(t) ∼ (4πt)−n/2 ∞ � k=0 tk/2Ak +logt ∞ � k=0 tk/2Hk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='21) However, Seeley [27] has shown that for Zaremba problem the logarithmic terms do not appear, that is, all Hk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='22) Therefore, the heat trace asymptotics has the canonical form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' However, the global heat kernel coefficients get a contribution from the codimension 2 subman- ifold Σ0 Ak = � M dvoltr adiag k + � Σ1 dvolb(1) k + � Σ2 dvolb(2) k + � Σ0 dvolck .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='23) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 11 12 It turns out [7] that the boundary conditions on the open sets Σ1 and Σ2 are not enough to fix the problem, and an additional boundary condition along the singular set Σ0 is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' This additional boundary condition can be considered formally as an extension of Dirichlet conditions from Σ1 to Σ0, (regular boundary condition) � √ρϕ�����Σ0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='24) where ρ is the normal geodesic distance to the singular set Σ0, or an extension of Neumann (or Robin) conditions from Σ2 to Σ0, (∂ρ −h)� √ρϕ�����Σ0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='25) where h is a real parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' However, strictly speaking the boundary condition on Σ0 does not follow from the boundary conditions on Σ1 and Σ2 and can be chosen rather arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The coefficients of the asymptotic expansion can be computed by constructing asymptotic solutions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' in the interior, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' in the thin shell near the Σ1 and Σ2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' in a thin strip close to the singular submanifold Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The trace of the heat kernel of the Zaremba boundary value problem has the fol- lowing asymptotic expansion as t → 0+ Θ(t) = (4πt)−n/2 � Nvol(M)+t1/2 √π 2 N [vol(Σ2)−vol(Σ1)] +t �� M dvol �N 6 R−trQ � + N 3 � Σ1 dvolK + � Σ2 dvol �N 3 K +2trS �� +α π 4Nvol(Σ0)+O � t3/2�� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='26) where α = −1 for the boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='24) and α = 7 for the boundary con- dition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 12 13 4 Non-Laplace Type Operators We consider a non-Laplace type operator of the form L = −∇µaµν∇ν + Q, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) where aµν is an endomorphism-valued symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' If we assume, in addi- tion, that the leading symbol of the operator L is positive then the operator L is self-adjoint and elliptic and, therefore, has the same canonical heat trace expan- sion of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) Θ(t) ∼ (4πt)−n/2 ∞ � k=0 tk/2Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) For manifolds without boundary one can get the asymptotic expansion of the heat kernel as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let ξ ∈ T ∗M be a covector and ⟨ξ, x⟩ = ξµxµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' It is easy to see that exp(−i⟨ξ, x⟩)Lexp(i⟨ξ, x⟩) = H + K + L, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) where H(x,ξ) = aµν(x)ξµξν (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) is the leading symbol of the operator L and K is the first order operator defined by K = −iξµ �aµν∇ν +∇νaµν�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) Then the asymptotics of the heat trace as t → 0 are given by Θ(t) ∼ (4πt)−n/2 � M dx � Rn dξ πn/2tr exp � −H − √ tK −tL � I , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) By using the Volterra series for the heat semigroup we get exp � −H − √ tK −tL � = e−H −t1/2 1 � 0 dτ1e−(1−τ1)HKe−τ1H +t � 1 � 0 dτ2 τ2 � 0 dτ1e−(1−τ2)HKe−(τ2−τ1)HKe−τ1H − − 1 � 0 dτ1e−(1−τ1)HLe−τ1H � +O(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 13 14 Since K is linear in ξ the term proportional to t1/2 vanishes after integration over ξ and we obtain the first two coefficients of the asymptotic expansion of the heat kernel diagonal in the form A0 = � M dx � Rn dξ πn/2tr exp�−H(x,ξ)�, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) A1 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) A2 = � M dx � Rn dξ πn/2 tr � 1 � 0 dτ2 τ2 � 0 dτ1e−(1−τ2)HKe−(τ2−τ1)HKe−τ1H − − 1 � 0 dτ1e−(1−τ1)HLe−τ1H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) Since there is no Riemannian metric, the spectral invariants of a non-Laplace type operator are not expressed in terms of the invariants of the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' It is very important to develop the corresponding differential-geometric language based on the non-scalar leading symbol of a non-Laplace type operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Boundary value problems for non-Laplace type operators are more compli- cated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For the Dirichlet boundary value problem the asymptotics are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let x = (r, ˆx) be local coordinates near the boundary where r is the normal geodesic distance to the boundary and ˆxi are local coordinates on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We decom- pose a covector ξ ∈ T ∗M as ξ = (ω, ˆξ) where ω is a real number and ˆξ ∈ T ∗∂M is a covector on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let Φ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' ˆx, ˆξ) be a function defined by Φ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' ˆx, ˆξ) = � R dω 2π � H(0, ˆx,ω, ˆξ)−λI �−1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) and Ψ(x,ξ) be a function defined by Ψ(x,ξ) = c+i∞ � c−i∞ dλ 2πie−λ ∂ ∂λ logdetΦ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' ˆx, ˆξ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) where c is a sufficiently large positive real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then the heat trace boundary coefficient A1 for the Dirichlet boundary conditions has the form A1 = − √π � ∂M d ˆx � Rn−1 dˆξ π(n−1)/2Ψ(ˆx, ˆξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 14 15 5 Non-perturbative Spectral Asymptotics Let M be a compact Riemannian manifold without boundary and V be a complex vector bundle over M realizing a representation of the group G ×U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let ϕ be a section of the bundle V and ∇ be the total connection on the bundle S (includ- ing the G-connection as well as the U(1)-connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then the commutator of covariant derivatives defines the curvatures [∇µ,∇ν]ϕ = (Rµν +iFµν)ϕ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) where Rµν is the curvature of the G-connection and Fµν is the curvature of the U(1)-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We assume that the U(1)-connection is parallel, that is, ∇µFαβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) This equation puts severe algebraic restriction on the curvature tensor RλαµνFλβ −RλβµνFλα = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) and, therefore, gives powerful restriction on the holonomy group of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For example, F could be a simplectic form of a K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) be the heat kernel of the Laplacian ∆ = gµν∇µ∇ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We rescale the curvature by F �→ 1 ε2 F, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) where ε > 0 is a small positive parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' let ∆ε be the rescaled Laplacian and Uε(εt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) be the rescaled heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then as ε → 0 there is the new asymptotic expansion of the off-diagonal heat kernel [20] Uε(ε2t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) ∼ U0(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) ∞ � k=0 εk−ntk/2bk(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) where U0(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = (4πt)−n/2∆1/2(x, x′)det � tiF sinh(tiF) �1/2 exp � − 1 4t ⟨u,tiF coth(tiF)u⟩ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) F is the matrix F = (Fµν) and uµ are normal coordinates with origin at x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Here the coefficients bk(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) are analytic functions of t that depend on F only in the hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 15 16 dimensionless combination tF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Of course, for t = 0 they are equal to the standard heat kernel coefficients, that is, bk(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x) = ak(x, x′), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) and, therefore, the odd-order coefficients at t = 0 vanish, b2k+1(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' More- over, the odd-order coefficients vanish also for any t on the diagonal x = x′, that is, bdiag 2k+1(t) = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) Then the asymptotic expansion of the heat kernel diagonal and the heat trace are Tr exp(ε2t∆ε) ∼ (4πt)−n/2 ∞ � k=0 εk−ntk/2Bk(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) where Bk(t) = � M dvol det � tiF sinh(tiF) �1/2 tr bdiag k (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) The coefficients Bk are new spectral invariants of the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' They are dif- ferential polynomials in the Riemann curvature tensor (and the curvature of the G-connection) and its derivatives with universal coefficients depending in a non- polynomial but analytic way on the curvature F, more precisely, on tF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We explicitly computed the coefficients bk (both off diagonal and the diagonal values) for k = 0,1,2,3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in F in the standard heat kernel coefficients adiag k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The first two non- zero coefficients have the form bdiag 0 (t) = I, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) bdiag 2 (t) = Jαβµν(t)RαβµνI + 1 2Hµν(t)Rµν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) where H(t) = coth(tiF)− 1 itF (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) and Jαβµν(t) is a more complicated tensor constructed from the matrix F that is analytic in t Jαβµν(t) = 1 6δα[µδβν] +O(t2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 16 17 6 Heat Determinant The existence of non-isometric isospectral manifolds demonstrates that the spec- trum alone does not determine the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' That is why it is worth studying new invariants that depend not only on the eigenvalues but also on the eigenfunc- tions, and, therefore, contain much more information about the geometry of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let {λk,ϕk}∞ k=1 be the eigenvalues and the eigensections of a Laplace type operator L acting on sections of a N-dimensional vector bundle V over a n- dimensional compact Riemannian manifold M without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let Φk l be the one-forms defined by Φk l = ⟨ϕk,Dϕl⟩ = � ϕk,∇µϕl � dxµ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) where ⟨·,·⟩ is the fiber inner product, and D = d+A is the covariant exterior deriva- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then the coefficients Ψk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='kn l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='ln = � M Φk1 l1 ∧···∧Φkn ln (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) measure the correlations between the eigensections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We define a new invariant (called the heat determinant) by K(t) = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' ∞ � k1,l1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=',kn,ln=1 exp�−t(λk1 +λl1 +···+λkn +λln)�����Ψk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='kn l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='ln ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) One can show that this invariant can be expressed directly in terms of the heat kernel U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′), K(t) = � M×M dxdx′ det � tr � U∗(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′)∇µ∇ν′U(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) We prove that there is an asymptotic expansion as t → 0 K(t) ∼ 1 2Nn(4π)−n2 � π 2n �n/2 t−n � n+ 1 2 � ∞ � k=0 tk/2Ck, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) where Ck = � M dvolck, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 17 18 and ck are differential polynomials in the Riemann curvature, the curvature of the bundle connection and the potential Q with some universal numerical coefficients that depend only on the dimensions n and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' On manifolds without boundary all odd-order coefficients vanish C2k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) In particular, C0 = vol(M) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) and the coefficients c2 and c4 are computed explicitly in our paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 7 Quantum Heat Traces We initiate the study of new invariants of second-order elliptic partial differential operators acting on sections of vector bundles over compact Riemannian mani- folds without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We draw a deep analogy between the spectral invariants of elliptic operators and the statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We consider an elliptic self-adjoint positive partial differential operator H and its square root, ω = H1/2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) which is an elliptic self-adjoint positive pseudo-differential operator of first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We interpret the classical heat trace Θ(β) = Tr exp(−βH) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) as the partition function for the Boltzman distribution with β = 1/T being the inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' By analogy, we define the relativistic heat trace Θr(β) = Tr exp(−βω) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) and the quantum heat traces Θb(β,µ) = Tr 1 exp[β(ω−µ)]−1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) Θf (β,µ) = Tr 1 exp[β(ω−µ)]+1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) where µ is a parameter that plays the role of the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We show that these new invariants can be reduced to some integrals of the classical heat trace and compute the high-temperature asymptotics of these invariants as β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 18 19 We introduce a function Aq of a complex variable q defined by the Mellin transform of the heat trace Aq = (4π)n/2 1 Γ(−q) � ∞ 0 dt t−q−1+n/2Θ(t) = (4π)n/2Γ �n 2 −q � Γ(−q) ζ �n 2 −q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) Then we show that for a positive operator H: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' the function Aq is an entire function of q, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' its values at non-negative integer points Ak, with k ∈ Z+, are equal to the standard heat trace coefficients, which are locally computable, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' while the values of the function Ak+1/2 at the half-integer points k + 1/2, with k a positive integer, as well as the values of its derivative A′ k = ∂qAq|q=k at the positive integer points k ∈ Z+ are new global invariants that are not locally computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We use the integral exp(−x) = (4π)−1/2 � ∞ 0 dt t−3/2exp � − 1 4t −tx2 � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) valid for x ≥ 0, to reduce the relativistic heat trace to the classical heat trace Θr(β) = (4π)−1/2 � ∞ 0 dt t−3/2exp � − 1 4t � Θ(tβ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) Next, we express the relativistic heat trace in terms of the function Aq via a Mellin- Barnes integral Θr(β) = 2(4π)−(n+1)/2 1 2πi c+i∞ � c−i∞ dqΓ(−q)Γ�−q+(n+1)/2��β 2 �2q−n Aq, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) We compute the asymptotics of the relativistic heat trace Θr(β) as β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We obtained in even dimension n = 2m, Θr(β) ∼ ∞ � k=0 β2k−2mb(1) k Ak + ∞ � k=0 β2k+1b(2) k Ak+m+1/2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 19 20 and in odd dimension n = 2m+1, Θr(β) ∼ ∞ � k=0 β2k−2m−1b(3) k Ak +logβ ∞ � k=0 β2k+1b(4) k Ak+m+1 + ∞ � k=0 β2k+1b(5) k A′ k+m+1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) and computed all numerical coefficients b(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Notice that the coefficients of the singular part containing the inverse powers of β and the logarithm are locally computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We express the quantum heat traces in terms of the classical one Θb, f (β,µ) = ∞ � 0 dt hb, f (t,βµ)Θ � tβ2� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) where hf (t,βµ) = (4π)−1/2t−3/2 ∞ � k=1 (−1)k+1kexp � −k2 4t +kβµ � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) hb(t,βµ) = (4π)−1/2t−3/2 ∞ � k=1 kexp � −k2 4t +kβµ � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) This gives the Mellin-Barnes representation of the quantum heat traces Θb, f (β,µ) = 2(4π)−(n+1)/2 1 2πi c+i∞ � c−i∞ dq Γ(−q)Γ�−q+(n+1)/2��β 2 �2q−n ×Fb, f (n−2q,βµ)Aq, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='15) where c < 0 and Fb(s,βµ) = ∞ � k=1 ekβµ ks = 1 Γ(s) � ∞ 0 dt ts−1 et−βµ −1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='16) F f (s,βµ) = ∞ � k=1 (−1)k+1ekβµ ks = 1 Γ(s) � ∞ 0 dt ts−1 et−βµ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='17) We compute their asymptotics as β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 20 21 For µ = 0 we obtain an asymptotic expansion as β → 0: in even dimension n = 2m, Θf (β,0) ∼ m � k=0 β2k−2mc(1) k Ak + ∞ � k=0 β2k+1c(2) k Ak+m+1/2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='18) Θb(β,0) = m � k=0 β2k−2mc(3) k Ak + ∞ � k=−1 β2k+1c(4) k Ak+m+1/2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='19) and in odd dimension n = 2m+1, Θf (β,0) ∼ ∞ � k=0 β2k−2m−1c(5) k Ak +logβ ∞ � k=0 β2k+1c(6) k Ak+m+1 + ∞ � k=0 β2k+1c(7) k A′ k+m+1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='20) Θb(β,0) = m−1 � k=0 β2k−2m−1c(8) k Ak +logβ ∞ � k=−1 β2k+1c(9) k Ak+m+1 + ∞ � k=−1 β2k+1c(10) k A′ k+m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='21) and computed all numerical coefficients c(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 8 Relative Spectral Invariants Let L± be two self-adjoint elliptic second-order partial differential operators acting on smooth sections of the vector bundle V with a positive definite scalar leading symbols of Laplace type, L± = −g−1/4 ± (∂i +A± i )g1/2 ± gi j ±(∂j +A± j )g−1/4 ± + Q±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) Here g± i j are two metrics, A± i are two connection one-forms and Q± are two endo- morphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' also gi j ± are the inverse metrics and g± = detg± i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We assume that V is a Clifford bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let D± be two self-adjoint elliptic first-order partial differential operators acting on smooth sections of the vector bundle V of Dirac type, D± = g1/4 ± iγaej ±a(∂j +A± j )g−1/4 ± +S ±, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) where γa are the Dirac matrices satisfying γaγb +γbγa = 2δabI, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 21 22 ei ±a are two orthonormal frames for the metrics gi j ± satisfying gi j ± = δabei ±aei ±b, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) A± j are two connections and S ± are two endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We require that the endomorphisms S ± commute with the Dirac matrices so that the square of the Dirac type operators, D2 ±, are operators of Laplace type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The spectral information about the operators L± and D± is contained in the classical heat traces Θ±(t) = Tr exp(−tL±), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) H±(t) = TrD± exp(−tD2 ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) The relative spectral information is contained in the relative spectral invariants Ψ(t, s) = Tr �exp(−tL+)−exp(−tL−)��exp(−sL+)−exp(−sL−)� (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) Φ(t, s) = Tr � D+ exp(−tD2 +)− D− exp(−tD2 −) �� D+ exp(−sD2 +)− D− exp(−sD2 −) � (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) which can be expressed in terms of the combined heat traces X(t, s) = Tr �exp(−tL+)exp(−sL−)� = � M×M dx dx′ tr �U+(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′)U−(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x′, x)�, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) Y(t, s) = Tr � D+ exp(−tD2 +)D− exp(−sD2 −) � = � M×M dx dx′ tr �D+U+(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′)D−U−(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x′, x)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) We study the asymptotics of the combined heat traces (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We define the time-dependent metric gi j = gi j(t, s) as the inverse of the matrix gi j = tgi j + + sgi j −, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) with t, s > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' throughout the paper we use the notation g = detgi j for the determi- nant of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Also, we define the time-dependent connection Ai = Ai(t, s) by Ai = gi j � tgjk + A+ k + sgjk − A− k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) and the vectors C± i = A± i −Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 22 23 Theorem 1 There are asymptotic expansions as ε → 0 X(εt,εs) ∼ (4πε)−n/2 ∞ � k=0 εkBk(t, s), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) Y(εt,εs) ∼ (4πε)−n/2 ∞ � k=0 εk−1Ck(t, s), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='15) where Bk(t, s) = � M dx g1/2(t, s)bk(t, s), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='16) Ck(t, s) = � M dx g1/2(t, s)ck(t, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='17) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The coefficients bk(t, s) and ck(t, s) are scalar invariants built polynomially from the covariant derivatives (defined with respect to the metric gi j and the connection Ai) of the metrics g± i j, the vectors C± i and the potentials Q± and S ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The coefficients bk(t, s) are homogeneous functions of t and s of degree k and the coefficients ck(t, s) are homogeneous functions of t and s of degree (k −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The first coefficients of the asymptotic expansion of the combined heat traces are b0(t, s) = N, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='18) c0(t, s) = N 1 2δabei +agi j(t, s)ej −b, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='19) where N = dimV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 9 Bogolyubov Invariant Further, let V be a twisted spin-tensor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let ξ and η be two self-adjoint anti-commuting involutive endomorphisms of the bundle V, ξ2 = η2 = I, ξ∗ = ξ, η∗ = η, ξη = −ηξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 23 24 Let D± be a self-adjoint elliptic first-order partial differential operators of Dirac type acting on smooth sections of the bundle V that anti-commute with η and commute with ξ, D±η = −ηD±, D±ξ = ξD±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) Suppose that the square of the operators D± are Laplace type operators D2 ± = H± (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) Then the operators D± + mη are Dirac type operators whose square are Laplace type operators (D± +mη)2 = H± +m2I (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) The bosonic and fermionic Bogolyubov invariants are determined by the fol- lowing traces Bb(β) = Tr � 1 exp(βω+)+1 − 1 exp(βω−)+1 �� 1 exp(βω+)−1 − 1 exp(βω−)−1 � , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) Bf(β) = β2Tr �(D+ +mη) sinh(βω+) − (D− +mη) sinh(βω−) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) where ω± = (H± +m2)1/2, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) The Bogolyubov invariants can be expressed in terms of the the relative spec- tral invariants Ψ(t, s) and Φ(t, s) defined in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) and (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let hb, f,0 be the func- tions defined by hf (t) = (4π)−1/2t−3/2 ∞ � k=1 (−1)k+1kexp � −k2 4t � , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) hb(t) = (4π)−1/2t−3/2 ∞ � k=1 kexp � −k2 4t � , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) h0(t) = (4π)−1/2t−3/2 ∞ � k=0 (2k +1)exp � −(2k +1)2 4t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) Then we obtain the heat trace representation for the Bogolyubov invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' For the bosonic case we get from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) Bb(β) = ∞ � 0 dt ∞ � 0 ds hf (t)hb(s)exp � −m2β2(s+t) � Ψ � β2t,β2s � , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 24 25 where Ψ(t, s) = Tr �exp(−tH+)−exp(−tH−)��exp(−sH+)−exp(−sH−)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) For the fermionic case we get from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) Bf(β) = ∞ � 0 dt ∞ � 0 ds h0(s)h0 (t)exp � −m2β2(s+t) � ×β2� Φ � β2t,β2s � +m2Ψ � β2t,β2s �� , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) where Φ(t, s) = Tr �D+ exp(−tH+)− D− exp(−tH−)��D+ exp(−sH+)− D− exp(−sH−)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) It is easy to see that the integrals for the Bogolyubov invariant converge both as t, s → 0 and (for sufficiently large m2) also as t, s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Now we can compute the asymptotics of Bogolyubov invariant as β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We obtain om odd dimension n = 2m+1 Bb(β) ∼ ∞ � k=0 β2k−nc(1) k + ∞ � k=0 β2kc(2) k , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='15) and in the even dimension n = 2m Bb(β) ∼ m−1 � k=0 β2k−nc(3) k + ∞ � k=0 β2kc(4) k +logβ2 ∞ � k=0 β2kc(5) k , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='16) Here c(i) k are some coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Notice that the coefficients c(1) k of the all odd powers of β as well as the coefficients c(3) k and c(5) k of the singular part and the logarithmic part are locally computable invariants whereas the coefficients c(2) k of the even non-negative powers of β as well as the coefficients c(4) k of the regular part are non-locally computable global invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The leading asymptotics have the form Bb(β) = β−nc(1) 0 +β−n+2c(1) 1 +O(β−n+4)+O(logβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='17) Similarly, for the Dirac operator we obtain: in the odd dimension n = 2m+1: Bf(β) ∼ ∞ � k=0 β2k−nd(1) k + ∞ � k=0 β2kd(2) k , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='18) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 25 26 and in the even dimension n = 2m Bf(β) ∼ m−1 � k=0 β2k−nd(3) k + ∞ � k=0 β2kd(4) k +logβ2 ∞ � k=0 β2kd(5) k , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='19) Here the coefficients d(1) k of the all odd powers of β as well as the coefficients d(3) k and d(5) k of the singular part and the logarithmic part are locally computable invariants whereas the coefficients d(2) k of the even non-negative powers of β as well as the coefficients d(4) k of the regular part are non-locally computable global invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The leading asymptotics have the form Bf (β) = β−nd(1) 0 +β−n+2d(1) 1 +O(β−n+4)+O(logβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='20) 10 Heat Semigroups on Weyl Algebra Let xi be the coordinates of the Euclidean space Rn and ∂i be the corresponding partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' More precisely, we consider the partial derivative operators acting on the space C∞ 0 (Rn) of smooth functions of compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Recall that this space is dense in the Hilbert space L2(Rn), which defines the extension of these operators to the whole Hilbert space L2(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Moreover, the partial deriva- tives are unbounded (essentially) anti-self-adjoint operators in this Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Then the Weyl algebra (the universal enveloping algebra of the Heisenberg alge- bra) is simply the ring of all differential operators with polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The operators ∇k are anti-self-adjoint operators of the form ∇k = ∂k − 1 2iRk jxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='1) Given a positive matrix g we introduce an operator called the Laplacian by ∆g = gi j∇i∇j, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='2) where gi j is the inverse matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' We consider two Laplacians ∆± defined with two different matrices R± i j and two different matrices g± i j and study the integral kernel of the product of the corre- sponding heat semigroups U(t, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = exp(t∆+)exp(s∆−)δ(x− x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='3) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 26 27 The (2n + 1) operators (∇1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=',∇n, x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=', xn,i) form the Lie algebra hn of the Heisenberg group H2n+1 with the commutation relations [∇k, xj] = δj k, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='4) [∇k,∇j] = iRk j , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='5) [∇k,i] = [xk,i] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='6) The universal enveloping algebra U(hn) of the Heisenberg algebra (called the Weyl algebra) is the set of all polynomials in these operators subject to these commuta- tion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' The operator exp⟨ξ,∇⟩ : C∞ 0 (Rn) → C∞ 0 (Rn) acts by �exp⟨ξ,∇⟩ f �(x) = exp � −1 2 ⟨ξ,iRx⟩ � f (x+ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='7) and is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' First, we show that the heat semigroup can be represented as a non-commutative Gaussian integral exp(t∆g) = (4π)−n/2Ω(t) � Rn dξexp � −1 4 ⟨ξ,D(t)ξ⟩ � exp⟨ξ,∇⟩ , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='8) where D(t) = (Di j) is a symmetric matrix defined by D(t) = iRcoth � tg−1iR � , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='9) and Ω(t) = det � g−1sinh(tg−1iR) g−1iR �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='10) Next, we consider two sets of such operators ∇+ i and ∇− j forming the Lie alge- bra [∇+ i ,∇+ j ] = iR+ i j, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='11) [∇− i ,∇− j ] = iR− i j, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='12) [∇+ i ,∇− j ] = iRi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='13) where Ri j = 1 2 � R+ i j +R− i j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='14) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 27 28 Then we compute the product of the semigroups exp(t∆+)exp(s∆−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' Let ∇i and Xi be the operators defined by ∇i = 1 2(∇+ i +∇− i ), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='15) Xi = ∇+ i −∇− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='16) Then exp(t∆+)exp(s∆−) = (4π)−n/2Ω(t, s)exp � X,D−1(t, s)X � (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='17) × � Rn dξ exp � −1 4 ⟨ξ,H(t, s)ξ⟩− 1 2 � ξ,ZT(t, s)D−1(t, s)X �� exp⟨ξ,∇⟩, where T±(t), D(t, s), Z(t, s) and H(t, s) are the matrices defined by T±(t) = D±(t)+iR, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='18) D(t, s) = D+(t)+ D−(s), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='19) Z(t, s) = D+(t)− D−(s)−2iR−, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='20) H(t, s) = 1 4 � D(t, s)−ZT(t, s)D−1(t, s)Z(t, s) � , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='21) and Ω(t, s) = �detT+(t)detT−(s) detD(t, s) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='22) Finally, we compute the convolution of the heat kernels U(t, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' x, x′) = (4π)−n/2(detB(t, s))1/2 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='23) ×exp � −1 4 ⟨x,A+(t, s)x⟩− 1 4 �x′,A−(t, s)x′�+ 1 2 �x,B(t, s)x′�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' � , where A±(t, s) and B(t, s) be matrices defined by A+(t, s) = D+(t)−T T + (t)D−1(t, s)T+(t), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='24) A−(t, s) = D−(s)−T T − (s)D−1(t, s)T−(s), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='25) B(t, s) = T+(t)D−1(t, s)T−(s), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='26) hkrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' January 23, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 1:22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFAT4oBgHgl3EQf4x5l/content/2301.08728v1.pdf'} +page_content=' 28 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a/b9FLT4oBgHgl3EQfYi8g/content/tmp_files/2301.12065v1.pdf.txt b/b9FLT4oBgHgl3EQfYi8g/content/tmp_files/2301.12065v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..072fd82aa5dcc4c6047dd4aa3d7b73475abd84b0 --- /dev/null +++ b/b9FLT4oBgHgl3EQfYi8g/content/tmp_files/2301.12065v1.pdf.txt @@ -0,0 +1,2080 @@ +Decentralized Entropic Optimal Transport for +Privacy-preserving Distributed Distribution Comparison +Xiangfeng Wang1 +Hongteng Xu2 +Moyi Yang1∗ +1School of Computer Science and Technology, East China Normal University +2Gaoling School of Artificial Intelligence, Renmin University of China +xfwang@sei.ecnu.edu.cn +hongtengxu@ruc.edu.cn +winnie yang@stu.ecnu.edu.cn +January 31, 2023 +Abstract +Privacy-preserving distributed distribution comparison measures the distance between the +distributions whose data are scattered across different agents in a distributed system and cannot +be shared among the agents. In this study, we propose a novel decentralized entropic optimal +transport (EOT) method, which provides a privacy-preserving and communication-efficient so- +lution to this problem with theoretical guarantees. In particular, we design a mini-batch random- +ized block-coordinate descent (MRBCD) scheme to optimize the decentralized EOT distance in +its dual form. The dual variables are scattered across different agents and updated locally and +iteratively with limited communications among partial agents. The kernel matrix involved in +the gradients of the dual variables is estimated by a distributed kernel approximation method, +and each agent only needs to approximate and store a sub-kernel matrix by one-shot commu- +nication and without sharing raw data. We analyze our method’s communication complexity +and provide a theoretical bound for the approximation error caused by the convergence error, +the approximated kernel, and the mismatch between the storage and communication protocols. +Experiments on synthetic data and real-world distributed domain adaptation tasks demonstrate +the effectiveness of our method. +1 +Introduction +Distribution comparison plays a central role in many machine learning problems, such as data +clustering (Hammouda & Karray, 2000), generative modeling (Bond-Taylor et al., 2021; Mattei & +Frellsen, 2019), domain adaptation (Ganin & Lempitsky, 2015; Farahani et al., 2020), etc. As a +valid metric for distributions, optimal transport (OT) distance (Villani, 2009) provides a powerful +solution to this task. Given two distributions, the optimal transport distance corresponds to the +minimum expectation of their underlying sample distance. The corresponding optimal joint distri- +bution of sample pairs takes these two distributions as its marginals. Mathematically, given two +probability measures in a compact space X, denoted as µ, γ ∈ P(X), the Kantorovich formulation +of the optimal transport distance between them is defined as +W(µ, γ) def. += +inf +π∈Π(µ,γ) +� +X 2 c(x, y) dπ(x, y), +(1) +∗The authors have equal contributions and are listed in alphabetical order of their last names. +1 +arXiv:2301.12065v1 [cs.LG] 28 Jan 2023 + +Source Domain +Target Domain +AC +BXicbVDJSgNBEO2JW4zbqEc9NAYhgoQZcTsGvXiMYBbIhFDT6SRNunuG7p5AGHLx4q948aCIV/Bm39jZzlo4oOCx3tVNULY8608bxvJ7O0vLK6 +l3PbWxube+4u3tVHSWK0AqJeKTqIWjKmaQVwyn9VhRECGntbB/O/ZrA6o0i+SDGca0KaArWYcRMFZquYe1VhoMQNFYMx7JUSEQySkOuiAEnLT +cvFf0JsCLxJ+RPJqh3HK/gnZEkGlIRy0bvhebJopKMIp6NckGgaA+lDlzYslSCobqaTL0b42Cpt3ImULWnwRP09kYLQeihC2ynA9PS8Nxb/8xq +J6Vw3UybjxFBJpos6CcmwuNIcJspSgwfWgJEMXsrJj1QIwNLmdD8OdfXiTVs6J/Wby4P8+XbmZxZNEBOkIF5KMrVEJ3qIwqiKBH9Ixe0Zvz5L +w4787HtDXjzGb20R84nz8ex5hUW"(µ, �) +Decentralized EOT 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+QhrLZbtqlm03Y3Qgl9Cd48 +aCIV3+RN/+N2zSCij4YeLw +3w8w8P+ZMacf5sJaWV1bX1 +gsbxc2t7Z3d0t5+W0WJLR +FIh7Jro8V5UzQlma024sK +Q59Tjv+5Grud+6pVCwSd3o +aUy/EI8ECRrA20m0/TAals +mNXHfeidoc28mQkbpbqSM +3V8qQozkovfeHEUlCKjThW +Kme68TaS7HUjHA6K/YTRWN +MJnhEe4YKHFLlpdmpM3Rsl +CEKImlKaJSp3ydSHCo1DX3 +TGWI9Vr+9ufiX10t0UPdSJ +uJEU0EWi4KEIx2h+d9oyCQ +lmk8NwUQycysiYywx0Sado +gnh61P0P2lXbLdmV2/Oyo3 +LPI4CHMIRnIAL59CAa2hC +wiM4AGe4Nni1qP1Yr0uWpe +sfOYAfsB6+wSe64Lµ +Distributed +Storage +Distributed +Storage +Figure 1: An illustration of the privacy-preserving distributed distribution comparison task and +the corresponding decentralized entropic optimal transport problem. To emphasis, the data in two +domains obey the distribution µ and γ, respectively, but are scattered among different agents. Each +agent merely contains a part of samples, and the local distribution is denoted as µi (or γj). +where c : X 2 �→ [0, ∞) denotes a continuous cost function, the joint distribution on the product +space X ×X is denoted as the coupling π, and Π(µ, γ) = {π ∈ P(X ×X) +�� ∀(A, B) ⊂ X ×X, π(A× +X) = µ(A), π(X × B) = γ(B)} denotes the marginal constraints of the coupling. The OT distance +is applicable even if the supports of the two distributions are non-overlapped. Therefore, leveraging +it to fit a model distribution to the data distribution (Arjovsky et al., 2017; Deshpande et al., 2018) +or transferring a source distribution to a target one (Courty et al., 2014; Damodaran et al., 2018) +usually leads to encouraging performance. +In practice, the optimal transport distance in equation 1 is often implemented with an en- +tropic regularizer (Cuturi, 2013; Blondel et al., 2018), which leads to a strictly-convex optimization +problem called entropic optimal transport (EOT): +Wε(µ, γ) def. += +inf +π∈Π(µ,γ) +� +X 2 c(x, y) dπ(x, y) − εH(π), +(2) +where H(π) = − +� +X×X π(x, y) log(π(x, y) − 1) dxdy denotes the entropy of π. Given the samples of +µ and γ, this problem can be solved more efficiently using alternate minimization schemes, e.g., the +Sinkhorn algorithm (Cuturi, 2013) and its stochastic version (Altschuler et al., 2017), the primal- +dual method (Blondel et al., 2018), and the block-coordinate descent method (Genevay et al., +2016). These algorithms are centralized in general – a central server records all pairwise costs, i.e., +C = [c(x, y)], and optimizes the objective accordingly. +However, the samples in many real-world applications are often large-scale and scattered across +different agents in a distributed and decentralized system (i.e., each agent only has limited storage +and computation power, and none works as a central server). In addition, sharing raw data directly +can be forbidden in this system because of privacy and security requirements. As illustrated in Fig- +ure 1, such a scenario leads to a challenging privacy-preserving distributed distribution comparison +task, in which each agent can access neither the whole sample sets nor the complete cost matrix +C. Accordingly, a decentralized method is required to solve the EOT problem with limited and +privacy-preserving communications. +In this study, we propose a novel decentralized entropic optimal transport method, which pro- +vides a theoretically-guaranteed solution to privacy-preserving distributed distribution comparison. +In particular, our method considers the dual form of the EOT problem, in which the dual objective +2 + +involves a kernel associated with the cost matrix, and the dual variables are scattered among the +agents. We approximate a sub-kernel matrix for each agent via one-shot communication and avoid +sharing raw data among the agents (Khanduri et al., 2021). Based on the approximated kernel, +the dual variables are optimized in a mini-batch randomized block-coordinate descent (MRBCD) +scheme (Zhao et al., 2014). Each agent stores and updates the dual variables corresponding to its +local data. The dual variables’ gradients are computed based on partial dual variables (rather than +raw data) from some randomly-selected agents. +Our method is privacy-preserving and communication-efficient because the communications do +not contain raw data, and the communication cost is independent of the data dimension. More- +over, as our main theoretical contribution, we make the first attempt to provide an error bound of +decentralized EOT distance caused by the convergence error, the kernel approximation error, and +the mismatch between the distributed system’s storage and communication protocols. Experiments +on synthetic data verify the effectiveness of our method and its robustness to hyperparameters and +communication protocols. Furthermore, we test our decentralized EOT in real-world distributed +domain adaptation tasks, demonstrating its usefulness for privacy-preserving distributed distribu- +tion comparison. +2 +Related Work +2.1 +Entropic optimal transport methods +Typically, the entropic optimal transport distance can be solved by the Sinkhorn algorithm (Sinkhorn +& Knopp, 1967; Cuturi, 2013; Benamou et al., 2015) (or its logarithmic variant (Chizat et al., +2018; Schmitzer, 2019) for improving numerical stability). Following the Sinkhorn algorithm, the +method in (Xie et al., 2020) computes the OT distance via an inexact proximal point algorithm, +which is equivalent to solving an EOT problem with a temporally-decayed entropic regularizer. The +Greenkhorn algorithm (Altschuler et al., 2017) works as a stochastic Sinkhorn algorithm with sig- +nificant computation efficiency. Besides the Sinkhorn algorithm, some other efficient algorithms are +developed, e.g., the Bregman alternating direction method of multipliers (Bregman ADMM) (Wang +& Banerjee, 2014a; Ye et al., 2017), smoothed semi-dual algorithm (Blondel et al., 2018), and con- +ditional gradient algorithm (Titouan et al., 2019). The work in (Genevay et al., 2016; Seguy et al., +2018) introduce stochastic optimization mechanisms into the large-scale optimal transport problem. +The above methods mainly focus on the EOT problems in centralized scenarios. The decen- +tralized EOT is seldom studied. Recently, the work in (Zhang & Zhu, 2019; Hughes & Chen, 2021) +proposes some ADMM-based decentralized algorithms for distributed resource allocation tasks. The +formulation of their tasks is relevant to an optimal transport problem rather than an EOT prob- +lem. Moreover, unlike our work, their methods neither apply any privacy-preserving mechanism +nor consider the mismatch between the distributed system’s storage and communication protocols. +Additionally, the decentralized EOT problem differs from the well-known distributed and decen- +tralized Wasserstein barycenter problems Staib et al. (2017); Uribe et al. (2018); Dvurechenskii +et al. (2018). In these barycenter problems, the samples of a distribution are still stored in a single +agent so that the (entropic) optimal transport distance between the distribution and the barycenter +can still be computed in a centralized manner. On the contrary, in our study, each agent can only +access a part of the distribution samples and cannot share them with other agents, so we need a +decentralized algorithm to compute the EOT distance. +3 + +2.2 +Distributed and decentralized optimization +Distributed and decentralized optimization methods can be broadly divided into primal and primal- +dual types (Nedic & Ozdaglar, 2009; Duchi et al., 2011). Primal methods usually refer to gradient- +based methods such as decentralized gradient descent (DGD) (Yuan et al., 2016; Lobel & Ozdaglar, +2010), EXTRA (Shi et al., 2015), etc. +For large-scale optimization tasks, the stochastic gra- +dient technique is often applied, e.g., the decentralized stochastic gradient descent method (D- +SGD) (Agarwal & Duchi, 2011) is generalized from the DGD, with significant computational effi- +ciency acceleration. The primal-dual approaches further employ dual variables to design distributed +optimization methods, which incorporate distributed dual decomposition (Terelius et al., 2011), +ADMM (Chang et al., 2015; Shi et al., 2014), etc. More discussions can refer to a series of survey +papers (Chang et al., 2020; Nedic, 2020; Assran et al., 2020). +Besides computational efficiency, communication efficiency is also required for distributed and +decentralized optimization. The most intuitive way to increase the communication efficiency is to +reduce the number of agents involved in communications (Smith et al., 2018), e.g., the random +node selection scheme (Arablouei et al., 2015; Mao et al., 2020) and the importance sampling +scheme (Chen et al., 2018; Liu et al., 2019). To reduce the bandwidth, the information for each +communication is often compressed by sparsification (Stich et al., 2018; Basu et al., 2019; Tang +et al., 2020) or quantization (Zhu et al., 2016; Alistarh et al., 2017; Zhang et al., 2019; Lu & De Sa, +2020). Recently, the distributed and decentralized optimization techniques have been utilized for +distributed deep learning (Tang et al., 2020), distributed edge AI system (Shi et al., 2020), federated +learning (Chen et al., 2021), etc. +As aforementioned, these applications often involve privacy- +preserving distributed distribution comparison, which motivates us to develop the decentralized +EOT method with privacy preservation and communication efficiency. +3 +Proposed Algorithm Framework +3.1 +Dual formulation of decentralized EOT +Suppose that there are I agents in the source domain storing the samples of µ and J agents in the +target domain storing the samples of γ, as illustrated in Figure 1. The distribution of the samples +in the i-th source agent (the j-th target agent) is denoted as µi (γj). Accordingly, the storage of +the samples in the agents can be captured by the following hierarchical model: +Agent selection: +i ∼ p, +j ∼ q, +Sample assignment: +Xi = {x(i) +n }Ni +n=1 ∼ µi, +Yj = {y(j) +m }Mj +m=1 ∼ γj, +(3) +where p = {pi}I +i=1 ∈ ∆I−1 and q = {qj}J +j=1 ∈ ∆J−1, respectively. pi (qj) represents the probability +of selecting the source agent i (the target agent j) to store the corresponding data. Xi = {x(i) +n }Ni +n=1 +and µi denote the samples stored in the agent i and the corresponding sample distribution. Yj = +{y(j) +m }Mj +m=1 and γj are denoted in the same way. Obviously, we have µ = � +i piµi, γ = � +j qjγj, +X = {xn}N +n=1 = ∪iXi, and Y = {ym}M +m=1 = ∪jYj. +• Storage Protocol. Sampling source and target agents independently from p and q is equiv- +alent to sampling the agent pairs from the distribution p ⊗ q = [piqj]. Here, we define p ⊗ q +as the storage protocol of the distributed system. +4 + +Taking the storage protocol and the Fenchel dual form of the EOT distance (Peyr´e & Cuturi, +2019) into account, we rewrite the EOT distance in equation 2 as follows: +Wε(µ, γ) = supu,v∈CX +� +X +u(x)dµ(x) + +� +X +v(y)dγ(y) − ε +� +X 2 e +u(x)+v(y)−c(x,y) +ε +dµ(x)dγ(y) += supu,v∈CX Ex∼µ,y∼γfε(x, y, u, v) += supu,v∈CX E(i,j)∼p⊗qEx∼µi,y∼γjfε(x, y, u, v). +(4) +Here, CX represents the set of continuous functions defined in X, u, v ∈ CX denote the dual functions, +which are also called Kantorovich potentials, and +fε(x, y, u, v) = u(x) + v(y) − εe +u(x)+v(y) +ε +e− c(x,y) +ε +� �� � +κ(x,y) +, +(5) +where κ(x, y) is a kernel function associated with the cost c(x, y). The second equation in equation 4 +indicates that the EOT problem can be modeled as an unconstrained expectation maximization +problem with respect to u and v (Genevay et al., 2016). The third equation in equation 4 is based +on the hierarchical model in equation 3. +• Communication Protocol. As shown in equation 4, computing the EOT distance requires +us to sample agent pairs based on the storage protocol. In practice, however, the sampling of +the agent pairs is determined by the communication protocol rather than the storage protocol +of the distributed system. Here, we define the communication protocol as the distribution of +the communicable agent pairs, denoted as E = [eij] ∈ {E ∈ RI×J ++ +| 1⊤ +I E1J = 1}. +Note that, the communication protocol can be mismatched with the storage protocol. +For +example, some systems do not allow multi-step routes and/or restrict the communication between +the agents to be directed, which may cause E ̸= p ⊗ q. +As a result, we actually approximate +Wε(µ, γ) by the following surrogate: +� +Wε(µ, γ) def. += +sup +u,v∈CX +E(i,j)∼EEx∼µi,y∼γjfε(x, y, u, v). +(6) +In theory, the gap between � +Wε(µ, γ) and Wε(µ, γ) is bounded under mild assumptions: +Theorem 1. Let µ = � +i piµi and γ = � +j qjγj be the two distributions in a distributed system with +I source agents and J target agents, whose storage and communication protocols are p ⊗ q = [piqj] +and E = [eij], respectively. Let maxi,j Wε(µi, γj) ≤ τ for some τ > 0 and � +i,j |eij − piqj| ≤ σ for +some σ > 0. We have +|� +Wε(µ, γ) − Wε(µ, γ)| ≤ τσ. +(7) +Theorem 1 indicates that as long as the mismatch between the storage and communication +protocols (i.e., σ) is small, we can approximate Wε(µ, γ) well by � +Wε(µ, γ).1 Ideally, � +Wε(µ, γ) = +Wε(µ, γ) when E = p ⊗ q. Fortunately, this ideal case can be available in many situations. For +example, in a distributed system built on a connected network and with a known storage protocol, +we can first select a source agent based on p and then select a target agent based on q (so that +E = p ⊗ q). In more general settings, we often can adjust the communication protocol, matching +it with the storage protocol as much as possible. +1Theorem 1 is valid for both continuous probability measures and sample-based discrete measures. +5 + +Given the samples of µ and γ, the problem in equation 6 becomes +max +u={u(i)}I +i=1∈RN +v={v(j)}J +j=1∈RM +Fε(u,v;K,E) +� +�� +� +I +� +i=1 +J +� +j=1 +eij +NiMj +fε(Kij, u(i), v(j)) +� +�� +� +f(i,j) +ε +, +(8) +where the dual functions u(x) and v(y) become the dual variables u ∈ RN and v ∈ RM, respectively. +The dual objective Fε(u, v; K, E) takes the kernel matrix K = [κ(xn, ym)] ∈ RN×M and the com- +munication protocol E as its hyperparameters. The dual objective is decomposable — for the agent +pair (i, j), f(i,j) +ε += �Ni +n=1 +�Mj +m=1 fε(κ(x(i) +n , y(j) +m ), u(i) +n , v(j) +m ) = u(i) +n + v(j) +m − ε exp( u(i) +n +v(j) +m +ε +)κ(x(i) +n , y(j) +m ) is +the corresponding local objective, in which Kij = [κ(x(i) +n , y(j) +m )] ∈ RNi×Mj is a block of K. More- +over, each local objective only involves a part of dual variables that correspond to the local samples +stored in the agents, e.g., the u(i) = [u(i) +n ] ∈ RNi in f(i,j) +ε +corresponds to the samples {x(i) +n }Ni +n=1 in +the source agent i. As a result, the dual variables can be scattered across different agents, and +accordingly, the gradient of u(i) can be formulated as follows: +∇u(i)Fε(u, v; K, E) = +J +� +j=1 +eij +NiMj +∇u(i)f(i,j) +ε += +J +� +j=1 +eij +NiMj +� +����� +�Mj +m=1 1 − e +u(i) +1 ++v(j) +m +ε +κ(x(i) +1 , y(j) +m ) +... +�Mj +m=1 1 − e +u(i) +Ni ++v(j) +m +ε +κ(x(i) +Ni, y(j) +m ) +� +����� +. +(9) +The gradient in equation 9 points out the challenges of the proposed decentralized EOT problem. +In particular, when computing the gradient of u(i), we need to transmit the raw data and the dual +variables of all the agents in the other domain to the agent i. The communication cost of this step +is high, especially for high-dimensional data. Moreover, sharing raw data results in the leakage of +private information. Facing the above challenges, in the following subsection, we will introduce a +decentralized EOT method to solve equation 8 in a privacy-preserving and communication-efficient +way. +3.2 +Proposed decentralized EOT method +The proposed decentralized EOT method consists of two steps: +i) leveraging a theoretically- +guaranteed method to approximate the kernel matrix without the share of raw data and ii) updat- +ing the dual variables locally and iteratively in a mini-batch randomized block-coordinate descent +(MRBCD) scheme. +Privacy-preserving kernel approximation. When the cost c(x, y) is Euclidean, the kernel +κ(x, y) in equation 5 is a special case of the following generalized inner product (GIP) kernel (Khan- +duri et al., 2021): +κ(x, y) = g(φ(x, y), ∥x∥, ∥y∥), +(10) +where φ(x, y) = arccos( ⟨x,y⟩ +∥x∥∥y∥), and g(φ, ∥x∥, ∥y∥) is a G-Lipschitz continuous function with respect +to φ. +For the GIP kernel, it is possible to approximate it without the share of raw data (Khanduri +et al., 2021). Denote D as the dimension of samples. Leveraging the random seed sharing method +in (Xu et al., 2021; Richards et al., 2020), we can sample P D-dimensional random variables from +a multivariate normal distribution, i.e., {ωℓ}P +ℓ=1 ∼ N(0, ID), and broadcast them to all the agents. +6 + +Algorithm 1 Privacy-preserving Kernel Approximation +1: Draw random variables {ωℓ ∈ RD}P +ℓ=1 ∼ N(0, ID) and broadcast them to all agents. +2: for Each source agent i ∈ {1, ..., I} do +3: +Construct Aµi via equation 11 and broadcast it to all target agents. +O(JNiP) +4: +If data is not normalized, broadcast {∥x(i) +n ∥}Ni +n=1 to all target agents. +O(JNi) +5: end for +6: for Each target agent j ∈ {1, ..., J} do +7: +Construct Aγj via equation 11 and broadcast it to all source agents. +O(IMjP) +8: +If data is not normalized, broadcast {∥y(j) +m ∥}Mj +m=1 to all source agents. +O(IMj) +9: end for +10: Construct { �Kij}J +j=1 for each source agent i and { �Kij}I +i=1 for each target agent j via equation 12. +Based on the random variables, we can construct a binary matrix for each agent. Take the source +agent i as an example. Given Ni samples {x(i) +n }Ni +n=1, we have +Aµi = [I(⟨ωℓ, x(i) +n ⟩ ≥ 0)] ∈ {0, 1}P×Ni, +(11) +where I(·) is an indicator, which outputs 1 if the input statement is true and outputs 0 otherwise. +As a result, for each agent pair (i, j), the kernel κ(x(i) +n , y(j) +m ) of their samples can be approximated +by +ˆκ(x(i) +n , y(j) +m ) = g( ˆψ(a(i) +n , a(j) +m ), ∥x(i) +n ∥, ∥y(j) +m ∥) = g +� +π +���1 − 2 +P ⟨a(i) +n , a(j) +m ⟩ +���, ∥x(i) +n ∥, ∥y(j) +m ∥ +� +, +(12) +where a(i) +n is the n-th column of Aµi and a(j) +m is the m-th column of Aγi. Based on equation 12, we +can obtain an approximated kernel matrix for an agent pair (i, j), i.e., �Kij = [ˆκ(x(i) +n , y(j) +m )]. This +approximation preserves data privacy because it only requires two constructed binary matrices and +the norms of samples. The scheme of the privacy-preserving kernel approximation method is shown +in Algorithm 1. The communication complexity of each step is shown in red. +As shown in Algorithm 1, by one-shot communication, each agent obtains the matrices A’s from +all the agents in the other domain. Accordingly, the overall communication complexity is O((IM + +JN)P). Note that, this complexity is independent with the sample dimension D, so it is suitable +for high-dimensional cases. Moreover, even if P = O(N), the practical communication cost can still +be tractable because the matrices A’s are binary and can be compressed before communication. +The precision of the proposed approximation is guaranteed under mild assumptions: +• Assumption 1. The kernel in the objective is a GIP kernel, i.e., κ(x, y) = g(φ(x, y), ∥x∥, ∥y∥) +and g is a G-Lipschitz continuous function with respect to φ. +• Assumption 2. Both the kernel κ and its approximation ˆκ are bounded, i.e., |κ(x, y)| ≤ b +and |ˆκ(x, y)| ≤ b for some b ≥ 1. +Theorem 2. Let K ∈ RN×M be the matrix defined by the GIP kernel in equation 10 and �K be the +approximation achieved via equation 12. Based on the assumptions 1-2, with probability at least +1 − δ, we have ∥K − �K∥ ≤ G(N + M) +�� +32π2 +P +log 2(N+M) +δ ++ 8π +3P log 2(N+M) +δ +� +. +Theorem 2 means that �K → K when P → ∞. P = O(ϵ−2) achieves an approximation error +ϵ > 0. Theorem 2 is based on the Lemma 4.1 in (Khanduri et al., 2021). More details can be found +at Appendix A. +7 + +Algorithm 2 MRBCD for Decentralized EOT Distance +1: For each source agent i and target agent j, construct { �Kij}J +j=1 and { �Kij}I +i=1 via Algorithm 1, +and initialize u(i) = 0 and v(j) = 0. +O((IM + JN)P) +2: Update dual variables: +O(TL( N +I + M +J )) +3: for t = 0, 1, · · · , T do +4: +Set the learning rate ηt = +η +√t+1. +5: +for An agent pair (i, j) ∼ E do +6: +Select L target agents JL ∼ +1 +∥E[i,:]∥1 E[i, :]. Send {v(j),t}j∈JL to the source agent i. +7: +u(i),t+1 ← u(i),t + ηt +� +j∈JL ∇u(i) ˆf(i,j),t +ε +8: +Select L source agents IL ∼ +1 +∥E[:,j]∥1 E[:, j]. Send {u(i),t}i∈IL to the target agent j +9: +v(j),t+1 ← v(j),t + ηt +� +i∈IL ∇v(j) ˆf(i,j),t +ε +10: +end for +11: end for +12: For an arbitrary source agent i, receive the optimal dual objectives {{ ˆf(i′,j) +ε +}J +j=1}i′̸=i from the +remaining agents in the source domain. +O(IJ) +13: Compute � +Wε(µ, γ) in the source agent i and broadcast it to all other agents. +O(IJ) +AB7n +icbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQi8cIx +gSJcxOJsmQmdlplcISz7CiwdFvPo93vwbJ8k +eNLGgoajqprsrSqSw6PvfXmFldW19o7hZ2tre2d +0r7x82jg1jDdYLGPTiqjlUmjeQIGStxLDqYokb +0aj26nfOLGilg/4DjhoaIDLfqCUXRSs6PSbhZ +MuWKX/VnIMskyEkFctS75a9OL2ap4hqZpNa2Az +/BMKMGBZN8UuqklieUjeiAtx3VHEbZrNzJ+TE +KT3Sj40rjWSm/p7IqLJ2rCLXqSgO7aI3Ff/z2in +2r8NM6CRFrtl8UT+VBGMy/Z30hOEM5dgRyoxwt +xI2pIYydAmVXAjB4svL5PGsGlxWL+7PK7WbPI4i +HMExnEIAV1CDO6hDAxiM4Ble4c1LvBfv3fuYtx +a8fOYQ/sD7/AFN04+Qµ1 +AB7n +icbVDJSgNBEK2JW4xb1KOXxiB4CjPB7Rj04jGCW +SAZQk+nJ2nS3TP0IoQhH+HFgyJe/R5v/o2dZA6 +a+KDg8V4VfWilDNtfP/bK6ytb2xuFbdLO7t7+w 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Plugging the +approximated kernel into equation 8, we denote the dual objective using the approximated kernel, +i.e., Fε(u, v; �K, E) = � +i,j eij ˆf(i,j) +ε +, where ˆf(i,j) +ε += fε( �Kij, u(i), v(j)). +Accordingly, we propose a +mini-batch randomized block coordinate descent scheme, computing the gradient of ˆf(i,j) +ε +based on +a batch of agents and optimizing the decentralized EOT problem iteratively. +Take a source agent i as an example. In the t-th iteration, the agent i receives the dual variables +{v(j),t}j∈JL from L target agents, where JL ⊂ {1, ..., J} denotes the set of the L target agents. In +practice, we sample JL based on the communication protocol E, i.e., JL ∼ +1 +∥Ei,:∥1 Ei,:, where Ei,: +is the i-th row of E. Then, the agent i computes a stochastic gradient � +j∈JL ∇u(i) ˆf(i,j),t +ε +(Zhao +et al., 2014) and update u(i) via +u(i),t+1 ← u(i),t + ηt +� +j∈JL ∇u(i) ˆf(i,j),t +ε +. +(13) +Here, ηt is the learning rate in the t-th iteration. We set ηt = +η +√t+1, where η is the initial learning +rate. The dual variables in the target agents can be updated in a similar way. Applying the above +steps iteratively till the dual variables converge, each source agent i can compute and store the local +8 + +dual objectives { ˆf(i,j) +ε +}J +j=1 based on the information received during the iterations. Accordingly, +the source agent i can compute the EOT distance � +Wε(µ, γ) by collecting {{ ˆf(i′,j) +ε +}J +j=1}i′̸=i from +other source agents. Finally, the source agent i broadcasts � +Wε(µ, γ) to all other agents. +Algorithm 2 shows the MRBCD scheme, in which the communication complexity per step is +given in red. In particular, when updating the dual variables, the communication cost per iteration +is � +i∈IL Ni+� +j∈JL Mj. When the numbers of samples in different agents are comparable, i.e., Ni = +O( N +I ) and Mj = O( M +J ), the communication complexity per iteration can be represented as O(L( N +I + +M +J )). Accordingly, the overall communication complexity for updating dual variables is O(TL( N +I + +M +J )), where T is the number of iterations. +When L = J, we compute the gradient ∇u(i)Fε = +�J +j=1 ∇u(i) ˆf(i,j) +ε +exactly, and Algorithm 2 becomes the classic randomized block-coordinate descent +(RBCD) (Nesterov, 2012). When L = 1, we only consider the exchange of dual variables between an +agent pair in each iteration. This setting is suitable for the agent with limited computation power +because each iteration only involves a pair of agent. This MRBCD scheme is privacy-preserving, +which only transmits dual variables, local dual objectives, and the approximated EOT distance. In +summary, Figure 2 illustrates the proposed decentralized EOT method. +Essentially, Algorithm 2 is a decentralized and mini-batch stochastic implementation of the +randomized block-coordinate descent (RBCD) method Nesterov (2012); Richt´arik & Tak´aˇc (2014); +Lu & Xiao (2015). Therefore, following the Theorem 3 in Wang & Banerjee (2014b), we analyze +the convergence error of Algorithm 2 as follows: +• Assumption 3. The gradient norm is bounded, i.e., ∥∇u,vFε∥2 ≤ R for some R > 0. +Lemma 3. For the problem in equation 8 with approximated kernel �K, let (u∗, v∗) ∈ C∗ be the +optimal solution in the optimal solution set and +�� +ut, vt�� +be the sequence generated by Algorithm 2. +Define R0 = minu,v∈C∗ ∥(u0, v0) − (u, v)∥2, ˆut = 1 +t +�t +ℓ=1 uℓ, and ˆvt = 1 +t +�t +ℓ=1 vℓ. Based on the +assumption 3, we have +E|Fε(ˆut, ˆvt; �K, E) − Fε(u∗, v∗; �K, E)| ≤ O +�IJ +� +( +√ +t + LFε)R2 +0 + +√ +tR2� +t +� +, +(14) +where LFε denotes the Lipschitz constant of the objective function Fε and the expectation is calcu- +lated with respect to the randomly-selected agents. +3.3 +Theoretical error bound of decentralized EOT +The proposed decentralized EOT method is theoretically-guaranteed. We take the convergence +error of the MRBCD algorithm, the optimal loss caused by the approximated kernel, and the +mismatch between the storage and communication protocols into account, deriving the bound of +the expected approximation error under mild assumptions. +In particular, let µ = � +i piµi and +γ = � +j qjγj be the two distributions in a distributed system with I source agents and J target +agents, whose storage and communication protocols are p ⊗ q = [piqj] and E = [eij], respectively. +We scatter N samples of µ to the source agents and M samples of γ to the target agents. The +kernel matrix of the samples is approximated via Algorithm 1, with the hyperparameter P. As a +consequence of Theorem 1, Theorem 2, and Lemma 3, we have +Theorem 4. For the problem in equation 8 with the kernel �K derived by Algorithm 1, let +�� +ut, vt�� +be the sequence generated by Algorithm 2. Define ˆut = 1 +t +�t +ℓ=1 uℓ, ˆvt = 1 +t +�t +ℓ=1 vℓ, and the gap be- +tween the storage and communication protocols as σ. Based on the assumptions 1-3, with probability +9 + +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +MRBCD (L=1) +MRBCD (L=2) +MRBCD (L=5) +MRBCD (L=10) +(a) Wε(N1, N2) +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +MRBCD (L=1) +MRBCD (L=2) +MRBCD (L=5) +MRBCD (L=10) +(b) Wε(GMM1, GMM2) +Figure 3: The results for the convergence of our MRBCD scheme. +at least 1 − δ, we have +E|Fε(ˆut, ˆvt; �K, E) − Wε(µ, γ)| ≤ O +�IJ +√ +t + (N + M) +� +1 +P log 2(N + M) +δ ++ σ +� +. +(15) +The proof of the theorem is shown in Appendix A. +4 +Numerical Experiments +To demonstrate the effectiveness of our decentralized EOT method, we analyze its performance +on synthetic data and apply it to distributed domain adaptation tasks. Representative results are +shown below. Implementation details and more results are in Appendix B. +4.1 +Analytic experiments on synthetic data +We first test the performance of the proposed MRBCD scheme and analyze the influence of the +hyperparameter L (i.e., the number of agents used to compute the gradients in each iteration). +In particular, we consider two synthetic datasets in this experiment: the first dataset contains +two 2D Gaussian distributions (N1, N2), each of which includes 2,000 samples, and the second one +contains two 2D Gaussian mixture models (GMM1, GMM2), each of which includes two Gaussian +components and 2,000 samples. For each dataset, we randomly scatter one distribution’s samples +to 10 source agents and the other distribution’s samples to 10 target agents, respectively. Following +the decentralized optimization work in (Zhang & Zhu, 2019; Hughes & Chen, 2021), we assume the +network of the agents to be connected, i.e., there always exists a route between two arbitrary agents. +Accordingly, the storage and communication protocols are uniform distributions. The kernel matrix +is approximated with P = 1000. We approximate Wε(N1, N2) and Wε(GMM1, GMM2) via our +MRBCD scheme, in which L ∈ {1, 2, 5, 10}, and compare the results with the ground truth achieved +by the centralized Sinkhorn algorithm Cuturi (2013). Figure 3 visualizes the convergence of our +scheme with different L’s. We can find that with the increase of L, the convergence is faster because +the gradients in each iteration are computed with less uncertainty. When L ≥ 2, our method can +approximate the EOT distance with a small gap after sufficient iterations. +10 + +� +� +� +� +� +�� +�����������)���)���(���� +���� +���� +���� +���� +���� +���� +���� +������()���� +����������) +����������� +������ +��� +��� +EOT Distance +(a) D = 10 +EOT Distance +� +� +� +� +� +�� +�����������)���)���(���� +���� +���� +���� +���� +���� +���� +���� +������()���� +����������) +����������� +������ +��� +��� +(b) D = 50 +Figure 4: The results corresponding to different kernel matrices. In this experiments, we set L = 5 +for our MRBCD scheme. +The influence of kernel approximation. The approximation errors shown in Figure 3 are +mainly caused by the approximated kernel (when L ≥ 2). To verify this claim, we decouple the in- +fluence of algorithmic convergence and that of kernel approximation, applying our MRBCD scheme +with the real kernel matrix and approximated kernel matrix, respectively. In this experiment, we +consider computing the EOT distance between two Gaussian distributions, each containing 2,000 +samples. By setting P ∈ {log N, D, N}, where N = 2000 is the number of samples per distribution +and D ∈ {10, 50} is the dimension of the samples, we approximate the kernel matrix with different +precision and apply our MRBCD accordingly. Experimental results are shown in Figure 4. When +applying the real kernel matrix, our MRBCD method converges quickly, with results close to the +ground truth. When P = N, the kernel matrix can be approximated with high accuracy, and thus, +the results are almost the same as those obtained under the real kernel matrix. When P = D, +although the performance of our method degrades, the gap between our result and the ground truth +can be small for high-dimensional cases. Note that, D ≈ log N when D = 10 and N = 2000, so the +corresponding results in Figure 4(a) are similar. +Remark. Figures 3 and 4 show an interesting phenomenon — although there are one hundred +agent pairs in the system, our method converges quickly within 50 iterations and approaches the +ground truth. One potential reason for this phenomenon is that all dual variables can be updated +based on few agent pairs, i.e., considering {(i, j)}I +i=1 for an arbitrary target agent j and {(i, j)}J +j=1 +for an arbitrary source agent i. Note that, estimating the EOT distance with few agent pairs is +equivalent to estimating the EOT distance with few pairwise costs. Following this direction, the +work in (Li et al., 2023) leverages the importance sparsification strategy to solve the primal-form +EOT problem with a sparse cost matrix. The theoretical connection between our method and the +importance sparsification is left as our future work. +The influence of communication protocol. +The communication protocol also impacts +our method, as shown in Theorem 4. In Figure 5(a), we apply our method with three different +communication protocols: i) the ideal communication protocol, i.e., E = p ⊗ q = [ 1 +IJ ], ii) a sparse +E, i.e., each source agent only communicate with five target agents (50% zeros in E), and iii) +a sparse and asymmetric E, i.e., setting the upper-triangular part of the sparse E to be all-zero +(directed communication). Experimental results show that the deterioration of the communication +environment leads to performance degradation. +11 + +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +Ideal Comm. Protocol +Sparse +Sparse+Asymmetric +(a) +� +�� +�� +�� +�� +�� +������������������ ����� +���� +���� +���� +���� +���� +���� +���� +��������� �� +���� ������� +������������ + � ������������� + � ������������� + � �������������� +EOT Distance +(b) +Figure 5: (a) The influence of communication protocol when computing Wε(GMM1, GMM2). We +set L = 2 and P = 1000 in this experiment. (b) The comparison for i.i.d. and non-i.i.d. data +storage methods when computing Wε(GMM1, GMM2). We set E = p ⊗ q = [ 1 +IJ ] and P = 1000 in +this experiment. +The influence of non-i.i.d. data storage. In previous experiments, we randomly scatter +the samples to different agents. Accordingly, for the agents in the same domain, their data are +independent and identically distributed (i.i.d.). In practice, however, the agents may store non- +i.i.d. data. For example, given two Gaussian mixture models, each of which contains five Gaussian +components, we i) make each agent store the data randomly sampled from the GMMs (i.e., i.i.d. +data storage) and ii) make each agent store the data from a single Gaussian component (i.e., non- +i.i.d. data storage). As shown in Figure 5(b), the non-i.i.d. data storage leads to the degradation of +the performance — the gaps between our results and the ground truth become more significant, and +the convergence curves are not stable. This phenomenon is because when computing the stochastic +gradients of the dual objective, we need to select L agents’ dual variables randomly. In the non-i.i.d. +case, the stochastic gradients we get are not unbiased estimations of the full-batch gradients unless +L equals the number of agents. As a result, compared to the i.i.d. case, we need to use more agents +to compute the gradients in the non-i.i.d. case. The results in Figure 5(b) verify our analysis — +increasing L can improve the performance in the non-i.i.d. case. +4.2 +Real-world distributed domain adaptation +Besides testing on synthetic data, we apply our method to real-world distributed domain adaptation +tasks in which the data in the source and target domains are scattered among different agents rather +than stored in the same server. Suppose we further protect the privacy of the target domain by +preventing the source domain from accessing the data of the target domain. The problem is even +more challenging in that case, and existing domain adaptation methods become inapplicable. +We conduct this experiment on two widely-used image datasets, USPS and MNIST (LeCun +et al., 1998), each of which has ten image categories corresponding to the digits from 0 to 9. We +follow the setting in (Courty et al., 2017a). Given 2,000 images of the MNIST domain and 1,800 +images of the USPS domain, we consider the adaptation in two directions: USPS→MNIST and +MNIST→USPS. We focus on the OT-based domain adaptation strategy (Courty et al., 2017b). +This strategy i) computes the (entropic) optimal transport distance between the source and target +12 + +Table 1: Comparisons on classification accuracy in distributed domain adaptation tasks +Structure +Method +USPS +MNIST +Preserve +→ MNIST +→ USPS +Privacy +Source only +1NN +0.385 +0.593 +Yes +EMD +0.544 +0.617 +No +Centralized +Sinkhorn +0.437 +0.620 +No +OT-LpL1 +0.490 +0.676 +No +Decentralized +MRBCDK +0.580 +0.681 +No +(Ours) +MRBCD � +K +0.522 +0.629 +Yes +domains, ii) maps the source samples to the target domain via the optimal coupling, and iii) trains +the 1-Nearest Neighbor (1NN) classifier based on the mapped data in a distributed manner. To +obtain the optimal coupling, we apply various methods, including our MRBCD scheme with real +or approximated kernel matrix, the earth mover distance (EMD) for OT distance, the Sinkhorn +algorithm for EOT distance, and the OT-LpL1 method in (Courty et al., 2014). +Experimental results in Table 1 show that without the information of the target domain, purely +training a 1NN classifier leads to unsatisfactory performance. The traditional centralized OT meth- +ods can improve classification accuracy. Still, they require a powerful central server to compute +the OT distance and need to access the raw data of the target domain. Our MRBCD scheme +outperforms the baselines when using the real kernel and achieves privacy preservation with toler- +able performance degradation when using the approximated kernel. 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Consensus-based distributed discrete optimal transport for decentralized +resource matching. IEEE Transactions on Signal and Information Processing over Networks, 5 +(3):511–524, 2019. +Zhao, T., Yu, M., Wang, Y., Arora, R., and Liu, H. Accelerated mini-batch randomized block +coordinate descent method. In NeurIPS, 2014. +Zhu, S., Hong, M., and Chen, B. Quantized consensus ADMM for multi-agent distributed opti- +mization. In ICASSP, 2016. +17 + +A +Delayed Proofs +A.1 +The proof of Theorem 1 +Proof. Let u1, v1 = arg supu,v∈C(X) E(i,j)∼E[Ex∼µi,y∼γj[fε(x, y, u, v)]] be the optimal dual functions +of Wε(µ, γ). Similarly, let u2, v2 be the optimal dual functions of � +Wε(µ, γ). We have +|� +Wε(µ, γ) − Wε(µ, γ)| ≤ +�� +i,j(eij − piqj)Ex∼µi,y∼γj[fε(x, y, u1, v1)] +if � +Wε(µ, γ) ≥ Wε(µ, γ) +� +i,j(piqj − eij)Ex∼µi,y∼γj[fε(x, y, u2, v2)] +if � +Wε(µ, γ) < Wε(µ, γ) +≤ +� +i,j |eij − piqj| maxu∈{u1,u2},v∈{v1,v2} Ex∼µi,y∼γj[fε(x, y, u, v)] +≤ +� +i,j |eij − piqj| supu,v∈CX Ex∼µi,y∼γj[fε(x, y, u, v)] += +� +i,j |eij − piqj|Wε(µi, γj) +≤ maxi,j Wε(µi, γj) +� +i,j |eij − piqj| +≤ τσ. +A.2 +The proof of Theorem 2 +Proof. The proof of this Theorem 2 is based on the Lemma 4.1 in (Khanduri et al., 2021). We +aims to approximately calculate a N × M kernel matrix K, which can be considered as a sub-block +matrix of the (N + M) × (N + M) full kernel matrix base on given N + M data samples. This +full kernel matrix can be denoted as Kfull ∈ R(N+M)×(N+M). The employed privacy-preserving +Kernel Approximation algorithm, i.e., Algorithm 1, can be considered as a partial version of +the Algorithm 1 in (Khanduri et al., 2021). As a result, we can utilize the theoretical result, i.e., +Lemma 4.1 in (Khanduri et al., 2021) to estimate the approximate level of �Kfull ∈ R(N+M)×(N+M) +obtained through the Algorithm 1 in (Khanduri et al., 2021). Based on the assumptions 1-2, with +probability at least 1 − δ, we have +∥Kfull − �Kfull∥ ≤ G(N + M) +�� +32π2 +P +log 2(N + M) +δ ++ 8π +3P log 2(N + M) +δ +� +. +For the reason that K is a sub-block matrix of Kfull, so as the approximated �K with respect to +�Kfull, Furthermore, based on the assumptions 1-2, with probability at least 1 − δ, we have +∥K − �K∥ ≤ ∥Kfull − �Kfull∥ ≤ G(N + M) +�� +32π2 +P +log 2(N + M) +δ ++ 8π +3P log 2(N + M) +δ +� +, +which indicates the result of this Theorem. +A.3 +The proof of Theorem 4 +Proof. Let ˆu∗, ˆu∗ = arg maxu,v Fε(u, v; �K, E) be the optimal solution of the problem equation 8 +given approximated kernel �K, and ˜u∗, ˜u∗ = arg maxu,v Fε(u, v; K, E) be the optimal solution of +the problem equation 8 given real kernel K. +In order to prove equation 15, we first establish +the connections with the algorithmic convergence error, the approximated kernel error and the +18 + +mismatch between the storage and communication protocols. In particular, applying the triangle +inequality, we have +E +���Fε(ˆut, ˆvt; �K, E) − Wε(µ, γ) +�� +� +≤E +���Fε(ˆut, ˆvt; �K, E) − Fε(ˆu∗, ˆv∗; �K, E) +�� +� ++ +��Fε(ˆu∗, ˆv∗; �K, E) − Fε(˜u∗, ˜v∗; K, E) +�� ++ +��Fε(˜u∗, ˜v∗; K, E) − Wε(µ, γ) +�� +=E +� ��Fε(ˆut, ˆvt; �K, E) − Fε(ˆu∗, ˆv∗; �K, E) +�� +� +�� +� +convergence error by Lemma 3 +� ++ +��Fε(ˆu∗, ˆv∗; �K, E) − Fε(˜u∗, ˜v∗; K, E) +�� +� +�� +� +approximated kernel error ++ +��� +Wε(µ, γ) − Wε(µ, γ) +�� +� +�� +� +gap by Theorem 1 +. +The first and third terms in the above equation have been analyzed in Lemma 3 and Theorem 1 +respectively. The key issue is to analyze the approximated kernel error term. +Each fε(x, y, u, v) is typically convex and Lipschitz continuous with respect to (u, v) Genevay +et al. (2016). The objective function Fε is Lipschitz continuous with respect to (u, v). Further based +on the definition of f(i,j) +ε +, it is obvious that f(i,j) +ε +is a liner function with respect to κ(x(i) +n , y(j) +m ). On +the whole, the objective function Fε can be considered as a linear function with respect to kernel +matrix K and thus also is Lipschitz continuous with respect to K. According to Dempe & Mehlitz +(2015)[Lemma 3.1], if we model K as the varible of the parametric optimization problem +φ(K) = max +u,v Fε +� +u, v; K, E +� +. +We can conclude that the optimal value function φ +� +K +� +with respect to K is Lκ-Lipschitz continuous, +i.e., +��Fε(ˆu∗, ˆv∗; �K, E) − Fε(˜u∗, ˜v∗; K, E) +�� = +��φ( �K) − φ(K) +�� ≤ Lκ +�� �K − K +��. +As a consequence of Theorem 1, Theorem 2, and Lemma 3, we have that for the problem +in equation 8 with the kernel �K derived by Algorithm 1, let +�� +ut, vt�� +be the sequence generated +by Algorithm 2. Define ˆut = 1 +t +�t +ℓ=1 uℓ, and ˆvt = 1 +t +�t +ℓ=1 vℓ. Based on the assumptions 1-3, with +probability at least 1 − δ, we have +E|Fε(ˆut, ˆvt; �K, E) − Wε(µ, γ)| +≤O +�IJ +� +( +√ +t + LFε)R2 +0 + +√ +tR2� +t +� ++ LκG(N + M) +�� +32π2 +P +log 2(N + M) +δ ++ 8π +3P log 2(N + M) +δ +� ++ τσ +≤O +�IJ +√ +t + (N + M) +� +1 +P log 2(N + M) +δ ++ σ +� +. +(16) +To emphasis, the second term in equation 16 has close relationship with the number of samples per +agent (i.e., Ni and Mj). We propose a worst case analysis, while it can be modified properly with +well-designed Ni and Mj. +19 + +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +MRBCD (I=J=5) +MRBCD (I=J=10) +MRBCD (I=J=20) +(a) Wε(N1, N2) +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +MRBCD (I=J=5) +MRBCD (I=J=10) +MRBCD (I=J=20) +(b) Wε(GMM1, GMM2) +Figure 6: The influence of the number of servers in each domain. +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +Ideal Comm. Protocol +Sparse +Sparse+Asymmetric +(a) L = 2 +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +Ideal Comm. Protocol +Sparse +Sparse+Asymmetric +(b) L = 5 +0 +10 +20 +30 +40 +50 +Number of Iterations (T) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +EOT Distance +Ground Truth +Ideal Comm. Protocol +Sparse +Sparse+Asymmetric +(c) L = 5 +Figure 7: (a, b) The robustness of our method to the communication protocol when computing +Wε(N1, N2). (c) The influence of communication protocol when computing Wε(GMM1, GMM2). +In this experiment, we set L = 5 and P = 1000. +B +Implementation Details and More Experimental Results +B.1 +Synthetic experiments +In order to gain insight into the effect of the number of servers on the algorithm, we set I, J ∈ +{5, 10, 20}, L = round(0.7I), and the number of samples to be N = M = 2000. From the experi- +mental results showed in Figure 6, we can find that as the number of servers increases, convergence +is faster at the beginning of the iteration and is accompanied by a more pronounced oscillation. +Additionally, Figure 7 shows more experimental results, visualizing the impacts of different +communication protocols. +B.2 +Real-world experiments +The distributed domain adaptation task is dedicated to solving the domain adaptation problem in +the case where both source and target domain data are scattered over different agents. The aim is to +use the label information available in the source domain X to learn a classifier f∗ that can be applied +20 + +Table 2: Summary of the domains used in the experiments +Problem +Domains +Datasets +#Samples +#Features +Abbr. +Digits +USPS +USPS +1, 800 +256 +U +MNIST +MNIST +2, 000 +256 +M +Objects +Art +Office-home +2, 427 +2, 048 +Ar +Clipart +Office-home +4, 365 +2, 048 +Cl +Product +Office-home +4, 439 +2, 048 +Pr +Real-World +Office-home +4, 357 +2, 048 +Rw +Table 3: Comparisons on classification accuracy in distributed domain adaptation tasks +Domains +Source only +Centralized +Decentralized (Ours) +1NN +EMD +Sinkhorn +OT-LpL1 +MRBCDK +MRBCD � +K +Ar→Cl +0.433 +0.471 +0.492 +0.490 +0.483 +0.458 +Ar→Pr +0.594 +0.642 +0.673 +0.633 +0.665 +0.639 +Ar→Rw +0.667 +0.677 +0.721 +0.686 +0.738 +0.705 +Cl→Ar +0.445 +0.504 +0.509 +0.478 +0.531 +0.509 +Cl→Pr +0.536 +0.647 +0.617 +0.642 +0.632 +0.606 +Cl→Rw +0.589 +0.638 +0.657 +0.664 +0.654 +0.618 +Pr→Ar +0.488 +0.516 +0.532 +0.494 +0.538 +0.506 +Pr→Cl +0.414 +0.455 +0.465 +0.450 +0.469 +0.425 +Pr→Rw +0.683 +0.707 +0.725 +0.714 +0.735 +0.704 +Rw→Ar +0.592 +0.611 +0.622 +0.605 +0.621 +0.598 +Rw→Cl +0.450 +0.498 +0.505 +0.509 +0.494 +0.463 +Rw→Pr +0.729 +0.749 +0.778 +0.770 +0.773 +0.736 +on the target domain Y without label information. Specifically, suppose we have the source domain +data Xi = {x(i) +n }Ni +n=1 associated with the class labels, and the target domain data Yj = {y(j) +m }Mj +m=1 +with unknown labels. Based on our algorithm MRBCD, each target agent j can obtain an optimal +coupling {Πij}I +i=1. Then, according to (Courty et al., 2017a), when the probability measures µ +and γ are uniform, we can derive the barycentric mapping as � +X = NΠY , where Π = [Πij] is the +complete coupling and � +X is the transported data of the source domain. Eventually, we can train +the 1NN classifier f given the transported data � +X and perform classification prediction on the +target domain data. +We further conduct experiments on the object recognition task. Here, we use the Office-home +dataset (Venkateswara et al., 2017). A summary of the properties of each domain used in this paper +is provided in Table2. The Office-home dataset contains around 15500 images coming from four +different domains: Art (artistic images in the form of sketches, paintings, etc), Clipart (collection +of clipart images), Product (images of objects without a background) and RealWorld (images of +objects captured with a regular camera). For this problem, all the experiments are based on pre- +trained ResNet-50 (He et al., 2015). We consider 12 transfer tasks for the Art (Ar), Clipart (Cl), +Product (Pr) and Real-World (Rw) domains for all combinations of source and target for the 4 +domains. The results are shown in the Table3. +Hyperparameter setting. As for the experimental setup, we scattered the source and target +domain data over four agents and set L = 7 and P = 10000. For all the EOT-based method, +we apply grid search, finding the optimal weight of regularizer ε ∈ {2, 1, 0.5, 0.1, 0.05}. For our +method, the learning rate is set in {1, 0.5, 0.1, 0.01, 0.001}. +21 + diff --git a/b9FLT4oBgHgl3EQfYi8g/content/tmp_files/load_file.txt b/b9FLT4oBgHgl3EQfYi8g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f56fab7522921faf14dc8b001689d0bee6501df --- /dev/null +++ b/b9FLT4oBgHgl3EQfYi8g/content/tmp_files/load_file.txt @@ -0,0 +1,1578 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf,len=1577 +page_content='Decentralized Entropic Optimal Transport for Privacy-preserving Distributed Distribution Comparison Xiangfeng Wang1 Hongteng Xu2 Moyi Yang1∗ 1School of Computer Science and Technology, East China Normal University 2Gaoling School of Artificial Intelligence, Renmin University of China xfwang@sei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='cn hongtengxu@ruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='cn winnie yang@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='cn January 31, 2023 Abstract Privacy-preserving distributed distribution comparison measures the distance between the distributions whose data are scattered across different agents in a distributed system and cannot be shared among the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' In this study, we propose a novel decentralized entropic optimal transport (EOT) method, which provides a privacy-preserving and communication-efficient so- lution to this problem with theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' In particular, we design a mini-batch random- ized block-coordinate descent (MRBCD) scheme to optimize the decentralized EOT distance in its dual form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' The dual variables are scattered across different agents and updated locally and iteratively with limited communications among partial agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' The kernel matrix involved in the gradients of the dual variables is estimated by a distributed kernel approximation method, and each agent only needs to approximate and store a sub-kernel matrix by one-shot commu- nication and without sharing raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' We analyze our method’s communication complexity and provide a theoretical bound for the approximation error caused by the convergence error, the approximated kernel, and the mismatch between the storage and communication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' Experiments on synthetic data and real-world distributed domain adaptation tasks demonstrate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' 1 Introduction Distribution comparison plays a central role in many machine learning problems, such as data clustering (Hammouda & Karray, 2000), generative modeling (Bond-Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' Mattei & Frellsen, 2019), domain adaptation (Ganin & Lempitsky, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' Farahani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=', 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' As a valid metric for distributions, optimal transport (OT) distance (Villani, 2009) provides a powerful solution to this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' Given two distributions, the optimal transport distance corresponds to the minimum expectation of their underlying sample distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' The corresponding optimal joint distri- bution of sample pairs takes these two distributions as its marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' Mathematically, given two probability measures in a compact space X, denoted as µ, γ ∈ P(X), the Kantorovich formulation of the optimal transport distance between them is defined as W(µ, γ) def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' = inf π∈Π(µ,γ) � X 2 c(x, y) dπ(x, y), (1) ∗The authors have equal contributions and are listed in alphabetical order of their last names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='12065v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='/FiXYlLZqpvycyLIwZi8h1CmyHZtGbiv95ndTGN2HGpEotlWS+KE45sgma/o76TFNi+dgRTDRztyIyxBoT6xIquRCxZeXS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content='fOsGlxVLx8uKrXbPI4iHMExnEIA1CDe6hDAwiM4Ble4c1T3ov37n3MWwtePnMIf+B9/gBQ3Y+Sµ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FLT4oBgHgl3EQfYi8g/content/2301.12065v1.pdf'} +page_content=' 0, so that ∥∆A(j) +2 k−1,2 k∥2, ∥∆A(j) +2 k,2 k−1∥ ≤ ε for each j = 1, 2, . . . , L and + +HODLR approximation of Hessians in ice sheet inverse problems +24 +k = 1, 2, . . . , 2j−1. For x ∈ RN we have +∥ +� +A − ˜ +A +� +x∥2 ≤ +L +� +j=1 +∥∆A(j) x∥2, +∥∆A(j) x∥2 += +� +� +� +� +2j−1 +� +k=1 +� +∥∆A(j) +2 k−1,2 k x(j) +2 k∥2 +2 + ∥∆A(j) +2 k,2 k−1 x(j) +2 k−1∥2 +2 +� +≤ +� +� +� +� +2j−1 +� +k=1 +� +ε2∥x(j) +2 k∥2 +2 + ε2∥x(j) +2 k−1∥2 +2 +� +, +∥∆A(j) x∥2 +≤ ε +� +� +� +� +2j−1 +� +k=1 +� +∥x(j) +2 k∥2 +2 + ∥x(j) +2 k−1∥2 +2 +� += ε∥x∥2, +∥ +� +A − ˜ +A +� +x∥2 ≤ ε L ∥x∥2, +∥A − ˜ +A∥2 +:= sup +x̸=0 +� +� +∥ +� +A − ˜ +A +� +x∥2 +∥x∥2 +� +� ≤ ε L. +8.3. Error analysis for posterior-covariance +Consider a symmetric matrix A ∈ RN×N, whose eigenvalues are bounded below +by a number greater than −1 and a symmetric approximant +˜ +A, with discrepancy +∆A = A − ˜ +A. We signify a generic eigenvalue of S by λ (S) so that s1 ≤ λ (S) ≤ s2 +indicates that all eigenvalues of S are bounded below by s1 and above by s2. Now we +provide a bound for the error of (I + A)−1− +� +I + ˜ +A +�−1 +, given that ∥∆A∥2 = ε, so that +one may assess the accuracy of an HODLR Gaussianized posterior covariance. When, as +in Section 3.2, A is the prior-preconditioned Hessian misfit, ∥ (I + A)−1− +� +I + ˜ +A +�−1 +∥2 +quantifies the discrepancy between an HODLR approximate Gaussianized posterior +covariance and the true Gaussianized posterior covariance. +(I + A)−1 − +� +I + ˜ +A +�−1 += (I + A)−1 − (I + A − ∆A)−1 = +(I + A)−1 − +� +(I + A) +� +I − (I + A)−1 ∆A +��−1 = +(I + A)−1 − +� +I − (I + A)−1 ∆A +�−1 (I + A)−1 = +� +I − +� +I − (I + A)−1 ∆A +�−1� +(I + A)−1 . +Given that ∥∆A∥2 = ε, we have +−ε ≤ λ (∆A) ≤ ε, +−ε∗ ≤ λ +� +(I + A)−1 ∆A +� +≤ ε∗, +ε∗ := ε(1 + λmin(A))−1, +1 + ε∗ ≥ λ +� +I − (I + A)−1 ∆A +� +≥ 1 − ε∗, + +HODLR approximation of Hessians in ice sheet inverse problems +25 +we next assume ε∗ < 1, so that the eigenvalues of I − (I + A)−1 ∆A are necessarily +positive and +(1 + ε∗)−1 ≤ λ +�� +I − (I + A)−1 ∆A +�−1� +≤ (1 − ε∗)−1 . +With this it follows that +∥ (I + A)−1 − +� +I + ˜ +A +�−1 +∥2/∥ (I + A)−1 ∥2 ≤ +� +1 − (1 + ε∗)−1� +∥ (I + A)−1 − +� +I + ˜ +A +�−1 +∥2/∥ (I + A)−1 ∥2 ≤ +ε∗ +1 + ε∗, +where, as before ε∗ = ∥∆A∥2/ (1 + λmin (A)). +Acknowledgments +The authors thank Trevor Hillebrand from Los Alamos National Laboratory for help +with setting up the Humboldt and Greenland ice-sheet grids and datasets. Support +for this work was provided by the National Science Foundation under Grant No. DMS- +1840265 and CAREER-1654311 and through the SciDAC project ProSPect, funded by +the U.S. Department of Energy (DOE) Office of Science, Advanced Scientific Computing +Research and Biological and Environmental Research programs. +This research used +resources of the National Energy Research Scientific Computing Center (NERSC), a +U.S. Department of Energy Office of Science User Facility operated under Contract No. +DE-AC02-05CH11231, under NERSC award ERCAP0020130. +Disclaimer +This paper describes objective technical results and analysis. +Any subjective views +or opinions that might be expressed in the paper do not necessarily represent the +views of the U.S. Department of Energy or the United States Government. +Sandia +National Laboratories is a multimission laboratory managed and operated by National +Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of +Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear +Security Administration under contract DE-NA-0003525. +References +[1] Isaac T, Petra N, Stadler G and Ghattas O 2015 Scalable and efficient algorithms for the +propagation of uncertainty from data through inference to prediction for large-scale problems, +with application to flow of the Antarctic ice sheet Journal of Computational Physics 296 348–368 +[2] Petra N, Martin J, Stadler G and Ghattas O 2014 A computational framework for infinite- +dimensional Bayesian inverse problems: Part II. 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https://nsidc.org/data/NSIDC-0478/versions/2 +[51] Perego M 2022 Large-scale PDE-constrained optimization for ice sheet model initialization SIAM +News Online +[52] Hillebrand T R, Hoffman M J, Perego M, Price S F and Howat I M 2022 The contribution +of Humboldt Glacier, North Greenland, to sea-level rise through 2100 constrained by recent +observations of speedup and retreat The Cryosphere Discussions 2022 1–33 + diff --git a/bNE2T4oBgHgl3EQfFQZd/content/tmp_files/load_file.txt b/bNE2T4oBgHgl3EQfFQZd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3dbf138e0205ae8251b9f27b69461193f80eedea --- /dev/null +++ b/bNE2T4oBgHgl3EQfFQZd/content/tmp_files/load_file.txt @@ -0,0 +1,736 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf,len=735 +page_content='Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization Tucker Hartland1, Georg Stadler2, Mauro Perego3, Kim Liegeois3, No´emi Petra1 1 Department of Applied Mathematics, University of California, Merced E-mail: thartland@ucmerced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='edu 2 Courant Institute of Mathematical Sciences, New York University 3 Center for Computing Research, Sandia National Laboratories Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Obtaining lightweight and accurate approximations of Hessian applies in inverse problems governed by partial differential equations (PDEs) is an essential task to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The O � N 3� computational complexity of dense linear algebraic routines such as that needed for sampling from Gaussian proposal distributions and Newton solves by direct linear methods, can be reduced to log- linear complexity by utilizing hierarchical off-diagonal low-rank (HODLR) matrix approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In this work, we show that a class of Hessians that arise from inverse problems governed by PDEs are well approximated by the HODLR matrix format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In particular, we study inverse problems governed by PDEs that model the instantaneous viscous flow of ice sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In these problems, we seek a spatially distributed basal sliding parameter field such that the flow predicted by the ice sheet model is consistent with ice sheet surface velocity observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We demonstrate the use of HODLR approximation by efficiently generating Hessian approximations that allow fast generation of samples from a Gaussianized posterior proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss-Newton data-misfit Hessian is more efficiently approximated by the HODLR matrix format than the low-rank (LR) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We then demonstrate that HODLR approximations can be favorable, when compared to global low-rank approximations, for large-scale problems by studying the data-misfit Hessian associated to inverse problems governed by the Stokes flow model on the Humboldt glacier and Greenland ice sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='03644v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='NA] 9 Jan 2023 HODLR approximation of Hessians in ice sheet inverse problems 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Introduction Model-based simulation of complex physical systems plays an essential role in understanding real world phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' These models are often characterized by partial differential equations (PDEs), and are typically subject to uncertainties stemming from unknown coefficient fields, constitutive laws, source terms, initial and/or boundary conditions, geometries, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' When observation data exist, these parameters can be estimated by solving an inverse problem governed by the underlying model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', PDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' It is well known that uncertainty is a fundamental feature of inverse problems, therefore in addition to inferring the parameters of interest, we need to quantify the uncertainty associated with this inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This uncertainty quantification can be done via Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Solving Bayesian inverse problems governed by complex PDEs can be extremely challenging due to high-dimensional parameter spaces that stem from discretization of infinite-dimensional parameter fields and the need to repeatedly solve the underlying PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' To overcome these computational challenges, it is essential to exploit problem structure, when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For example, the underlying PDE solution operator is often diffusive, that observation data may be sparse or only contain limited information about the parameter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' These particularities give rise to a low-rank structure in the second derivative of the data-misfit component of the inverse problem objective (or of the negative log likelihood), hereafter referred to as the data-misfit Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In previous work [1, 2] we exploited this low-rank structure in the context of inverse ice sheet flow problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' However, for cases when this “low-rank” is in fact large, as is the case for many inverse problems of practical interest, where the observation data are highly informative, low-rank approximation is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In this article, we exploit the local sensitivity of model predictions to parameters, which gives rise to an off-diagonal low-rank structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We do so by invoking hierarchical off-diagonal low- rank (HODLR) matrix approximations and detail how they can be used to reduce the computational cost to solve large-scale PDE-based inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Related work Global low-rank approximation of Hessians in inverse problems have been successfully utilized in [1, 3, 4, 5, 6], with deterministic and randomized methods [7, 5] being available to generate said approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' However, some problems, specifically those with highly informative observation data, are not amenable to global low-rank approximation, and thus other structure-exploiting strategies are needed such as those based on local translation invariance and localized sensitivities [8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here we focus on hierarchical low-rank methods for which convenient randomized methods are available [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Hierarchical matrices have been demonstrated in [13, 14] to be an effective means to approximate covariance matrices associated to large-scale Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In [15], hierarchical matrix approximations with general hierarchical partitioning patterns are utilized for the construction of explicit representations of Hessian inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In one of the examples studied, the authors find that the diffusivity of the parameter-to-PDE-solution HODLR approximation of Hessians in ice sheet inverse problems 3 map and the informativeness of the observation data impact whether the data-misfit Hessian is more suited for compression with hierarchical or global low-rank formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here, we build on this study and focus on a specific inverse problem arising in land ice modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Contributions The main contributions of this work are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (1) We motivate the use of HODLR compression for data-misfit Hessians in inverse problems governed by PDEs, and present a detailed study for large-scale ice sheet inverse problems, such as the Greenland ice sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (2) We describe a strategy that leverages the fast manipulation of HODLR matrices to efficiently generate approximate samples from a Gaussian posterior distribution for uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (3) We numerically study the influence of various problem setups on the off-diagonal low-rank structure of the data-misfit Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The results show the effectiveness of the HODLR approximation for various problem scales including for a Greenland ice sheet inverse problem, which has a discretized parameter dimension of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Preliminaries In this section, we summarize background material regarding the solution of discretizations of infinite-dimensional inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We also briefly review HODLR matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Specifically, we define HODLR matrices, list some of their properties and summarize the computational complexities of computing symmetric HODLR matrix approximations of symmetric operators that are only available through their application on vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We refer to [16, 17] for a more thorough discussion of hierarchical matrices and to [12] for more detail on HODLR matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Bayesian Inverse Problems A means to account for uncertainty in parametric inference is to employ the Bayesian approach to inverse problems [18, 19, 20], which takes as input observation data d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', the data, prior knowledge of the parameter and a model for the likelihood of data conditional to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Prior knowledge of the discretized parameter β is typically determined by the expertise of domain scientists and mathematically encoded in a probability density function πprior (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The likelihood π (d|β) involves the data uncertainty and the mathematical model for the parameter-to-observable process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The solution of a Bayesian inverse problem is a probability density function for the discretized parameter β, that is conditioned on the observation data according to Bayes formula πpost (β) = π (β|d) ∝ πprior (β) π (d|β) , which provides a formal expression for the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here, “∝” means equal up to a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For a problem with Gaussian prior N � β, Γprior � HODLR approximation of Hessians in ice sheet inverse problems 4 and data noise η described by the zero mean Gaussian N (0, Γnoise), πpost(·) has the following form πpost (β) ∝ exp � −1 2∥F(β) − d∥2 Γ−1 noise − 1 2∥β − β∥2 Γ−1 prior � , (1) where F is the parameter-to-observable map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The notation ∥ · ∥A means that the norm is weighted with the positive-definite matrix A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', ∥v∥A = √ v⊤Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The parameter- to-PDE-solution map is typically nonlinear, and consequently the posterior distribution is not a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' One characteristic of the posterior distribution is the point at which it is maximized, or equivalently the point which minimizes the negative log-posterior, the so-called maximum a posteriori (MAP) point, β⋆ := arg minβ J(β) := 1 2∥F(β) − d∥2 Γ−1 noise + 1 2∥β − β∥2 Γ−1 prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (2) A means to compute the MAP point is to employ a (Gauss) Newton method for optimization [21], which critically relies on the availability of the (Gauss-Newton) Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Since, J is defined implicitly in terms of the parameter-to-observable map, which involves a PDE solution operator, we utilize the adjoint method [22, 23, 24] to compute it’s gradient and Hessian-applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' To fully explore posterior distributions, Markov chain Monte-Carlo (MCMC) techniques [25, 26] can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Such techniques require a proposal distribution that ideally approximates the posterior and is easily sampled from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' One method to generate a Gaussian proposal distribution is through the Laplace approximation of the posterior about βk (or around the MAP point) ˜πpost (β, βk) ∝ exp � −1 2⟨β − µk, Hk (β − µk)⟩ℓ2 � , µk = βk − H−1 k gk, where gk, Hk are the gradient and Hessian of the log-posterior J(β) at βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Another MCMC sampling approach is the generalized preconditioned Crank-Nicholson (gpCN) method [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' An attractive choice for the preconditioner is the Hessian at the MAP point, [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For these and other MCMC samplers, one typically needs to apply the inverse Hessian H−1 k or its square root H−1/2 k repeatedly and efficiently, which also motivates the study presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In particular, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 we discuss how HODLR approximations can be used for the fast application of the Hessian square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Symmetric HODLR Matrices A HODLR matrix A ∈ RN×N, is a matrix equipped with a depth L ∈ N, hierarchical partitionings of the index set I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , N} into continguous subsets and low-rank off-diagonal blocks defined by the partition, which is described in greater detail in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The block rank-structure of a HODLR matrix for various hierarchical depths is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' An HODLR matrix must satisfy two additional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 5 Figure 1: Rank-structure of a matrix A with hierarchical depths L = 1 (left), L = 2 (middle) and L = 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Off-diagonal blocks are assumed to be low-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (i) The depth of the hierarchical partitioning scales with the logarithm of the size of the matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', L = O (log N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (ii) The maximum rank of each hierarchical level ℓ off-diagonal block, rℓ, is bounded above by a number r that is independent of the problem size N, for each level ℓ max 1≤ℓ≤L rℓ ≤ r = O (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Such matrices are referred to as data-sparse since the low-rank blocks allow for them to be represented computationally with less than O (N 2) floating point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In particular, the storage of an HODLR matrix is O (N log N), O(N log N) flops are needed to compute a HODLR matrix-vector product [7], and O(N log2 N) flops are required for direct methods to compute an inverse HODLR matrix-vector product [30], as well as square root and inverse square root matrix-vector products [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Compression We aim to generate HODLR approximations of data-misfit Hessians in inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For large-scale problems, the data-misfit Hessian is available only as a matrix-free operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In order to construct HODLR approximations of symmetric matrix-free operators, we employ previously developed randomized linear algebraic routines which only require the matrix-free action on a limited number of random vectors with specified null entries, referred to as structured random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The Hessian action on these structured random vectors is used to sample row and column spaces of off- diagonal Hessian submatrices and allow for randomized approximate truncated singular value decompositions of the aforementioned off-diagonal submatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' More details can be found in the appendix, see Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For the results that we present in Section 5 a rank-adaptive symmetric matrix- free [32, 33], hierarchical compression algorithm is utilized, that is based on [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' A similar algorithm is presented in [34], wherein the hierarchical partitioning is more general and the low-rank blocks have nested bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The rank-adaptivity provides a high probability means of resolving the off-diagonal blocks to a desired level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' By utilizing available matrix-vector product information and the Rayleigh quotient, a rank adaptive relative tolerance algorithm is made possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 6 Computational Cost of Generating HODLR Approximations The number of matrix- vector products ζ, needed to compress a symmetric matrix using d oversampling vectors, into a level L HODLR matrix with off-diagonal ranks {rℓ}L ℓ=1 is given by ζ = 2 (⟨r⟩ + d) L + N/2L, where ⟨r⟩ := 1 L L � ℓ=1 rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (3) Equation 3 can be understood from Algorithm 2 in Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1, as for each level ℓ one needs to compute rℓ + d Hessian vector products, in order to compute Y (line 7 of Algorithm 2) and rℓ + d Hessian vector products to compute Z (line 14 of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The remaining N/2L Hessian vector products arise from the need to determine the diagonal subblocks, which is detailed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We note that with an adaptive procedure to determine an approximate basis Q, such as that in [33], for a block matrix column space, the cost is reduced to ζadaptive = 2 (⟨r⟩ + d/2) L + N/2L but with the additional computational burden of extra orthogonalization routine calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We note that ζ = O(log N) matrix-vector products are needed to generate an HODLR approximation of a matrix with HODLR structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For sufficiently large problems HODLR compression is not expected to be more computationally efficient than global low-rank (LR) compression, as ζLR = r + d, the number of matrix-vector products to generate a rank r compression by the single-pass algorithm [12] with d oversampling vectors is independent of the problems size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' However, for problems of substantial size, we observe that the HODLR format does offer computational savings (see Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR matrices in inverse problems governed by PDEs Here, we illustrate why data-misfit Hessians in inverse problems governed by PDEs may contain numerically low-rank off-diagonal blocks, describe how one can permute parameters to expose this HODLR structure, and show how HODLR approximations can be leveraged to draw samples from Gaussian approximations of Bayesian posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Motivation Consider the following data-misfit cost functional Jmisfit (β) := 1 2∥F(β) − d∥2 Γ−1 noise, with F(β) = Bu, where B linearly maps the PDE solution u = u(β), for the spatially-distributed parameter field β, to the model predictions associated to the data d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Moreover, Γnoise is the covariance matrix describing the Gaussian noise of the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For illustration purposes, we assume that the parameter function β is defined on a region Γ1 and the data d is observed on a region Γ2, which may or may not be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' These quantities are related through the solution of the governing PDE and the measurement operator B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The characteristics of this relation depends on properties of the governing PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In the following, we assume that a spatially (or temporally) localized perturbation HODLR approximation of Hessians in ice sheet inverse problems 7 perturbation ψi sensitivity cone, δu δβ(β)(ψi) Γ2 Γ1 Figure 2: Sketch illustrating a case where the influence of changes in the parameter β on the PDE solution u in Γ2 is focused in a small area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' To illustrate this, we show a sensitivity cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', the PDE solution u is predominantly impacted in a cone about the support of the localized parameter perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' in the β field leads to a predominantly localized effect in the PDE solution u, and thus the model predictions Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This property is illustrated in Figure 2, where we use a sensitivity cone to illustrate the influence of a local perturbation in β, defined over Γ1, on the PDE solution u in Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' It is well known that for an elliptic PDE, local perturbations influence the solution globally, but depending on the geometry of the domain and the equation, this global effect may rapidly decay outside a subset of Γ2 that captures the main effects of the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For instance, in a problem as in Figure 2, the influence of perturbations in β on u is likely to become more localized when the distance between Γ1 and Γ2 decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We next discuss the relationship between properties of the PDE as discussed above and off-diagonal blocks in the Hessian matrix (or its Gauss-Newton variant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The data- misfit Hessian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', the Hessian of the data-misfit part of the cost functional, can be derived using the adjoint method [22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' we find that the HODLR structure of the data-misfit Hessian is most easily seen by studying a formal expression of it in terms of the first and second order sensitivities δu/δβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' δ2u/δβ2 δ2 δβ2Jmisfit (β) (β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' β2) = (Bu − d)⊤ Γ−1 noise � Bδ2u δβ2 (β) (β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' β2) � + � Bδu δβ (β) (β1) �⊤ Γ−1 noise � Bδu δβ (β) (β2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' where δu/δβ (β) (β1) is the first variation [35] of u with respect to β in direction β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' and δ2u/δβ2 (β) (β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' β2) is the second variation of u with respect to β in directions β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' δu δβ (β) (β1) := � d dϵu (β + ϵβ1) � ϵ=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' δ2u δβ2 (β) (β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' β2) := � d dϵ δu δβ (β + ϵβ2) (β1) � ϵ=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 8 Upon discretizing β with finite elements we obtain the following formal expression for the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' j)-entry of the data-misfit Hessian Hmisfit and of the Gauss-Newton data-misfit Hessian HGN misfit (Hmisfit)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j = δ2 δβ2 � Jmisfit (β) � (ψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' ψj) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (4) � HGN misfit � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j = � Bδu δβ (β) (ψi) �⊤ Γ−1 noise � Bδu δβ (β) (ψj) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (5) where {ψj}N j=1 is a basis for the nodal finite-element space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' which is used to approximate β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' When sensitivities are predominantly local as discussed above and when the support of two finite element basis functions ψi, ψj are well separated, the terms � Bδu δβ (β) (ψi) �⊤ Γ−1 noise � Bδu δβ (β) (ψj) � and B �δ2u δβ2 (β) (ψi, ψj) � , are rather small (assuming diagonally dominant noise covariance matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', due to Bδu/δβ(β)(ψi) having small values when Bδu/δβ(β)(ψj) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Now, let I, J be disjoint index subsets of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , N}, then the entries in the matrix block {(Hmisfit)i∈I,j∈J } of the data-misfit Hessian are relatively small whenever ∪i∈Isupp (ψi) and ∪j∈J supp (ψj) are well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Such Hessian blocks are well suited for approximation by low-rank matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' When the degrees of freedom corresponding to the finite element basis functions ψi are ordered such that I, J are contiguous, (Hmisfit)I,J is an off-diagonal subblock of Hmisfit and Hmisfit tends to have HODLR structure as defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The Gauss-Newton data-misfit Hessian may have HODLR structure for the same reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In both cases, the order of the basis functions and thus the degrees of freedom influence this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Ideally, one wants an order that maintains locality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', consecutive indices correspond to basis functions that are close to each other, and as a consequence, basis function with significantly different indices are far from each other such that the corresponding off-diagonal blocks have small entries and can be well approximated using a low-rank matrix approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We defer to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 for a discussion of methods and numerical experiments regarding the order of the degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Application of HODLR structure for fast sampling of Gaussian posterior approximations In [2], the following expressions of the Gaussianized posterior covariance are provided, Γpost = � Hmisfit + Γ−1 prior �−1 = Γ1/2 prior (H′ misfit + I)−1 Γ⊤/2 prior, H′ misfit := Γ⊤/2 priorHmisfitΓ1/2 prior, Γ1/2 post = Γ1/2 prior (H′ misfit + I)−1/2 , where the matrix square-root A1/2 is such that A = A1/2 � A1/2�⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For Bayesian inverse problems with a parameter field that is distributed spatially over a bounded subset of HODLR approximation of Hessians in ice sheet inverse problems 9 Rm, m = 2, 3, a reasonable choice is to use the square of an inverse elliptic PDE operator for the prior covariance [20], which permits a means of obtaining a symmetric square root of Γprior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In previous works such as [1, 3, 4, 5, 6], the prior-preconditioned data- misfit Hessian H′ misfit, was approximated by global low-rank compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This strategy provides an efficient means of approximating the posterior covariance matrix in inverse problems with data sets that contain sufficiently small amounts of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here we propose to exploit HODLR problem structure and generate approximate posterior covariance matrices by HODLR approximations of the prior-preconditioned data-misfit ˜ H′ misfit, see Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3 for an analysis on how such an approximation impacts the accuracy of the approximate posterior covariance ˜Γpost = Γ1/2 prior � ˜ H′ misfit + I �−1 Γ⊤/2 prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' A symmetric square-root factorization of ˜ H′ misfit+I is then generated with O � N log2 N � flops [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The symmetric factorization allows for a O (N log N) means of applying both the square root and inverse square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Bayesian inverse ice sheet problems The simulation of the dynamics of ice sheets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', the Greenland or Antarctic ice sheets) is an important component of coupled climate simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Such simulations require estimation of a present state of the ice that is consistent with available observations, a process sometimes referred to as model initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This estimation problem can be formulated either as a deterministic inverse problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', as nonlinear least squares optimization governed by PDEs) or as a Bayesian inverse problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', as a statistical problem which aims to characterize a distribution of states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The latter approach, while more expensive, provides uncertainty estimates in addition to determining a best parameter fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Ice sheet dynamics [36] is typically governed by nonlinear Stokes equations or simplifications thereof, such as the first-order equations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Generally, the most uncertain component in ice sheet simulations is the basal boundary condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', how the ice sheet interacts with the rock, sand, water or a mix thereof at its base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Estimating an ice sheet’s effective boundary condition from velocity observations on the top surface, the ice sheet’s geometry and a model for its dynamics is thus an important problem that can mathematically formulated as an inverse problem [1, 38, 39, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We summarize the formulation of this inverse problem next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' As common in the literature, we use a snapshot optimization approach, where all the data are assumed to be collected over a short period of time during which changes in the ice geometry are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We denote the bounded domain covered by ice by Ω ⊂ Rm, m ∈ {2, 3}, and the basal, lateral and top parts of the domain boundary ∂Ω by Γb, Γl, and Γt, as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The governing equations are nonlinear incompressible Stokes equations whose solution is the ice flow velocity u : Ω → Rm and the pressure p : Ω → R given as HODLR approximation of Hessians in ice sheet inverse problems 10 follows: −∇ · σu = ρg in Ω, (6) ∇ · u = 0 in Ω, (7) σun = 0 on Γt, (8) u · n = 0 and T (σun + exp (β) u) = 0 on Γb, (9) along with additional lateral boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here, β is a basal sliding parameter field, ρg the body force density, where ρ is the mass density of the ice and g the acceleration due to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Equation 6 describes the conservation of momentum, 7 the conservation of mass, and 8 are stress-free boundary conditions for the top surface (the ice-air interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In normal direction, Equation 9 states a non-penetration condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', the ice cannot flow into the rock/sand layer which supports it (here n denotes the outward unit normal to the boundary ∂Ω and T the tangential operator, T v = v − n(n⊤v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In tangential direction, Equation 9 specifies a tangential sliding condition that relates the fraction of tangential sliding and tangential stress through the (logarithmic) basal sliding field β = β(x), x ∈ Γb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We employ Glen’s flow law [42], a constitutive law for ice that relates the stress tensor σu and the strain rate tensor ˙εu = 1 2 � ∇u + ∇u⊤� , σu = 2η (u) ˙εu − Ip, with η (u) = 1 2A−1/n ˙ε 1−n 2n II , (10) where η is the effective viscosity, I is the unit matrix, ˙εII = tr ( ˙ε2 u) is the second invariant of the strain rate tensor, A is a flow rate factor, and n is Glen’s exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Ice is typically modeled using n ≈ 3, which corresponds to a shear-thinning constitutive relation, here we use n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' As discussed above, the parameter containing the largest uncertainty is the (logarithmic) basal sliding field β = β(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Thus, it is usually the parameter inferred from (typically, satellite) observation data d, here in the form of surface velocity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Using an appropriate point observation operator B that extracts point data from the solution u of the governing equations 6-9, and assuming additive observation errors η, the relationship between model and data is now of the typical form d = Bu + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (11) Assuming that the observation errors η and the prior for the parameter field β follow Gaussian distributions, we are in the framework of Bayesian inverse problems summarized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Example I: Two-dimensional ISMIP-HOM benchmark We first study the prospects of compressing the Gauss-Newton data-misfit Hessian in a problem inspired by the ISMIP-HOM collection of ice sheet simulation benchmark problems [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This problem set was used to explore inverse ice sheet problems in [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' After a short description of the problem setup, we present results such as the MAP HODLR approximation of Hessians in ice sheet inverse problems 11 point estimate β⋆ and approximate Gaussianized posterior samples using an HODLR compression of the posterior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Then, we study the impact that various problem features have on the suitability of the Gauss-Newton data-misfit Hessian for compression to the HODLR and global low-rank formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Problem setup This problem setup consists of a rectangular piece of ice on a slope, as sketched in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This simple example allows us to study the influence of the domain aspect ratio, the number of observations and the level of mesh refinement on the properties of the Gauss-Newton data-misfit Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The domain has a width of W = 104 [m] and a height of H = 102 [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Periodic boundary conditions are employed along the lateral boundaries such that the setup models an infinite slab of ice on a slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The governing equations and other boundary conditions are as discussed in Equations 6-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The Stokes equations are discretized using Taylor-Hood finite elements on a mesh of 256 × 10 rectangles, each subdivided into two triangles, for the domain length [0, W) and height [0, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' To compute a MAP estimate, we generate synthetic surface velocity data using the “true” logarithmic basal sliding field, βtrue (x) := log � 1 200 + 1 100 sin � 2πx W �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Given this basal sliding field, we solve Equations 6-9, extract the tangential velocity component at 100 uniformly distributed points on the top boundary Γt, and add 1% relative Gaussian noise to each data point, resulting in the synthetic data d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' It remains to define the prior distribution for the parameter field β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The average value of βtrue is used as constant prior mean β (x) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='73315 ≈ 1 W � W 0 βtrue (s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The prior covariance matrix Γprior is a discretization of the covariance PDE operator C := (δI − γ∆)−1, with γ = 6 × 102 and δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='4 × 10−3, with Robin boundary conditions [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' These values are chosen in order to provide a relatively large prior correlation length of 103 [m] [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Next, we summarize the computation of the MAP point and the compression of the Gauss-Newton data-misfit Hessian matrix at the MAP point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' MAP point and HODLR Gaussianized posterior The nonlinear optimization problem for finding the MAP estimate is solved using an inexact Gauss-Newton minimization method with backtracking linesearch [21], where the linear systems are iteratively solved by the conjungate gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The resulting MAP point is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The MAP parameter field β⋆, closely resembles the true parameter βtrue, which is a consequence of the large amount of available data and relatively small noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Next, we use the Gaussianized posterior distribution with a compressed prior- preconditioned data-misfit Hessian H′ misfit to generate approximate samples from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Upon construction of the HODLR compression of the prior- preconditioned data-misfit Hessian (details and comparisons can be found below in HODLR approximation of Hessians in ice sheet inverse problems 12 x Γb z θ H Γl Γl W Γt Figure 3: Schematic of two-dimensional slab of ice used for Example I in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The blue circles show representative (random) measurement locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The angle θ is the slope of the ice slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 0 2,500 5,000 7,500 10,000 4 5 6 7 8 x β reconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' truth 0 2,500 5,000 7,500 10,000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5 x T u|z=H reconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' data Figure 4: Shown for Example I are on the left the MAP point β⋆ (red) and the truth basal sliding parameter βtrue (black) used to generate synthetic observations of the tangential velocity component on the upper surface Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Shown on the right are noisy synthetic observations (black dots) used for computing the MAP point and the associated tangential surface velocity reconstruction (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3), we draw samples from the HODLR Gaussianized posterior as outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In Figure 5, we compare the mean, pointwise standard deviation and samples from the prior and the posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' As expected, we find that the data updates our belief about the spatially distributed parameter field and reduces the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In particular, the 2σ bounds on the one-dimensional point marginals σ (x), σi = [Γi,i]−1/2 of the Gaussianized posterior and the prior distributions are shown, in order to verify that the samples are largely contained within two standard deviations of their respective means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The prior-preconditioned data-misfit Hessian H′ misfit, is compressed using a relative tolerance of 10−6, that is ∥H′ misfit − ˜ H′ misfit∥2/∥H′ misfit∥2 ≤ 10−6, with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 13 Figure 5: Results for Example I: Two random samples (red), mean β (blue) and boundaries of the region R = {(x, y) such that 0 ≤ x ≤ W and β(x) − 2σ(x) ≤ y ≤ β(x)+2σ(x)} (dashed black) are shown for the prior (left) and a HODLR Gaussianized posterior using the scheme described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Dependence of Hessian block spectra on problem setting Next, we study how problem features impact the numerical suitability of using global low-rank and HODLR compressions to approximate the Gauss-Newton data-misfit Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In this and subsequent sections we measure the cost to generate the matrix compression in terms of Hessian vector products, which we also describe as Hessian applies, as each said vector product requires two linearized PDE solves and thus dominates the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We use the result of Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2, to claim ε absolute error in a level L HODLR approximation, when there is no more than ε/L absolute error in each off-diagonal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' What is particular to this section, is that adaptive single-pass and HODLR algorithms are used to generate global low-rank and HODLR approximations, based on absolute tolerance criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The absolute tolerance algorithmic input is scaled by the largest global low-rank singular value in order to report relative approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We note that additional errors are neglected in the reported approximation error such as that incurred in the peeling process [11, 12] and additional approximation assumptions in the single-pass algorithm, both of which are not expected to be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Influence of aspect ratio Here, we vary the aspect ratio of the domain φ = H/W, where H and W are the domain height and width respectively, in order to study how it influences the block spectra of the Gauss-Newton data-misfit Hessian and ultimately the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Figure 6 shows that the global spectrum is more sensitive to changes in the relative length scale φ than the spectra of the off-diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Low- rank approximations of the off-diagonal blocks become computationally cheaper as φ decreases as a result of the sensitivity cones becoming increasingly localized as the ice sheet thickness decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Global low-rank approximations become more expensive as φ decreases, a result of the data being more informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We note that realistic problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' such as the Humboldt glacier and the Greenland ice sheet studied later in Section 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 9 8 VV 6 5 4 3 0 2000 4000 6000 8000 10000 x[m]9 8 AMN 7 6 5 4 3 0 2000 4000 6000 8000 10000 x[m]HODLR approximation of Hessians in ice sheet inverse problems 14 10−8 10−6 10−4 10−2 0 25 50 75 100 125 ∥HGN misfit − ˜ HGN misfit∥2/∥HGN misfit∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' approximation error Computaitonal cost (Hessian applies) HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/200 HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/100 HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/50 HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/25 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/200 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/100 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/50 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' φ = 1/25 Figure 6: Comparison of HODLR and global low-rank (LR) compression costs of the Gauss-Newton data-misfit Hessian HGN misfit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' for Example I with ice sheet aspect ratio φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This figure shows that for low aspect ratios, HODLR becomes more efficient than global low-rank for medium levels of target accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' have small aspect ratios and are thus expected to have data-misfit Hessians that are less amenable to global low-rank approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Influence of the parameter dimension We now vary the level of mesh discretization refinement in order to study the influence of data informativeness, through the discretized parameter dimension N = dim(β), on the computational cost to generate HODLR and global low-rank approximations of the Gauss-Newton data-misfit Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The hierarchical depth L is incremented for every doubling of the discretized parameter dimension, in order that the hierarchical depth scales with the logarithm of the size of the Hessian matrix, a condition described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Figure 7 provides computational evidence of the claim made in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2, that the number of applies needed to hierarchically compress an operator with HODLR structure is O (log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' On the contrary, the number of applies to generate the global low-rank approximation is rather insensitive to the level of mesh refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Influence of the data dimension Figure 8 shows that the global rank grows with the number of observations points and thus global low-rank compression tends to be less efficient for problems with strongly informative observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The rate of spectral decay of the (Gauss-Newton) data-misfit Hessian is related to the degree of ill-posedness of the unregularized inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' As the number of observations increases, these associated model predictions are increasingly sensitive to small scale variations in the basal sliding field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Thus, more data, generally makes the data set more informative about the parameter and the (Gauss-Newton) data-misfit Hessian have a weaker rate of spectral decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 15 10−8 10−6 10−4 10−2 0 25 50 75 100 125 150 ∥HGN misfit − ˜ HGN misfit∥2/∥HGN misfit∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' approximation error Computaitonal cost (Hessian applies) HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' dim (β) = 128 HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' dim (β) = 256 HODLR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' dim (β) = 512 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' dim (β) = 128 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' dim (β) = 256 LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' dim (β) = 512 Figure 7: Dependence of HODLR and global low-rank (LR) compression costs of the Gauss-Newton data-misfit Hessian on dim (β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' the dimension of the discretized logarithmic basal sliding field for Example I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The cost of global low-rank compression is almost constant, while the cost of HODLR compression increases as the mesh is refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 10−8 10−7 10−6 10−5 10−4 10−3 10−2 0 25 50 75 100 125 150 175 200 ∥HGN misfit − ˜ HGN misfit∥2/∥HGN misfit∥2, approximation error Computational cost (Hessian applies) HODLR, dim(d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='0 × 102 HODLR, dim(d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5 × 102 HODLR, dim(d) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='0 × 102 LR, dim(d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='0 × 102 LR, dim(d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5 × 102 LR, dim(d) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='0 × 102 Figure 8: Dependence of HODLR and global low-rank (LR) compression costs of the Gauss-Newton data-misfit Hessian on dim(d), the data dimension, for Example I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The computational cost for global low-rank approximation increases with the number of observations, while the cost for HODLR compression is rather insensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Example II: Humboldt glacier and Greenland ice sheet Here, we study the scalability of the proposed methods using large-scale ice sheet problems which are typically used in climate simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Namely, we focus on the Humboldt glacier in North-West Greenland, and the entire Greenland ice sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For these simulations, we use the ice sheet model MALI, [46], which relies on HODLR approximation of Hessians in ice sheet inverse problems 16 Albany, [47], a C++ multi-physics library for the implementation of the so-called first-order approximation of Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This first-order approximation is based on scaling arguments motivated by the shallow nature of ice sheets and uses the incompressibility condition to reduce the unknows to the horizontal velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We use PyAlbany [48] a convenient Python interface to the Albany package, which in turn builds upon Trilinos [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Albany is designed to support parallel and scalable finite- element discretized PDE solvers and various analysis capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Details about the parameter, state, data dimensions as well as the number of cores and hierarchical levels used in the computations is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Humboldt Greenland dim(β) 11 608 320 116 dim(u) 255 376 7 042 552 dim(d) 23 216 640 232 # of cores 120 2 048 L 8 10 Table 1: Problem specifications for the Humboldt glacier and Greenland ice-sheet problems (Example II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Dimension of the discretized parameter field dim(β), dimension of the discretized velocity field dim(u), dimension of the observations dim(d), processors employed for computations and L depth of HODLR hierarchical partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The following study is partially motivated by findings made in the Section 5, namely that the role of the aspect ratio between the vertical and horizontal directions (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3) influences the ability to use global low-rank compression and favors HODLR compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We generate HODLR and global low-rank approximations and then based on the computed spectra, Equation 3 and ζLR = r + d, we estimate the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Additionally, we study how the ordering of the degrees of freedom impacts the spectral decay for off-diagonal blocks of the data-misfit Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We present results for both, the Humboldt glacier, which expands about 4 × 102 [km] laterally, and the Greenland ice sheet, which expands about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='8 × 103 [km].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The ice is at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='4 [km] thick, resulting in approximate aspect ratios of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5×10−3 for Humboldt and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='9×10−3 for Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We use a nonuniform triangulation of the Greenland ice sheet, with mesh size ranging from 1 to 10 [km], and we then extrude it in the vertical direction, obtaining a 3D mesh having 10 layers of prismatic elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The velocity observations at the top surface of the Greenland ice sheet are obtained from satellite observations [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The MAP basal sliding field and the temperature fields are obtained as part of the initialization process, using a numerical optimization approach to match the ice velocity observations and constrained by the first-order flow model coupled with a temperature model [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Additional details about the mesh geometries and data, in particular regarding the Humboldt glacier, can be found in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In Figure 9, we show the observed surface velocity d in [m/yr], the MAP estimates HODLR approximation of Hessians in ice sheet inverse problems 17 Figure 9: Data and MAP estimates for Example II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Shown are the surface velocity observation data (left), and the reconstructed surface velocity field (middle) that is based on the MAP estimate of the logarithmic basal sliding field (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Top row is for the Humboldt glacier and bottom row for the Greenland ice sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' of the logarithmic basal sliding field β⋆ (exp(β⋆) is in [kPa yr/m]) and surface velocity in [m/yr] generated by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR compressability We next generate global low-rank approximations of a Greenland and Humboldt data- misfit Hessian as well as low-rank approximations of various off-diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Plots of the estimated singular values are provided in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We observe that the spectrum of the Greenland ice sheet decays substantially slower than the one for the Humboldt glacier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Besides the different sizes of these two discretized problems, this is also due to the different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Having estimated singular values of the data-misfit Hessians and the appropriate off-diagonal blocks, one is able to estimate computational costs to compress them into the global low-rank and HODLR matrix formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The computational cost as a function of Hessian approximation target accuracy is given in Figure 11, wherein it is demonstrated that the HODLR compression format can offer a favorable means to approximate data-misfit Hessians for large-scale inverse problems governed by complex ice-sheet models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='8 58 5800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='8 58 580-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='425 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='210000 1000 100 — 1010000 1000 100 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='65 --1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='125 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='9HODLR approximation of Hessians in ice sheet inverse problems 18 1,000 2,000 3,000 4,000 10−5 10−3 10−1 101 j, singular value number σj, singular value 50 100 150 200 10−10 10−8 10−6 10−4 10−2 100 j, singular value number ℓ = 1 Humboldt ℓ = 1 GIS ℓ = 2 Humboldt ℓ = 2 GIS ℓ = 3 Humboldt ℓ = 3 GIS Figure 10: Singular values of the data-misfit Hessian (left figure) and various off-diagonal blocks of the data-misfit Hessian (right figure) for Example II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The color-scheme in the right most figure is consistent with Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' On the left, the singular values of the Humboldt and Greenland data-misfit Hessians are shown using a solid and dash-dotted line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' On the right, we show the singular values of the upper most blocks, that is A(ℓ) 1,2 as defined in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Approximation error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Computational cost (Hessian applies) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Approximation error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='LR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='HODLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Figure 11: Estimated computational costs (measured by the number of Hessian applies) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='to compress the Humboldt glacier (left) and Greenland ice-sheet (right) data-misfit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Hessians into the global low-rank (LR) and hierarchical off-diagonal low-rank (HODLR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='formats as a function of the approximation error ∥Hmisfit − ˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Hmisfit∥2/∥Hmisfit∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Impact of parameter degree of freedom ordering We seek to ensure that the off-diagonal blocks, determined by the hierarchical partitioning described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2, of the data-misfit Hessian are low-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For HODLR approximation of Hessians in ice sheet inverse problems 19 this reason, the nodes {xi}i associated to the degrees of freedom (dofs) are ordered according to a kd-tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', a recursive hyperplane splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The ordering provided by the kd-tree is such that the (i, j)-entry of the distance matrix Di,j = ∥xi − xj∥2, is typically small whenever |i−j| is small, that is the dof ordering preserves some notion of locality (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In particular, a sparse permutation matrix B, is determined, whose action reorders the dofs from the default ordering provided by the finite element discretization to that specified by the kd-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The data-misfit Hessian with respect to the kd-tree ordering, Hkd misfit := BHmisfitB⊤, is then amenable to HODLR compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Subsequently, B⊤ ˜ Hkd misfitB is an approximation of the data-misfit Hessian with respect to the default ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The dof ordering has no impact on a matrix’s global numerical rank but does indeed impact the numerical rank of its numerous submatrices that are defined by a fixed partitioning scheme, such as the off-diagonal blocks of an HODLR matrix (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here, we study the HODLR compressibility of the Humboldt glacier data-misfit Hessian by comparing the rate of decay of an off-diagonal block’s singular values using the default ordering provided by Albany and the ordering obtained by a kd-tree recursive hyperplane splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' As observed in Figure 12, the rate at which the singular values of the level-1 off-diagonal block decay, strongly depends on the dof ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This is because the ordering given by the kd-tree better preserves locality, and as a consequence, by the argument provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1, the singular values decay much faster when using the kd-tree ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The kd-tree ordering therefore provides a substantially computationally cheaper means to generate an HODLR approximation of the data-misfit Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Figure 12 also shows distance matrices for the default and kd- tree bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' These show the improved locality for the kd-orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Note that data-misfit Hessian matrices are expected to follow a similar structure as these distance matrices, which explains why the former’s off-diagonal blocks can be compressed more effectively in the kd-order than in the default order of dofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Conclusion In this work, we motivated why data-misfit Hessians which arise from a class of inverse problems governed by PDEs have HODLR matrix structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR matrices can efficiently be inverted and factorized, operations needed for solving inverse problems governed by PDEs by Newton’s method, for constructing Gaussian approximations and for Markov chain Monte Carlo sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We study inverse ice sheet problems, for which, under certain regimes, HODLR matrices provide a more computationally efficient approximation format than the global low-rank matrix format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' These problems are those with highly informative data and small aspect ratio ice sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' While global low-rank matrices are favorable for large discretized parameter dimension and small data dimension, we find that HODLR matrices can offer computational savings for large-scale inverse problems such as a Greenland ice sheet inverse problem with satellite observational data and a discretized parameter dimension that exceeds 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 20 50 100 150 200 250 300 350 400 450 500 10−11 10−9 10−7 10−5 10−3 10−1 j, singular value number σj, singular value default basis kd-tree basis Figure 12: Singular values of the hierarchical level 1 off-diagonal block, A(1) 1,2, of the Humboldt glacier data-misfit Hessian, when expressed in a kd-tree basis and the default basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Shown also are heat maps of the distance matrices Di,j = ∥xi − xj∥2, wherein the nodes {xi}i, associated to the finite element degrees of freedom have been ordered according to a default standard and a kd-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For future work, we believe that the computational cost can be reduced further by utilizing hierarchical matrix partitionings that satisfy a strong admissibility condition [17], as they are better suited to exploit data-misfit Hessian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' However, generating a hierarchical matrix approximation with such a partitioning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', by the peeling method [11, 12], requires substantially more Hessian vector products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Ultimately, to further reduce the computational cost of Hessian approximations in inverse problems governed by PDEs, exploiting further problem structure will be essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Randomized Compression Algorithms Here, for completeness we outline the randomized matrix-free double-pass global low- rank and HODLR compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The essential ideas of the randomized double-pass low-rank algorithm [32] are (i) the application of a vector ω with random entries to a matrix A, yields a vector y = Aω, which is likely aligned with the dominant left singular vectors of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (ii) a matrix Q, whose columns are nearly aligned with the dominant left singular vectors of A, can be used to construct an accurate low-rank approximation HODLR approximation of Hessians in ice sheet inverse problems 21 ˜ A = QQ⊤A of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The double-pass randomized SVD algorithm is presented in Algorithm 1 and does not significantly differ from that in [32], specifically it is lines 7, 8 and 9 that are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This minor modification frees us from the need to compute a (parallel) singular value decomposition (SVD) of a (distributed) N ×k matrix, such as Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here, we only need to compute an SVD of the smaller k×k matrix RZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' In the distributed memory parallelism setting of Section 6, this algorithmic modification allows us to only require the invocation of serial SVD routines, as RZ, which is typically small, is available on each processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Algorithm 1 Double-pass randomized SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Input: A ∈ RN×N, r ∈ N desired rank and oversampling parameter d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Output: low-rank approximation ˜ A of A 1: k = r + d 2: Ω = randn(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Initiate random matrix} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3: Y = AΩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Sample column space} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='4: QY = orthog(Y ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Orthogonalize column samples} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='5: Z = A⊤QY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Sample row space} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='6: QZ = orthog(Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Orthogonalize row samples} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='7: RZ = Q⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='ZZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Compress row samples} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='8: RZ = ˆV Σ ˆU ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{SVD of k × k compressed row sample matrix} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='9: V = QZ ˆV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Project row space information} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='10: U = QY ˆU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Project column space information} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='A = UΣV ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Form low-rank approximation} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='The ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='randomized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='hierarchical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='off-diagonal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='low-rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='proceeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='compressing off-diagonal blocks by the double-pass algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' The larger off-diagonal blocks are compressed prior to the compression of smaller off-diagonal blocks, via a peeling procedure [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here, both A and ˜ A are assumed to be symmetric as we seek compression of symmetric operators and computation of symmetric approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' HODLR approximation of Hessians in ice sheet inverse problems 22 Algorithm 2 Symmetric matrix-free randomized HODLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Input: symmetric A ∈ RN×N, hierarchical depth L ∈ N, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , rL desired ranks of the off-diagonal blocks at each hierarchical depth and oversampling parameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Output: symmetric HODLR approximation ˜ A of A 1: for ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , L do 2: kℓ = rℓ + d 3: Ω = zeros(N, kℓ) 4: for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , 2ℓ−1 do 5: Ω(I(ℓ) 2j , :) = randn(|I(ℓ) 2j |, kℓ) {Initiate structured random matrix} 6: end for 7: Y = � A − �ℓ−1 j=1 A(j)� Ω {Sample off-diagonal block column spaces} 8: for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , 2ℓ−1 do 9: Y (j) = zeros(N, kℓ) 10: Y (j)(I(ℓ) 2j−1, :) = Y (I(ℓ) 2j−1, :) 11: Q(j) Y = orthog(Y (j)) {Orthogonalize column samples of the level ℓ off-diagonal blocks} 12: end for 13: QY = �2ℓ−1 j=1 Q(j) Y {Row space sampling matrix} 14: Z = � A − �ℓ−1 j=1 A(j)� QY {Sample off-diagonal block row spaces} 15: for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 2ℓ−1 do 16: Z(j) = Z(I(ℓ) 2j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' :) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Q(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Z = orthog(Z(j)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Orthogonalize row samples of the level ℓ off-diagonal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='blocks} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='R(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Z = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Q(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='�⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Z(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Compress level ℓ off-diagonal block row samples} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='R(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Z = ˆV (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1Σ(ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ˆU (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{SVD of kℓ × kℓ compressed row sample matrix} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='V (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 = Q(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Z ˆV (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Project row space information} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='U (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 = Q(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Y ˆU (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='{Project column space information} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='22: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='V (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j = U (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='23: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='U (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j = V (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='24: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='Σ(ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j = Σ(ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2j−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='25: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='26: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='A(ℓ) = �2ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j=1 U (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j Σ(ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='V (ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='�⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='27: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='28: obtain block diagonal D of A by sampling A − �L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j=1 A(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='29: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='A = D + �L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='ℓ=1 A(ℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='HODLR approximation of Hessians in ice sheet inverse problems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Global HODLR approximation error from the accumulation of block low-rank off-diagonal approximation errors Let A be a N × N matrix and consider the following partitioning A(1) = � 0 A(1) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 A(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' A(2) = � � � � � 0 A(2) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 0 0 A(2) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1 0 0 0 0 0 0 A(2) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='4 0 0 A(2) 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3 0 � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' D = � � � � � A(2) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='1 0 0 0 0 A(2) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2 0 0 0 0 A(2) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3 0 0 0 0 A(2) 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='4 � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' where A(ℓ) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='j is the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' j) block of a 2ℓ × 2ℓ block partitioning of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' where 1 ≤ ℓ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' A(ℓ) contains all blocks A(ℓ) i,j such that |i − j| = 1 and D contains the diagonal blocks A(L) i,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Above, we show the decomposition A = �L ℓ=1 A(ℓ) + D for L = 2 hierarchical depth but in the following analysis L is a arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Let x ∈ RN, then Ax = L � j=1 A(j)x + Dx, A(1)x = � A(1) 1,2x(1) 2 A(1) 2,1x(1) 1 � , x = � x(1) 1 x(2) 2 � , A(j)x = � � � � � � � � A(j) 1,2x(j) 2 A(j) 2,1x(j) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' A(j) 2j−1,2jx(j) 2j A(j) 2j,2j−1x2j−1 � � � � � � � � , x = � � � � � � � x(j) 1 x(j) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' x(j) 2j−1 x(j) 2j � � � � � � � , from which we obtain the following expression ∥A(j)x∥2 2 = 2j−1 � k=1 � ∥A(j) 2 k−1,2 kx(j) 2 k∥2 2 + ∥A(j) 2 k,2 k−1x(j) 2 k−1∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Now assume that ˜ A is an HODLR approximation of A, whose diagonal D is equal to the diagonal of A so that � A − ˜ A � = L � j=1 ∆A(j), ∆A(j) := � A(j) − ˜ A(j)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Here we assume each off-diagonal block has been approximated to some absolute tolerance ε > 0, so that ∥∆A(j) 2 k−1,2 k∥2, ∥∆A(j) 2 k,2 k−1∥ ≤ ε for each j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , L and HODLR approximation of Hessians in ice sheet inverse problems 24 k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' , 2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' For x ∈ RN we have ∥ � A − ˜ A � x∥2 ≤ L � j=1 ∥∆A(j) x∥2, ∥∆A(j) x∥2 = � � � � 2j−1 � k=1 � ∥∆A(j) 2 k−1,2 k x(j) 2 k∥2 2 + ∥∆A(j) 2 k,2 k−1 x(j) 2 k−1∥2 2 � ≤ � � � � 2j−1 � k=1 � ε2∥x(j) 2 k∥2 2 + ε2∥x(j) 2 k−1∥2 2 � , ∥∆A(j) x∥2 ≤ ε � � � � 2j−1 � k=1 � ∥x(j) 2 k∥2 2 + ∥x(j) 2 k−1∥2 2 � = ε∥x∥2, ∥ � A − ˜ A � x∥2 ≤ ε L ∥x∥2, ∥A − ˜ A∥2 := sup x̸=0 � � ∥ � A − ˜ A � x∥2 ∥x∥2 � � ≤ ε L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Error analysis for posterior-covariance Consider a symmetric matrix A ∈ RN×N, whose eigenvalues are bounded below by a number greater than −1 and a symmetric approximant ˜ A, with discrepancy ∆A = A − ˜ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' We signify a generic eigenvalue of S by λ (S) so that s1 ≤ λ (S) ≤ s2 indicates that all eigenvalues of S are bounded below by s1 and above by s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Now we provide a bound for the error of (I + A)−1− � I + ˜ A �−1 , given that ∥∆A∥2 = ε, so that one may assess the accuracy of an HODLR Gaussianized posterior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' When, as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='2, A is the prior-preconditioned Hessian misfit, ∥ (I + A)−1− � I + ˜ A �−1 ∥2 quantifies the discrepancy between an HODLR approximate Gaussianized posterior covariance and the true Gaussianized posterior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' (I + A)−1 − � I + ˜ A �−1 = (I + A)−1 − (I + A − ∆A)−1 = (I + A)−1 − � (I + A) � I − (I + A)−1 ∆A ��−1 = (I + A)−1 − � I − (I + A)−1 ∆A �−1 (I + A)−1 = � I − � I − (I + A)−1 ∆A �−1� (I + A)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Given that ∥∆A∥2 = ε, we have −ε ≤ λ (∆A) ≤ ε, −ε∗ ≤ λ � (I + A)−1 ∆A � ≤ ε∗, ε∗ := ε(1 + λmin(A))−1, 1 + ε∗ ≥ λ � I − (I + A)−1 ∆A � ≥ 1 − ε∗, HODLR approximation of Hessians in ice sheet inverse problems 25 we next assume ε∗ < 1, so that the eigenvalues of I − (I + A)−1 ∆A are necessarily positive and (1 + ε∗)−1 ≤ λ �� I − (I + A)−1 ∆A �−1� ≤ (1 − ε∗)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' With this it follows that ∥ (I + A)−1 − � I + ˜ A �−1 ∥2/∥ (I + A)−1 ∥2 ≤ � 1 − (1 + ε∗)−1� ∥ (I + A)−1 − � I + ˜ A �−1 ∥2/∥ (I + A)−1 ∥2 ≤ ε∗ 1 + ε∗, where, as before ε∗ = ∥∆A∥2/ (1 + λmin (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Acknowledgments The authors thank Trevor Hillebrand from Los Alamos National Laboratory for help with setting up the Humboldt and Greenland ice-sheet grids and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Support for this work was provided by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' DMS- 1840265 and CAREER-1654311 and through the SciDAC project ProSPect, funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Department of Energy (DOE) Office of Science, Advanced Scientific Computing Research and Biological and Environmental Research programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Department of Energy Office of Science User Facility operated under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' DE-AC02-05CH11231, under NERSC award ERCAP0020130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Disclaimer This paper describes objective technical results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Department of Energy or the United States Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=', for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' References [1] Isaac T, Petra N, Stadler G and Ghattas O 2015 Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Series A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Mathematical and Physical Sciences 228 519–538 [43] Pattyn F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Perichon L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Aschwanden A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' Breuer B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfFQZd/content/2301.03644v1.pdf'} +page_content=' de Smedt B,' metadata={'source': 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b/bdFIT4oBgHgl3EQfmSv9/content/tmp_files/2301.11309v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..55895d56f98c4785e9849bb305aa3800eabd7805 --- /dev/null +++ b/bdFIT4oBgHgl3EQfmSv9/content/tmp_files/2301.11309v1.pdf.txt @@ -0,0 +1,2393 @@ +SemSup-XC: Semantic Supervision for Zero and Few-shot +Extreme Classification +Pranjal Aggarwal 1 Ameet Deshpande 2 Karthik Narasimhan 2 +1Indian Institute of Technology, Delhi +2Department of Computer Science, Princeton University +pranjal2041@gmail.com, {asd, karthikn}@cs.princeton.edu +Abstract +Extreme classification (XC) involves predicting +over large numbers of classes (thousands to mil- +lions), with real-world applications like news arti- +cle classification and e-commerce product tagging. +The zero-shot version of this task requires gener- +alization to novel classes without additional su- +pervision. In this paper, we develop SemSup-XC, +a model that achieves state-of-the-art zero-shot +and few-shot performance on three XC datasets +derived from legal, e-commerce, and Wikipedia +data. To develop SemSup-XC, we use automati- +cally collected semantic class descriptions to rep- +resent classes and facilitate generalization through +a novel hybrid matching module that matches in- +put instances to class descriptions using a combi- +nation of semantic and lexical similarity. Trained +with contrastive learning, SemSup-XC signifi- +cantly outperforms baselines and establishes state- +of-the-art performance on all three datasets con- +sidered, gaining up to 12 precision points on zero- +shot and more than 10 precision points on one- +shot tests, with similar gains for recall@10. Our +ablation studies highlight the relative importance +of our hybrid matching module and automatically +collected class descriptions.1 +1. Introduction +Extreme classification (XC) studies the problem of predict- +ing over a large space of classes, ranging from thousands +to millions (Agrawal et al., 2013; Bengio et al., 2019; Bha- +tia et al., 2015; Chang et al., 2019; Lin et al., 2014; Jiang +et al., 2021). This paradigm has multiple real-world ap- +plications including movie and product recommendation, +1Code +and +demo +are +available +at +https:// +github.com/princeton-nlp/semsup-xc +and +https://huggingface.co/spaces/Pranjal2041/ +SemSup-XC/. +search-engines, and e-commerce product tagging. In many +of these applications, models are required to handle the ad- +dition of new classes on a regular basis, which has been +the subject of recent work on zero-shot and few-shot ex- +treme classification (ZS-XC and FS-XC) (Gupta et al., 2021; +Xiong et al., 2022; Simig et al., 2022). These setups are +challenging because of (1) the presence of a large number of +fine-grained classes which are often not mutually exclusive, +(2) limited or no labeled data per class, and (3) increased +computational expense and model size due to the large la- +bel space. While the aforementioned works have tried to +tackle the latter two issues, they lack a semantically rich +representation of classes, and instead rely on class names or +hierarchies to represent them. +In +this +work, +we +leverage +semantic +supervision +(SEMSUP) (Hanjie et al., 2022) for developing mod- +els for extreme classification. +SemSup-XC represents +classes using multiple diverse class descriptions, which +allows it to generalize naturally to novel classes when +provided with corresponding descriptions. +However, +SEMSUP as proposed in (Hanjie et al., 2022) cannot be +naively applied to XC for several reasons: (1) SEMSUP +requires encoding descriptions of all classes for each +training batch, which is prohibitively computationally +expensive for large label spaces, (2) it uses semantic +similarity only at the sentence level between the instance +and label description, and (3) it requires human intervention +to collect descriptions, which is expensive for extremely +large label spaces we are dealing with. +We remedy these deficiencies by developing SemSup-XC, a +model that scales to large class spaces in XC and establishes +a new state-of-the-art using three innovations. First, we use +a novel hybrid lexical-semantic similarity model (Hybrid- +Match) that combines semantic similarity of sentences with +relaxed lexical-matching between all token pairs. Second, +we propose SEMSUP-WEB – an automatic pipeline with pre- +cise heuristics to discover high-quality descriptions. Finally, +we use a contrastive learning objective that samples a fixed +number of negative label descriptions, improving computa- +arXiv:2301.11309v1 [cs.CL] 26 Jan 2023 + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Labels +Query a +search engine and +apply heuristic filtering +Pick the positive classes and +sample = 2 negative classes. +κ +1 +Automatic class +description collection +Axe +… +Book +CD +Paper +2 +Contrastive learning + A tool to cut wood. +🪓 + Written or printed work. +📚 + A digital data storage format. +📀 + A sheet material for writing. +📝 +... +Axe +Book +CD +Paper +... +Positive class +Negative class +3 +Lexico-semantic matching +(SemSup-XC) +Input: Tagore wrote a gem +Label: Book +Instance +[CLS] +Tagore +wrote +a +gem +[CLS] +Written +or +printed +work +Class descriptions corresponding +to positive and negative classes. +Dot +Dot +SemSup-XC computes the logits using: +(a) semantic similarity ([CLS]) and +(b) relaxed lexical similarity (wrote, written). +Logits +(a) +(b) +Negative class +Figure 1: Our model SemSup-XC achieves state-of-the-art performance on zero-shot and few-shot extreme classification +through three innovations – 1 large-scale automated class description collection with heuristic filtering to improve semantic +understanding of classes, 2 contrastive learning to make training faster by over 99% when compared to the previous work +(SEMSUP), and 3 a novel lexico-semantic matching model building called Hybrid-Match to utilize both semantic similarity +at the sentence level and contextual lexico-semantic similarity at the token level. +tion speed by up to 99% when compared to SEMSUP. +SemSup-XC achieves state-of-the-art performance on three +diverse XC datasets from legal (EURLex), e-commerce +(AmazonCat), and wiki (Wikipedia) domains, across zero- +shot (ZS-XC), generalized zero-shot (GZS-XC) and few- +shot (FS-XC) settings. For example, on ZS-XC, SemSup- +XC outperforms the next best baseline by 5 − 12 precision +points over all datasets and all metrics. On FS-XC, SemSup- +XC consistently outperforms baselines by over 10 P@1 +points on the EURLex and AmazonCat datasets. Surpris- +ingly, SemSup-XC even outperforms larger unsupervised +language models like T5 and Sentence Transformers (e.g., +by over 30 P@1 points on EURLex) which are pre-trained +on much larger web-scale corpora. Our ablation studies +dissect the importance of each component in SemSup-XC, +and shows the importance of our proposed hybrid matching +module. Qualitative error analysis of our model (Table 5) +shows that it predicts diverse correct classes that are appli- +cable to the instance at times, whereas other models either +predict incorrect classes or suffer a mode collapse. +2. Methodology +2.1. Background +Extreme Classification +Extreme classification deals with +prediction over large label spaces (thousands to millions +classes) and multiple correct classes per instance (multi- +label) (Agrawal et al., 2013; Bhatia et al., 2015; Babbar +& Schölkopf, 2017; Gupta et al., 2021; Xiong et al., 2022; +Simig et al., 2022). Zero-shot extreme classification (ZS- +XC) is a variant where models are evaluated on unseen +classes not encountered during training. We evaluate both +on (1) zero-shot (ZS), where the model is tested only on +unseen classes and (2) generalized zero-shot (G-ZS), where +the model is tested on a combined set of train and unseen +classes. We also consider few-shot extreme classification +(FS-XC), where a small number of supervised examples +(e.g., 5) are available for unseen classes. The heavy tailed +distribution of a large number of fine-grained classes in XC +poses efficiency and performance challenges. +Zero-shot classification +Zero-shot classification is usu- +ally performed by matching instances to auxiliary infor- +mation corresponding to classes, like their class name or +attributes (Larochelle et al., 2008; Dauphin et al., 2014). Re- +cently, Hanjie et al. (2022) proposed the use of multiple class +descriptions to endow the model with a holistic semantic +understanding of the class from different viewpoints. SEM- +SUP uses an (1) input encoder (fIE) to encode the instance +and an (2) output encoder (gOE) to encode class descriptions, +and makes predictions by measuring the compatibility of +the input and output representations. Formally, let xi be +the input instance, dj ∈ Dj be one sampled description +of class j, fIE (xi) , gOE (dj) ∈ Rd be the input and output +representation respectively. For the multi-label XC setting, + +Hou'shethaomyyomes +The mufh. The rohh +Thetorulhmahens.ThSemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +the probability of picking the jth class is: +SEMSUP := P(yj = 1|xi) = σ (gOE (dj)⊺ · fIE (xi)) +(1) +Challenges with large class spaces +Hanjie et al. (2022)’s +method cannot be directly applied to XC for several reasons. +First, they use a bi-encoder model (Bai et al., 2009) which +measures only the semantic similarity between the input +instance and the class description at the sentence level. How- +ever, instances and descriptions often share lexical terms +with the same or similar meaning and lemma (e.g., picture +and photo), which is not exploited by their method. Second, +their class description collection pipeline requires human +intervention, which is not feasible for the large number of +classes in XC datasets. And third, they fine-tune the label +encoder by encoding descriptions for all classes for every +batch of instances, making it computationally infeasible for +large label spaces because of GPU memory constraints. +2.2. SemSup-XC: Improved ZS extreme classification +SemSup-XC addresses the aforementioned challenges using: +(1) a novel hybrid semantic-lexical similarity model for +improved performance, (2) an automatic class description +discovery pipeline with accurate heuristics for garnering +high-quality class descriptions, and (3) contrastive learning +with negative samples for improved computational speed . +2.2.1. HYBRID LEXICO-SEMANTIC SIMILARITY MODEL +SEMSUP’s bi-encoder architecture measures only the seman- +tic similarity of the input instance and class description at +sentence level. However, semantic similarity ignores lexical +matching of shared words which exhibit strong evidence of +compatibility. (eg., Input: It was cold and flavorful and De- +scription: Ice cream is a cold dessert). A recently proposed +information-retrieval model, COIL (Gao et al., 2021a), al- +leviates this by incorporating lexical similarity by adding +the dot product of contextual representations corresponding +to common tokens between the query and document. But +COIL has the drawback that semantically similar tokens +(e.g., “pictures” and “photos”) and words with the same +lemma (e.g., walk and walking) are treated as dissimilar +tokens, despite being commonly used interchangeably. (eg., +Input: Capture the best moments in high quality pictures +and Class description: A camera is used to take photos.) +We propose Hybrid-Match to exploit such token similar- +ity. We create clusters of tokens based on: 1) the BERT +token-embedding similarity (Rajaee & Pilehvar, 2021) is +higher than a threshold or 2) if tokens share the same lemma, +which results in tokens like “photo”, “picture”, and “pic- +tures” being in the same cluster. We provide implementation +details on clustering in Appendix C. In addition to semantic +similarity, Hybrid-Match uses these clusters to for relaxed +lexical-matching by computing the dot-product of contex- +tual representations of “similar” tokens in the input and +description, as judged by the clusters. For cases where an +input token has several similar tokens in the description, +we choose the description token with the max dot product +with the former. Formally let xi = (xi1, . . . , xin) be the +input instance with n tokens, dj = (dj1, . . . , djm) class +jth descriptions with m tokens, vxi +cls and vdj +cls be the [CLS] +representations of the input and description, and vxi +k and +vdj +l +be the representation of the kth and lth token of xi and +dj respectively. Let CL(w) denote the cluster membership +of the token w, with CL(wi) = CL(wj) implying that the +tokens are similar. Then probability of class yj is: +Hybrid-Match := P(yj = 1|xi) = σ +� +vxi +cls +⊺ · vdj +cls ++ +n +� +k=1 +max +l∈{1,...,m}, CL(xik)=CL(djl) +� +vxi +k +⊺vdj +l +�� +(2) +2.2.2. SEMSUP-WEB: AUTOMATIC COLLECTION OF +HIGH-QUALITY DESCRIPTIONS +We create a completely automatic pipeline for collecting de- +scriptions which includes sub-routines for removing spam, +advertisements, and irrelevant descriptions, and we de- +tail the list of heuristics used in Appendix B. These sub- +routines contain precise rules to remove irrelevant descrip- +tions, for example by removing sentences with too many +special characters (usually spam), descriptions with click- +bait phrases (usually advertisements), ones with multiple +interrogative phrases (usually people’s comments), small +descriptions (usually titles) and so on (Appendix B). Further +in Wikipedia, querying search engine for labels return no +useful results, since labels are very specific (eg., Fencers +at the 1984 Summer Olympics) . Therefore, we design a +multi-stage approach, where we first break label names into +relevant constituents and query each of them individually +(see Appendix B.3). In addition to web-scraped label de- +scriptions, we utilize label-hierarchy information if provided +by the dataset (EURLex and AmazonCat), which allows us +to encode properties about parent and children classes wher- +ever present. Further details for hierarchy are present in +Appendix B.2. As we show in the ablation study (§ 4.3), +label descriptions that we collect automatically provide sig- +nificant performance boosts. +2.2.3. TRAINING USING CONTRASTIVE LEARNING +For datasets with a large number of classes (large |C|), it +is not computationally feasible to encode class descriptions +for all classes for every batch. We draw inspiration from +contrastive learning (Hadsell et al., 2006) and sample a sig- +nificantly smaller number of negative classes to train the +model. For an instance xi, consider two partitions of the la- +bels Yi = {yi1, . . . , yiC|yij ∈ {0, 1}}, with Y + +i +containing +the positive classes (yij = 1) and Y − +i +containing the nega- + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Dataset +Documents +Labels +Ntrain +Ntest +Ntest(ZS-XC) +|Yseen | +|Yunseen | +EURLex-4.3K +45 K +6 K +5.3 K +3,136 +1,057 +AmazonCat-13K +1.1M +307K +268 K +6,830 +6,500 +Wikipedia-1M +2.3M +2.7M +2.2M +495,107 +776,612 +Table 1: Dataset statistics along with information about +zero-shot (ZS-XC) splits. +tive classes (yij = 0). SemSup-XC caps the total number +of class descriptions being encoded for this instance to K +by using all the positive classes (|Y + +i |) and sampling only +K−|Y + +i | negative classes. Intuitively, our training objective +incentivizes the representations of the instance and positive +classes to be similar while simultaneously making them dis- +similar to the negative classes. To improve learning, rather +than picking negative labels at random, we sample hard neg- +atives that are lexically similar to positive classes. A typical +dataset we consider (AmazonCat) has |C|= 13, 000 and K +≈ 1000, which leads to SemSup-XC being 12000 +13000 = 92.3% +faster than SEMSUP. Mathematically, the following is the +training objective, where N is the train dataset size. +LSemSup-XC = +1 +N · K +� +i +� � +yk∈Y + +i +LBCE (P (yk = 1|xi) , yk) ++ +K−|Y + +i | +� +yl∼Y − +i +, l=1 +LBCE (P(yl = 0|xi), yl) +� +(3) +For each batch, a class description is randomly sampled +for each class (dl +j ∈ Dj), thus allowing the model to see all +the descriptions in Dj over the course of training, with the +same sampling strategy during evaluation. We refer readers +to appendix D for additional details. +3. Experimental Setup +Datasets +We evaluate our model on three diverse public +datasets. They are, EURLex-4.3K (Chalkidis et al., 2019) +which is legal document classification dataset with 4.3K +classes, AmazonCat-13K (McAuley & Leskovec, 2013) +which is an e-commerce product tagging dataset including +Amazon product descriptions and titles with 13K categories, +and Wikipedia-1M (Gupta et al., 2021) which is an article +classification dataset made up of 5 million Wikipedia articles +with over 1 million categories. We provide detailed statistics +about the number of instances and classes in train and test +set in Table 1. See Appendix E.1 for details on split creation. +Baselines +We perform extensive experiments with several +baselines, which can be divided into unsupervised (first +three) and supervised which are fine-tuned on the datasets +we consider (the remaining four). 1) TF-IDF performs +a nearest neighbour match between the sparse tf-idf fea- +tures of the input and class description. 2) T5 (Raffel et al., +2019) is a large sequence-to-sequence model which has +been pre-trained on 750GB unsupervised data and further +fine-tuned on MNLI (Williams et al., 2018). We evaluate +the model as an NLI task where labels are ranked based +on the likelihood of entailment to the input document. For +computational efficiency, we evaluate T5 only on top 50 +labels shortlisted by TF-IDF on each instance. 3) Sen- +tence Transformer (Reimers et al., 2019) is a semantic +text similarity model fine-tuned using a contrastive learn- +ing objective on over 1 billion sentence pairs. We rank the +labels based on the similarity between input and descrip- +tion embeddings. The latter two baselines use significantly +more data than SemSup-XC and T5 has 9× the parame- +ters. The aforementioned baselines are unsupervised and +not fine-tuned on our datasets. The following baselines are +previously proposed supervised models and are fine-tuned +on the datasets we consider. 4) ZestXML (Gupta et al., +2021) learns a highly sparsified linear transformation ( W ) +which projects sparse input features close to corresponding +positive label features. At inference, for each input instance +xi, label lj is scored based on the formula sij = lT +j Wxi. +5) MACLR (Xiong et al., 2022) is a bi-encoder based model +pre-trained on two self-supervised learning tasks to improve +extreme classification—Inverse Cloze Task (Lee et al., 2019) +and SimCSE (Gao et al., 2021b), and we fine-tune it on the +datasets considered. 6) GROOV (Simig et al., 2022) is a T5 +model that learns to generate labels given an input instance. +7) SPLADE (Formal et al., 2021) is a state-of-the-art sparse +neural retreival model that learns label/document sparse +expansion via a Bert masked language modelling head. +We evaluate baselines under two different settings – by pro- +viding either class names or class descriptions as auxillary +information, and use the label hierarchy in both settings. +The version of baselines which use our class descriptions +are strictly comparable to SemSup-XC models. See Ap- +pendix A.2 for additional details. +SemSup-XC implementation details +We use the Bert- +base model (Devlin et al., 2019b) as the backbone for the +input encoder and Bert-small model (Turc et al., 2019) for +the output encoder. SemSup-XC follows the model archi- +tecture described in Section 2.2 (Hybrid-Match) and we +use contrastive learning (Hadsell et al., 2006) to train our +models. During training, we randomly sample K − |Y + +i | +negatives for each instance, where K is the number of labels +for the instance. At inference, to improve computational +efficiency, we precompute the output representations of la- +bel descriptions and shortlist top 1000 labels based on the +TF-IDF scores. We use the AdamW optimizer (Loshchilov +& Hutter, 2019) and tune our hyperparameters using grid +search on the respective validation set. See Appendix A.1 +for more details. + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Model +EURLex-4.3K +AmazonCat-13K +Wikipedia-1M +ZS-XC +GZS-XC +ZS-XC +GZS-XC +ZS-XC +GZS-XC +P@1 +R@10 +P@1 +R@10 +P@1 +R@10 +P@1 +R@10 +P@1 +R@10 +P@1 +R@10 +Baselines with Class Names +Unsupervised Baselines +TF-IDF +44.0 +55.8 +53.4 +41.2 +18.7 +21.0 +21.5 +14.7 +14.5 +18.3 +14.4 +14.7 +T5 (Raffel et al., 2019) +7.2 +29.2 +10.4 +23.0 +2.5 +10.5 +3.2 +10.2 +8.2 +23.6 +4.2 +15.1 +Sent. Transformer (Reimers et al., 2019) +16.6 +23.2 +20.9 +42.0 +18.2 +25.0 +21.1 +17.9 +7.8 +13.3 +5.2 +9.1 +Supervised Baselines +ZestXML (Gupta et al., 2021) +24.7 +46.4 +84.9 +60.2 +15.6 +24.4 +87.6 +54.2 +15.8 +20.8 +26.3 +17.2 +SPLADE (Formal et al., 2021) +20.2 +24.4 +52.3 +34.2 +17.2 +28.7 +75.8 +41.3 +14.3 +17.8 +20.3 +22.4 +MACLR (Xiong et al., 2022) +24.9 +42.1 +60.7 +55.2 +36.0 +54.4 +46.0 +46.9 +29.8 +41.7 +28.0 +32.7 +GROOV (Simig et al., 2022) +1.2 +7.0 +84.1 +49.4 +0.0 +2.4 +87.4 +47.9 +6.0 +15.4 +31.4 +29.0 +Baselines with SemSup-XC scraped Class Descriptions +Unsupervised Baselines +TF-IDF +43.7 +50.4 +57.2 +39.5 +17.4 +20.8 +21.1 +15.0 +9.2 +12.5 +9.1 +10.3 +T5 (Raffel et al., 2019) +5.0 +24.8 +3.3 +8.1 +2.8 +7.7 +3.2 +4.2 +3.7 +13.4 +3.4 +13.2 +Sent. Transformer (Reimers et al., 2019) +15.9 +31.1 +18.8 +25.5 +15.2 +22.2 +16.0 +18.4 +19.6 +22.5 +14.2 +16.6 +Supervised Baselines +ZestXML (Gupta et al., 2021) +22.6 +44.6 +84.2 +60.7 +5.4 +24.8 +76.9 +50.7 +10.6 +14.1 +20.9 +17.9 +SPLADE (Formal et al., 2021) +20.7 +22.0 +45.1 +32.9 +16.9 +28.9 +77.0 +42.0 +8.2 +11.1 +20.7 +22.4 +MACLR (Xiong et al., 2022) +20.9 +37.9 +60.3 +53.8 +18.4 +22.3 +36.5 +23.8 +30.7 +41.9 +28.1 +33.6 +GROOV (Simig et al., 2022) +0.3 +0.6 +80.2 +18.1 +0.0 +0.0 +84.5 +23.5 +0.5 +0.2 +7.0 +1.5 +SEMSUP-XC (Our Model) +SEMSUP-XC +49.3 +62.4 +87.0 +62.9 +48.2 +72.9 +88.6 +71.6 +36.5 +38.5 +33.7 +34.1 +Table 2: Zero-shot (ZS-XC) and generalized zero-shot (GZS-XC) results for all models on three XC benchmarks. SemSup- +XC significantly outperforms state-of-the-art models on both precision (P@) and recall (R@) metrics across the board. +Evaluation setting and metrics +We evaluate all models +on three different settings: Zero-shot classification (ZS-XC) +on a set of unseen classes, generalized zero-shot classifi- +cation (GZS-XC) on a combined set of seen and unseen +classes, and few-shot classification (FS-XC) on a set of +classes with minimal amounts of supervised data (1 to 20 +examples per class). For all three settings, we train on in- +put instances of seen classes. We use Precision@K and +Recall@K as our evaluation metrics, as is standard practice. +Precision@K measures how accurate the top-K predictions +of the model are, and Recall@K measures what fraction of +correct labels are present in the top-K predictions, and they +are mathematically defined as P@k = +1 +k +� +i∈rankk(ˆy) yi +and R@k = +1 +� +i yi +� +i∈rankk(ˆy) yi, where rankk(ˆy) is the +set of top-K predictions. We average the metrics over test +instances. +4. Results +4.1. Zero-shot extreme classification +For the zero-shot scenario, we compare SemSup-XC with +baselines which use class descriptions and counterparts +which use class names as auxiliary information. We provide +label hierarchy as additional supervision in both cases. We +compare SemSup-XC with the best variant of each baseline. +Table 2 shows that SemSup-XC significantly outperforms +baselines on almost all datasets and metrics, under both zero- +shot (ZS-XC) and generalized zero-shot (GZS-XC) settings. +On ZS-XC, SemSup-XC outperforms MACLR by over 24, +12, and 6 P@1 points on the three datasets respectively, even +though MACLR uses XC specific pre-training. SemSup- +XC also outperforms GROOV (e.g., over 48 P@1 points on +EURLex) which uses a generative T5 model pre-trained on +significantly more data than our BERT backbone. GROOV’s +unconstrained output space might be one of the reasons for +its worse performance. SemSup-XC’s semantic understand- +ing of instances and labels stands out against ZestXML +which uses sparse non-contextual features with the former +consistently scoring twice as higher compared to the lat- +ter. SemSup-XC consistently outperforms SPLADE (For- +mal et al., 2021), a state-of-the-art information retrieval +method. This shows that the straightforward application +of IR baselines on XC, even when they are fine-tuned, un- +derperforms. This is likely because of the multi-label and +fine-grained nature of classes coupled with a heavy-tailed +distribution. While TF-IDF is competitive with deep base- +lines, SemSup-XC’s hybrid lexico-semantic similarity mod- + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +(a) EURLex-4.3K +(b) AmazonCat-13K +Figure 2: Few-Shot P@1 for different Values of K on EURLex and AmazonCat. SemSup-XC starts off significantly higher +and for EURLex maintains the gap for larger values of K to the second best model, MACLR (Xiong et al., 2022). For +AmazonCat, SemSup-XC maintains similar leads for most baselines, while being at par with Light XML (Jiang et al., 2021). +ule (Hybrid-Match) can perform fine-grained lexical and +semantic matching and outperforms both sparse and deep +methods on all datasets. SemSup-XC also outperforms the +unsupervised baselines T5 and Sentence-Transformer, even +though they are pre-trained on significantly larger amounts +of data than BERT (T5 use 50× compared our base model). +SemSup-XC also achieves higher recall on EURLex and +AmazonCat datasets, beating the best performing baselines +by 6, 18 R@10 points respectively, while being only 3 +R@10 points less on Wikipedia. SemSup-XC is also the +best model for GZS-XC. The margins of improvement are +1-2 P@1 points, which are smaller only because GZS-XC +includes seen labels during evaluation, which are usually +large in number. Table 6 in Appendix E contains additional +results with more methods and metrics. Our results show +that SemSup-XC is able to utilize the semantic and lexical +information in class descriptions to improve performance +significantly, while other baselines hardly improve when +using descriptions instead of class names. +4.2. Few-shot extreme classification +We now consider the FS-XC setup, where new classes added +at evaluation time have a small number of labeled instances +each (K). We evaluate on four settings – K ∈ {1, 5, 10, 20} +and all baselines other than ZestXML, which cannot be used +for FS-XC (See Appendix F). Further, we omit evaluation +on Wikipedia, since it has ≈ 10 training examples per label, +which is insufficient to study the effect of increasing values +of K. For the sake of completeness, we also include zero- +shot performance (ZS-XC, K = 0) and report results in +Figure 2. Detailed results for other metrics (showing the +same trend as P@1) and implementation details regarding +creation of the few-shot splits are in appendix F. +Similar to the ZS-XC case, SemSup-XC outperforms all +baselines for all values of K on EURLex. +For Ama- +zonCat, SemSup-XC outperforms all baselines other than +Light XML. Light XML is significantly outperformed for +K = {0, 1} and matches for K = {5, 10, 20}. In com- +parison to MACLR and GROOV, SemSup-XC consistently +outperforms by large margins (eg., 12 & 27P@1 points +for EURLex) across all values of K. SemSup-XC’s zero- +shot performance is higher than even the few-shot scores +of MACLR and GROOV that have access to K = 20 la- +beled samples on AmazonCat, which further strengthens +the model’s applicability to the XC paradigm. Moreover, +adding a few labeled examples seems to be more effective in +EURLex than AmazonCat, with the performance difference +between K = 1 and K = 20 being 22 and 12 P@1 points re- +spectively. This, along with the fact that performance seems +to plateau for both datasets, suggests that SemSup-XC learns +label semantics better for AmazonCat than EURLex, due to +its larger label space with rich descriptions. +4.3. Analysis +We dissect the performance of SemSup-XC by conducting +ablation studies on model components and label descriptions +and further provide qualitative analysis on EURLex and +AmazonCat for the zero-shot extreme classification setting +(ZS-XC) in the following sections. +Ablating components of SemSup-XC +SemSup-XC’s +use of the Hybrid-Match model and semantically rich de- + +70 +60 +Precision@ +50 +SemSup-XC (Ours) +30 +一 +TF-IDF +20 +MACLR (Xiong et al., 2022) +GROOV (Simig et al., 2022) +10 +LightXML (Jiang et al., 2021) +0 +5 +10 +0 +20 +K Instances Per Label70 +60 +Precision@ +50 +40 +30 +20 +10 +0 +5 +10 +20 +0 +K Instances Per LabelSemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Method +Components +EURLex-4.3K +AmazonCat-13K +Auxillary +Hierarchy +Exact +Hybrid +P@1 +P@5 +R@10 +P@1 +P@5 +R@10 +Information +Match +Match +Ablating Label Descriptions +SemSup-XC +Descriptions + + + +44.7 +20.9 +57.4 +48.2 +27.0 +72.9 +Replace descriptions with names +Names + + + +45.4 +20.6 +57.0 +43.9 +25.4 +69.7 +Remove hierarchy +Descriptions + + + +30.2 +14.2 +40.2 +21.7 +13.6 +40.3 +Ablating Model Architecture Components +SemSup-XC +Descriptions + + + +44.7 +20.9 +57.4 +48.2 +27.0 +72.9 +Replace Hybrid with Exact Lexical Matching +Descriptions + + + +42.6 +19.3 +53.7 +45.8 +25.5 +69.2 +Remove all lexical matching +Descriptions + + + +11.9 +8.9 +29.4 +37.3 +22.0 +60.6 +Table 3: Component-wise Model Analysis of SemSup-XC for ZS-XCon EURLex and AmazonCat. Each component +contributes to the final performance, with lexical-matching playing an important role. +Method +EURLex-4.3K +AmazonCat-13K +P@1 +P@5 +R@10 +P@1 +P@5 +R@10 +SemSup-XC +44.7 +20.9 +57.4 +48.2 +27.0 +72.9 ++ Augmentation +45.5 +21.6 +59.0 +47.8 +26.8 +72.6 +Table 4: Description augmentation helps boost performance +for ZS-XC on EURLex, but does not help on AmazonCat, +which is a significantly larger dataset (3× labels). This +demonstrates SemSup-XC’s out-of-the-box performance, +since augmentation is unnecessary for larger label spaces. +scriptions enables it to outperform all baselines considered, +and we analyze the importance of each component in Ta- +ble 3. As our base model (first row) we consider SemSup- +XC without ensembling it with TF-IDF. We note that +the SemSup-XC base model is the best performing vari- +ant for both datasets and on all metrics other than P@1 +for EURLex, for which it is only 0.5 points lower. Web +scraped class descriptions are important because removing +them decreases both precision and recall scores (e.g., P@1 +is lower by 4 points on AmazonCat) on all settings consid- +ered. We see bigger improvements with AmazonCat, which +is the dataset with larger number of classes (13K), which +substantiates the need for semantically rich descriptions +when dealing with a large number of fine-grained classes. +Label hierarchy information is similarly crucial, with large +performance drops on both datasets in its absence (e.g., 26 +P@1 points on AmazonCat), thus showing that access to +structured hierarchy information leads to better semantic +representations of labels. +On the modeling side, we observe that relaxed and exact +lexical matching, which are components of Hybrid-Match, +are important, with their absence leading to 3 and 11 P@1 +point degradation on AmazonCat. Even for EURLex, hy- +brid lexical matching improves performance by 33 P@1 +points when compared to a model with no lexical matching. +This highlights that our proposed model Hybrid-Match’s +hybrid semantic-lexical approach significantly improves per- +formance on XC datasets. +Augmenting Label Descriptions +The previous result +showed the importance of class descriptions, and we explore +the effect of augmenting them to increase their diversity and +quantity (See Table 4). We use the easy data augmenta- +tion (EDA) method (Wei & Zou, 2019) for augmentations. +Specifically, we apply random word deletion, random word +swapping, random insertion, and synonym replacement each +with a probability of 0.5 on each description, and add the +augmented descriptions to the original ones. We notice that +augmentation improves performance on EURLex by 1, 1, +and 2 P@1, P@5, and R@10 points respectively, suggesting +that augmentation can be a viable way to increase the quan- +tity of descriptions. On AmazonCat, augmentation has no +effect on the performance and rather slightly hurts it (e.g., +0.4 P@1 points). Given that AmazonCat has 3× the number +of labels in EURLex, we believe this shows SemSup-XC’s +effectiveness in capturing the label semantics in the presence +of a larger number of classes, rendering data augmentation +redundant. However, we believe that data augmentation +might be a simple tool to boost performance on smaller +datasets with lesser labels or descriptions. +Qualitative analysis +We present a qualitative analysis of +the performance SemSup-XC’s predictions in Table 5 com- +pared to MACLR. Examples are instances where SemSup- +XC outperforms MACLR, highlighting the strengths of our +method, with correct predictions in bold. In the first ex- +ample, while MACLR predicts five labels which are all +similar, SemSup-XC is able to predict diverse labels while +getting the correct label in five predictions. In the second +example, SemSup-XC realizes the content of the document +is a story and hence predicts literature & fiction, whereas +MACLR predicts classes based on the content of the story +instead. This shows the nuanced understanding of the label +space that SemSup-XC has learned. In the third example, + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Input Document +Top 5 Predictions +SEMSUP-XC +MACLR +Start-Up: A Technician’s Guide. In addition to being an excellent stand-alone +self-instructional guide, ISA recommends this book to prepare for the Start-Up +Domain of CCST Level I, II, and III examinations. +test preparation +vocational tests +schools & teaching +graduate preparation +new +test prep & study guides +used and rental textbooks +testing +software +vocational +Homecoming (High Risk Books). When Katey Bruscke’s bus arrives in her unnamed +hometown, she finds the scenery blurred, "as if my hometown were itself surfacing +from beneath a black ocean." At ... +literature & fiction +friendship +thriller & suspense +mothers & children +thrillers +drugs +genre fiction +coming of age +general +braille +Rolls RM65 MixMax 6x4 Mixer. The new RM65b HexMix is a single rack space unit +featuring 6 channels of audio mixing, each with an XLR Microphone Input and +1/4ünbalanced Line Input. A unique ... +studio recording equipment +powered mixers +powered mixers +hand mixers +home audio +mixers & accessories +musical instruments +mixers +speaker parts & components +mixer parts +Table 5: Sample predictions from SemSup-XC (our model) compared to MACLR (Xiong et al., 2022). Bold represents +correct predictions. Qualitative analysis shows that SemSup-XC can understand the document at a higher level than baselines +like MACLR. The second example poses an especially interesting case where SemSup-XC is able to understand that the +document is a fiction book, whereas MACLR tries to parse the story itself and predicts all labels incorrectly. +SemSup-XC shows a deep understanding of the label space +by predicting "studio recording equipment" even though +the document has no explicit mention of the words studio, +recording or equipment. For same example, MACLR fails +as it predicts labels like powered mixers because of the +presence of the word mixer. These examples show that +SemSup-XC’s understanding of how different fine-grained +classes are related and how instances refer to them is better +than the baselines considered. We list more such examples +in Appendix H. +5. Related Work +Extreme +classification +Extreme +classification +(XC) (Agrawal et al., 2013) studies multi-class and +multi-label classification problems over numerous classes +(thousands to millions). Traditionally, studies have used +sparse bag-of-words features of input documents (Bhatia +et al., 2015; Chang et al., 2019; Lin et al., 2014), simple +one-versus-all binary classifiers (Babbar & Schölkopf, +2017; Yen et al., 2017; Jain et al., 2019; Dahiya et al., +2021a), and tree-based methods which utilize the label +hierarchy (Prabhu et al., 2018; Wydmuch et al., 2018; +Khandagale et al., 2020). Recently, neural-network (NN) +based contextual dense-features have improved accuracies. +Studies have experimented with convolutional neural +networks (Liu et al., 2017), Transformers (Chang et al., +2020; Jiang et al., 2021; Zhang et al., 2021), attention-based +networks (You et al., 2019), and shallow networks (Medini +et al., 2019; Mittal et al., 2021; Dahiya et al., 2021b). While +the aforementioned works show impressive performance +when the labels during training and testing are the same, +they do not consider the practical zero-shot classification +scenario with unseen test labels. +Zero-shot +classification +Zero-shot +classification +(ZS) (Larochelle et al., 2008) aims to predict unseen classes +not encountered during training by utilizing auxiliary +information like class names or prototypes. Multiple works +have attempted ZS for text (Dauphin et al., 2014; Nam +et al., 2016; Wang et al., 2018; Pappas & Henderson, +2019; Hanjie et al., 2022), however, they face performance +degradation and are computationally expensive due to +XC’s large label space. +ZestXML (Gupta et al., 2021) +was the first study to attempt ZS extreme classification +by projecting non-contextual bag-of-words input features +close to corresponding label features using a sparsified +linear transformation. Subsequent works have used NNs +to generate contextual representations (Xiong et al., 2022; +Simig et al., 2022; Zhang et al., 2022; Rios & Kavuluru, +2018), with MACLR (Xiong et al., 2022) adding an XC +specific pre-training step and GROOV (Simig et al., 2022) +using a sequence-to-sequence model to predict novel labels. +However, these works use only label names to represent +classes (e.g., the word “car”), which lack semantic informa- +tion. We use semantically rich descriptions (Hanjie et al., +2022), which coupled with our modeling innovations (§ 2.2) +achieves state-of-the-art performance on ZS-XC. +6. Conclusion +We tackle the task of zero-shot extreme classification (XC) +which involves very large label spaces, by using 1) Hybrid- +Match, which incorporates both semantic similarity at the + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +sentence level and relaxed lexical similarity at the token +level, 2) contrastive learning to make training efficient, and +3) semantically rich class descriptions to gain a better un- +derstanding of the label space. We achieve state-of-the-art +results on three standard XC benchmarks and significantly +outperform prior work. Our various ablation studies and +qualitative analyses demonstrate the relative importance of +our modeling choices. Future work can further improve +description quality, and given the strong performance of +Hybrid-Match, can experiment with better architectures to +further push the boundaries of this practical task. +Acknowledgements +This work was supported by a grant from the Chadha Center +for Global India at Princeton University. We thank Jens +Tuyls, Khanh Nguyen and other members of the Princeton- +NLP group for comments on the draft. +References +Agrawal, R., Gupta, A., Prabhu, Y., and Varma, M. Multi- +label learning with millions of labels: recommending +advertiser bid phrases for web pages. Proceedings of +the 22nd international conference on World Wide Web, +2013. +Babbar, R. and Schölkopf, B. +Dismec: +Distributed +sparse machines for extreme multi-label classification. +Proceedings of the Tenth ACM International Conference +on Web Search and Data Mining, 2017. +Bai, B., Weston, J., Grangier, D., Collobert, R., Sadamasa, +K., Qi, Y., Chapelle, O., and Weinberger, K. 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In Proceedings of the ACM Web Conference 2022, +pp. 3162–3173, 2022. + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Appendices +A. Training Details +A.1. Hyperparameter Tuning +We tune the learning rate, batch_size using grid search. For +the EURLex dataset, we use the standard validation split for +choosing the best parameters. We set the input and output +encoder’s learning rate at 5e−5 and 1e−4, respectively. We +use the same learning rate for the other two datasets. We +use batch_size of 16 on EURLex and 32 on AmazonCat +and Wikipedia. For Eurlex, we train our zero-shot model +for fixed 2 epochs and the generalized zero-shot model for +10 epochs. For the other 2 datasets, we train for a fixed 1 +epoch. For baselines, we use the default settings as used in +respective papers. +Training +All of our models are trained end-to-end. We use the pre- +trained BERT model (Devlin et al., 2019a) for encoding +input documents, and Bert-Small model (Turc et al., 2019) +for encoding output descriptions. For efficiency in train- +ing, we freeze the first two layers of the output encoder. +We use contrastive learning to train our models and sample +hard negatives based on TF-IDF features. All implemen- +tation was done in PyTorch and Huggingface transformer +and experiments were run NVIDIA RTX2080 and NVIDIA +RTX3090 gpus. +A.2. Baselines +We use the code provided by ZestXML, MACLR and +GROOV for running the supervised baselines. We employ +the exact implementation of TF-IDF as used in ZestXML. +We evaluate T5 as an NLI task (Xue et al., 2021). We sepa- +rately pass the names of each of the top 100 labels predicted +by TF-IDF, and rank labels based on the likelihood of entail- +ment. We evaluate Sentence-Transformer by comparing the +similarity between the emeddings of input document and +the names of the top 100 labels predicted by TF-IDF. Splade +is a sparse neural retreival model that learns label/document +sparse expansion via a Bert masked language modelling +head. We use the code provided by authors for running +the baselines. We experiment with various variations and +pretrained models, and find splade_max_CoCodenser pre- +trained model with low sparsity(λd = 1e−6 & λq = 1e−6) +to be performing the best. +B. Label descriptions from the web +B.1. Automatically scraping label descriptions from the +web +We mine label descriptions from web in an automated +end-to-end pipeline. We make query of the form ‘what +is ’(or component name in case of Wikipedia) +on duckduckgo search engine. Region is set to United +States(English), and advertisements are turned off, with +safe search set to moderate. We set time range from 1990 +uptil June 2019. On average top 50 descriptions are scraped +for each query. To further improve the scraped descriptions, +we apply a series of heuristics: +• We remove any incomplete sentences. Incomplete sen- +tences do not end in a period or do not have more than +one noun, verb or auxiliary verb in them. +Eg: Label = Adhesives ; Removed Sentence = What +is the best glue or gel for applying +• Statements with lot of punctuation such as semi-colon +were found to be non-informative. Descriptions with +more than 10 non-period punctuations were removed. +Eg: +Label += +Plant +Cages +& +Supports +; +Removed Description = Plant Cages & Supports. My +Account; Register; Login; Wish List (0) Shopping Cart; +Checkout $ USD $ AUD THB; R$ BRL $ CAD $ CLP +$ ... +• We used regex search to identify urls and currencies +in the text. Most of such descriptions were spam and +were removed. +Eg: +Label += +Accordion +Accessories +; +Removed Description += +Buy +Accordion +Acces- +sories Online, with Buy Now & Pay Later and Rental +Options. Free Shipping on most orders over $250. +Start Playing Accordion Accessories Today! +• Descriptions with small sentences(<5 words) were re- +moved. +Eg: Label = Boats ; Removed Description = Boats for +Sale. Buy A Boat; Sell A Boat; Boat Buyers Guide; +Boat Insurance; Boat Financing ... +• Descriptions with more than 2 interrogative sentences +were filtered out. +Eg: Label = Shower Curtains ; Removed Descrip- +tion = So you’re interested...why? you’re starting a +company that makes shower curtains? or are you just +fooling around? Wiki User 2010-04 +• We mined top frequent n-grams from a sample of +scraped descriptions, and based on it identified n-grams +which were commonly used in advertisements. Exam- +ples include: ‘find great deals’, ‘shipped by’. +Label = Boat compasses ; Removed Description = +Shop and read reviews about Compasses at West Ma- +rine. Get free shipping on all orders to any West +Marine Store near you today. +• We further remove obscene words from the datasets +using an open-source library (Friedland, 2013). + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +• We also run a spam detection model (Grandury, 2021) +on the descriptions and remove those with a confidence +threshold above 0.9. +Eg: Label = Phones ; Removed Description = Check +out the Phones page at — the world’s +leading music technology and instrument retailer! +• Additionally, most of the sentences in first person, were +found to be advertisements, and undetected by previ- +ous model. We remove descriptions with more than 3 +first person words (such as I, me, mine) were removed. +Eg: Label = Alarm Clocks ; Removed Description = +We selected the best alarm clocks by taking the neces- +sary, well, time. We tested products with our families, +waded our way through expert and real-world user +opinions, and determined what models lived up to man- +ufacturers’ claims. ... +B.2. Post-Processing +We further add hierarchy information in a natural language +format to the label descriptions for AmazonCat and EURLex +datasets. Precisely, we follow the format of ‘key is value.’ +with each key, value pair represented in new line. Here key +belongs to the set { ‘Description’, ‘Label’, ‘Alternate Label +Names’, ‘Parents’, ‘Children’ }, and the value corresponds +to comma separated list of corresponding information from +the hierarchy or scraped web description. For example, +consider the label ‘video surveillance’ from EURLex dataset. +We pass the text: +‘Label is video surveillance. +Description is . +Parents are video communications. +Alternate Label Names are camera surveillance, security +camera surveillance.’ +to the output encoder. +For Wikipedia, label hierarchy is not present, so we only +pass the description along with the name of label. +B.3. Wikipedia Descriptions +When labels are fine-grained, as in the Wikipedia dataset, +making queries for the full label name is not possible. For +example, consider the label ‘Fencers at the 1984 Summer +Olympics’ from Wikipedia categories; querying for it would +link to the same category on Wikipedia itself. Instead, we +break the label names into separate constituents using a +dependency parser. Then for each constituent(‘Fencers’ +and ‘Summer Olympics’), we scrape descriptions. +No +descriptions are scraped for constituents labelled by +Named-Entity Recognition(‘1984’), and their NER tag +is directly used. Finally, all the scraped descriptions are +concatenated in a proper format and passed to the output +encoder. +B.4. De-Duplication +To ensure no overlap between our descriptions and input +documents, we used SuffixArray-based exact match algo- +rithm (Lee et al., 2022) with a minimum threshold of 60 +characters and removed the matched descriptions. +C. Hybrid-Match +We propose Hybrid-Match to exploit token similarity. We +create clusters of tokens based on: 1) the BERT token- +embedding similarity (Rajaee & Pilehvar, 2021) is higher +than a threshold or 2) if tokens share the same lemma. +Specifically, first tokens with BERT embedding cosine sim- +ilarity greater than 0.6 are put into same cluster. In the +second stage if two different tokens share the same lemma, +but are in different clusters, their clusters are merged. In +model, a mask is created of size (Q * LQ * D * LD), where +Q is the number of label descriptions, LQ is the max length +of all label descriptions, D is the number of documents, and +LD is the length of label descriptions. Here a entry of 1 +means that corresponding token in label description and +input share the same cluster, else it is set to 0. +D. Contrastive Learning +During training, for both EURLex and AmazonCat, we +randomly sample 1000 − |Y + +i | negative labels for each +input document. For Wikipedia, we precompute the top +1000 labels for each input based on TF-IDF scores. We +then randomly sample 1000 − |Y + +i | negative labels for each +document. At inference time, we evaluate our models on +all labels for both EURLex and AmazonCat. However, +even evaluation on millions of labels in Wikipedia is not +computationally tractable. Therefore, we evaluate only on +top 1000 labels predicted by TF-IDF for each input. +E. Full results for zero-shot classification +E.1. Split Creation +For EURLex, and AmazonCat, we follow the same pro- +cedure as detailed in GROOV (Simig et al., 2022). We +randomly sample k labels from all the labels present in +train set, and consider the remaining labels as unseen. For +EURLex we have roughly 25%(1057 labels) and for Ama- +zonCat roughly 50%(6500 labels) as unseen. For Wikipedia, +we use the standard splits as proposed in ZestXML (Gupta +et al., 2021). +E.2. Results +Table 6 contains complete results for ZS-XCacross the three +datasets, including additional baselines and metrics. + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Method +Precision - ZSL +Recall - ZSL / GZSL +Precision - GZSL +@1 +@3 +@5 +R@10 +R@10 +@1 +@3 +@5 +Eurlex-4.3K +TF-IDF +44.0(±0.0) +26.9(±0.0) +19.6(±0.0) +55.8(±0.0) +41.2(±0.0) +53.4(±0.0) +35.2(±0.0) +28.0(±0.0) +T5 +7.2(±0.0) +7.1(±0.0) +7.0(±0.0) +29.2(±0.0) +23.0(±0.0) +10.4(±0.0) +11.0(±0.0) +11.2(±0.0) +Sentence Transformer +15.9(±0.0) +10.8(±0.0) +9.1(±0.0) +31.1(±0.0) +25.5(±0.0) +18.8(±0.0) +15.7(±0.0) +11.9(±0.0) +ZestXML +9.6(±0.0) +7.3(±0.0) +6.5(±0.0) +25.7(±0.0) +54.8(±0.0) +84.8(±0.0) +64.8(±0.0) +48.9(±0.0) +ZestXML + TF-IDF +24.7(±0.0) +17.7(±0.0) +14.4(±0.0) +46.4(±0.0) +54.2(±0.0) +84.9(±0.0) +65.7(±0.0) +50.3(±0.0) +MACLR +24.9(±0.6) +16.6(±0.2) +13.4(±0.2) +42.1(±0.5) +55.2(±1.3) +60.7(±1.3) +49.1(±1.9) +41.1(±1.3) +GROOV +1.2 (±0.1) +2.6 (±0.4) +2.6 (±0.3) +7.0 (±0.9) +49.4 (±0.1) +84.1 (±0.1) +61.5 (±0.2) +45.3 (±0.1) +SemSup-XC-Hier +45.4(±0.2) +28.1(±0.1) +20.6(±0.2) +57.0(±0.2) +65.6(±1.0) +86.4(±0.2) +69.2(±0.2) +54.2(±0.4) +SemSup-XC +44.7(±2.3) +27.9(±1.1) +20.9(±0.6) +57.4(±3.4) +65.2(±0.3) +87.1(±0.1) +68.5(±0.1) +53.7(±0.1) +SemSup-XC + TF-IDF +49.3(±0.9) +31.2(±0.8) +23.1(±0.3) +62.4(±0.8) +62.9(±1.1) +87.0(±0.1) +67.6(±0.1) +51.6(±0.6) +Amazon-13K +TF-IDF +18.7(±0.0) +11.5(±0.0) +8.5(±0.0) +21.0(±0.0) +14.7(±0.0) +21.5(±0.0) +14.4(±0.0) +11.1(±0.0) +T5 +2.5(±0.0) +2.8(±0.0) +3(±0.0) +10.5(±0.0) +10.2(±0.0) +3.2(±0.0) +4.2(±0.0) +4.9(±0.0) +Sentence Transformer +15.2(±0.0) +10.5(±0.0) +8.3(±0.0) +22.2(±0.0) +16.0(±0.0) +18.4(±0.0) +13.4(±0.0) +11.0(±0.0) +ZestXML +12.7(±0.0) +8.9(±0.0) +7.1(±0.0) +21.2(±0.0) +52.5(±0.0) +87.9(±0.0) +58.6(±0.0) +41.5(±0.0) +ZestXML + TF-IDF +15.6(±0.0) +11.1(±0.0) +8.8(±0.0) +24.4(±0.0) +54.2(±0.0) +87.6(±0.0) +59.0(±0.0) +42.3(±0.0) +MACLR +36.0(±0.6) +23.5(±0.4) +18.0(±0.4) +54.4(±0.8) +46.9(±0.4) +46.0(±0.3) +33.7(±0.3) +27.2(±0.3) +GROOV +0.0(±0.0) +0.3(±0.0) +0.5(±0.0) +2.4 (±0.2) +47.9(±0.3) +87.4(±0.5) +55.8(±0.8) +38.8(±0.5) +SemSup-XC-Hier +43.9(±0.4) +31.5(±0.7) +25.4(±0.3) +69.7(±0.6) +71.5(±0.3) +88.4(±0.3) +65.0(±0.5) +50.2(±0.3) +SemSup-XC +48.2(±0.5) +33.9(±0.2) +27.0(±0.5) +72.9(±0.5) +71.6(±0.3) +88.6(±0.1) +65.3(±0.3) +51.2(±0.1) +Wikipedia-1M +TF-IDF +14.5(±0.0) +7.7(±0.0) +5.5(±0.0) +18.3(±0.0) +14.7(±0.0) +14.4(±0.0) +8.5(±0.0) +6.5(±0.0) +T5 +8.2(±0.0) +7.6(±0.0) +6.7(±0.0) +23.6(±0.0) +15.1(±0.0) +4.2(±0.0) +4.5(±0.0) +4.4(±0.0) +Sentence Transformer +19.6(±0.0) +11.1(±0.0) +7.9(±0.0) +22.5(±0.0) +16.6(±0.0) +14.2(±0.0) +9.1(±0.0) +7.0(±0.0) +ZestXML +12.9(±0.0) +8.0(±0.0) +6.0(±0.0) +20.0(±0.0) +25.7(±0.0) +26.7(±0.0) +18.8(±0.0) +14.6(±0.0) +ZestXML + TF-IDF +15.8(±0.0) +8.9(±0.0) +6.4(±0.0) +20.8(±0.0) +26.3(±0.0) +30.6(±0.0) +22.2(±0.0) +17.2(±0.0) +MACLR +29.8(±0.8) +17.8(±0.6) +13.2(±0.4) +41.7(±1.3) +32.7(±0.6) +28.0(±0.3) +18.3(±0.3) +14.4(±0.4) +GROOV +6.0(±0.1) +5.8(±0.3) +5.0(±0.4) +15.4(±0.2) +29.0(±0.2) +31.4(±0.1) +24.9(±0.1) +19.1(±0.0) +SemSup-XC +34.6(±0.2) +18.9(±0.2) +13.1(±0.3) +37.9(±0.4) +33.0(±0.3) +29.8(±0.1) +22.0(±0.2) +17.0(±0.4) +SemSup-XC + TF-IDF +36.5(±0.3) +19.5(±0.2) +13.4(±0.5) +38.5(±0.3) +34.1(±0.3) +33.7(±0.4) +23.4(±0.3) +17.7(±0.2) +Table 6: +Comparison of SEMSUP-XC with other supervised and unsupervised baselines. Our method consistently +outperfoms all methods across all datasets. +Model +Device +Throughput +Storage +P@1 +(Inputs/s) +(GB) +(ZSL) +SemSup-XC +1 GPU +46.2 +17.9 +36.5 +MACLR +1 GPU +77.8 +4.6 +29.8 +GROOV +1 GPU +8.9 +0.4 +6.0 +ZestXML +16 CPUs +2371 +1.8 +15.8 +Table 7: Computational Efficiency of SemSup-XC and base- +lines on Wikipedia dataset. We have comparable throughput +to dense baselines while requiring higher storage but with +substantial performance gains. +F. Full results for few-shot classification +F.1. Split Creation +We iteratively select k instances of each label in train docu- +ments. If a label has more than k documents associated with +it, we drop the label from training(such labels are not sam- +pled as either positives or negatives) for the extra documents. +We refer to these labels as neutral labels for convenience. +Because of such labels, loss functions of dense methods +need to be modified accordingly. For ZestXML, this is not +possible because it directly learns a transformation over the +whole dataset, and individual labels for particular instances +cannot be masked as neutral. +F.2. Models +We use MACLR, GROOV, Light XML as baselines. We +initialize the weights from the corresponding pre-trained +models in the GZSL setting. We use the default hyperpa- +rameters for baselines and SEMSUP models. As discussed in +the previous section, neutral labels are not provided at train +time for MACLR and GROOV baselines. However, since +Light XML uses a final fully-connected classification layer, + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +we cannot selectively remove them for a particular input. +Therefore, we mask the loss for labels which are neutral to +the documents. We additionally include scores for TF-IDF, +but since it is a fully unsupervised method, only zero-shot +numbers are included. +F.3. Results +The full results for few-shot classification are present in +Table 8. +G. Computational Efficiency +Extreme Classification necessitates that the models scale +well in terms of time and memory efficiency with labels at +both train and test times. SemSup-XC uses contrastive learn- +ing for efficiency at train time. During inference, SemSup- +XC predicts on top 1000 shortlists by TF-IDF, thereby +achieving sub-linear time. Further, contextualized tokens +for label descriptions are computed only once and stored in +memory-mapped files, thus decreasing computational time +significantly. Overall, our computational complexity can +be represented by O(TIE ∗ N + TOE ∗ |Y |+k ∗ N ∗ Tlex), +where TIE, TOE represent the time taken by input encoder +and output encoder respectively, N is the total number of +input documents, |Y| is the number of all labels, k indicates +the shortlist size and |Tlex| denotes the time in soft-lexical +computation between contextualized tokens of documents +and labels. In our experiments, TIE ∗ N >> TOE ∗ |Y | +and TIE ≈ Tlex ∗ k. Thus effectively, computational com- +plexity is approximately equal to O(TIE ∗ N), which is in +comparison to other SOTA extreme classification methods. +To ensure efficiency at inference time, similar to training, +SemSup-XC predicts on top of 1000 labels shortlisted by +TF-IDF. Table 7 shows that SemSup-XC’s throughput is +comparable to deep baselines (MACLR and GROOV) while +demonstrating much better performance. While ZestXML is +significantly faster, SemSup-XC’s P@1 is 2× higher. While +SemSup-XC’s storage is higher, 17 GB of space on modern- +day hard drives is trivial, especially given that the dataset has +over a million labels. SemSup-XC’s Hybrid-Match module +requires contextualized representations for every token in +the description, which contributes to the majority of the +storage. This shows that SemSup-XC provides the best +throughput-performance trade-off while having practical +storage requirements. +H. Qualitative Analysis +Table 9 shows multiple qualitative examples for which our +model outperforms the next baseline MACLR. The exam- +ples were chosen so as to increase diversity of input doc- +ument’s topic, number of correct predictions and relative +improvement over baseline. All examples are on Amazon- +Cat dataset. + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Method +EURLex-4.3K +AmazonCat-13K +P@1 +P@5 +R@10 +P@1 +P@5 +R@10 +1-shot +SemSup-XC +50.8(±0.9) +21.4(±0.7) +57.9(±2.9) +49.6(±0.4) +25.2(±0.2) +66.3(±0.7) +MACLR +39.2(±0.7) +17.3(±0.5) +50.9(±1.0) +38.6(±0.6) +19.7(±0.1) +56.4(±0.8) +MACLR with Descriptions +38.5(±0.4) +17.0(±0.5) +49.1(±0.8) +36.3(±0.4) +18.3(±0.7) +52.4(±0.6) +GROOV +17.5(±0.6) +4.2(±0.2) +9.4(±0.4) +11.4(±0.8) +4.3(±0.4) +9.1(±0.9) +GROOV with Descriptions +1.1(±0.2) +0.4(±0.2) +1.3(±0.3) +4.0(±0.3) +0.9(±0.1) +1.9(±0.4) +Light XML +12.5(±1.9) +6.3(±0.9) +19.5(±2.9) +7.5(±0.7) +7.0(±0.3) +25.1(±0.3) +5-shot +SemSup-XC +63.3(±0.3) +26.3(±0.3) +67.6(±0.1) +51.8(±0.1) +26.1(±0.2) +70.0(±0.8) +MACLR +51.2(±0.4) +23.0(±0.2) +63.8(±0.7) +42.4(±0.3) +21.6(±0.4) +61.4(±0.1) +MACLR with Descriptions +52.5(±0.4) +22.9(±0.5) +62.1(±0.4) +39.3(±0.5) +19.9(±0.3) +57.4(±0.6) +GROOV +43.1(±1.0) +14.2(±0.5) +33.3(±1.1) +24.2(±0.8) +9.7(±0.6) +19.4(±1.5) +GROOV with Descriptions +6.0(±0.6) +1.3(±0.4) +3.5(±0.6) +17.6(±0.5) +3.5(±0.2) +9.5(±0.3) +Light XML +52.7(±0.1) +23.7(±0.0) +62.6(±0.3) +50.9(±0.7) +24.7(±0.4) +64.3(±0.4) +10-shot +SemSup-XC +68.5(±0.1) +28.3(±0.5) +71.9(±2.1) +56.8(±0.3) +27.7(±0.4) +69.8(±0.5) +MACLR +56.1(±0.1) +25.4(±0.0) +69.4(±0.1) +43.9(±0.2) +22.3(±0.0) +62.9(±0.3) +MACLR with Descriptions +57.0(±0.2) +26.7(±0.3) +69.7(±0.3) +41.6(±0.3) +21.4(±0.2) +60.3(±0.5) +GROOV +46.9(±0.6) +18.0(±0.1) +43.2(±0.2) +29.6(±0.4) +12.8(±0.1) +29.3(±0.4) +GROOV with Descriptions +10.1(±0.4) +2.0(±0.1) +4.8(±0.4) +21.4(±0.5) +4.5(±0.5) +13.4(±0.4) +Light XML +61.6(±0.6) +27.1(±0.3) +71.0(±0.4) +57.7(±0.3) +27.5(±0.3) +69.3(±0.7) +20-shot +SemSup-XC +72.6(±0.2) +30.8(±0.1) +78.2(±0.4) +61.7(±0.5) +29.9(±0.2) +73.9(±0.3) +MACLR +59.0(±0.3) +27.2(±0.0) +73.2(±0.2) +47.6(±0.2) +24.1(±0.1) +66.9(±0.3) +MACLR with Descriptions +60.6(±0.5) +27.4(±0.4) +73.2(±0.4) +47.2(±0.3) +23.5(±0.2) +66.3(±0.2) +GROOV +53.3(±1.3) +21.5(±0.8) +52.7(±2.2) +35.7(±0.2) +15.6(±0.2) +39.5(±0.3) +GROOV with Descriptions +16.4(±0.3) +3.6(±0.4) +10.0(±0.6) +29.0(±0.3) +6.4(±0.4) +18.1(±0.4) +Light XML +67.8(±0.5) +30.4(±0.1) +76.6(±0.0) +63.1(±0.2) +29.8(±0.3) +73.6(±0.3) +Table 8: Detailed table for few-shot results. SemSup-XC outperforms all other baselines with significant margins for +k = 1, 5, &10 shot settings. For 20-shot we perform almost at par with fully supervised method of Light XML, which +otherwise performs poorly for zero-shot and lower values of k in few shot setting. + +SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification +Input Document +Top 5 Predictions +SEMSUP-XC +MACLR +Start-Up: A Technician’s Guide. In addition to being an excellent stand-alone +self-instructional guide, ISA recommends this book to prepare for the Start-Up +Domain of CCST Level I, II, and III examinations. +test preparation +vocational tests +schools & teaching +graduate preparation +new +test prep & study guides +used and rental textbooks +testing +software +vocational +Homecoming (High Risk Books). When Katey Bruscke’s bus arrives in her unnamed +hometown, she finds the scenery blurred, "as if my hometown were itself surfacing +from beneath a black ocean." 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The idea was as simple as it was ... +girls +sport sandals +clothing +shoes +sandals +athletic +sneakers +mountaineering boots +outdoor +sandals +Table 9: Examples of class predictions from SemSup-XC (our model) compared to MACLR (Xiong et al., 2022). Bold +represents correct predictions. + diff --git a/bdFIT4oBgHgl3EQfmSv9/content/tmp_files/load_file.txt b/bdFIT4oBgHgl3EQfmSv9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f372c7ac55ad91a6c3d60856d20065ab606c3dd --- /dev/null +++ b/bdFIT4oBgHgl3EQfmSv9/content/tmp_files/load_file.txt @@ -0,0 +1,2074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf,len=2073 +page_content='SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Pranjal Aggarwal 1 Ameet Deshpande 2 Karthik Narasimhan 2 1Indian Institute of Technology, Delhi 2Department of Computer Science, Princeton University pranjal2041@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='com, {asd, karthikn}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='edu Abstract Extreme classification (XC) involves predicting over large numbers of classes (thousands to mil- lions), with real-world applications like news arti- cle classification and e-commerce product tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The zero-shot version of this task requires gener- alization to novel classes without additional su- pervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In this paper, we develop SemSup-XC, a model that achieves state-of-the-art zero-shot and few-shot performance on three XC datasets derived from legal, e-commerce, and Wikipedia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' To develop SemSup-XC, we use automati- cally collected semantic class descriptions to rep- resent classes and facilitate generalization through a novel hybrid matching module that matches in- put instances to class descriptions using a combi- nation of semantic and lexical similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Trained with contrastive learning, SemSup-XC signifi- cantly outperforms baselines and establishes state- of-the-art performance on all three datasets con- sidered, gaining up to 12 precision points on zero- shot and more than 10 precision points on one- shot tests, with similar gains for recall@10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Our ablation studies highlight the relative importance of our hybrid matching module and automatically collected class descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Introduction Extreme classification (XC) studies the problem of predict- ing over a large space of classes, ranging from thousands to millions (Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Bha- tia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This paradigm has multiple real-world ap- plications including movie and product recommendation, 1Code and demo are available at https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='com/princeton-nlp/semsup-xc and https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='co/spaces/Pranjal2041/ SemSup-XC/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' search-engines, and e-commerce product tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In many of these applications, models are required to handle the ad- dition of new classes on a regular basis, which has been the subject of recent work on zero-shot and few-shot ex- treme classification (ZS-XC and FS-XC) (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' These setups are challenging because of (1) the presence of a large number of fine-grained classes which are often not mutually exclusive, (2) limited or no labeled data per class, and (3) increased computational expense and model size due to the large la- bel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' While the aforementioned works have tried to tackle the latter two issues, they lack a semantically rich representation of classes, and instead rely on class names or hierarchies to represent them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In this work, we leverage semantic supervision (SEMSUP) (Hanjie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) for developing mod- els for extreme classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC represents classes using multiple diverse class descriptions, which allows it to generalize naturally to novel classes when provided with corresponding descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' However, SEMSUP as proposed in (Hanjie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) cannot be naively applied to XC for several reasons: (1) SEMSUP requires encoding descriptions of all classes for each training batch, which is prohibitively computationally expensive for large label spaces, (2) it uses semantic similarity only at the sentence level between the instance and label description, and (3) it requires human intervention to collect descriptions, which is expensive for extremely large label spaces we are dealing with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We remedy these deficiencies by developing SemSup-XC, a model that scales to large class spaces in XC and establishes a new state-of-the-art using three innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' First, we use a novel hybrid lexical-semantic similarity model (Hybrid- Match) that combines semantic similarity of sentences with relaxed lexical-matching between all token pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Second, we propose SEMSUP-WEB – an automatic pipeline with pre- cise heuristics to discover high-quality descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Finally, we use a contrastive learning objective that samples a fixed number of negative label descriptions, improving computa- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='11309v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='CL] 26 Jan 2023 SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Labels Query a search engine and apply heuristic filtering Pick the positive classes and sample = 2 negative classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' κ 1 Automatic class description collection Axe … Book CD Paper 2 Contrastive learning A tool to cut wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 🪓 Written or printed work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 📚 A digital data storage format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 📀 A sheet material for writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 📝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Axe Book CD Paper .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Positive class Negative class 3 Lexico-semantic matching (SemSup-XC) Input: Tagore wrote a gem Label: Book Instance [CLS] Tagore wrote a gem [CLS] Written or printed work Class descriptions corresponding to positive and negative classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Dot Dot SemSup-XC computes the logits using: (a) semantic similarity ([CLS]) and (b) relaxed lexical similarity (wrote, written).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Logits (a) (b) Negative class Figure 1: Our model SemSup-XC achieves state-of-the-art performance on zero-shot and few-shot extreme classification through three innovations – 1 large-scale automated class description collection with heuristic filtering to improve semantic understanding of classes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 2 contrastive learning to make training faster by over 99% when compared to the previous work (SEMSUP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' and 3 a novel lexico-semantic matching model building called Hybrid-Match to utilize both semantic similarity at the sentence level and contextual lexico-semantic similarity at the token level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' tion speed by up to 99% when compared to SEMSUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC achieves state-of-the-art performance on three diverse XC datasets from legal (EURLex), e-commerce (AmazonCat), and wiki (Wikipedia) domains, across zero- shot (ZS-XC), generalized zero-shot (GZS-XC) and few- shot (FS-XC) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For example, on ZS-XC, SemSup- XC outperforms the next best baseline by 5 − 12 precision points over all datasets and all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' On FS-XC, SemSup- XC consistently outperforms baselines by over 10 P@1 points on the EURLex and AmazonCat datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Surpris- ingly, SemSup-XC even outperforms larger unsupervised language models like T5 and Sentence Transformers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', by over 30 P@1 points on EURLex) which are pre-trained on much larger web-scale corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Our ablation studies dissect the importance of each component in SemSup-XC, and shows the importance of our proposed hybrid matching module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Qualitative error analysis of our model (Table 5) shows that it predicts diverse correct classes that are appli- cable to the instance at times, whereas other models either predict incorrect classes or suffer a mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Background Extreme Classification Extreme classification deals with prediction over large label spaces (thousands to millions classes) and multiple correct classes per instance (multi- label) (Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Babbar & Schölkopf, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zero-shot extreme classification (ZS- XC) is a variant where models are evaluated on unseen classes not encountered during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We evaluate both on (1) zero-shot (ZS), where the model is tested only on unseen classes and (2) generalized zero-shot (G-ZS), where the model is tested on a combined set of train and unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We also consider few-shot extreme classification (FS-XC), where a small number of supervised examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 5) are available for unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The heavy tailed distribution of a large number of fine-grained classes in XC poses efficiency and performance challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zero-shot classification Zero-shot classification is usu- ally performed by matching instances to auxiliary infor- mation corresponding to classes, like their class name or attributes (Larochelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Dauphin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Re- cently, Hanjie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' (2022) proposed the use of multiple class descriptions to endow the model with a holistic semantic understanding of the class from different viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SEM- SUP uses an (1) input encoder (fIE) to encode the instance and an (2) output encoder (gOE) to encode class descriptions, and makes predictions by measuring the compatibility of the input and output representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Formally, let xi be the input instance, dj ∈ Dj be one sampled description of class j, fIE (xi) , gOE (dj) ∈ Rd be the input and output representation respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=" For the multi-label XC setting, Hou'shethaomyyomes The mufh." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The rohh Thetorulhmahens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='ThSemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification the probability of picking the jth class is: SEMSUP := P(yj = 1|xi) = σ (gOE (dj)⊺ · fIE (xi)) (1) Challenges with large class spaces Hanjie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' (2022)’s method cannot be directly applied to XC for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' First, they use a bi-encoder model (Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2009) which measures only the semantic similarity between the input instance and the class description at the sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' How- ever, instances and descriptions often share lexical terms with the same or similar meaning and lemma (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', picture and photo), which is not exploited by their method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Second, their class description collection pipeline requires human intervention, which is not feasible for the large number of classes in XC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' And third, they fine-tune the label encoder by encoding descriptions for all classes for every batch of instances, making it computationally infeasible for large label spaces because of GPU memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Improved ZS extreme classification SemSup-XC addresses the aforementioned challenges using: (1) a novel hybrid semantic-lexical similarity model for improved performance, (2) an automatic class description discovery pipeline with accurate heuristics for garnering high-quality class descriptions, and (3) contrastive learning with negative samples for improved computational speed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' HYBRID LEXICO-SEMANTIC SIMILARITY MODEL SEMSUP’s bi-encoder architecture measures only the seman- tic similarity of the input instance and class description at sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' However, semantic similarity ignores lexical matching of shared words which exhibit strong evidence of compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Input: It was cold and flavorful and De- scription: Ice cream is a cold dessert).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' A recently proposed information-retrieval model, COIL (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021a), al- leviates this by incorporating lexical similarity by adding the dot product of contextual representations corresponding to common tokens between the query and document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' But COIL has the drawback that semantically similar tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', “pictures” and “photos”) and words with the same lemma (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', walk and walking) are treated as dissimilar tokens, despite being commonly used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Input: Capture the best moments in high quality pictures and Class description: A camera is used to take photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=') We propose Hybrid-Match to exploit such token similar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We create clusters of tokens based on: 1) the BERT token-embedding similarity (Rajaee & Pilehvar, 2021) is higher than a threshold or 2) if tokens share the same lemma, which results in tokens like “photo”, “picture”, and “pic- tures” being in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We provide implementation details on clustering in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In addition to semantic similarity, Hybrid-Match uses these clusters to for relaxed lexical-matching by computing the dot-product of contex- tual representations of “similar” tokens in the input and description, as judged by the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For cases where an input token has several similar tokens in the description, we choose the description token with the max dot product with the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Formally let xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' , xin) be the input instance with n tokens, dj = (dj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' , djm) class jth descriptions with m tokens, vxi cls and vdj cls be the [CLS] representations of the input and description, and vxi k and vdj l be the representation of the kth and lth token of xi and dj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Let CL(w) denote the cluster membership of the token w, with CL(wi) = CL(wj) implying that the tokens are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Then probability of class yj is: Hybrid-Match := P(yj = 1|xi) = σ � vxi cls ⊺ · vdj cls + n � k=1 max l∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=',m}, CL(xik)=CL(djl) � vxi k ⊺vdj l �� (2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SEMSUP-WEB: AUTOMATIC COLLECTION OF HIGH-QUALITY DESCRIPTIONS We create a completely automatic pipeline for collecting de- scriptions which includes sub-routines for removing spam, advertisements, and irrelevant descriptions, and we de- tail the list of heuristics used in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' These sub- routines contain precise rules to remove irrelevant descrip- tions, for example by removing sentences with too many special characters (usually spam), descriptions with click- bait phrases (usually advertisements), ones with multiple interrogative phrases (usually people’s comments), small descriptions (usually titles) and so on (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Further in Wikipedia, querying search engine for labels return no useful results, since labels are very specific (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Fencers at the 1984 Summer Olympics) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Therefore, we design a multi-stage approach, where we first break label names into relevant constituents and query each of them individually (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In addition to web-scraped label de- scriptions, we utilize label-hierarchy information if provided by the dataset (EURLex and AmazonCat), which allows us to encode properties about parent and children classes wher- ever present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Further details for hierarchy are present in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' As we show in the ablation study (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3), label descriptions that we collect automatically provide sig- nificant performance boosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' TRAINING USING CONTRASTIVE LEARNING For datasets with a large number of classes (large |C|), it is not computationally feasible to encode class descriptions for all classes for every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We draw inspiration from contrastive learning (Hadsell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2006) and sample a sig- nificantly smaller number of negative classes to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For an instance xi, consider two partitions of the la- bels Yi = {yi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' , yiC|yij ∈ {0, 1}}, with Y + i containing the positive classes (yij = 1) and Y − i containing the nega- SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Dataset Documents Labels Ntrain Ntest Ntest(ZS-XC) |Yseen | |Yunseen | EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K 45 K 6 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 K 3,136 1,057 AmazonCat-13K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1M 307K 268 K 6,830 6,500 Wikipedia-1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2M 495,107 776,612 Table 1: Dataset statistics along with information about zero-shot (ZS-XC) splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' tive classes (yij = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC caps the total number of class descriptions being encoded for this instance to K by using all the positive classes (|Y + i |) and sampling only K−|Y + i | negative classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Intuitively, our training objective incentivizes the representations of the instance and positive classes to be similar while simultaneously making them dis- similar to the negative classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' To improve learning, rather than picking negative labels at random, we sample hard neg- atives that are lexically similar to positive classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' A typical dataset we consider (AmazonCat) has |C|= 13, 000 and K ≈ 1000, which leads to SemSup-XC being 12000 13000 = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3% faster than SEMSUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Mathematically, the following is the training objective, where N is the train dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' LSemSup-XC = 1 N · K � i � � yk∈Y + i LBCE (P (yk = 1|xi) , yk) + K−|Y + i | � yl∼Y − i , l=1 LBCE (P(yl = 0|xi), yl) � (3) For each batch, a class description is randomly sampled for each class (dl j ∈ Dj), thus allowing the model to see all the descriptions in Dj over the course of training, with the same sampling strategy during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We refer readers to appendix D for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Experimental Setup Datasets We evaluate our model on three diverse public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' They are, EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K (Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) which is legal document classification dataset with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K classes, AmazonCat-13K (McAuley & Leskovec, 2013) which is an e-commerce product tagging dataset including Amazon product descriptions and titles with 13K categories, and Wikipedia-1M (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) which is an article classification dataset made up of 5 million Wikipedia articles with over 1 million categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We provide detailed statistics about the number of instances and classes in train and test set in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 for details on split creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Baselines We perform extensive experiments with several baselines, which can be divided into unsupervised (first three) and supervised which are fine-tuned on the datasets we consider (the remaining four).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 1) TF-IDF performs a nearest neighbour match between the sparse tf-idf fea- tures of the input and class description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 2) T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) is a large sequence-to-sequence model which has been pre-trained on 750GB unsupervised data and further fine-tuned on MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We evaluate the model as an NLI task where labels are ranked based on the likelihood of entailment to the input document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For computational efficiency, we evaluate T5 only on top 50 labels shortlisted by TF-IDF on each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 3) Sen- tence Transformer (Reimers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) is a semantic text similarity model fine-tuned using a contrastive learn- ing objective on over 1 billion sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We rank the labels based on the similarity between input and descrip- tion embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The latter two baselines use significantly more data than SemSup-XC and T5 has 9× the parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The aforementioned baselines are unsupervised and not fine-tuned on our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The following baselines are previously proposed supervised models and are fine-tuned on the datasets we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 4) ZestXML (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) learns a highly sparsified linear transformation ( W ) which projects sparse input features close to corresponding positive label features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' At inference, for each input instance xi, label lj is scored based on the formula sij = lT j Wxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 5) MACLR (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) is a bi-encoder based model pre-trained on two self-supervised learning tasks to improve extreme classification—Inverse Cloze Task (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) and SimCSE (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021b), and we fine-tune it on the datasets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 6) GROOV (Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) is a T5 model that learns to generate labels given an input instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 7) SPLADE (Formal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) is a state-of-the-art sparse neural retreival model that learns label/document sparse expansion via a Bert masked language modelling head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We evaluate baselines under two different settings – by pro- viding either class names or class descriptions as auxillary information, and use the label hierarchy in both settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The version of baselines which use our class descriptions are strictly comparable to SemSup-XC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' See Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC implementation details We use the Bert- base model (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019b) as the backbone for the input encoder and Bert-small model (Turc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) for the output encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC follows the model archi- tecture described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 (Hybrid-Match) and we use contrastive learning (Hadsell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2006) to train our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' During training, we randomly sample K − |Y + i | negatives for each instance, where K is the number of labels for the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' At inference, to improve computational efficiency, we precompute the output representations of la- bel descriptions and shortlist top 1000 labels based on the TF-IDF scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use the AdamW optimizer (Loshchilov & Hutter, 2019) and tune our hyperparameters using grid search on the respective validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Model EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K AmazonCat-13K Wikipedia-1M ZS-XC GZS-XC ZS-XC GZS-XC ZS-XC GZS-XC P@1 R@10 P@1 R@10 P@1 R@10 P@1 R@10 P@1 R@10 P@1 R@10 Baselines with Class Names Unsupervised Baselines TF-IDF 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 Sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Transformer (Reimers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 Supervised Baselines ZestXML (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 SPLADE (Formal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 MACLR (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 46.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 SEMSUP-XC (Our Model) SEMSUP-XC 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1 Table 2: Zero-shot (ZS-XC) and generalized zero-shot (GZS-XC) results for all models on three XC benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup- XC significantly outperforms state-of-the-art models on both precision (P@) and recall (R@) metrics across the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Evaluation setting and metrics We evaluate all models on three different settings: Zero-shot classification (ZS-XC) on a set of unseen classes, generalized zero-shot classifi- cation (GZS-XC) on a combined set of seen and unseen classes, and few-shot classification (FS-XC) on a set of classes with minimal amounts of supervised data (1 to 20 examples per class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For all three settings, we train on in- put instances of seen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use Precision@K and Recall@K as our evaluation metrics, as is standard practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Precision@K measures how accurate the top-K predictions of the model are, and Recall@K measures what fraction of correct labels are present in the top-K predictions, and they are mathematically defined as P@k = 1 k � i∈rankk(ˆy) yi and R@k = 1 � i yi � i∈rankk(ˆy) yi, where rankk(ˆy) is the set of top-K predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We average the metrics over test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zero-shot extreme classification For the zero-shot scenario, we compare SemSup-XC with baselines which use class descriptions and counterparts which use class names as auxiliary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We provide label hierarchy as additional supervision in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We compare SemSup-XC with the best variant of each baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Table 2 shows that SemSup-XC significantly outperforms baselines on almost all datasets and metrics, under both zero- shot (ZS-XC) and generalized zero-shot (GZS-XC) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' On ZS-XC, SemSup-XC outperforms MACLR by over 24, 12, and 6 P@1 points on the three datasets respectively, even though MACLR uses XC specific pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup- XC also outperforms GROOV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', over 48 P@1 points on EURLex) which uses a generative T5 model pre-trained on significantly more data than our BERT backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' GROOV’s unconstrained output space might be one of the reasons for its worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC’s semantic understand- ing of instances and labels stands out against ZestXML which uses sparse non-contextual features with the former consistently scoring twice as higher compared to the lat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC consistently outperforms SPLADE (For- mal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021), a state-of-the-art information retrieval method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This shows that the straightforward application of IR baselines on XC, even when they are fine-tuned, un- derperforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This is likely because of the multi-label and fine-grained nature of classes coupled with a heavy-tailed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' While TF-IDF is competitive with deep base- lines, SemSup-XC’s hybrid lexico-semantic similarity mod- SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification (a) EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K (b) AmazonCat-13K Figure 2: Few-Shot P@1 for different Values of K on EURLex and AmazonCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC starts off significantly higher and for EURLex maintains the gap for larger values of K to the second best model, MACLR (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For AmazonCat, SemSup-XC maintains similar leads for most baselines, while being at par with Light XML (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' ule (Hybrid-Match) can perform fine-grained lexical and semantic matching and outperforms both sparse and deep methods on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC also outperforms the unsupervised baselines T5 and Sentence-Transformer, even though they are pre-trained on significantly larger amounts of data than BERT (T5 use 50× compared our base model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC also achieves higher recall on EURLex and AmazonCat datasets, beating the best performing baselines by 6, 18 R@10 points respectively, while being only 3 R@10 points less on Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC is also the best model for GZS-XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The margins of improvement are 1-2 P@1 points, which are smaller only because GZS-XC includes seen labels during evaluation, which are usually large in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Table 6 in Appendix E contains additional results with more methods and metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Our results show that SemSup-XC is able to utilize the semantic and lexical information in class descriptions to improve performance significantly, while other baselines hardly improve when using descriptions instead of class names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Few-shot extreme classification We now consider the FS-XC setup, where new classes added at evaluation time have a small number of labeled instances each (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We evaluate on four settings – K ∈ {1, 5, 10, 20} and all baselines other than ZestXML, which cannot be used for FS-XC (See Appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Further, we omit evaluation on Wikipedia, since it has ≈ 10 training examples per label, which is insufficient to study the effect of increasing values of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For the sake of completeness, we also include zero- shot performance (ZS-XC, K = 0) and report results in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Detailed results for other metrics (showing the same trend as P@1) and implementation details regarding creation of the few-shot splits are in appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Similar to the ZS-XC case, SemSup-XC outperforms all baselines for all values of K on EURLex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For Ama- zonCat, SemSup-XC outperforms all baselines other than Light XML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Light XML is significantly outperformed for K = {0, 1} and matches for K = {5, 10, 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In com- parison to MACLR and GROOV, SemSup-XC consistently outperforms by large margins (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 12 & 27P@1 points for EURLex) across all values of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC’s zero- shot performance is higher than even the few-shot scores of MACLR and GROOV that have access to K = 20 la- beled samples on AmazonCat, which further strengthens the model’s applicability to the XC paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Moreover, adding a few labeled examples seems to be more effective in EURLex than AmazonCat, with the performance difference between K = 1 and K = 20 being 22 and 12 P@1 points re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This, along with the fact that performance seems to plateau for both datasets, suggests that SemSup-XC learns label semantics better for AmazonCat than EURLex, due to its larger label space with rich descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Analysis We dissect the performance of SemSup-XC by conducting ablation studies on model components and label descriptions and further provide qualitative analysis on EURLex and AmazonCat for the zero-shot extreme classification setting (ZS-XC) in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Ablating components of SemSup-XC SemSup-XC’s use of the Hybrid-Match model and semantically rich de- 70 60 Precision@ 50 SemSup-XC (Ours) 30 一 TF-IDF 20 MACLR (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) GROOV (Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) 10 LightXML (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) 0 5 10 0 20 K Instances Per Label70 60 Precision@ 50 40 30 20 10 0 5 10 20 0 K Instances Per LabelSemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Method Components EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K AmazonCat-13K Auxillary Hierarchy Exact Hybrid P@1 P@5 R@10 P@1 P@5 R@10 Information Match Match Ablating Label Descriptions SemSup-XC Descriptions \x13 \x13 \x13 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 Replace descriptions with names Names \x13 \x13 \x13 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 Remove hierarchy Descriptions \x17 \x13 \x13 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 Ablating Model Architecture Components SemSup-XC Descriptions \x13 \x13 \x13 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 Replace Hybrid with Exact Lexical Matching Descriptions \x13 \x13 \x17 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 Remove all lexical matching Descriptions \x13 \x17 \x17 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 Table 3: Component-wise Model Analysis of SemSup-XC for ZS-XCon EURLex and AmazonCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Each component contributes to the final performance, with lexical-matching playing an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Method EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K AmazonCat-13K P@1 P@5 R@10 P@1 P@5 R@10 SemSup-XC 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 + Augmentation 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 Table 4: Description augmentation helps boost performance for ZS-XC on EURLex, but does not help on AmazonCat, which is a significantly larger dataset (3× labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This demonstrates SemSup-XC’s out-of-the-box performance, since augmentation is unnecessary for larger label spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' scriptions enables it to outperform all baselines considered, and we analyze the importance of each component in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' As our base model (first row) we consider SemSup- XC without ensembling it with TF-IDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We note that the SemSup-XC base model is the best performing vari- ant for both datasets and on all metrics other than P@1 for EURLex, for which it is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 points lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Web scraped class descriptions are important because removing them decreases both precision and recall scores (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', P@1 is lower by 4 points on AmazonCat) on all settings consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We see bigger improvements with AmazonCat, which is the dataset with larger number of classes (13K), which substantiates the need for semantically rich descriptions when dealing with a large number of fine-grained classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Label hierarchy information is similarly crucial, with large performance drops on both datasets in its absence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 26 P@1 points on AmazonCat), thus showing that access to structured hierarchy information leads to better semantic representations of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' On the modeling side, we observe that relaxed and exact lexical matching, which are components of Hybrid-Match, are important, with their absence leading to 3 and 11 P@1 point degradation on AmazonCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Even for EURLex, hy- brid lexical matching improves performance by 33 P@1 points when compared to a model with no lexical matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This highlights that our proposed model Hybrid-Match’s hybrid semantic-lexical approach significantly improves per- formance on XC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Augmenting Label Descriptions The previous result showed the importance of class descriptions, and we explore the effect of augmenting them to increase their diversity and quantity (See Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use the easy data augmenta- tion (EDA) method (Wei & Zou, 2019) for augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Specifically, we apply random word deletion, random word swapping, random insertion, and synonym replacement each with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 on each description, and add the augmented descriptions to the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We notice that augmentation improves performance on EURLex by 1, 1, and 2 P@1, P@5, and R@10 points respectively, suggesting that augmentation can be a viable way to increase the quan- tity of descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' On AmazonCat, augmentation has no effect on the performance and rather slightly hurts it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 P@1 points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Given that AmazonCat has 3× the number of labels in EURLex, we believe this shows SemSup-XC’s effectiveness in capturing the label semantics in the presence of a larger number of classes, rendering data augmentation redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' However, we believe that data augmentation might be a simple tool to boost performance on smaller datasets with lesser labels or descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Qualitative analysis We present a qualitative analysis of the performance SemSup-XC’s predictions in Table 5 com- pared to MACLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Examples are instances where SemSup- XC outperforms MACLR, highlighting the strengths of our method, with correct predictions in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In the first ex- ample, while MACLR predicts five labels which are all similar, SemSup-XC is able to predict diverse labels while getting the correct label in five predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In the second example, SemSup-XC realizes the content of the document is a story and hence predicts literature & fiction, whereas MACLR predicts classes based on the content of the story instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This shows the nuanced understanding of the label space that SemSup-XC has learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In the third example, SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Input Document Top 5 Predictions SEMSUP-XC MACLR Start-Up: A Technician’s Guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In addition to being an excellent stand-alone self-instructional guide, ISA recommends this book to prepare for the Start-Up Domain of CCST Level I, II, and III examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' test preparation vocational tests schools & teaching graduate preparation new test prep & study guides used and rental textbooks testing software vocational Homecoming (High Risk Books).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' When Katey Bruscke’s bus arrives in her unnamed hometown, she finds the scenery blurred, "as if my hometown were itself surfacing from beneath a black ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='" At .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' literature & fiction friendship thriller & suspense mothers & children thrillers drugs genre fiction coming of age general braille Rolls RM65 MixMax 6x4 Mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The new RM65b HexMix is a single rack space unit featuring 6 channels of audio mixing, each with an XLR Microphone Input and 1/4ünbalanced Line Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' A unique .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' studio recording equipment powered mixers powered mixers hand mixers home audio mixers & accessories musical instruments mixers speaker parts & components mixer parts Table 5: Sample predictions from SemSup-XC (our model) compared to MACLR (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Bold represents correct predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Qualitative analysis shows that SemSup-XC can understand the document at a higher level than baselines like MACLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The second example poses an especially interesting case where SemSup-XC is able to understand that the document is a fiction book, whereas MACLR tries to parse the story itself and predicts all labels incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC shows a deep understanding of the label space by predicting "studio recording equipment" even though the document has no explicit mention of the words studio, recording or equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For same example, MACLR fails as it predicts labels like powered mixers because of the presence of the word mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' These examples show that SemSup-XC’s understanding of how different fine-grained classes are related and how instances refer to them is better than the baselines considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We list more such examples in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Related Work Extreme classification Extreme classification (XC) (Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2013) studies multi-class and multi-label classification problems over numerous classes (thousands to millions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Traditionally, studies have used sparse bag-of-words features of input documents (Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2014), simple one-versus-all binary classifiers (Babbar & Schölkopf, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Yen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Dahiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021a), and tree-based methods which utilize the label hierarchy (Prabhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Wydmuch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Khandagale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Recently, neural-network (NN) based contextual dense-features have improved accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Studies have experimented with convolutional neural networks (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2017), Transformers (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021), attention-based networks (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019), and shallow networks (Medini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Dahiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' While the aforementioned works show impressive performance when the labels during training and testing are the same, they do not consider the practical zero-shot classification scenario with unseen test labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zero-shot classification Zero-shot classification (ZS) (Larochelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2008) aims to predict unseen classes not encountered during training by utilizing auxiliary information like class names or prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Multiple works have attempted ZS for text (Dauphin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Nam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Pappas & Henderson, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Hanjie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022), however, they face performance degradation and are computationally expensive due to XC’s large label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' ZestXML (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021) was the first study to attempt ZS extreme classification by projecting non-contextual bag-of-words input features close to corresponding label features using a sparsified linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Subsequent works have used NNs to generate contextual representations (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Rios & Kavuluru, 2018), with MACLR (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) adding an XC specific pre-training step and GROOV (Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) using a sequence-to-sequence model to predict novel labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' However, these works use only label names to represent classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', the word “car”), which lack semantic informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use semantically rich descriptions (Hanjie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022), which coupled with our modeling innovations (§ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) achieves state-of-the-art performance on ZS-XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Conclusion We tackle the task of zero-shot extreme classification (XC) which involves very large label spaces, by using 1) Hybrid- Match, which incorporates both semantic similarity at the SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification sentence level and relaxed lexical similarity at the token level, 2) contrastive learning to make training efficient, and 3) semantically rich class descriptions to gain a better un- derstanding of the label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We achieve state-of-the-art results on three standard XC benchmarks and significantly outperform prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Our various ablation studies and qualitative analyses demonstrate the relative importance of our modeling choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Future work can further improve description quality, and given the strong performance of Hybrid-Match, can experiment with better architectures to further push the boundaries of this practical task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Acknowledgements This work was supported by a grant from the Chadha Center for Global India at Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We thank Jens Tuyls, Khanh Nguyen and other members of the Princeton- NLP group for comments on the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' References Agrawal, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Dhillon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', and Xing, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Ppdsparse: A parallel primal-dual sparse method for extreme classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' You, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Dai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Mamitsuka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', and Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Attentionxml: Label tree-based attention-aware deep model for high-performance extreme multi-label text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Chang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', and Dhillon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Fast multi-resolution transformer fine-tuning for extreme multi-label text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:7267–7280, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Shen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Xie, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Hao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', and Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Metadata-induced con- trastive learning for zero-shot multi-label text classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In Proceedings of the ACM Web Conference 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' 3162–3173, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Training Details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Hyperparameter Tuning We tune the learning rate, batch_size using grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For the EURLex dataset, we use the standard validation split for choosing the best parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We set the input and output encoder’s learning rate at 5e−5 and 1e−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use the same learning rate for the other two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use batch_size of 16 on EURLex and 32 on AmazonCat and Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For Eurlex, we train our zero-shot model for fixed 2 epochs and the generalized zero-shot model for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For the other 2 datasets, we train for a fixed 1 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For baselines, we use the default settings as used in respective papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Training All of our models are trained end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use the pre- trained BERT model (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019a) for encoding input documents, and Bert-Small model (Turc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2019) for encoding output descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For efficiency in train- ing, we freeze the first two layers of the output encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use contrastive learning to train our models and sample hard negatives based on TF-IDF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' All implemen- tation was done in PyTorch and Huggingface transformer and experiments were run NVIDIA RTX2080 and NVIDIA RTX3090 gpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Baselines We use the code provided by ZestXML, MACLR and GROOV for running the supervised baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We employ the exact implementation of TF-IDF as used in ZestXML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We evaluate T5 as an NLI task (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We sepa- rately pass the names of each of the top 100 labels predicted by TF-IDF, and rank labels based on the likelihood of entail- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We evaluate Sentence-Transformer by comparing the similarity between the emeddings of input document and the names of the top 100 labels predicted by TF-IDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Splade is a sparse neural retreival model that learns label/document sparse expansion via a Bert masked language modelling head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use the code provided by authors for running the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We experiment with various variations and pretrained models, and find splade_max_CoCodenser pre- trained model with low sparsity(λd = 1e−6 & λq = 1e−6) to be performing the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Label descriptions from the web B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Automatically scraping label descriptions from the web We mine label descriptions from web in an automated end-to-end pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We make query of the form ‘what is ’(or component name in case of Wikipedia) on duckduckgo search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Region is set to United States(English), and advertisements are turned off, with safe search set to moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We set time range from 1990 uptil June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' On average top 50 descriptions are scraped for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' To further improve the scraped descriptions, we apply a series of heuristics: We remove any incomplete sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Incomplete sen- tences do not end in a period or do not have more than one noun, verb or auxiliary verb in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Adhesives ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Sentence = What is the best glue or gel for applying Statements with lot of punctuation such as semi-colon were found to be non-informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Descriptions with more than 10 non-period punctuations were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Plant Cages & Supports ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Description = Plant Cages & Supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' My Account;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Register;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Login;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Wish List (0) Shopping Cart;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Checkout $ USD $ AUD THB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' R$ BRL $ CAD $ CLP $ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We used regex search to identify urls and currencies in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Most of such descriptions were spam and were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Accordion Accessories ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Description = Buy Accordion Acces- sories Online, with Buy Now & Pay Later and Rental Options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Free Shipping on most orders over $250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Start Playing Accordion Accessories Today!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Descriptions with small sentences(<5 words) were re- moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Boats ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Description = Boats for Sale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Buy A Boat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Sell A Boat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Boat Buyers Guide;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Boat Insurance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Boat Financing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Descriptions with more than 2 interrogative sentences were filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Shower Curtains ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Descrip- tion = So you’re interested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' you’re starting a company that makes shower curtains?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' or are you just fooling around?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Wiki User 2010-04 We mined top frequent n-grams from a sample of scraped descriptions, and based on it identified n-grams which were commonly used in advertisements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Exam- ples include: ‘find great deals’, ‘shipped by’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Label = Boat compasses ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Description = Shop and read reviews about Compasses at West Ma- rine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Get free shipping on all orders to any West Marine Store near you today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We further remove obscene words from the datasets using an open-source library (Friedland, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification We also run a spam detection model (Grandury, 2021) on the descriptions and remove those with a confidence threshold above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Phones ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Description = Check out the Phones page at — the world’s leading music technology and instrument retailer!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Additionally, most of the sentences in first person, were found to be advertisements, and undetected by previ- ous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We remove descriptions with more than 3 first person words (such as I, me, mine) were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Eg: Label = Alarm Clocks ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Removed Description = We selected the best alarm clocks by taking the neces- sary, well, time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We tested products with our families, waded our way through expert and real-world user opinions, and determined what models lived up to man- ufacturers’ claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Post-Processing We further add hierarchy information in a natural language format to the label descriptions for AmazonCat and EURLex datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Precisely, we follow the format of ‘key is value.’ with each key, value pair represented in new line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Here key belongs to the set { ‘Description’, ‘Label’, ‘Alternate Label Names’, ‘Parents’, ‘Children’ }, and the value corresponds to comma separated list of corresponding information from the hierarchy or scraped web description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For example, consider the label ‘video surveillance’ from EURLex dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We pass the text: ‘Label is video surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Description is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Parents are video communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Alternate Label Names are camera surveillance, security camera surveillance.’ to the output encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For Wikipedia, label hierarchy is not present, so we only pass the description along with the name of label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Wikipedia Descriptions When labels are fine-grained, as in the Wikipedia dataset, making queries for the full label name is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For example, consider the label ‘Fencers at the 1984 Summer Olympics’ from Wikipedia categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' querying for it would link to the same category on Wikipedia itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Instead, we break the label names into separate constituents using a dependency parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Then for each constituent(‘Fencers’ and ‘Summer Olympics’), we scrape descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' No descriptions are scraped for constituents labelled by Named-Entity Recognition(‘1984’), and their NER tag is directly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Finally, all the scraped descriptions are concatenated in a proper format and passed to the output encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' De-Duplication To ensure no overlap between our descriptions and input documents, we used SuffixArray-based exact match algo- rithm (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022) with a minimum threshold of 60 characters and removed the matched descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Hybrid-Match We propose Hybrid-Match to exploit token similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We create clusters of tokens based on: 1) the BERT token- embedding similarity (Rajaee & Pilehvar, 2021) is higher than a threshold or 2) if tokens share the same lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Specifically, first tokens with BERT embedding cosine sim- ilarity greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 are put into same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In the second stage if two different tokens share the same lemma, but are in different clusters, their clusters are merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In model, a mask is created of size (Q * LQ * D * LD), where Q is the number of label descriptions, LQ is the max length of all label descriptions, D is the number of documents, and LD is the length of label descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Here a entry of 1 means that corresponding token in label description and input share the same cluster, else it is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Contrastive Learning During training, for both EURLex and AmazonCat, we randomly sample 1000 − |Y + i | negative labels for each input document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For Wikipedia, we precompute the top 1000 labels for each input based on TF-IDF scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We then randomly sample 1000 − |Y + i | negative labels for each document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' At inference time, we evaluate our models on all labels for both EURLex and AmazonCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' However, even evaluation on millions of labels in Wikipedia is not computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Therefore, we evaluate only on top 1000 labels predicted by TF-IDF for each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Full results for zero-shot classification E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Split Creation For EURLex, and AmazonCat, we follow the same pro- cedure as detailed in GROOV (Simig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We randomly sample k labels from all the labels present in train set, and consider the remaining labels as unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For EURLex we have roughly 25%(1057 labels) and for Ama- zonCat roughly 50%(6500 labels) as unseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For Wikipedia, we use the standard splits as proposed in ZestXML (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Results Table 6 contains complete results for ZS-XCacross the three datasets, including additional baselines and metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Method Precision - ZSL Recall - ZSL / GZSL Precision - GZSL @1 @3 @5 R@10 R@10 @1 @3 @5 Eurlex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K TF-IDF 44.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) ZestXML + TF-IDF 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) MACLR 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) GROOV 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) SemSup-XC 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) SemSup-XC + TF-IDF 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) Table 6: Comparison of SEMSUP-XC with other supervised and unsupervised baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Our method consistently outperfoms all methods across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Model Device Throughput Storage P@1 (Inputs/s) (GB) (ZSL) SemSup-XC 1 GPU 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5 MACLR 1 GPU 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 GROOV 1 GPU 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0 ZestXML 16 CPUs 2371 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8 Table 7: Computational Efficiency of SemSup-XC and base- lines on Wikipedia dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We have comparable throughput to dense baselines while requiring higher storage but with substantial performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Full results for few-shot classification F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Split Creation We iteratively select k instances of each label in train docu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' If a label has more than k documents associated with it, we drop the label from training(such labels are not sam- pled as either positives or negatives) for the extra documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We refer to these labels as neutral labels for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Because of such labels, loss functions of dense methods need to be modified accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For ZestXML, this is not possible because it directly learns a transformation over the whole dataset, and individual labels for particular instances cannot be masked as neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Models We use MACLR, GROOV, Light XML as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We initialize the weights from the corresponding pre-trained models in the GZSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We use the default hyperpa- rameters for baselines and SEMSUP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' As discussed in the previous section, neutral labels are not provided at train time for MACLR and GROOV baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' However, since Light XML uses a final fully-connected classification layer, SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification we cannot selectively remove them for a particular input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Therefore, we mask the loss for labels which are neutral to the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' We additionally include scores for TF-IDF, but since it is a fully unsupervised method, only zero-shot numbers are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Results The full results for few-shot classification are present in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Computational Efficiency Extreme Classification necessitates that the models scale well in terms of time and memory efficiency with labels at both train and test times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC uses contrastive learn- ing for efficiency at train time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' During inference, SemSup- XC predicts on top 1000 shortlists by TF-IDF, thereby achieving sub-linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Further, contextualized tokens for label descriptions are computed only once and stored in memory-mapped files, thus decreasing computational time significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Overall, our computational complexity can be represented by O(TIE ∗ N + TOE ∗ |Y |+k ∗ N ∗ Tlex), where TIE, TOE represent the time taken by input encoder and output encoder respectively, N is the total number of input documents, |Y| is the number of all labels, k indicates the shortlist size and |Tlex| denotes the time in soft-lexical computation between contextualized tokens of documents and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In our experiments, TIE ∗ N >> TOE ∗ |Y | and TIE ≈ Tlex ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Thus effectively, computational com- plexity is approximately equal to O(TIE ∗ N), which is in comparison to other SOTA extreme classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' To ensure efficiency at inference time, similar to training, SemSup-XC predicts on top of 1000 labels shortlisted by TF-IDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Table 7 shows that SemSup-XC’s throughput is comparable to deep baselines (MACLR and GROOV) while demonstrating much better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' While ZestXML is significantly faster, SemSup-XC’s P@1 is 2× higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' While SemSup-XC’s storage is higher, 17 GB of space on modern- day hard drives is trivial, especially given that the dataset has over a million labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC’s Hybrid-Match module requires contextualized representations for every token in the description, which contributes to the majority of the storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' This shows that SemSup-XC provides the best throughput-performance trade-off while having practical storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Qualitative Analysis Table 9 shows multiple qualitative examples for which our model outperforms the next baseline MACLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The exam- ples were chosen so as to increase diversity of input doc- ument’s topic, number of correct predictions and relative improvement over baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' All examples are on Amazon- Cat dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Method EURLex-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3K AmazonCat-13K P@1 P@5 R@10 P@1 P@5 R@10 1-shot SemSup-XC 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) MACLR 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8) MACLR with Descriptions 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) GROOV 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9) GROOV with Descriptions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) Light XML 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 5-shot SemSup-XC 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8) MACLR 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) MACLR with Descriptions 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) GROOV 43.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) GROOV 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) GROOV with Descriptions 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) Light XML 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7) 20-shot SemSup-XC 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) MACLR 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='9(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) MACLR with Descriptions 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) GROOV 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3(±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='7(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) GROOV with Descriptions 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4) Light XML 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='5) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='4(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='0) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='1(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='2) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='8(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='6(±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='3) Table 8: Detailed table for few-shot results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC outperforms all other baselines with significant margins for k = 1, 5, &10 shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' For 20-shot we perform almost at par with fully supervised method of Light XML, which otherwise performs poorly for zero-shot and lower values of k in few shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' SemSup-XC: Semantic Supervision for Zero and Few-shot Extreme Classification Input Document Top 5 Predictions SEMSUP-XC MACLR Start-Up: A Technician’s Guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' In addition to being an excellent stand-alone self-instructional guide, ISA recommends this book to prepare for the Start-Up Domain of CCST Level I, II, and III examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' test preparation vocational tests schools & teaching graduate preparation new test prep & study guides used and rental textbooks testing software vocational Homecoming (High Risk Books).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' When Katey Bruscke’s bus arrives in her unnamed hometown, she finds the scenery blurred, "as if my hometown were itself surfacing from beneath a black ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='" At the conclusion of new novelist Gussoff’s "day-in-the-life-of" first-person narrative, the reader feels equally blurred by the relentless .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' literature & fiction friendship thriller & suspense mothers & children thrillers drugs genre fiction coming of age general braille Rolls RM65 MixMax 6x4 Mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' The new RM65b HexMix is a single rack space unit featuring 6 channels of audio mixing, each with an XLR Microphone Input and 1/4ünbalanced Line Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' A unique feature of the 1/4¨line inputs is they may be internally reconfigured to operate as Inserts for the Microphone Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFIT4oBgHgl3EQfmSv9/content/2301.11309v1.pdf'} +page_content=' Each channel, in .' metadata={'source': 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b/edFAT4oBgHgl3EQf7R4U/content/tmp_files/2301.08743v1.pdf.txt @@ -0,0 +1,861 @@ +Cosmological implications of an interacting dark energy model with the matter fields +Keshav Ram Mishra,∗ Shibesh Kumar Jas Pacif +,† Rajesh Kumar,‡ and Kazuharu Bamba +§ +(Dated: January 24, 2023) +In this paper, we have studied an interacting dark energy model. We have assumed the gravita- +tional interaction between the matter fields i.e. bewteen barotropic fluid and the dark energy. The +dark energy evolution within the framework of spatially homogeneous and isotropic Friedmann- +Robertson-Walker space-time. Therefore, we examine the cosmic evolution from the perspective of +interacting scenario by selecting a suitable ansatz for the scale factor resulting from a parametrization +of Hubble parameter. The evolution of the cosmological parameters are discussed in some details +in the considered interacting scenario by calculating parameters and quantities such as deceleration +parameter, energy density, pressure, equation of state (EoS) etc. Also, we have performed some cos- +mological tests and analysis in support of our obtained interacting model. Finally, we reconstruct the +potential of the scalar field and refute the refined swampland conjecture using the equation of state +of dark energy and the relationship between energy density and pressure with the scalar field and +potential, and then thoroughly describe the findings. +I. +INTRODUCTION +One of the major issues in theoretical physics and more generally, in cosmology, over the past decades is to deter- +mine the enigmatic nature of the two dominant components in the universe namely, dark energy and dark matter. +This greatest open challenge in cosmology nowadays is the physical reason underpinning the late-time cosmic +speed-up. The physical mechanism has been revealed by explicate the various statistical observational data sets [1], +[2], [3], [4], [5], [6], [7], [8], [9], [10]. Many models of dark energy postulate the existence of an extra, unknown field +that is responsible for the rapid expansion of the Universe named dark energy, but some viable hypotheses include +infra-red modification to the theory of general relativity (for some reviews, see [11], [12], [13]). Dark energy accounts +for around 68% of the overall energy density of the Universe, and hence consequently controls the evolution of the +current cosmos. +The cosmological constant Λ is the most straightforward dark energy possibility, with equation of state (EoS) +parameter ωΛ = pΛ +ρΛ = −1. The ΛCDM model sides well with the current cosmological observations [14], and its +parameters have been well-established with impressive precision using latest observational data. On the other hand, +ΛCDM model has always been beset by obvious theoretical objections such as fine-tuning and cosmic-coincidence +issues [15], [16]. As a result, various expansions to the underlying ΛCDM cosmology have received a lot of atten- +tion, including the possibility of vacuum energy interacting with ΛCDM [17], [18], [19], [20], [21], [22], [23], [24], +[25], [26]. Due to the significant difference between the theoretical and observational predictions on the value of Λ +and to alleviate the tension between the ΛCDM and the observational data [27] advised to modify the gravity and +move toward the alternating theories of gravity with an enhancement in the goodness of fit [28], [29], [30]. As a +replacement for ΛCDM model, the rolling scalar field has been intensively introduced into the literature [31], [32], +[33]. Resultantly, the significance of seeing the scalar field as a DE candidate in understanding the unification of +cosmic acceleration (early and late time) has earned high acceptance [11]. +In addition to the shortcomings of the standard model of cosmology, certain observational results are still +puzzling. For instance, in some studies, ΛCDM model has been modified by connecting a dynamical dark energy +model with H2.34 data. Work in [34], [35] reveals how to evaluate a range of models that allow for the evolution +of dark energy models based on BAO, CMB, and SN data, as well as how to match dynamical DE models with +∗ krmgkp1995@gmail.com +† shibesh.math@gmail.com +‡ rkmath09@gmail.com +§ bamba@sss.fukushima-u.ac.jp +arXiv:2301.08743v1 [gr-qc] 19 Jan 2023 + +2 +H2.34 data. In some observations of BOSS data [36], the observed value of Hubble parameter H at redshift z = 2.34 +is 222 ± 7kms−1Mpc−1 which falls below the ΛCDM model’s prediction. [37], [38] have some additional similar +work. The goal of this study is to create a cosmological model in a manner that incorporates new phenomenological +types of interaction between dark energy and dark matter. We use the well-known technique of parametrization +that explains the cosmic acceleration nicely (since we are interested in the subject of accelerated expansion of the big +scale cosmos in this work) and also alleviates the tension between the standard ΛCDM model with the observational +datasets [39], [40]. In this sense, some theories come up with DM and DE interaction, disclosing new properties of +both components. +Due to a lack of understanding of the nature of the dark sector, numerous ways have been developed to +divulge the physical features of DM and DE. Searching for models that could provide an alternate means of inves- +tigating the nature of the dark sector and, ideally, allowing for differentiation between the many theoretical models +is still on. In this sense, some theories propose that dark matter and dark energy interact, revealing new properties +of both (for a recent review see [41]). Indeed, numerous interactions have been proposed, including non-minimally +coupled theories [42, 43], where the complete Lagrangian contains a specific interaction term. +Dark matter and dark energy interaction is a potential mechanism that states that in particle physics or more +theoretical ground, any two matter fields can interact with each other. This particular phenomenological theory has +piqued the cosmology community’s curiosity due to various conceivable outcomes. Interacting models of DM and +DE are an equivalent description of the dark sector of the Universe that has been extensively researched and are +motivated by a viable explanation to the so called coincidence and cosmological constant concerns as in the interac- +tion model dark energy decay into dark matter [44], [45], [46], [47], [48]. Some studies make fair assumptions about +a DM-DE relationship to help ease or overcome the conundrum of the coincidence problem. Because the nature of +DE and DM is unknown to us, and interaction is allowed in field theory [49]. The energy densities of DE and DM +are allied in this way, and both become dynamical, allowing them to be comparable and reducing the coincidence. +Numerous work [50], [49], [51], [52], [53] have already been presented in this regard that highlight the relation +between DE and DM interaction. For a more comprehensive list of references on interaction evidence discovered so +far and the discussion of theoretical and cosmological features, see [54]. It has been identified that allowing for in- +teraction can shift the dark energy EoS from quintessence to phantom, implying that the dark energy EoS parameter +is imposed with an effective quintom type of nature. In phantom scalar field models, phantom crossing line causes +instabilities with negative kinetic correction term. In addition to this, the interaction theory has been found to be +particularly effective in addressing the Hubble constant H0 gap between global and local measurements. +Since there is no such overarching rule for recruiting interaction functions, therefore, a variety of linear and +non-linear functions are presented in the literature and one can select some new functions apparently. Generally, +it is tedious to discuss the dynamics of the interacting model using a non-linear interaction function, hence these +models are quite uncommon in literature [46], [55], [56]. Nonetheless, non-linear interacting models are always +fascinating to explore the dynamics of the Universe to see if we can glean any further information from it. As a +result of the motivation by the interacting scenarios, we examine an interacting model in this paper by adopting a +model-independent approach. +The work of this paper has been organized as follows: Sect. I provides a brief introduction to general relativity +and current status of the some hot cosmological problems addressed. In Sect. II, we consider the spacetime metric +and formulated the Einstein. In Sect. III, solution of the field equations are obtained using a parametrization scheme. +In Sect. IV, we have discussed the energy conditions for our model. We have discussed the stability of the model +through velocity of sound in Sect. V. In Sect. VI, we have discussed the swampland conjecture and finally we have +concluded our results in Sect. VII. + +3 +II. +FIELD EQUATIONS IN AN INTERACTING SCENARIO +We start our analysis will go over the metric given in the form of homogeneous, isotropic, and spherically sym- +metric spatially Robertson-Walker geometry of the form +ds2 = −dt2 + a(t)2 +� +dr2 +1 − κr2 + r2(dθ2 + sin2 θdφ2) +� +. +(1) +Here, a(t) denote the scale factor of the Universe: a function of cosmic time t and c = 1. The curvature constant, +denoted by the term κ, assumes the values 0, −1, 1 showing flat, open, or closed geometry respectively. +Let us introduce the Einstein field equations (EFEs) for the Friedmann Robertson-Walker metric in general theory +of relativity as +Rµν − 1 +2 Rgµν = Tµν, +(2) +where LHS of the above equation denotes the geometry of the Universe and RHS represents the energy momen- +tum tensors of the various components in the Universe i.e. radiation, baryons, dark matter, and dark energy. The +independent field equations will be, +3M2 +pl +� +H2 + ka−2� += ρr + ρb + ρm + ρd = ρtotal, +(3) +− M2 +pl +� +2 ¨a +a + H2 + ka−2 +� += pr + pb + pm + pd = ptotal. +(4) +Here, an overhead dot represents the time derivative and H = ˙a +a is the Hubble parameter. ρr, ρb, ρm, ρd, pr, pb, pm, +and pd denote the energy densities and corresponding pressures of various components of the Universe. According +to the Bianchi identity, G;j +ij = 0 leads to T;j +ij = 0, which gives us the following continuity equation. +˙ρtotal + 3H(ρtotal + ptotal) = 0. +(5) +As a result of the discussion of motivation in the introduction, let us consider the interacting scenario in this article. +We consider the gravitational interaction is between two major dominating components i.e. dark matter and dark +energy, which yields the equations: +˙ρr + 3H(ρr + pr) = 0, +(6) +˙ρb + 3H(ρb + pb) = 0, +(7) +˙ρm + 3H(ρm + pm) = Q(t), +(8) +and +˙ρd + 3H(ρd + pd) = −Q(t), +(9) +where Q(t) represents the energy exchange rate between the two dark sectors in these equations, with Q(t) > 0 +implying a transfer of energy from the dark matter to the dark energy sector, and Q(t) < 0 implying the opposite. +The energy transfer rate between dark and light sector is determined by the interaction term Q. However, the exact +nature of it is still unknown to us. To investigate the topic of dark sector interaction, certain probable forms of Q +should be assumed. In this paper, we look at the the form +Q = 3γHρm, +(10) + +4 +where γ is a coupling constant. The interaction term is exactly proportional to Hubble parameter H to make the +preceding equations (8), (9) to hold the continuity law, since it is needed that the interaction term be proportional to +the inverse unit of time. Therefore, We have chosen the form of Q = 3γHρm to reflect this feature. +Let us define the equation of state parameter ω for various components of the Universe as ωi = pi +ρi . Then, from the +field equations (6), (7), (8) and (9) together (10) yield the solution, +ρr = C1a−4, +(11) +ρb = C2a−3, +(12) +ρm = C3a3γ−3, +(13) +and +˙ρde + 3H (1 + ωde) ρde = −3C3γHa3γ−3, +(14) +where C1, C2 and C3 are integrating constants. Let us now define the redshift z = a0 +a − 1. We use the standard lore +and normalize scale factor a0 = 1, henceforth. Equations (11), (12) and (13) can now be written as, +ρr = C1 (1 + z)4 , ρb = C2 (1 + z)3 , ρm = C3 (1 + z)3−3γ , +(15) +and the expressions of ρde and pde can be derived in terms of geometrical parameters as, +pd = M2 +pl +�� +2q − 1 +� +H2 − k (1 + z)2� +− 1 +3C1 (1 + z)4 , +(16) +ρd = 3M2 +pl +� +H2 + k (1 + z)2� +− +� +C1 (1 + z)4 + C2 (1 + z)3 + C3 (1 + z)3−3γ� +. +(17) +Let us now define the density parameter (Ω) as Ωi = ρi +ρc , where ρc = 3M2 +plH2 stands for the critical density and +the suffix i = r, b, m (radiation, baryon and dark matter). Then, we have +Ωr = Ωr0 (1 + z)4 +� H0 +H +�2 +, Ωb = Ωb0 (1 + z)3 +� H0 +H +�2 +, Ωm = Ωm0 (1 + z)3−3γ +� H0 +H +�2 +. +(18) +Here, the suffix 0 stands for the values of the cosmological parameters at present time (t = t0 or z = 0). The +friedmann equation (3) can be expressed in terms of density parameter as +Ωd = (1 + Ωk) − +� H0 +H +�2 � +Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� +, +(19) +with the understanding of k(1+z)2 +H2 += +k +a2H2 = Ωk. Finally, we have +pd = M2 +pl +�� +2q − 1 +� +H2 − k (1 + z)2� +− H2 +0Ωr0 (1 + z)4 , +(20) +ρde = 3M2 +pl +� +H2 + k (1 + z)2� +− H2 +0 +� +Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� +, +(21) +and + +5 +ωd = 1 +3 +� +2q − 1 +� +H2 − k (1 + z)2 − H2 +0Ωr0 (1 + z)4 +H2 + k (1 + z)2 − H2 +0 +� +Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� +(22) +It is now evident from the calculations above that we have a fully deterministic solution of the field equations for +a known form of the scale factor, Hubble parameter, or deceleration parameter, and we can explain the features of +the model. +The model independent manner analysis of the dark energy model is one of the easiest techniques among the +few explored in the literature to find deterministic solutions. A functional form of any geometrical or physical +parameter known as cosmological parametrization is a model independent way serves as a natural way to discuss +the cosmological dynamics of the model. Following the same, we take care of this scenario in our present study by +adopting an appropriate parametrization of the Hubble parameter H used by Singh [57], Banerjee et al. [58], Nagpal +et al. [59], Pacif et al. [60], and Mandal et al. [61]: +H(a) = α(1 + a−n), +(23) +where α > 0 and n > 1 are constants (call them as model parameters), which is to be constrained through some +observational datasets. In the next section, we shall discuss the dynamics of the model with the considered cosmo- +logical parametrization. +III. +THE MODEL +The form of Hubble parameter considered in equation (23) provides a smooth dynamics of expansion of the Uni- +verse from a decelerating phase to an accelerating one [59]. Equation (23) readily yields, the explicit form of scale +factor as, +a(t) = (enαt − 1) +1 +n + c. +(24) +Using the initial big bang condition (at t = 0, a = 0), which make the constant of integration c, zero. We can +establish the t − z relationship as t(z) = +1 +nα log +� +1 + (1 + z)−n� +and we can write, +H(z) = α(1 + (1 + z)n), +(25) +or +H(z) = H0 +2 (1 + (1 + z)n). +(26) +The constant α is of the order of H0 and α = H0 +2 . The deceleration parameter q(z) comes out to be +q(z) = (n − 1) (1 + z)n − 1 +1 + (1 + z)n +(27) +The detailed dynamical behaviour of these geomertical parameters are discussed in Nagpal et al. [59], Pacif et +al. [60], and Mandal et al. [61] in different scenarios in classical and in a modified theory. Here, in this paper, we +are trying to discuss the physical dynamics of an interacting model of dark energy with this parametrization. We +can observe from equations (26) and (27) that the expressions contain only one model parameter n. A suitable value +of the n would provide the evolution of different cosmological parameters in our model, which can be obtained by +constraining it with any cosmological datasets. In the references [59] and [61], the authors have found the constrained +values of n = 1.43, to which we are going to use here for our further analysis. Together with equations (26) and (27), +we obtain the physical parameters as follows: +Ωr = Ωr0 +4(1 + z)4 +(1 + (1 + z)n)2 , +(28) + +6 +0 +2 +4 +6 +8 +10 +0 +50 +100 +150 +z +ρm/3Mpl2H02 +0 +2 +4 +6 +8 +10 +0 +10 +20 +30 +40 +50 +60 +z +ρb/3Mpl +2H0 +2 +0 +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +z +ρr/3Mpl2H02 +(a) +(b) +(c) +FIG. 1. The plots of energy densities ρm, ρb and ρr for the model. The plots clearly indicate the faster decrease of the radiation than the baryonic matter and faster +than the dark matter. +Ωb = Ωb0 +4(1 + z)3 +(1 + (1 + z)n)2 , +(29) +Ωm = Ωm0 +4(1 + z)−3(γ−1) +(1 + (1 + z)n)2 , +(30) +Ωd = (1 + Ωk) − +4 +� +Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� +(1 + (1 + z)n)2 +. +(31) +Equation (31) is verified with Ωd0 = 1 + Ωk − (Ωr0 + Ωb0 + Ωm0). Equations (20), (21) and (22) can be written with +the help of equations (26) and (27) as, +pd +M2 +plH2 +0 += +� +� +�(2n − 3) (1 + z)n − 3 +� � +1 + (1 + z)n� +4 +− Ωk +� +1 + (1 + z)n�2 +4 +− Ωr0 (1 + z)4 +� +� , +(32) +ρd +3M2 +plH2 +0 += +� +� +� +1 + (1 + z)n�2 +4 ++ Ωk +� +1 + (1 + z)n�2 +4 +− +� +Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� +� +� , +(33) +and +ωd = 1 +3 +�(2n − 3) (1 + z)n − 3 +� � +1 + (1 + z)n� − Ωk +� +1 + (1 + z)n�2 − 4Ωr0 (1 + z)4 +� +1 + (1 + z)n�2 + Ωk +� +1 + (1 + z)n�2 − 4 +� +Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� +(34) +IV. +ENERGY CONDITIONS +There are several prevalent energy conditions (ECs) in general relativity that set restrictions to prevent some re- +gions having negative energy density. In other words, ECs are considered a valid generalisation of the energy +momentum tensor to the whole Universe, where energy density can never be negative. The plausibility of numerous + +7 +0 +2 +4 +6 +8 +10 +0 +200 +400 +600 +800 +z +ρde/3Mpl2H02 +0 +2 +4 +6 +8 +10 +-60 +-50 +-40 +-30 +-20 +-10 +0 +z +pde/Mpl2H02 +0 +2 +4 +6 +8 +10 +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +z +ωde +(a) +(b) +(c) +FIG. 2. The plots of ρd, pd and ωd for the model. +significant singularity problems involving black holes, and wormholes, and many others are extensively examined +using ECs. The energy conditions can mainly be defined in two ways: (i) geometrically, where ECs are well ex- +pressed in terms of Ricci tensor or Weyl tensor, and (ii) physically, where ECs are expressed as a function of the +physical world. The ECs are defined as follows. +• Weak energy condition (WEC) ⇔ ρtotal ≥ 0, ρtotal + ptotal ≥ 0, +• Null energy condition (NEC) ⇔ ρtotal + ptotal ≥ 0, +• Strong energy condition (SEC) ⇔ ρtotal + 3ptotal ≥ 0, +• Dominant energy condition (DEC) ⇔ ρtotal ≥ 0 , ρtotal ≥ |ptotal| . +Additionally, the energy density (ρ) and isotropic pressure (p) of the model can be expressed in terms of potential +energy (V(φ)) and scalar field φ respectively. Thus the point-wise ECs in GR are defined as: +• WEC ⇔ V(φ) ≥ +˙φ2 +2 , +• NEC: ∀ V(φ), +• SEC ⇔ V(φ) ≤ ˙φ2, +• DEC ⇔ V(φ) ≥ 0. +The examination of various energy conditions to determine whether weak, null, strong and dominant ECs are +satisfied in the given scenarios is another crucial topic covered here. We plot some statistics for interacting scenarios +and explore various energy conditions for our model. +V. +VELOCITY OF SOUND +As it is widely known that, the stability of linear perturbations is a significant test for the sustainability of a cos- +mological model. The need that the speed of sound (C2 +s ) be sufficiently less than 1 to prevent unintended oscillations +in the matter power spectrum imposes a strict limitation, nevertheless. We, here, plot the graph of velocity of sound +(C2 +s ) for our obtained models using suitable choice of the appropriate parameters, as illustrated in the following +figure: + +8 +0 +2 +4 +6 +8 +10 +0 +100 +200 +300 +400 +500 +z +NEC +0 +2 +4 +6 +8 +10 +-400 +-300 +-200 +-100 +0 +z +SEC +0 +2 +4 +6 +8 +10 +0 +500 +1000 +1500 +z +DEC +(a) +(b) +(c) +FIG. 3. The plots of NEC, SEC and DEC for the model. +0 +2 +4 +6 +8 +10 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +z +Cs +2 +FIG. 4. The plots of C2 +s for the model. +VI. +SWAMPLAND CONJECTURE +In the present era, there is a growing interest in linking modern cosmological models with quantum gravity theory, +which can easily distinguish between the effective field models to the nominal string theory. In order to choose field +potentials properly, Swampland criteria have just been established. Therefore, in this section, we shall shed light +on how the refined swampland programme conjecture and dark energy are related. The swampland conjecture, as +we are aware, questions a number of structures, including the physics of black holes, inflation, and other cosmolog- +ical ideas. The connection of practical cosmological models with gravity’s quantum theory, which can distinguish +between the effective field models associated to the nominal string landscape, has recently attracted considerable +interest. Swampland criteria were recently established to help with field potential selection. Excitement was specifi- +cally sparked by the fact that precise de Sitter solutions with positive cosmological constants do not match with the +string landscape, making it impossible to link high energy arrangements to fundamental theories. + +9 +The swampland criteria of the string theory bound with the quintessence model of DE were recently derived [62], +[63]. Numerous works followed the work in [62], [63] in approaching the Swampland criterion in the connection +of DE’s quintessence models, for instance [64] - [75]. The validity of the f(R) theory was recently studied by the re- +searcher in relation to the Swampland conjecture [64]. When a field theory contradicts quantum gravity, the Swamp- +land requirements constrain and regulate it [76]. Although these hypotheses are still in their infancy and address +some issues, they are rapidly evolving and changing in some situations like warm inflation [77]. We therefore choose +to make a connection between these hypotheses and dark energy. In subsequent works, we’ll endeavour to develop +this concept and make a more substantial connection between the swampland programme, dark energy, dark matter, +and in action. Researchers take into account several models for dark energy, and studied their cosmological applica- +tions, but the nature of the concept is still unclear. +Here, we investigate the cosmological restrictions put forth by two string Swampland criteria. These criteria include +a lower restriction on φV +V when V > 0 and an upper bound on the range that scalar fields can travel. The major +disadvantage of these two criteria seems that they generally in conflict with inflationary theories. We discover that +certain quintessence models can meet these restrictions at the same time when we apply the same criterion to dark +energy in the current epoch. We contend that the universe will experience a phase change within a few Hubble +times if the two Swampland requirements are true. These conditions intensify the drive for upcoming dark energy +equation of state ω measurements and equivalence principle tests for dark matter. With this motivation, we propose, +the refined swampland conjecture in the below form. +Conjecture 1. |V′| +V > C1Mpl +Conjecture 2. |V′′| +V +< −C2M2 +pl +As we have already noted that the notion to forecast the acceleration in the Universe is to fill it with an exotic +type of matter that satisfies the inequality 1 + 3ω < 0. The energy that causes the acceleration, according to the +measurements, satisfies ω ≃ −1. The development of the EoS parameter ω for our model was already covered in +earlier sections, thus to get an appropriate matter field that produces unusual behaviour and is capable of exhibiting +repulsive effects whose cause is given by dark energy (scalar field) we will consider the following equations, +ρ = 1 +2 +˙φ2 + V(φ), +(35) +p = 1 +2 +˙φ2 − V(φ). +(36) +where the terms 1 +2 ˙φ2 and V(φ) refer to the kinetic energy (KE) and potential energy (PE) of the scalar field. So the +term ω = ω(t) i.e. can no more be treated as a constant. The quintessence or phantom model is consistent with the +observations provided ω ≃ −1. Thus, we need ˙φ2 << V(φ) i.e. the KE of φ is insignificant in comparison to the PE. +In this study, we consider that φ is the only source of DE with V(φ), so one can consider energy density and pressure +of scalar field as ρφ and pφ respectively for flat FLRW space-time under Barrow’s scheme [78] using Eqs. (35) and +(36) as +ρ = 1 +2 +˙φ2 + V(φ) = ρφ, +(37) +p = 1 +2 +˙φ2 − V(φ) = pφ. +(38) +The KE and PE can be obtained by solving the Eqs. (37) and (38). Fig. 4 demonstrates the potential energy V(φ) +plots w.r.t. scalar field φ for the same considered values of model parameters as we have taken in Fig. 1, 2. From Fig. +4, we notice that the potential V(φ) is present in the interval −1 < φ < 0 and V(φ) ≃ 0 at φ ≃ 0. Therefore, we can +predict that the scalar field φ is the only source of DE with potential V(φ). Thus we conclude that our model is an +accelerating dark energy model. +To understand the nature of dark energy, we have considered its only source of energy as scalar field φ, which +plays the role of quintessence model, therefore in this regard we have analysed our model innthe connection with + +10 +-1 +0 +1 +2 +3 +0 +50 +100 +150 +200 +250 +z +|ϕ| +-1 +0 +1 +2 +3 +4 +5 +0 +20 +40 +60 +80 +z +V(ϕ) +0 +1 +2 +3 +4 +0.0 +0.5 +1.0 +1.5 +2.0 +ϕ +V(ϕ) +(a) +(b) +(c) +FIG. 5. The plots of scalar field φ, potential V(φ) w.r.t. z and scalar filed correspondence V(φ) vs φ for the model. +0 +2 +4 +6 +8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +z +C1 +0 +2 +4 +6 +8 +0.0 +0.5 +1.0 +1.5 +z +C2 +FIG. 6. The plots of swampland conjecture 1 and 2 for the model. +Swampland conjecture. Conjecture 1 of this criteria is linked with scalar field φ, whereas conjecture 2 is connected +withscalar potential V. We have composed the behavior of scalar field with respect to redshift for our interacting +model in the figure below. +The below figure illustrates the behavior of refined swampland conjecture’s components plotted for two distinct +potentials connected to an interacting scenario for various parameters of the universe. We are aware that each +of the swampland conjectures in the iterature has positive values of unit order and the component C2, is smaller +than component C1. The interacting model demonstrates satisfaction of both conjectures, as shown in figure. The +Swampland criterion, which has been proposed for a consistent theory of gravity in this regard and which satisfies +the requirement |V′| +V +> C1 ≈ O(1), demonstrates that this dark energy model accords better with the Swampland +criteria. +VII. +CONCLUSION +We have examined an interacting dark energy cosmological model in the classical theory of gravity. The interact- +ing scenario discussed here provides an insight into the evolution of cosmological parameters. In order to find a + +11 +consistent solution, we have considered a parametrization of Hubble parameter, which provide a smooth transition +from early deceleration to present aceleration and discuss the late evolution of the Universe. In order to discuss the +Universe’s evolution in particular the late-time behavior, all the concerned cosmological parameters were written in +terms of redsfift z after establishing the (t − z) relationship. All the cosmological parameters concerned with one +model parameter n only that could be constrained from any cosmological datasets but we have used the constrained +value of n = 1.43 as found in our earlier works [59], [61] for our analysis here. The evolutionary profiles of energy +densities of radiation, baryonic & dark matter, ark energy, pressure and equation of state parameter are shown in +figures 1 and 2. We can observe the faster decrease of the radiation energy than the baryonic matter, which is faster +than the dark matter. Panel 2(b) highlights the profile of pressure of dark energy which is more negative in the past, +increases in a concave upward way and finally tends to zero in the future. The negative pressure of dark energy +in due course of evolution favours the standard lore. Figure 2(c) shows the redshift evolution of equation of state +parameter, which is approaching to −1 in the far future assuming the negative values in the expected ranges ar- +round z = 0. 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B 310, 743 (1988). + diff --git a/edFAT4oBgHgl3EQf7R4U/content/tmp_files/load_file.txt b/edFAT4oBgHgl3EQf7R4U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7a9cc4a5b5cfbfac4ad8705d24d679a5947c249 --- /dev/null +++ b/edFAT4oBgHgl3EQf7R4U/content/tmp_files/load_file.txt @@ -0,0 +1,856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf,len=855 +page_content='Cosmological implications of an interacting dark energy model with the matter fields Keshav Ram Mishra,∗ Shibesh Kumar Jas Pacif ,† Rajesh Kumar,‡ and Kazuharu Bamba § (Dated: January 24, 2023) In this paper, we have studied an interacting dark energy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We have assumed the gravita- tional interaction between the matter fields i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' bewteen barotropic fluid and the dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The dark energy evolution within the framework of spatially homogeneous and isotropic Friedmann- Robertson-Walker space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Therefore, we examine the cosmic evolution from the perspective of interacting scenario by selecting a suitable ansatz for the scale factor resulting from a parametrization of Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The evolution of the cosmological parameters are discussed in some details in the considered interacting scenario by calculating parameters and quantities such as deceleration parameter, energy density, pressure, equation of state (EoS) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Also, we have performed some cos- mological tests and analysis in support of our obtained interacting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Finally, we reconstruct the potential of the scalar field and refute the refined swampland conjecture using the equation of state of dark energy and the relationship between energy density and pressure with the scalar field and potential, and then thoroughly describe the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' INTRODUCTION One of the major issues in theoretical physics and more generally, in cosmology, over the past decades is to deter- mine the enigmatic nature of the two dominant components in the universe namely, dark energy and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' This greatest open challenge in cosmology nowadays is the physical reason underpinning the late-time cosmic speed-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The physical mechanism has been revealed by explicate the various statistical observational data sets [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Many models of dark energy postulate the existence of an extra, unknown field that is responsible for the rapid expansion of the Universe named dark energy, but some viable hypotheses include infra-red modification to the theory of general relativity (for some reviews, see [11], [12], [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Dark energy accounts for around 68% of the overall energy density of the Universe, and hence consequently controls the evolution of the current cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The cosmological constant Λ is the most straightforward dark energy possibility, with equation of state (EoS) parameter ωΛ = pΛ ρΛ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The ΛCDM model sides well with the current cosmological observations [14], and its parameters have been well-established with impressive precision using latest observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' On the other hand, ΛCDM model has always been beset by obvious theoretical objections such as fine-tuning and cosmic-coincidence issues [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' As a result, various expansions to the underlying ΛCDM cosmology have received a lot of atten- tion, including the possibility of vacuum energy interacting with ΛCDM [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Due to the significant difference between the theoretical and observational predictions on the value of Λ and to alleviate the tension between the ΛCDM and the observational data [27] advised to modify the gravity and move toward the alternating theories of gravity with an enhancement in the goodness of fit [28], [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' As a replacement for ΛCDM model, the rolling scalar field has been intensively introduced into the literature [31], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Resultantly, the significance of seeing the scalar field as a DE candidate in understanding the unification of cosmic acceleration (early and late time) has earned high acceptance [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In addition to the shortcomings of the standard model of cosmology, certain observational results are still puzzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' For instance, in some studies, ΛCDM model has been modified by connecting a dynamical dark energy model with H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='34 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Work in [34], [35] reveals how to evaluate a range of models that allow for the evolution of dark energy models based on BAO, CMB, and SN data, as well as how to match dynamical DE models with ∗ krmgkp1995@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='com † shibesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='com ‡ rkmath09@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='com § bamba@sss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='fukushima-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='jp arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='08743v1 [gr-qc] 19 Jan 2023 2 H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='34 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In some observations of BOSS data [36], the observed value of Hubble parameter H at redshift z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='34 is 222 ± 7kms−1Mpc−1 which falls below the ΛCDM model’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [37], [38] have some additional similar work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The goal of this study is to create a cosmological model in a manner that incorporates new phenomenological types of interaction between dark energy and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We use the well-known technique of parametrization that explains the cosmic acceleration nicely (since we are interested in the subject of accelerated expansion of the big scale cosmos in this work) and also alleviates the tension between the standard ΛCDM model with the observational datasets [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In this sense, some theories come up with DM and DE interaction, disclosing new properties of both components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Due to a lack of understanding of the nature of the dark sector, numerous ways have been developed to divulge the physical features of DM and DE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Searching for models that could provide an alternate means of inves- tigating the nature of the dark sector and, ideally, allowing for differentiation between the many theoretical models is still on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In this sense, some theories propose that dark matter and dark energy interact, revealing new properties of both (for a recent review see [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Indeed, numerous interactions have been proposed, including non-minimally coupled theories [42, 43], where the complete Lagrangian contains a specific interaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Dark matter and dark energy interaction is a potential mechanism that states that in particle physics or more theoretical ground, any two matter fields can interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' This particular phenomenological theory has piqued the cosmology community’s curiosity due to various conceivable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Interacting models of DM and DE are an equivalent description of the dark sector of the Universe that has been extensively researched and are motivated by a viable explanation to the so called coincidence and cosmological constant concerns as in the interac- tion model dark energy decay into dark matter [44], [45], [46], [47], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Some studies make fair assumptions about a DM-DE relationship to help ease or overcome the conundrum of the coincidence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Because the nature of DE and DM is unknown to us, and interaction is allowed in field theory [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The energy densities of DE and DM are allied in this way, and both become dynamical, allowing them to be comparable and reducing the coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Numerous work [50], [49], [51], [52], [53] have already been presented in this regard that highlight the relation between DE and DM interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' For a more comprehensive list of references on interaction evidence discovered so far and the discussion of theoretical and cosmological features, see [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' It has been identified that allowing for in- teraction can shift the dark energy EoS from quintessence to phantom, implying that the dark energy EoS parameter is imposed with an effective quintom type of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In phantom scalar field models, phantom crossing line causes instabilities with negative kinetic correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In addition to this, the interaction theory has been found to be particularly effective in addressing the Hubble constant H0 gap between global and local measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Since there is no such overarching rule for recruiting interaction functions, therefore, a variety of linear and non-linear functions are presented in the literature and one can select some new functions apparently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Generally, it is tedious to discuss the dynamics of the interacting model using a non-linear interaction function, hence these models are quite uncommon in literature [46], [55], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Nonetheless, non-linear interacting models are always fascinating to explore the dynamics of the Universe to see if we can glean any further information from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' As a result of the motivation by the interacting scenarios, we examine an interacting model in this paper by adopting a model-independent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The work of this paper has been organized as follows: Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' I provides a brief introduction to general relativity and current status of the some hot cosmological problems addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' II, we consider the spacetime metric and formulated the Einstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' III, solution of the field equations are obtained using a parametrization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' IV, we have discussed the energy conditions for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We have discussed the stability of the model through velocity of sound in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' VI, we have discussed the swampland conjecture and finally we have concluded our results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' FIELD EQUATIONS IN AN INTERACTING SCENARIO We start our analysis will go over the metric given in the form of homogeneous, isotropic, and spherically sym- metric spatially Robertson-Walker geometry of the form ds2 = −dt2 + a(t)2 � dr2 1 − κr2 + r2(dθ2 + sin2 θdφ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (1) Here, a(t) denote the scale factor of the Universe: a function of cosmic time t and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The curvature constant, denoted by the term κ, assumes the values 0, −1, 1 showing flat, open, or closed geometry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Let us introduce the Einstein field equations (EFEs) for the Friedmann Robertson-Walker metric in general theory of relativity as Rµν − 1 2 Rgµν = Tµν, (2) where LHS of the above equation denotes the geometry of the Universe and RHS represents the energy momen- tum tensors of the various components in the Universe i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' radiation, baryons, dark matter, and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The independent field equations will be, 3M2 pl � H2 + ka−2� = ρr + ρb + ρm + ρd = ρtotal, (3) − M2 pl � 2 ¨a a + H2 + ka−2 � = pr + pb + pm + pd = ptotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (4) Here, an overhead dot represents the time derivative and H = ˙a a is the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' ρr, ρb, ρm, ρd, pr, pb, pm, and pd denote the energy densities and corresponding pressures of various components of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' According to the Bianchi identity, G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='j ij = 0 leads to T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='j ij = 0, which gives us the following continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' ˙ρtotal + 3H(ρtotal + ptotal) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (5) As a result of the discussion of motivation in the introduction, let us consider the interacting scenario in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We consider the gravitational interaction is between two major dominating components i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' dark matter and dark energy, which yields the equations: ˙ρr + 3H(ρr + pr) = 0, (6) ˙ρb + 3H(ρb + pb) = 0, (7) ˙ρm + 3H(ρm + pm) = Q(t), (8) and ˙ρd + 3H(ρd + pd) = −Q(t), (9) where Q(t) represents the energy exchange rate between the two dark sectors in these equations, with Q(t) > 0 implying a transfer of energy from the dark matter to the dark energy sector, and Q(t) < 0 implying the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The energy transfer rate between dark and light sector is determined by the interaction term Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' However, the exact nature of it is still unknown to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' To investigate the topic of dark sector interaction, certain probable forms of Q should be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In this paper, we look at the the form Q = 3γHρm, (10) 4 where γ is a coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The interaction term is exactly proportional to Hubble parameter H to make the preceding equations (8), (9) to hold the continuity law, since it is needed that the interaction term be proportional to the inverse unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Therefore, We have chosen the form of Q = 3γHρm to reflect this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Let us define the equation of state parameter ω for various components of the Universe as ωi = pi ρi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Then, from the field equations (6), (7), (8) and (9) together (10) yield the solution, ρr = C1a−4, (11) ρb = C2a−3, (12) ρm = C3a3γ−3, (13) and ˙ρde + 3H (1 + ωde) ρde = −3C3γHa3γ−3, (14) where C1, C2 and C3 are integrating constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Let us now define the redshift z = a0 a − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We use the standard lore and normalize scale factor a0 = 1, henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Equations (11), (12) and (13) can now be written as, ρr = C1 (1 + z)4 , ρb = C2 (1 + z)3 , ρm = C3 (1 + z)3−3γ , (15) and the expressions of ρde and pde can be derived in terms of geometrical parameters as, pd = M2 pl �� 2q − 1 � H2 − k (1 + z)2� − 1 3C1 (1 + z)4 , (16) ρd = 3M2 pl � H2 + k (1 + z)2� − � C1 (1 + z)4 + C2 (1 + z)3 + C3 (1 + z)3−3γ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (17) Let us now define the density parameter (Ω) as Ωi = ρi ρc , where ρc = 3M2 plH2 stands for the critical density and the suffix i = r, b, m (radiation, baryon and dark matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Then, we have Ωr = Ωr0 (1 + z)4 � H0 H �2 , Ωb = Ωb0 (1 + z)3 � H0 H �2 , Ωm = Ωm0 (1 + z)3−3γ � H0 H �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (18) Here, the suffix 0 stands for the values of the cosmological parameters at present time (t = t0 or z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The friedmann equation (3) can be expressed in terms of density parameter as Ωd = (1 + Ωk) − � H0 H �2 � Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� , (19) with the understanding of k(1+z)2 H2 = k a2H2 = Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' we have pd = M2 pl �� 2q − 1 � H2 − k (1 + z)2� − H2 0Ωr0 (1 + z)4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (20) ρde = 3M2 pl � H2 + k (1 + z)2� − H2 0 � Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (21) and 5 ωd = 1 3 � 2q − 1 � H2 − k (1 + z)2 − H2 0Ωr0 (1 + z)4 H2 + k (1 + z)2 − H2 0 � Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� (22) It is now evident from the calculations above that we have a fully deterministic solution of the field equations for a known form of the scale factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Hubble parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' or deceleration parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' and we can explain the features of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The model independent manner analysis of the dark energy model is one of the easiest techniques among the few explored in the literature to find deterministic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' A functional form of any geometrical or physical parameter known as cosmological parametrization is a model independent way serves as a natural way to discuss the cosmological dynamics of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Following the same, we take care of this scenario in our present study by adopting an appropriate parametrization of the Hubble parameter H used by Singh [57], Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [58], Nagpal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [59], Pacif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [60], and Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [61]: H(a) = α(1 + a−n), (23) where α > 0 and n > 1 are constants (call them as model parameters), which is to be constrained through some observational datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In the next section, we shall discuss the dynamics of the model with the considered cosmo- logical parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' THE MODEL The form of Hubble parameter considered in equation (23) provides a smooth dynamics of expansion of the Uni- verse from a decelerating phase to an accelerating one [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Equation (23) readily yields, the explicit form of scale factor as, a(t) = (enαt − 1) 1 n + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (24) Using the initial big bang condition (at t = 0, a = 0), which make the constant of integration c, zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We can establish the t − z relationship as t(z) = 1 nα log � 1 + (1 + z)−n� and we can write, H(z) = α(1 + (1 + z)n), (25) or H(z) = H0 2 (1 + (1 + z)n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (26) The constant α is of the order of H0 and α = H0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The deceleration parameter q(z) comes out to be q(z) = (n − 1) (1 + z)n − 1 1 + (1 + z)n (27) The detailed dynamical behaviour of these geomertical parameters are discussed in Nagpal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [59], Pacif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [60], and Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [61] in different scenarios in classical and in a modified theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Here, in this paper, we are trying to discuss the physical dynamics of an interacting model of dark energy with this parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We can observe from equations (26) and (27) that the expressions contain only one model parameter n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' A suitable value of the n would provide the evolution of different cosmological parameters in our model, which can be obtained by constraining it with any cosmological datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In the references [59] and [61], the authors have found the constrained values of n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='43, to which we are going to use here for our further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Together with equations (26) and (27), we obtain the physical parameters as follows: Ωr = Ωr0 4(1 + z)4 (1 + (1 + z)n)2 , (28) 6 0 2 4 6 8 10 0 50 100 150 z ρm/3Mpl2H02 0 2 4 6 8 10 0 10 20 30 40 50 60 z ρb/3Mpl 2H0 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='7 z ρr/3Mpl2H02 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots of energy densities ρm, ρb and ρr for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots clearly indicate the faster decrease of the radiation than the baryonic matter and faster than the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Ωb = Ωb0 4(1 + z)3 (1 + (1 + z)n)2 , (29) Ωm = Ωm0 4(1 + z)−3(γ−1) (1 + (1 + z)n)2 , (30) Ωd = (1 + Ωk) − 4 � Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� (1 + (1 + z)n)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (31) Equation (31) is verified with Ωd0 = 1 + Ωk − (Ωr0 + Ωb0 + Ωm0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Equations (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (21) and (22) can be written with the help of equations (26) and (27) as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' pd M2 plH2 0 = � � �(2n − 3) (1 + z)n − 3 � � 1 + (1 + z)n� 4 − Ωk � 1 + (1 + z)n�2 4 − Ωr0 (1 + z)4 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (32) ρd 3M2 plH2 0 = � � � 1 + (1 + z)n�2 4 + Ωk � 1 + (1 + z)n�2 4 − � Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (33) and ωd = 1 3 �(2n − 3) (1 + z)n − 3 � � 1 + (1 + z)n� − Ωk � 1 + (1 + z)n�2 − 4Ωr0 (1 + z)4 � 1 + (1 + z)n�2 + Ωk � 1 + (1 + z)n�2 − 4 � Ωr0 (1 + z)4 + Ωb0 (1 + z)3 + Ωm0 (1 + z)3−3γ� (34) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' ENERGY CONDITIONS There are several prevalent energy conditions (ECs) in general relativity that set restrictions to prevent some re- gions having negative energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In other words, ECs are considered a valid generalisation of the energy momentum tensor to the whole Universe, where energy density can never be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plausibility of numerous 7 0 2 4 6 8 10 0 200 400 600 800 z ρde/3Mpl2H02 0 2 4 6 8 10 60 50 40 30 20 10 0 z pde/Mpl2H02 0 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 z ωde (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots of ρd, pd and ωd for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' significant singularity problems involving black holes, and wormholes, and many others are extensively examined using ECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The energy conditions can mainly be defined in two ways: (i) geometrically, where ECs are well ex- pressed in terms of Ricci tensor or Weyl tensor, and (ii) physically, where ECs are expressed as a function of the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The ECs are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Weak energy condition (WEC) ⇔ ρtotal ≥ 0, ρtotal + ptotal ≥ 0, Null energy condition (NEC) ⇔ ρtotal + ptotal ≥ 0, Strong energy condition (SEC) ⇔ ρtotal + 3ptotal ≥ 0, Dominant energy condition (DEC) ⇔ ρtotal ≥ 0 , ρtotal ≥ |ptotal| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Additionally, the energy density (ρ) and isotropic pressure (p) of the model can be expressed in terms of potential energy (V(φ)) and scalar field φ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Thus the point-wise ECs in GR are defined as: WEC ⇔ V(φ) ≥ ˙φ2 2 , NEC: ∀ V(φ), SEC ⇔ V(φ) ≤ ˙φ2, DEC ⇔ V(φ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The examination of various energy conditions to determine whether weak, null, strong and dominant ECs are satisfied in the given scenarios is another crucial topic covered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We plot some statistics for interacting scenarios and explore various energy conditions for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' VELOCITY OF SOUND As it is widely known that, the stability of linear perturbations is a significant test for the sustainability of a cos- mological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The need that the speed of sound (C2 s ) be sufficiently less than 1 to prevent unintended oscillations in the matter power spectrum imposes a strict limitation, nevertheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We, here, plot the graph of velocity of sound (C2 s ) for our obtained models using suitable choice of the appropriate parameters, as illustrated in the following figure: 8 0 2 4 6 8 10 0 100 200 300 400 500 z NEC 0 2 4 6 8 10 400 300 200 100 0 z SEC 0 2 4 6 8 10 0 500 1000 1500 z DEC (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots of NEC, SEC and DEC for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 z Cs 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots of C2 s for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' SWAMPLAND CONJECTURE In the present era, there is a growing interest in linking modern cosmological models with quantum gravity theory, which can easily distinguish between the effective field models to the nominal string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In order to choose field potentials properly, Swampland criteria have just been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Therefore, in this section, we shall shed light on how the refined swampland programme conjecture and dark energy are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The swampland conjecture, as we are aware, questions a number of structures, including the physics of black holes, inflation, and other cosmolog- ical ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The connection of practical cosmological models with gravity’s quantum theory, which can distinguish between the effective field models associated to the nominal string landscape, has recently attracted considerable interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Swampland criteria were recently established to help with field potential selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Excitement was specifi- cally sparked by the fact that precise de Sitter solutions with positive cosmological constants do not match with the string landscape, making it impossible to link high energy arrangements to fundamental theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 9 The swampland criteria of the string theory bound with the quintessence model of DE were recently derived [62], [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Numerous works followed the work in [62], [63] in approaching the Swampland criterion in the connection of DE’s quintessence models, for instance [64] - [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The validity of the f(R) theory was recently studied by the re- searcher in relation to the Swampland conjecture [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' When a field theory contradicts quantum gravity, the Swamp- land requirements constrain and regulate it [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Although these hypotheses are still in their infancy and address some issues, they are rapidly evolving and changing in some situations like warm inflation [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We therefore choose to make a connection between these hypotheses and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In subsequent works, we’ll endeavour to develop this concept and make a more substantial connection between the swampland programme, dark energy, dark matter, and in action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Researchers take into account several models for dark energy, and studied their cosmological applica- tions, but the nature of the concept is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Here, we investigate the cosmological restrictions put forth by two string Swampland criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' These criteria include a lower restriction on φV V when V > 0 and an upper bound on the range that scalar fields can travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The major disadvantage of these two criteria seems that they generally in conflict with inflationary theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We discover that certain quintessence models can meet these restrictions at the same time when we apply the same criterion to dark energy in the current epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We contend that the universe will experience a phase change within a few Hubble times if the two Swampland requirements are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' These conditions intensify the drive for upcoming dark energy equation of state ω measurements and equivalence principle tests for dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' With this motivation, we propose, the refined swampland conjecture in the below form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' |V′| V > C1Mpl Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' |V′′| V < −C2M2 pl As we have already noted that the notion to forecast the acceleration in the Universe is to fill it with an exotic type of matter that satisfies the inequality 1 + 3ω < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The energy that causes the acceleration, according to the measurements, satisfies ω ≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The development of the EoS parameter ω for our model was already covered in earlier sections, thus to get an appropriate matter field that produces unusual behaviour and is capable of exhibiting repulsive effects whose cause is given by dark energy (scalar field) we will consider the following equations, ρ = 1 2 ˙φ2 + V(φ), (35) p = 1 2 ˙φ2 − V(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (36) where the terms 1 2 ˙φ2 and V(φ) refer to the kinetic energy (KE) and potential energy (PE) of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' So the term ω = ω(t) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' can no more be treated as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The quintessence or phantom model is consistent with the observations provided ω ≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Thus, we need ˙φ2 << V(φ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' the KE of φ is insignificant in comparison to the PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In this study, we consider that φ is the only source of DE with V(φ), so one can consider energy density and pressure of scalar field as ρφ and pφ respectively for flat FLRW space-time under Barrow’s scheme [78] using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (35) and (36) as ρ = 1 2 ˙φ2 + V(φ) = ρφ, (37) p = 1 2 ˙φ2 − V(φ) = pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (38) The KE and PE can be obtained by solving the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' (37) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 4 demonstrates the potential energy V(φ) plots w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' scalar field φ for the same considered values of model parameters as we have taken in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 4, we notice that the potential V(φ) is present in the interval −1 < φ < 0 and V(φ) ≃ 0 at φ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Therefore, we can predict that the scalar field φ is the only source of DE with potential V(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Thus we conclude that our model is an accelerating dark energy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' To understand the nature of dark energy, we have considered its only source of energy as scalar field φ, which plays the role of quintessence model, therefore in this regard we have analysed our model innthe connection with 10 1 0 1 2 3 0 50 100 150 200 250 z |ϕ| 1 0 1 2 3 4 5 0 20 40 60 80 z V(ϕ) 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 ϕ V(ϕ) (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots of scalar field φ, potential V(φ) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' z and scalar filed correspondence V(φ) vs φ for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='2 z C1 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='5 z C2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The plots of swampland conjecture 1 and 2 for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Swampland conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Conjecture 1 of this criteria is linked with scalar field φ, whereas conjecture 2 is connected withscalar potential V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We have composed the behavior of scalar field with respect to redshift for our interacting model in the figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The below figure illustrates the behavior of refined swampland conjecture’s components plotted for two distinct potentials connected to an interacting scenario for various parameters of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We are aware that each of the swampland conjectures in the iterature has positive values of unit order and the component C2, is smaller than component C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The interacting model demonstrates satisfaction of both conjectures, as shown in figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The Swampland criterion, which has been proposed for a consistent theory of gravity in this regard and which satisfies the requirement |V′| V > C1 ≈ O(1), demonstrates that this dark energy model accords better with the Swampland criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' CONCLUSION We have examined an interacting dark energy cosmological model in the classical theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The interact- ing scenario discussed here provides an insight into the evolution of cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In order to find a 11 consistent solution, we have considered a parametrization of Hubble parameter, which provide a smooth transition from early deceleration to present aceleration and discuss the late evolution of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' In order to discuss the Universe’s evolution in particular the late-time behavior, all the concerned cosmological parameters were written in terms of redsfift z after establishing the (t − z) relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' All the cosmological parameters concerned with one model parameter n only that could be constrained from any cosmological datasets but we have used the constrained value of n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='43 as found in our earlier works [59], [61] for our analysis here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The evolutionary profiles of energy densities of radiation, baryonic & dark matter, ark energy, pressure and equation of state parameter are shown in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We can observe the faster decrease of the radiation energy than the baryonic matter, which is faster than the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Panel 2(b) highlights the profile of pressure of dark energy which is more negative in the past, increases in a concave upward way and finally tends to zero in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' The negative pressure of dark energy in due course of evolution favours the standard lore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Figure 2(c) shows the redshift evolution of equation of state parameter, which is approaching to −1 in the far future assuming the negative values in the expected ranges ar- round z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' We have discussed different energy conditions for interacting models and shown them with graphical representations and also we have discussed the velocity of sound for our obtained model using suitable choice of appropriate parameters and is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Further we reconstructed the potential of the scalar field and challenged the refined swampland conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Acknowledgement SKJP thank IUCAA, Pune for hospitality, where a large part of this paper was written.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='022 [astro-ph/0503075].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Sasaki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Valiviita and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Wands, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=', Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' and Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' 594 (2016) A14 [arXiv: 1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content='01590].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFAT4oBgHgl3EQf7R4U/content/2301.08743v1.pdf'} 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Tokyo University of Agriculture and Technology, 2–24–16 Nakacho, Koganei, +Tokyo, 184–8588 Japan +2Graduate School of Engineering, Osaka University, 2–1 Yamada-oka, Suita, Osaka 565–0871, Japan +3Faculty of Engineering, Tokyo University of Science, 6–8–1 Niijuku, Katsushika-ku, Tokyo, 125–8585 Japan +*yukito.onodera@ynlb.org +Abstract: +This paper experimentally demonstrates 512 color shift keying (CSK) signal +transmission for optical camera communication (OCC). We achieved error-free operation +with a CMOS image sensor module and a multi-label classification neural network-based +equalizer. © 2023 The Author(s) +1. +Introduction +Optical Camera Communication (OCC) is a strong option for the next-generation optical wireless communica- +tion. It leverages a complementary metal-oxide-semiconductor (CMOS) image sensor for a data receiver, and +widespread commercial devices with embedded cameras can be employed as receiver devices. OCC provides +cost-efficient and license-free communication channels without using radio waves. To enhance throughput has +been one of the significant research topics for OCC, because the data rate is mainly limited by the exposure time +and the frame rate of the receiver camera. In addition, a flicker-free operation is an essential requirement. +Color-Shift Keying (CSK) has attracted attention for increasing the data rate in OCC. CSK is a VLC modulation +scheme recommended by the IEEE 802.15.7 task group. The color space chromaticity diagram defined by CIE +1931 (Fig. 1 (a)) is generally used for CSK. CIE 1931 diagram maps all colors that are perceivable by human eyes +to the (x,y) color space. It is then converted to the emission intensity of the RGB-LED transmitter. However, an +optical camera generally has spectral sensitivity characteristics as exemplified in Fig. 1 (b) and RGB components +are extracted through color filters. This nature causes crosstalk among colors which should be canceled via digital +signal processing. As related works, 8-CSK data transmission over 4 cm was presented in [1]. 16-CSK over 80 +cm using a quadrichromatic LED was reported in [2]. In [3], 16-digital CSK over 100 cm was achieved based +on IEEE 802.15.7 CSK constellations. Tri-LEDs based 32-CSK over 3 cm was demonstrated in [4, 5]. More +high-modulation schemes have only been investigated in theory and computer simulation [6,7]. The above recent +achievements are summarized in Fig. 1 (c). +This paper presents the first demonstration of 512-CSK signal error-free demodulation using Sony’s IMX530 +CMOS image sensor and 50-mm optical lens. The nonlinear crosstalk is compensated with neural equalization +from the received CMOS image sensor raw data. +(a) CIE1931 color space [8]. +(b) Spectral sensitivity [9]. +Distance [cm] +M-CSK +100 +200 +300 +400 +8 +16 +32 +64 +512 +0 +H.-W. Chen, 2019 [1] +N. Murata, 2016 [3] +P.Hu,2019 [4] +R. Singh, 2014 [5] +C. Zhu, 2016 [2] +This work +256 +128 +Simulation +Experiment +(c) Related work. +Fig. 1: Overview of OCC-CSK. +arXiv:2301.01599v1 [eess.IV] 29 Dec 2022 + +0.9 +520 +0.8 +540 +0.7 +560 +0.6 +500- +0.5 +580 +y +0.4 +600 +620 +0.3 +490 +700 +0.2 +480 +0.1 +470 +0.0 +460° +380 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +x0,9 +0,8 +0,7 +0.6 +Relative +0,5 +0,4 +0,3 +0,2 +0,1 +0 +400 +500 +600 +700 +800 +900 +1000 +Wavelength [nm]Fig. 2: Receiver configuration and experiment setup. +2. +Receiver Configuration +Fig. 2 shows the receiver block diagram. We employed the camera system provided by Sony Semiconductor +Solutions Corporations as a receiver which has a 12-bit resolution for respective RGB sensitivity. It can also output +purely raw image data without demosaicing, denoising, and white balancing. An optical signal is transmitted +through 8×8 LED planar array and received via 50 mm optical lens. The raw data of the image sensor are converted +from RGB color space to CIE 1931 format for CSK demodulation using the following formula; +� +x +y +� += +� +0.4124 +0.3576 +0.1805 +0.2126 +0.7152 +0.0722 +�� +� +R +G +B +� +�. +(1) +They are then transferred to the multi-label neural network (NN)-based equalizer to erase the nonlinear crosstalk +between color channels. The numbers of input units, hidden units, and output units are 2(= x,y), Nu, M, respec- +tively. M corresponds to the number of bits per symbol. In this paper, we employ 512-CSK, so M = log2 512 = 9 +units. The number of hidden layers is Nh. Log likelihood ratio (LLR) is calculated from the posterior probability +distribution p(1|x,y) obtained with the multi-label NN [10]. Finally, LLR is input to the low-density parity-check +(LDPC) decoder. Symbols for 512-CSK are sequentially placed with bits in a triangular manner, starting from the +vertex of blue (x = 0.1805,y = 0.0722). +3. +Experiment Results +Specifications of the camera and lens are described in Fig. 2. We used 8 × 8 LED planar array and its panel size +is 6.5 cm. The number of LEDs was varied from 1×1 to 8×8 in order to evaluate BER characteristics based +on the area occupied by LEDs in the captured image (light intensity). The transmission distance was 4 m. The +experiments were conducted in a dark room to avoid contamination by external light. The main parameters for +NN are also listed in Fig. 2. The numbers of hidden layers and units are evaluation variables to be optimized. +15000 samples were used for NN training with 5000 epochs. The batch size was set to 4096. Parameters for LDPC +conforms to Digital Video Broadcasting–Satellite2 (DVB-S.2). The codeword length is 64800 bits per block and +3 blocks are transmitted for BER calculation. BER performance of 512-CSK is evaluated using various code rates +from 1/4 to 9/10. +Fig. 3 summarizes the measured BER performance with the number of turned-on LEDs for 512-CSK. Examples +of flash images for LED array from 2×2 to 8×8 are shown in Fig. 3 (a). It can be confirmed that the reception +accuracy would vary on the number of light sources. Received constellation map for 512-CSK is visualized in +Fig. 3 (b). The slight cracks can be observed due to the quantization level; 512-CSK symbols is expressed by +100 steps in each RGB light emission and the receiver with 12 quantization bits can recognize such fine-grained +signals. Figs. 3 (c)–(e) plots uncoded BER performances with the number of turned-on LEDs to optimize NN +parameters. Legends in each figure indicate the number of units. NN structured by 256 units and 5 hidden layers +can achieve the lowest BER performance with avoiding overfitting. Finally, Fig. 3 (f) presents LDPC-coded BER +with the number of turned-on LEDs based on the above optimized NN parameters. If the receiver can capture 5×5 + +.... +4 m +.... +Camera +RGB-to-CSK +FF +FF +Nu +ReLU +Sigmoid +.... +system +M +conversion +Arduino +Uno +512 +LDPC +BER +LLR +LED +test +decode +calc. +Camera +system0 +20 +40 +60 +Number of turned-on LEDs +10 -4 +10 -3 +10 -2 +10 -1 +10 0 +BER +8 +16 +32 +64 +128 +256 +512 +1024 +0 +20 +40 +60 +Number of turned-on LEDs +10 -4 +10 -3 +10 -2 +10 -1 +10 0 +BER +8 +16 +32 +64 +128 +256 +512 +1024 +0 +20 +40 +60 +Number of turned-on LEDs +10 -4 +10 -3 +10 -2 +10 -1 +10 0 +BER +8 +16 +32 +64 +128 +256 +512 +1024 +0 +20 +40 +60 +Number of turned-on LEDs +10 -6 +10 -4 +10 -2 +10 0 +BER +Pre-FEC +2/5 +3/5 +3/4 +5/6 +9/10 +(b) CSK constellation map. +(a) Flash images. +(c) 4 hidden layers. +(d) 5 hidden layers. +(e) 6 hidden layers. +(f) Post-FEC BER. +0 +2 ×2 +3 ×3 +4 ×4 +6 ×6 +8 ×8 +Fig. 3: Experiment results. +LEDs, sufficiently low BER is attained even with a higher coding rate overheaded by up to 10%. It should be noted +that we transmitted about 2×106 bits which guarantees BER below 10−6 even though the results show BER= 0. +As a result, we have experimentally verified that our proposed scheme significantly outperforms existing CSK in +terms of modulation order as well as transmission distance. +4. +Conclusion +This paper demonstrated the first experimental results of 512-CSK for OCC. We achieved error-free transmission +over 4 meters using high-resolution raw data from Sony’s IMX530 CMOS image sensor, neural network based +equalization, and LDPC forward error correction. This work significantly contributes to enhancing the OCC data +rate for next-generation visible light communication. +Acknowledgment +A part of this work was supported by Sony Semiconductor Solutions Corporation, JST Presto Grant Number +JPMJPR2137, and JST START Project Promotion Type (Supporting Small Business Innovation Research (SBIR) +Phase 1), Grant Number JPMJST2260, Japan. +References +1. H.-W. Chen, S.-S. Wen, X.-L. Wang, M.-Z. Liang, M.-Y. Li, Q.-C. Li, and Y. Liu, “Color-shift keying for optical camera +communication using a rolling shutter mode,” IEEE Photonics J. 11, 1–8 (2019). +2. X. Liang, M. Yuan, J. Wang, Z. Ding, M. Jiang, and C. Zhao, “Constellation design enhancement for color-shift keying +modulation of quadrichromatic LEDs in visible light communications,” J. Light. Technol. 35, 3650–3663 (2017). +3. N. Murata, Y. Kozawa, and Y. Umeda, “Digital color shift keying with multicolor LED array,” IEEE Photonics J. 8, +1–13 (2016). +4. P. Hu, P. H. Pathak, X. Feng, H. Fu, and P. Mohapatra, “Colorbars: Increasing data rate of led-to-camera communica- +tion using color shift keying,” in proceedings of the 11th ACM conference on Emerging Networking experiments and +technologies, (2015), pp. 1–13. +5. P. Hu, P. H. Pathak, H. Zhang, Z. Yang, and P. Mohapatra, “High speed led-to-camera communication using color shift +keying with flicker mitigation,” IEEE Trans. on Mob. Comput. 19, 1603–1617 (2019). +6. R. Singh, T. O’Farrell, and J. P. David, “An enhanced color shift keying modulation scheme for high-speed wireless +visible light communications,” J. Light. Technol. 32, 2582–2592 (2014). +7. C. Zhu, Y. Huo, J. Jiang, H. Sun, C. Dong, R. Zhang, and L. Hanzo, “Hierarchical colour-shift-keying aided layered +video streaming for the visible light downlink,” IEEE Access 4, 3127–3152 (2016). +8. CIE, Commission Internationale de lEclairage Proc. (Cambridge University Press, 1931). +9. “Fsm-imx530 +datasheet,” +https://www.framos.com/wp-content/uploads/media/pdf/23/ed/61/FSM-IMX530 -V1A- +Datasheet v1-1c Brief.pdf. +10. O. Shental and J. Hoydis, ““machine llrning”: Learning to softly demodulate,” in 2019 IEEE Globecom Workshops (GC +Wkshps), (2019), pp. 1–7. + +0.6 +0.4 +0.2 +0 +0.2 +0.4 +0.6 \ No newline at end of file diff --git a/gNAzT4oBgHgl3EQfof3Q/content/tmp_files/load_file.txt b/gNAzT4oBgHgl3EQfof3Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dcf5373de1d668f6529cca65c7db1a64fd47560d --- /dev/null +++ b/gNAzT4oBgHgl3EQfof3Q/content/tmp_files/load_file.txt @@ -0,0 +1,257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf,len=256 +page_content='First Demonstration of 512-Color Shift Keying Signal Demodulation Using Neural Equalization for Optical Camera Communication Yukito Onodera1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='*,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Japan 3Faculty of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Tokyo University of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 6–8–1 Niijuku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Katsushika-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 125–8585 Japan yukito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='onodera@ynlb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='org Abstract: This paper experimentally demonstrates 512 color shift keying (CSK) signal transmission for optical camera communication (OCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' We achieved error-free operation with a CMOS image sensor module and a multi-label classification neural network-based equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' © 2023 The Author(s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Introduction Optical Camera Communication (OCC) is a strong option for the next-generation optical wireless communica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' It leverages a complementary metal-oxide-semiconductor (CMOS) image sensor for a data receiver, and widespread commercial devices with embedded cameras can be employed as receiver devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' OCC provides cost-efficient and license-free communication channels without using radio waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' To enhance throughput has been one of the significant research topics for OCC, because the data rate is mainly limited by the exposure time and the frame rate of the receiver camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' In addition, a flicker-free operation is an essential requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Color-Shift Keying (CSK) has attracted attention for increasing the data rate in OCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' CSK is a VLC modulation scheme recommended by the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='7 task group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The color space chromaticity diagram defined by CIE 1931 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 1 (a)) is generally used for CSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' CIE 1931 diagram maps all colors that are perceivable by human eyes to the (x,y) color space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' It is then converted to the emission intensity of the RGB-LED transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' However, an optical camera generally has spectral sensitivity characteristics as exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 1 (b) and RGB components are extracted through color filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' This nature causes crosstalk among colors which should be canceled via digital signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' As related works, 8-CSK data transmission over 4 cm was presented in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 16-CSK over 80 cm using a quadrichromatic LED was reported in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' In [3], 16-digital CSK over 100 cm was achieved based on IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='7 CSK constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Tri-LEDs based 32-CSK over 3 cm was demonstrated in [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' More high-modulation schemes have only been investigated in theory and computer simulation [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The above recent achievements are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' This paper presents the first demonstration of 512-CSK signal error-free demodulation using Sony’s IMX530 CMOS image sensor and 50-mm optical lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The nonlinear crosstalk is compensated with neural equalization from the received CMOS image sensor raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (a) CIE1931 color space [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (b) Spectral sensitivity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Distance [cm] M-CSK 100 200 300 400 8 16 32 64 512 0 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Chen, 2019 [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Murata, 2016 [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='Hu,2019 [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Singh, 2014 [5] C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 2: Receiver configuration and experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Receiver Configuration Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 2 shows the receiver block diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' We employed the camera system provided by Sony Semiconductor Solutions Corporations as a receiver which has a 12-bit resolution for respective RGB sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' It can also output purely raw image data without demosaicing, denoising, and white balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' An optical signal is transmitted through 8×8 LED planar array and received via 50 mm optical lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The raw data of the image sensor are converted from RGB color space to CIE 1931 format for CSK demodulation using the following formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' � x y � = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='4124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='3576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='1805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='2126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='7152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='0722 �� � R G B � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (1) They are then transferred to the multi-label neural network (NN)-based equalizer to erase the nonlinear crosstalk between color channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The numbers of input units, hidden units, and output units are 2(= x,y), Nu, M, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' M corresponds to the number of bits per symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' In this paper, we employ 512-CSK, so M = log2 512 = 9 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The number of hidden layers is Nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Log likelihood ratio (LLR) is calculated from the posterior probability distribution p(1|x,y) obtained with the multi-label NN [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Finally, LLR is input to the low-density parity-check (LDPC) decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Symbols for 512-CSK are sequentially placed with bits in a triangular manner, starting from the vertex of blue (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='1805,y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='0722).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Experiment Results Specifications of the camera and lens are described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' We used 8 × 8 LED planar array and its panel size is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The number of LEDs was varied from 1×1 to 8×8 in order to evaluate BER characteristics based on the area occupied by LEDs in the captured image (light intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The transmission distance was 4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The experiments were conducted in a dark room to avoid contamination by external light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The main parameters for NN are also listed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The numbers of hidden layers and units are evaluation variables to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 15000 samples were used for NN training with 5000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The batch size was set to 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Parameters for LDPC conforms to Digital Video Broadcasting–Satellite2 (DVB-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The codeword length is 64800 bits per block and 3 blocks are transmitted for BER calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' BER performance of 512-CSK is evaluated using various code rates from 1/4 to 9/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3 summarizes the measured BER performance with the number of turned-on LEDs for 512-CSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Examples of flash images for LED array from 2×2 to 8×8 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' It can be confirmed that the reception accuracy would vary on the number of light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Received constellation map for 512-CSK is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' The slight cracks can be observed due to the quantization level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 512-CSK symbols is expressed by 100 steps in each RGB light emission and the receiver with 12 quantization bits can recognize such fine-grained signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3 (c)–(e) plots uncoded BER performances with the number of turned-on LEDs to optimize NN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Legends in each figure indicate the number of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' NN structured by 256 units and 5 hidden layers can achieve the lowest BER performance with avoiding overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3 (f) presents LDPC-coded BER with the number of turned-on LEDs based on the above optimized NN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' If the receiver can capture 5×5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='. 4 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='. Camera RGB-to-CSK FF FF Nu ReLU Sigmoid .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='. system M conversion Arduino Uno 512 LDPC BER LLR LED test decode calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Camera system0 20 40 60 Number of turned-on LEDs 10 -4 10 -3 10 -2 10 -1 10 0 BER 8 16 32 64 128 256 512 1024 0 20 40 60 Number of turned-on LEDs 10 -4 10 -3 10 -2 10 -1 10 0 BER 8 16 32 64 128 256 512 1024 0 20 40 60 Number of turned-on LEDs 10 -4 10 -3 10 -2 10 -1 10 0 BER 8 16 32 64 128 256 512 1024 0 20 40 60 Number of turned-on LEDs 10 -6 10 -4 10 -2 10 0 BER Pre-FEC 2/5 3/5 3/4 5/6 9/10 (b) CSK constellation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (a) Flash images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (c) 4 hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (d) 5 hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (e) 6 hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' (f) Post-FEC BER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 0 2 ×2 3 ×3 4 ×4 6 ×6 8 ×8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 3: Experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' LEDs, sufficiently low BER is attained even with a higher coding rate overheaded by up to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' It should be noted that we transmitted about 2×106 bits which guarantees BER below 10−6 even though the results show BER= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' As a result, we have experimentally verified that our proposed scheme significantly outperforms existing CSK in terms of modulation order as well as transmission distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Conclusion This paper demonstrated the first experimental results of 512-CSK for OCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' We achieved error-free transmission over 4 meters using high-resolution raw data from Sony’s IMX530 CMOS image sensor, neural network based equalization, and LDPC forward error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' This work significantly contributes to enhancing the OCC data rate for next-generation visible light communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Acknowledgment A part of this work was supported by Sony Semiconductor Solutions Corporation, JST Presto Grant Number JPMJPR2137, and JST START Project Promotion Type (Supporting Small Business Innovation Research (SBIR) Phase 1), Grant Number JPMJST2260, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAzT4oBgHgl3EQfof3Q/content/2301.01599v1.pdf'} +page_content=' Chen, S.' metadata={'source': 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b/h9E1T4oBgHgl3EQffgSU/content/tmp_files/2301.03219v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4bf7a7ff0fa9d69f629ca55603899e14897dddbc --- /dev/null +++ b/h9E1T4oBgHgl3EQffgSU/content/tmp_files/2301.03219v1.pdf.txt @@ -0,0 +1,300 @@ +arXiv:2301.03219v1 [math.RA] 9 Jan 2023 +Formal Matrix Rings: Isomorphism Problem +P.A. Krylov1, A.A. Tuganbaev2 +Abstract. We consider the isomorphism problem for formal matrix rings +over a given ring. Principal factor matrices of such rings play an important +role in this case. The work is supported by Russian Scientific Foundation, +project 23-21-00375 (P.A. Krylov) and project 22-11-00052 (A.A. Tugan- +baev). +Key words: formal matrix ring, principal factor matrix +MSC Classification. 16R99; 16D10 +1 +Introduction +Formal (or generalized) matrix rings over a given ring attract a lot of atten- +tion from specialists. It is natural, since such rings regularly appear in ring +theory. In addition, they play an important role in the study of a number +of classes of Artinian rings and algebras (see [3], [4]). They also serve as a +source of varied examples for general ring theory. A number of aspects of the +theory of formal matrix rings are presented in the book [8]. +There is one interesting type of formal matrix rings. In the case of 2 × 2 +matrices, they appeared in [6]. In the case of n × n matrices, they appeared +in [11]. We mean formal matrix rings over a given ring R (or we say «with +values in R»). This means that a concrete formal matrix ring has the same +ring R on all positions. +The class of such rings is a direct expansion of +an ordinary ring M(n, R) of n × n matrices. However, properties of formal +matrix rings over the ring R may differ greatly from properties of the ring +M(n, R). In [8, Chapter 4], some questions are raised on formal matrix rings +over a ring R and three problems on formal matrix rings are formulated at +the beginning of this chapter. In [8, Sections 4.3–4.5], these problems are +solved for some types of formal matrix rings over the ring R. This paper is +devoted to one of these three problems. Namely, this is isomorphism problem +III. This problem is considered in [1], [2], [5], [12], [13], [14]. +In this paper, we consider only associative rings with non-zero identity ele- +ment. If R is a ring, then M(n, R) is an ordinary ring of all n × n matrices +1National Research Tomsk State University, e-mail: krylov@math.tsu.ru . +2National Research University «MPEI», Lomonosov Moscow State University; e-mail: +tuganbaev@gmail.com . +1 + +with values in the ring R. The prime radical of an arbitrary ring S is denoted +by P(S). +2 +Formal Matrix Rings over a Given Ring +We briefly recall the definition of a formal matrix ring. We fix a positive +integer n ≥ 2. Let R1, . . . , Rn be rings and let Mij be Ri-Rj-bimodules with +Mii = Ri, i, j = 1, . . . , n. +Let’s assume that for any subscripts i, j, k = +1, . . . , n, we have an Ri-Rk-bimodule homomorphism Mij ⊗Rj Mjk → Mik. +We denote by K the set of all n × n matrices with values in bimodules +Mij. The set K forms a ring with respect to standard matrix operations +of addition and multiplication. +Matrices are multiplies by the use of the +above-mentioned bimodule homomorphisms. The ring K is called a formal +(or generalized) matrix ring of order n. The ring K is of the following form: +K = + + + + +R1 +M12 +. . . +M1n +M21 +R2 +. . . +M2n +. . . +. . . +. . . +. . . +Mn1 +Mn2 +. . . +Rn + + + + . +Let R be some ring. If K is a formal matrix ring such that Mij = R for all i +and j, then K is called a formal matrix ring over the ring R or a formal matrix +ring with values in the ring R. Such rings can be defined directly. Namely, +let {sijk | i, j, k = 1, . . . , n} be some set of central elements of the ring R +satisfying relations +siik = 1 = sikk, sijk · sikℓ = sijℓ · sjkℓ +(1) +for all subscripts i, j, k, ℓ = 1, . . . , n. For arbitrary n × n matrices A = (aij) +and B = (bij) with values in R, we define a new multiplication, by setting +AB = C = (cij), where cij = +n +� +k=1 +sikjaikbkj. +As a result, we obtain a ring which is denoted by K or M(n, R, Σ), where +Σ is the set of all elements sijk. The set Σ is called a factor system and its +elements are called factors of the ring K. If all sijk are equal to 1, then we +obtain the ordinary matrix ring M(n, R). +The main relations (1) imply the following relations: +siji = sjij = sijℓ · sjiℓ = sℓij · sℓji. +(2) +2 + +It is useful to rewrite the last relation in (2) in the form of three relations +which follow from each other by permutation of subscripts: +siji = sjij = sijk · sjik = skij · skji, +sjkj = skjk = sjki · skji = sijk · sikj, +siki = skik = sikj · skij = sjik · sjki. +(3) +Let τ be a permutation of degree n. If Σ = {sijk} is some factor system, +then we set tijk = sτ(i)τ(j)τ(k). Then {tijk} is a factor system, as well, since +it satisfies relations (1). We denote it by τΣ. Consequently, there exists a +formal matrix ring M(n, R, τΣ). The rings M(n, R, Σ) and M(n, R, τΣ) are +isomorphic to each other under the correspondence A → τA, where A = (aij) +and τA = (aτ(i)τ(j)). +We can associate several matrices with a given ring M(n, R, Σ). We set S = +(siji) and Sk = (sikj) for every k = 1, . . . , n. These matrices are called factor +matrices of the ring M(n, R, Σ). The matrix S is symmetrical. Following +[5], we call it a principal factor matrix. In [5] matrices (sijk) and (skij) are +also used for k = 1, . . . , n. It is clear that the matrices τS and τSk are the +corresponding factor matrices for the ring M(n, R, τΣ). +Sometimes, it is possible to choose a permutation τ such that the principal +factor matrix τS of the ring M(n, R, τΣ) obtains a specific, simpler and more +convenient form. We briefly present three corresponding cases. The first case +slightly generalizes considerations in [8, Section 4.3]; see Lemmas 4.3.1, 4.3.2 +and the paragraph after the proof of Lemma 4.3.2. +Let Σ be a factor system such that every factor sijk is either non-invertible +or equal to 1. Under this assumption, additional relations appear between +factors sijk. For example, with the use of relations (3), it is easy to verify +that the following lemma is true. +Lemma 2.1. Let i, j, k be pairwise distinct subscripts. Then elements siji, +siki and sjkj satisfy only one of the following three conditions. +1) All three elements are equal to 1. +2) Some two elements of these three elements are non-invertible and the +third element is equal to 1. +3) All three these elements are non-invertible. +On the set {1, . . . , n}, we define a binary relation ∼, by setting i ∼ j ⇔ +siji = 1. Lemma 2.1 implies the following assertion. +3 + +Lemma 2.2. The relation ∼ is an equivalence relation. +Let’s write the final result at the moment. +Lemma 2.3. There exists a permutation τ such that the matrix τS can be +presented in a block form such that blocks, consisting of 1s, stay on the main +diagonal and non-invertible elements stay on all remaining positions. +Proof. +Let τ be a permutation such that in the upper row consists of +numbers 1, . . . , n in a natural order. The bottom row consists of equivalence +classes of the relation ∼ which are arranged in random order. In every class, +numbers are also arranged in random order. The matrix τS has the structure +specified in the lemma. □ +Two other cases were considered later. One of these cases is considered in +[7] and [9, Lemma 12.1, Lemma 12.2] and another case is considered in [5, +Lemma 4.1]. Moreover, no restrictions are imposed on factors. However, the +ring R is assumed to be commutative in [7] and [9]. But in this context it +doesn’t matter, since factors are central elements. +Similarly to the above, omitting details, we can say that it turns out that +[7] and [9] deal with the situation where always exists a permutation τ such +that the matrix τS has the following block structure: the blocks on the main +diagonal are filled with non-zero-divisors, and the remaining blocks are filled +with zero-divisors. +In [5, Lemma 4.1], it is proved that there exists a permutation τ such that +the corresponding blocks on the main diagonal of the matrix τS consist of in- +vertible elements and all remaining blocks consist of non-invertible elements. +The analogues of Lemma 2.1 and Lemma 2.2 are also true. +Of course, the three situations outlined can be combined within the frame- +work of some general approach. +An interesting important class of formal matrix rings is formed by the rings +M(n, R, Σ) which factor systems consist of 1s and some central element s. In +[5], such systems Σ are called binary systems. We denote the corresponding +ring M(n, R, Σ) by M(n, R, s) and agree to call it the (s1)-ring of formal +matrices. +The rings M(n, R, s), where s2 ̸= 1 and s2 ̸= s, are studied in [8, Sections 4.3, +4.4] and [5]. Connections of such rings with crossed matrix rings are known; +see [4]. In [5, Lemma 4.7], it is proved that a principal factor matrix of the +ring M(n, R, s), where s2 ̸= 1 and s2 ̸= s, uniquely determines all remaining +factor matrices. This result is very useful. +We obtain another quite specific type of rings if we set s = 0. All factor +4 + +matrices of such rings are (01)-matrices. +Papers [7], [9] and [10] contain +various material on automorphism groups of formal matrix rings. +We return to an arbitrary ring M(n, R, s). If the element s is invertible, then +we have an isomorphism M(n, R, s) ∼= M(n, R). Therefore, we can assume +that the element s is not invertible. Then it follows from Lemma 2.3 that +there exists a permutation τ with the following property: the matrix τS can +be divided into blocks in such a way that the diagonal blocks are filled with +1s, and all other blocks are filled with the element s. In such a situation, we +say that the matrix τS has a canonical form, and the matrix S is reduced to +a canonical form. +Remark 2.4. Of course, the canonical form is determined up to permuta- +tion of blocks on the main diagonal and the corresponding permutation of +remaining blocks. +3 +Isomorphism Problem for Formal Matrix +Rings +[8, Section 4.1] contains the isomorphism problem III for formal matrix rings: +• When do two factor systems define isomorphic formal matrix rings? +We consider this problem for (s1)-formal matrix rings. +We say that some ring R satisfies (n, m)-condition if we have m = n for any +positive integers n and m such that the rings M(n, R) and M(m, R) are +isomorphic to each other. For example, the (n, m)-condition holds if the ring +R is either commutative, or local, or is a left (right) principal ideal domain. +A ring S is said to be indecomposable if 0 and 1 are only central idempotents +of S. +Theorem 3.1. Let the factor ring R/P(R) be indecomposable and satisfies +(n, m)-condition and s ∈ P(R). Let K1 and K2 be two (s1)-formal matrix +rings with principal factor matrices S and T, respectively. +The following +assertions hold. +1. If the rings K1 and K2 are isomorphic to each other, then the matrices S +and T have the same canonical forms. +2. If s2 ̸= 1 and s2 ̸= s, then the converse is also true. +Proof. 1. By Lemma 2.3, we can assume that the matrices S and T are +5 + +presented in the canonical form. Let’s assume that K1 ∼= K2. Then there +exists a ring isomorphism +γ : K1/P(K1) → K2/P(K2). +The structure of the prime radicals P(K1) and P(K2) is known (see [8, Corol- +lary 4.2.2] and the paragraph after the corollary). We also know the block +structure of the matrices S and T. With the use this information, we obtain +relations +K1/P(K1) = P1 × . . . × Pk and K2/P(K2) = Q1 × . . . × Qℓ, +where k (resp., ℓ) is the number of blocks on the main diagonal of the ma- +trix S (resp., T). In addition, all Pi and Qj are full matrix rings of some +orders. Since the ring R/P(R) is indecomposable, all the rings Pi/P(Pi) and +Qj/P(Qj) are indecomposable. +Here, we remark that there is an analogue of [9, Lemma 9.6] (or [10, Lemma +6.1]) on automorphisms of direct products of indecomposable rings for iso- +morphisms between direct products of indecomposable rings. +Therefore, +k = ℓ and there exists a permutation τ of degree k such that the restric- +tion γ to Pi is an isomorphism Pi → Qτ(i), i = 1, . . . , k. Consequently, the +canonical forms of the matrices S and T coincide. +2. Let {sijk} (resp., {tijk}) be the set of all factors of the ring K1 (resp., K2). +As noted earlier, factors of the form siji, i.e., elements of the the principal +factor matrix S, determine all remaining factors sijk; the same is true for +factors of the ring K2. Thus, sijk = tijk for all i, j, k. Therefore, we have the +relation K1 = K2. □ +Corollary 3.2. The factor rings K1/P(K1) and K2/P(K2) are isomorphic +to each other if and only if the matrices S and T have the same canonical +form. +Remark 3.3. In [5, Theorem 4.12], a result, which is similar to Theorem +3.1, is proved for a left Artinian ring R. +The introduction to this article mentions isomorphism problem III from [8, +Section 4.1]. The following open question is a partial case of this problem. +Open question. Let s and t be two central elements of the ring R. When +is the isomorphism M(n, R, s) ∼= M(n, R, t) true? +A similar question for some other rings M(n, R, Σ) is considered in [8, Section +4.5]. +6 + +References +[1] A. N. Abyzov and D. T. Tapkin. Formal matrix rings and their isomor- +phisms // Sib. Mat. Zh. – 2015. – Vol. 56, no. 6. P. 955–967. +[2] A. N. Abyzov and D. T. Tapkin. On certain classes of formal matrix +rings // Russian Mathematics. – 2015. – Vol. 59, no. 3. – P. 1–12. +[3] Auslander M., Reiten I., Smalø S.O. Representation Theory of Artin +Algebras. – Cambridge University Press, Cambridge, 1995. +[4] Baba Y., Oshiro K. Classical Artinian Rings and Related Topics. – +World Scientific, New Jersey–London–Singapore, 2009. +[5] Chen W., Deng G., Su H. On the Binary System of Factors of Formal +Matrix Rings // Czech. Math. J. – 2020. – Vol. 70. – P. 693–709. +[6] Krylov P.A. Isomorphisms of generalized matrix rings // Algebra and +Logic. – 2008. – Vol. 47, no. 4. – P. 258–262. +[7] Krylov P.A., Norbosambuev T.D. Automorphisms of formal matrix al- +gebras // Sib. Mat. Zh. – 2018. – Vol. 59, no. 5. – P. 1116–1127. +[8] Krylov P.A., Tuganbaev A.A. Formal Matrices, Springer-Verlag, Berlin, +2017. +[9] Krylov P.A., Tuganbaev A.A. Automorphism groups of formal matrix +rings // Journal of Mathematical Sciences (Springer). – 2021. – Vol. 258, +no. 2. – P. 222–249. +[10] Krylov +P.A., +Tuganbaev +A.A. +Automorphisms +of +For- +mal +Matrix +Rings +// +arXiv:2204.13332 +[math.RA]. +– +2022. +https://doi.org/10.48550/arXiv.2204.13332 +[11] Tang G., Zhou Y. A class of formal matrix rings // Linear Algebra Appl. +– 2013. – Vol. 438, no. 12. – P. 4672–4688. +[12] D. T. Tapkin. Formal matrix rings and a generalization of an incidence +algebra (Russian) // Chebyshev. Sb. – 2015. – Vol. 16, no. 3. – P. 442– +449. +[13] D. T. Tapkin. Isomorphisms of formal matrix incidence rings // Russian +Mathematics. – 2017. – Vol. 61. – P. 73–79. +[14] D. T. Tapkin. Isomorphisms of formal matrix rings with zero trace ideals +// Sib. Math. Zh. – 2018. – Vol. 59, no. 3. – P. 523–535. +7 + diff --git a/h9E1T4oBgHgl3EQffgSU/content/tmp_files/load_file.txt b/h9E1T4oBgHgl3EQffgSU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e99384601d08d0babc34ae4a6f3d136464a00d9b --- /dev/null +++ b/h9E1T4oBgHgl3EQffgSU/content/tmp_files/load_file.txt @@ -0,0 +1,408 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf,len=407 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='03219v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='RA] 9 Jan 2023 Formal Matrix Rings: Isomorphism Problem P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Krylov1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Tuganbaev2 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We consider the isomorphism problem for formal matrix rings over a given ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Principal factor matrices of such rings play an important role in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The work is supported by Russian Scientific Foundation, project 23-21-00375 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Krylov) and project 22-11-00052 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Tugan- baev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Key words: formal matrix ring, principal factor matrix MSC Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 16R99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 16D10 1 Introduction Formal (or generalized) matrix rings over a given ring attract a lot of atten- tion from specialists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' It is natural, since such rings regularly appear in ring theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In addition, they play an important role in the study of a number of classes of Artinian rings and algebras (see [3], [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' They also serve as a source of varied examples for general ring theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' A number of aspects of the theory of formal matrix rings are presented in the book [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' There is one interesting type of formal matrix rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In the case of 2 × 2 matrices, they appeared in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In the case of n × n matrices, they appeared in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We mean formal matrix rings over a given ring R (or we say «with values in R»).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' This means that a concrete formal matrix ring has the same ring R on all positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The class of such rings is a direct expansion of an ordinary ring M(n, R) of n × n matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' However, properties of formal matrix rings over the ring R may differ greatly from properties of the ring M(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [8, Chapter 4], some questions are raised on formal matrix rings over a ring R and three problems on formal matrix rings are formulated at the beginning of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [8, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='5], these problems are solved for some types of formal matrix rings over the ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' This paper is devoted to one of these three problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Namely, this is isomorphism problem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' This problem is considered in [1], [2], [5], [12], [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In this paper, we consider only associative rings with non-zero identity ele- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If R is a ring, then M(n, R) is an ordinary ring of all n × n matrices 1National Research Tomsk State University, e-mail: krylov@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='tsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='ru .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 2National Research University «MPEI», Lomonosov Moscow State University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' e-mail: tuganbaev@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='com .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 1 with values in the ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The prime radical of an arbitrary ring S is denoted by P(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 2 Formal Matrix Rings over a Given Ring We briefly recall the definition of a formal matrix ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We fix a positive integer n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , Rn be rings and let Mij be Ri-Rj-bimodules with Mii = Ri, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let’s assume that for any subscripts i, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n, we have an Ri-Rk-bimodule homomorphism Mij ⊗Rj Mjk → Mik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We denote by K the set of all n × n matrices with values in bimodules Mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The set K forms a ring with respect to standard matrix operations of addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Matrices are multiplies by the use of the above-mentioned bimodule homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The ring K is called a formal (or generalized) matrix ring of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The ring K is of the following form: K = \uf8eb \uf8ec \uf8ec \uf8ed R1 M12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' M1n M21 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' M2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Mn1 Mn2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Rn \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let R be some ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If K is a formal matrix ring such that Mij = R for all i and j, then K is called a formal matrix ring over the ring R or a formal matrix ring with values in the ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Such rings can be defined directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Namely, let {sijk | i, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n} be some set of central elements of the ring R satisfying relations siik = 1 = sikk, sijk · sikℓ = sijℓ · sjkℓ (1) for all subscripts i, j, k, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' For arbitrary n × n matrices A = (aij) and B = (bij) with values in R, we define a new multiplication, by setting AB = C = (cij), where cij = n � k=1 sikjaikbkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' As a result, we obtain a ring which is denoted by K or M(n, R, Σ), where Σ is the set of all elements sijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The set Σ is called a factor system and its elements are called factors of the ring K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If all sijk are equal to 1, then we obtain the ordinary matrix ring M(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The main relations (1) imply the following relations: siji = sjij = sijℓ · sjiℓ = sℓij · sℓji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' (2) 2 It is useful to rewrite the last relation in (2) in the form of three relations which follow from each other by permutation of subscripts: siji = sjij = sijk · sjik = skij · skji, sjkj = skjk = sjki · skji = sijk · sikj, siki = skik = sikj · skij = sjik · sjki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' (3) Let τ be a permutation of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If Σ = {sijk} is some factor system, then we set tijk = sτ(i)τ(j)τ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Then {tijk} is a factor system, as well, since it satisfies relations (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We denote it by τΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Consequently, there exists a formal matrix ring M(n, R, τΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The rings M(n, R, Σ) and M(n, R, τΣ) are isomorphic to each other under the correspondence A → τA, where A = (aij) and τA = (aτ(i)τ(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We can associate several matrices with a given ring M(n, R, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We set S = (siji) and Sk = (sikj) for every k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' These matrices are called factor matrices of the ring M(n, R, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The matrix S is symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Following [5], we call it a principal factor matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [5] matrices (sijk) and (skij) are also used for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' It is clear that the matrices τS and τSk are the corresponding factor matrices for the ring M(n, R, τΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Sometimes, it is possible to choose a permutation τ such that the principal factor matrix τS of the ring M(n, R, τΣ) obtains a specific, simpler and more convenient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We briefly present three corresponding cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The first case slightly generalizes considerations in [8, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' see Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2 and the paragraph after the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let Σ be a factor system such that every factor sijk is either non-invertible or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Under this assumption, additional relations appear between factors sijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' For example, with the use of relations (3), it is easy to verify that the following lemma is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let i, j, k be pairwise distinct subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Then elements siji, siki and sjkj satisfy only one of the following three conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 1) All three elements are equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 2) Some two elements of these three elements are non-invertible and the third element is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 3) All three these elements are non-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' On the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n}, we define a binary relation ∼, by setting i ∼ j ⇔ siji = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1 implies the following assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 3 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The relation ∼ is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let’s write the final result at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' There exists a permutation τ such that the matrix τS can be presented in a block form such that blocks, consisting of 1s, stay on the main diagonal and non-invertible elements stay on all remaining positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let τ be a permutation such that in the upper row consists of numbers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , n in a natural order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The bottom row consists of equivalence classes of the relation ∼ which are arranged in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In every class, numbers are also arranged in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The matrix τS has the structure specified in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' □ Two other cases were considered later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' One of these cases is considered in [7] and [9, Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1, Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2] and another case is considered in [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Moreover, no restrictions are imposed on factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' However, the ring R is assumed to be commutative in [7] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' But in this context it doesn’t matter, since factors are central elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Similarly to the above, omitting details, we can say that it turns out that [7] and [9] deal with the situation where always exists a permutation τ such that the matrix τS has the following block structure: the blocks on the main diagonal are filled with non-zero-divisors, and the remaining blocks are filled with zero-divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1], it is proved that there exists a permutation τ such that the corresponding blocks on the main diagonal of the matrix τS consist of in- vertible elements and all remaining blocks consist of non-invertible elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The analogues of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2 are also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Of course, the three situations outlined can be combined within the frame- work of some general approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' An interesting important class of formal matrix rings is formed by the rings M(n, R, Σ) which factor systems consist of 1s and some central element s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [5], such systems Σ are called binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We denote the corresponding ring M(n, R, Σ) by M(n, R, s) and agree to call it the (s1)-ring of formal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The rings M(n, R, s), where s2 ̸= 1 and s2 ̸= s, are studied in [8, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='4] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Connections of such rings with crossed matrix rings are known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='7], it is proved that a principal factor matrix of the ring M(n, R, s), where s2 ̸= 1 and s2 ̸= s, uniquely determines all remaining factor matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' This result is very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We obtain another quite specific type of rings if we set s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' All factor 4 matrices of such rings are (01)-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Papers [7], [9] and [10] contain various material on automorphism groups of formal matrix rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We return to an arbitrary ring M(n, R, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If the element s is invertible, then we have an isomorphism M(n, R, s) ∼= M(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Therefore, we can assume that the element s is not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Then it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3 that there exists a permutation τ with the following property: the matrix τS can be divided into blocks in such a way that the diagonal blocks are filled with 1s, and all other blocks are filled with the element s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In such a situation, we say that the matrix τS has a canonical form, and the matrix S is reduced to a canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Of course, the canonical form is determined up to permuta- tion of blocks on the main diagonal and the corresponding permutation of remaining blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 3 Isomorphism Problem for Formal Matrix Rings [8, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1] contains the isomorphism problem III for formal matrix rings: When do two factor systems define isomorphic formal matrix rings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We consider this problem for (s1)-formal matrix rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We say that some ring R satisfies (n, m)-condition if we have m = n for any positive integers n and m such that the rings M(n, R) and M(m, R) are isomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' For example, the (n, m)-condition holds if the ring R is either commutative, or local, or is a left (right) principal ideal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' A ring S is said to be indecomposable if 0 and 1 are only central idempotents of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let the factor ring R/P(R) be indecomposable and satisfies (n, m)-condition and s ∈ P(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let K1 and K2 be two (s1)-formal matrix rings with principal factor matrices S and T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If the rings K1 and K2 are isomorphic to each other, then the matrices S and T have the same canonical forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' If s2 ̸= 1 and s2 ̸= s, then the converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3, we can assume that the matrices S and T are 5 presented in the canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let’s assume that K1 ∼= K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Then there exists a ring isomorphism γ : K1/P(K1) → K2/P(K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The structure of the prime radicals P(K1) and P(K2) is known (see [8, Corol- lary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2] and the paragraph after the corollary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' We also know the block structure of the matrices S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' With the use this information, we obtain relations K1/P(K1) = P1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' × Pk and K2/P(K2) = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' × Qℓ, where k (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=', ℓ) is the number of blocks on the main diagonal of the ma- trix S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=', T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In addition, all Pi and Qj are full matrix rings of some orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Since the ring R/P(R) is indecomposable, all the rings Pi/P(Pi) and Qj/P(Qj) are indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Here, we remark that there is an analogue of [9, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='6] (or [10, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1]) on automorphisms of direct products of indecomposable rings for iso- morphisms between direct products of indecomposable rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Therefore, k = ℓ and there exists a permutation τ of degree k such that the restric- tion γ to Pi is an isomorphism Pi → Qτ(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Consequently, the canonical forms of the matrices S and T coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let {sijk} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=', {tijk}) be the set of all factors of the ring K1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=', K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' As noted earlier, factors of the form siji, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=', elements of the the principal factor matrix S, determine all remaining factors sijk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' the same is true for factors of the ring K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Thus, sijk = tijk for all i, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Therefore, we have the relation K1 = K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The factor rings K1/P(K1) and K2/P(K2) are isomorphic to each other if and only if the matrices S and T have the same canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' In [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='12], a result, which is similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1, is proved for a left Artinian ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The introduction to this article mentions isomorphism problem III from [8, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' The following open question is a partial case of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Let s and t be two central elements of the ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' When is the isomorphism M(n, R, s) ∼= M(n, R, t) true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' A similar question for some other rings M(n, R, Σ) is considered in [8, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' 6 References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Abyzov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Tapkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Formal matrix rings and their isomor- phisms // Sib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQffgSU/content/2301.03219v1.pdf'} +page_content=' – 2015.' metadata={'source': 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a/rdFQT4oBgHgl3EQfuDap/content/tmp_files/2301.13394v1.pdf.txt b/rdFQT4oBgHgl3EQfuDap/content/tmp_files/2301.13394v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..796a71fcdb86694ef90e85295d86d0bcc53ea005 --- /dev/null +++ b/rdFQT4oBgHgl3EQfuDap/content/tmp_files/2301.13394v1.pdf.txt @@ -0,0 +1,754 @@ +arXiv:2301.13394v1 [astro-ph.HE] 31 Jan 2023 +MNRAS 000, 1–5 (2021) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +Spectro-polarimetric view of bright atoll source GX 9+9 using IXPE and +AstroSat +Rwitika Chatterjee1⋆, Vivek K. Agrawal1, Kiran M. Jayasurya1, Tilak Katoch2 +1Space Astronomy Group, ISITE Campus, U. R. Rao Satellite Center, ISRO, Bengaluru 560037, India +2Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We have carried out the first spectro-polarimetric study of the bright NS-LMXB GX 9+9 using IXPE and AstroSat observations. +We report a significant detection of polarization of 1.7 ± 0.4% over the 2 − 8 keV energy band, with a polarization angle of +63◦ ± 7◦. The polarization is found to be energy-dependent, with a 3σ polarization degree consistent with null polarization in +2−4 keV, and 3.2% in 4−8 keV. Typical of the spectra seen in NS-LMXBs, we find that a combination of soft thermal emission +from the accretion disc and Comptonized component from the optically thick corona produces a good fit to the spectra. We +also attempt to infer the individual polarization of these components, and obtain a 3σ upper limit of ∼ 11% on the polarization +degree of the thermal component, and constrain that of the Comptonized component to ∼ 3%. We comment on the possible +corona geometry of the system based on our results. +Key words: accretion, accretion discs - polarization - X-rays: binaries - X-rays: individual: GX 9+9 +1 INTRODUCTION +Atoll and Z-sources are luminous low-mass X-ray binaries +(LMXBs) where a neutron star accretes matter from a low-mass +companion through Roche-lobe overflow. These two types of +LMXBs trace different patterns on their colour-colour diagram +(CCD, Hasinger & van der Klis 1989). Atoll sources exhibit high +variability over timescales ranging from milliseconds to years, and +depending on their X-ray luminosity, may be found either in the high +soft state (HSS) or low hard state (LHS). +X-ray +spectra +of +the +NS-LMXBs +are +complex +and +re- +quire multiple components to describe them. A combination +of multi-temperature disc blackbody and Comptonized emis- +sion (Di Salvo et al. 2000a, 2002; Agrawal & Sreekumar 2003; +Tarana et al. 2008; Agrawal & Misra 2009; Agrawal et al. 2023) has +been widely used to describe the X-ray spectra of the atoll and Z- +sources. Here, the softer component from the disc is directly ob- +served, and the Comptonized emission may come from the boundary +layer (BL) or spreading layer (SL) near the surface of the neutron +star (‘eastern model’). An alternative scenario is that the accretion +disc is Comptonized and blackbody emission comes from the neu- +tron star surface (‘western model’), and a combination of blackbody +plus a Comptonized component is adopted for describing the spec- +tra of these sources (Piraino et al. 2000, 2007; Di Salvo et al. 2000b, +2001; Barret & Olive 2002; Wang et al. 2019). +GX 9+9 is an NS-LMXBs, classified as an atoll source. It was dis- +covered by a sounding rocket experiment on 1967 July 7 (Bradt et al. +1968). Hertz & Wood (1988) reported a 4.19 ± 0.02 h orbital pe- +riod using scanning mode of HEAO A-1. Several models have been +used to describe the spectrum of this source, such as power-law +⋆ E-mail: rwitika@ursc.gov.in +with exponential cutoff (Church & Baluci´nska-Church 2001), black- +body plus Comptonization model (Kong et al. 2006), and multicolor +disc (MCD) plus Comptonized blackbody emission from the SL +(Savolainen et al. 2009). Göˇgü¸s et al. (2007) described the RXTE- +PCA spectrum of this source by a double blackbody plus a broad iron +line at 6 keV. Iaria et al. (2020) reported the presence of a blurred re- +flection component using 0.3 − 40 keV spectrum of the source. +Spectral modeling of such sources helps to understand the nature +of the coronal plasma (e.g. temperature, optical depth). However, +these models are degenerate with respect to the geometry and lo- +cation of corona, and polarization information, if available, can be +used to constrain them. Recent polarimetric studies of several NS- +LMXBs such as Sco X-1 (Long et al. 2022), Cyg X-2 (Farinelli et al. +2023) and GS 1826 − 238 (Capitanio et al. 2022) have revealed the +capability of polarization studies in inferring the source geometry +and emission properties. In this letter, we present the first spectro- +polarimetric study of the bright atoll source GX 9+9 using IXPE ob- +servations and archival AstroSat data. +2 OBSERVATIONS AND DATA ANALYSIS +2.1 AstroSat +The source GX 9+9 was observed from 2020 July 25 to 2020 July 27 +with AstroSat for a net exposure time of 56 ks. To understand the +spectral nature of the source, we analyzed data from the Soft X-ray +Telescope (SXT, 0.3 − 8 keV) and Large Area X-ray Proportional +Counter (LAXPC, 3 − 60 keV), which were operated in photon- +counting mode and event analysis mode respectively. +© 2021 The Authors + +2 +Rwitika et al. +500 +600 +700 +800 +Counts/s +0.9 +1 +1.1 +1.2 +Soft−colour +0 +5×104 +105 +0.4 +0.5 +0.6 +0.7 +Hard−colour +Time (s) +Figure 1. LAXPC 3−25 keV count rate (top), soft-colour (middle) and hard- +colour (bottom) during the AstroSat observation. Bin time of 16 s has been +used to generate the light curve and hardness plots. + 0.4 + 0.42 + 0.44 + 0.46 + 0.48 + 0.5 + 0.52 + 0.54 + 0.56 + 0.58 + 0.6 + 0.85 + 0.9 + 0.95 + 1 + 1.05 + 1.1 + 1.15 + 1.2 +B1 +B2 +B3 +B4 +Hard-Colour +Soft-Colour +Figure 2. CCD of GX 9+9 , with different parts denoted by labels B1, B2, +B3 and B4. Each data points corresponds to 256 s of integration time. +HEASOFT FTOOL1 XSELECT was used to create the image, spec- +tra and light curves from the Level-2 data of SXT. The SXT count rate +during the observation was above the pileup limit. Hence, we used +an annular region excluding the central 2′, and with outer radius 12′ +to create light curves and spectra. ARF file was created using the task +sxtARFModule2. LAXPC light curves and spectra were generated by +the latest version of the LAXPC analysis software LaxpcSoft3. The +background spectra of the dark sky was provided by the instrument +team and was used for the analysis. +Figure 1 shows the variation of the LAXPC 3 − 25 keV count rate, +soft-color and hard-colour as a function of time, over the AstroSat +observation. We defined soft-colour as the ratio of count rates in +5 − 7 keV and 3 − 5 keV, and hard colour as the ratio of count rates +in 10 − 20 keV and 7 − 10 keV energy bands. We also produced the +CCD of the source, which is shown in Figure 2. We note that the +source is in the ‘banana’ state during the observation. We divided +the CCD into 4 sections and denote them by B1, B2, B3 and B4. +We created spectra for the four sections and performed joint +1 http://heasarc.gsfc.nasa.gov/ftools +2 http://www.tifr.res.in/∼astrosat_sxt/dataanalysis.html +3 http://www.tifr.res.in/∼astrosat_laxpc/LaxpcSoft.html +56500 +57500 +58500 +59500 +MJD +100 +150 +200 +250 +300 +350 +Count rate (mcrab) +Astrosat +IXPE +Jul 2013 +Apr 2016 +Jan 2019 +Oct 2021 +Date +Figure 3. MAXI/GSC 2−20 keV light curve of GX 9+9 from 2012 till present +day. The epochs of the AstroSat and IXPE observations are marked. The +horizontal line indicates the mean count rate of 220 mcrab. +spectral fitting of each section using the 0.7 − 7 keV spectra from +SXT and 4 − 25 keV spectra from LAXPC4 using XSPEC5 (Arnaud +1996) v12.12.1. We found that a Comptonized emission component +(Zdziarski et al. 1996 nthcomp in XSPEC) plus a thermal compo- +nent, either described by a single temperature blackbody or an MCD +(bbodyrad and diskbb respectively), modified by ISM absorption +(tbabs), give a satisfactory fit to the 0.7 − 25 keV spectra of the +source6. The seed photons for Comptonization are assumed to follow +a blackbody distribution (inp_type parameter set to 0 in nthcomp), +when the thermal component comes from the accretion disc (here- +after, model 1). However, multi-temperature disc blackbody is used +for seed photons (inp_type = 1) when thermal component is de- +scribed by bbodyrad (hereafter, model 2). We note here that model 1 +is reminiscent of the ‘eastern’ type model where thermal component +from the disc is directly observed and the Comptonized emission +comes from a shell-like corona around the neutron star/SL, whereas +model 2 resembles a ‘western’-like scenario, with the corona as a +slab or wedge above the accretion disc. +2.2 IXPE +The Imaging X-ray Polarimetry Explorer (IXPE, Weisskopf et al. +2022), launched on 2021 December 9, consists of three identical +telescopes each consisting of a mirror module assembly with a +polarization-sensitive imaging X-ray detector at the focus. IXPE ob- +served GX 9+9 on 2022 October 9 for about 92 ks of net exposure +time. In Figure 3, we show the long-term MAXI/GSC light curve of +the source and note that the source is stable and remains in the HSS +over the period of both AstroSat and IXPE observations. +We analyzed the processed Level-2 data7 using IXPEOBSSIM soft- +ware v30.0.0 (Baldini et al. 2022). Source was defined as a circular +region of radius 60" centered at the intensity centroid, and back- +ground as an annular region with the same center and inner and +outer radii as 180" and 240" respectively. Source and background +events were filtered using XPSELECT task and various binning algo- +rithms were applied using XPBIN task. The average count rate was +4 Beyond 25 keV, the source is weak and the background dominates. +5 https://heasarc.gsfc.nasa.gov/xanadu/xspec/ +6 We also need two edge components at 2.4 keV and 8.7 keV (instrumental +features) to obtain a good fit. +7 Publicly available on the HEASARC archive +MNRAS 000, 1–5 (2021) + +Spectro-polarimetric view of GX 9+9 +3 +Table 1. Best-fit parameters of model 1 to SXT+LAXPC spectra. NH: hy- +drogen column density (1022 cm−2); Γ: photon index; τ: optical depth; kTin: +inner disc temperature (keV); kTe: plasma temperature (keV). Ndbb and Nnth +are the normalizations of the diskbb and nthcomp components. 90% uncer- +tainties are quoted. +Parameters +B1 +B2 +B3 +B4 +Average +NH +0.17±0.01 0.16±0.01 0.15±0.01 0.16±0.01 0.17±0.01 +kTin +0.68±0.05 0.65±0.08 0.77±0.09 0.96±0.06 0.65±0.04 +Ndbb +345±75 +373±127 +221±62 +108±18 +405±64 +Γ +3.46±0.24 3.10±0.19 3.11±0.24 2.62±0.09 3.14±0.14 +kTe +7.55±1.78 5.12±0.59 5.56±1.51 4.20±0.35 5.58±0.82 +Seed kT a 1.05±0.05 1.03±0.06 1.11±0.06 1.24±0.13 1.05±0.03 +Nnth +0.15±0.02 0.16±0.02 0.12±0.02 0.10±0.01 0.15±0.01 +τb +2.88±0.72 3.92±0.48 3.76±0.77 +5.5±0.47 +3.72±0.69 +χ2/DOF +598/654 +656/654 +608/654 +683/654 +628/654 +a Blackbody seed photons +b Assuming spherical corona +13 cts/s with the background contribution ≲ 0.2%. The polarization +parameters (polarization degree PD and polarization angle PA) were +extracted in the energy bands 2 − 4 keV, 4 − 8 keV and 2 − 8 keV +using the model-independent PCUBE algorithm (Kislat et al. 2015). +For calculating the uncertainties, PCUBE assumes that the PD and +PA are independent whereas they are not actually so. Hence, the con- +tours of the joint measurement of PD and PA represent the uncertain- +ties more appropriately. +We also performed a spectro-polarimetric model-dependent fit +(see e.g. Strohmayer 2017) of the data using XSPEC. Source and +background spectra corresponding to the Stokes parameters I, Q and +U were extracted using algorithms PHA1, PHA1Q and PHA1U re- +spectively, and the latest response files (v12) were used in spec- +tral fitting. The best-fit models obtained from the joint SXT and +LAXPC fits (see Section 3.1), modified by a constant polarization +(polconst8 in XSPEC), was used in the fitting. +3 RESULTS +3.1 Spectral properties +We show the best fit spectral parameters obtained by fitting the dif- +ferent sections of the atoll track using model 1 in Table 1. As the +source moves along the banana track from B1 to B4, the spectrum +becomes harder, with the photon index decreasing from 3.5 to 2.6. +The temperature at the inner radius of the accretion disc increases +from 0.7 keV to ∼ 1 keV, and the seed photon temperature ranges +from 1 keV to 1.2 keV. The temperature of the corona varies in the +range 4.2 keV to 7.6 keV. The corona is optically thick, with τ in the +range ∼ 2.9 − 5.5. +It is evident from Figures 1 and 2 that soft-colour and hard-colour +vary significantly and can trace a significant portion of the banana +track on the timescale of a few hours. We also note that the spectral +properties show only subtle variations as the source moves along +the CCD. The PD and PA have been calculated using ∼ 90 ks of +8 Assumes a constant PD and PA over the energy range of interest. +10−3 +0.01 +0.1 +Photons cm−2 s−1 keV−1 +1 +10 +2 +5 +20 +−2 +0 +2 +χ +Energy (keV) +Figure 4. (Top) Unfolded average X-ray spectrum (SXT:black, LAXPC: red) +of GX 9+9 , fitted with model 1. (Bottom) Residuals in units of σ. +Table 2. Polarimetric parameters obtained from PCUBE in different energy +bands. Values reported are for all three DUs combined. Associated 1σ uncer- +tainties are quoted. +Parameter +2 − 4 keV +4 − 8 keV +2 − 8 keV +Q/I (%) +−0.75 ± 0.38 +−1.36 ± 0.72 +−0.96 ± 0.39 +U/I (%) +0.57 ± 0.38 +2.89 ± 0.72 +1.36 ± 0.39 +PD (%) +0.94 ± 0.38 +3.19 ± 0.72 +1.66 ± 0.39 +PA (deg) +71.4 ± 11.5 +57.6 ± 6.4 +62.6 ± 6.7 +IXPE observations (Section 3.2) and it is difficult to identify the lo- +cation of the source on the CCD during the observations. Hence, we +also constructed an average spectrum across all the sections of the +CCD (B1 to B4), which can be described by a combination of disc +blackbody with kTin ∼ 0.6 keV and 5.6 keV corona (see Table 1 and +Figure 4). We note that model 2 is able to fit the spectra equally well, +and the average spectrum can also, equivalently, be described by a +1.4 keV blackbody and 0.8 keV disc (seed) photons Comptonized +by a 3.7 keV corona. +3.2 Spectro-polarimetric properties +The results obtained using PCUBE for the three identical detector +units (DUs) combined, in 2 − 4 keV, 4 − 8 keV and 2 − 8 keV energy +bands, are summarized in Table 2. The normalized stokes param- +eters of each DU and all three DUs combined are also presented +graphically in Figure 5. In 2 − 4 keV, the measured PD is below the +minimum detectable polarization (MDP) at the 99% level. However, +in 4−8 keV as well as over the entire IXPE energy range of 2−8 keV, +we report a significant detection of polarization (4.4σ and 4.2σ re- +spectively), with PD = 3.2% and 1.7%, and PA = 57.6◦ ± 6.4◦ and +62.6◦ ± 6.7◦ respectively. +To verify the robustness of the polarization parameters ob- +tained with PCUBE, we also followed a model-dependent spectro- +polarimetric approach. We simultaneously fitted the Stokes I, Q and +U spectra of all three DUs with model 1 (Section 3.1), multiplied +by polconst. Owing to statistics and limited energy band of IXPE, +the parameters NH, kTin, Γ and kTe were frozen to the best fit values +obtained by fitting the average spectrum. We performed the spectral +fits for the three energy bands as defined earlier, and the results are +shown in Table 3 and Figures 6a and 6b. The obtained PD and PA +values are consistent with those obtained via PCUBE, within uncer- +tainties. The contours, although not entirely coincident, are in good +agreement with each other. +MNRAS 000, 1–5 (2021) + +4 +Rwitika et al. +−6 +−4 +−2 +0 +2 +4 +6 +Q/I (%) +−6 +−4 +−2 +0 +2 +4 +6 +U/I (%) +2 +− +4 keV +DU1 +DU2 +DU3 +All DUs +−6 +−4 +−2 +0 +2 +4 +6 +Q/I (%) +−6 +−4 +−2 +0 +2 +4 +6 +U/I (%) +4 +− +8 keV +DU1 +DU2 +DU3 +All DUs +−6 +−4 +−2 +0 +2 +4 +6 +Q/I (%) +−6 +−4 +−2 +0 +2 +4 +6 +U/I (%) +2 +− +8 keV +DU1 +DU2 +DU3 +All DUs +Figure 5. Normalised stokes parameters (Q/I and U/I) obtained using PCUBE in the energy bands 2 − 4 keV (left), 4 − 8 keV (middle) and 2 − 8 keV (right), +for DU1 (orange), DU2 (blue), DU3 (green) and for the three DUs combined (black). In each panel, the red star corresponds to null polarization. +0° +30° +60° +90° +120° +150° +180° +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +N +S +E +2 +− +4 keV +4 +− +8 keV +XSPEC pol. +68.27% +PCUBE pol. +68.27% +XSPEC pol. +68.27% +PCUBE pol. +68.27% +(a) +0° +30° +60° +90° +120° +150° +180° +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +N +S +E +XSPEC pol. +68.27% +95.45% +99.73% +PCUBE pol. +68.27% +95.45% +99.73% +(b) +0° +30° +60° +90° +120° +150° +180° +0 +1 +2 +3 +4 +5 +6 +7 +8 +N +S +E +diskbb pol. +68.27% +95.45% +99.73% +nthcomp pol. +68.27% +95.45% +99.73% +(c) +Figure 6. (a) Results of PCUBE (star, dashed lines) and XSPEC (triangle, solid lines) in the 2 − 4 keV (blue) and 4 − 8 keV energy bands. 1σ contours are +plotted (b) Results of PCUBE (star, dashed lines) and XSPEC (triangle, solid lines) over 2 − 8 keV. 1σ (blue), 2σ (red) and 3σ (green) contours are shown. +(c) Individual polconst for diskbb (triangle, blue contours) and star, green contours. In all three panels, the IXPE spectrum has been fitted with the +parameters of the average spectrum from AstroSat, fitted with model 1. See text for details. +Table 3. Spectro-polarimetric parameters obtained by simultaneously fitting +Stokes spectra in XSPEC in different energy bands using model 1. Results +using single polconst as well as two polconst (one for each spectral com- +ponent) are reported. Uncertainties, upper limits quoted are at 3σ level. +Parameter +Single polconst +Multiple polconst* +2 − 4 keV +4 − 8 keV +2 − 8 keV +diskbb +nthcomp +PD (%) +< 2.03 +2.77 ± 1.92 +1.38 ± 0.99 +< 10.98 +2.96 ± 2.01 +PA (◦) +... +62.1 ± 21.9 +68.4 ± 23.3 +... +66.4 ± 21.4 +χ2/DOF +400/434 +839/884 +1281/1334 +1274/1332 +* Fitted over 2 − 8 keV +Motivated +by +the +energy-dependence +of +the +PD, +we +attempted +to +compute +the +PD +corresponding +to +each +spectral +component, +and +hence +we +fitted +the +model +tbabs*(polconst*diskbb+polconst*nthcomp) to the spectra. +The results are shown in Table 3 and Figure 6c. The polarization +of diskbb could be constrained at only the 1σ level, with PD = +4.49% and PA = 153.2◦. However, the polarization of nthcomp +is well-constrained, with PD = 2.96% and PA = 66.4◦, roughly +perpendicular to that of diskbb. +We also repeated the spectro-polarimetric fitting using model 2, +with the spectral parameters frozen to the best-fit values correspond- +ing to the average spectrum. Using a single polconst to describe +the emission, we obtained equivalent fits as with model 1, with sim- +ilar PD and PA values in 2 − 4 keV, 4 − 8 keV and 2 − 8 keV energy +bands. However, we were not able to constrain the polarization prop- +erties of the individual model components (bbodyrad and nthcomp) +at the 3σ level. +4 DISCUSSION +In this letter, we attempt to understand the geometry of the X-ray +emitting region and radiative processes in the bright atoll X-ray bi- +nary GX 9+9 using IXPE and AstroSat data. As in the case of sev- +eral NS-LMXBs, it has been observed that X-ray spectral fits of this +source with respect to different models are degenerate and produce +equivalent fits to its spectrum. In particular, it is of interest to deter- +mine the location and geometry of the Comptonizing corona, and po- +larization information along with spectroscopy can potentially dis- +tinguish between the models. +In the present work, we found that the spectrum of GX 9+9 can be +described by an optically thick Comptonized emission plus a thermal +component. The thermal component can be represented by either a +MCD (model 1) or blackbody (model 2), and both these approaches +produce statistically good descriptions of the data. To break the spec- +tral degeneracy in the NS-LMXBs, sensitive polarization measure- +MNRAS 000, 1–5 (2021) + +Spectro-polarimetric view of GX 9+9 +5 +ments were long-awaited and thanks to the superior quality of polari- +metric data from IXPE, it is now possible to attempt to distinguish +between the different spectral models, and also provide an important +probe of the geometry of the corona (see e.g. Farinelli et al. 2023). +An analysis of the IXPE data of GX 9+9 reveals that the 2−8 keV +X-ray emission of the source is polarized, with PD = 1.7% and PA +∼ 63◦. The polarization is higher in the 4−8 keV (3.2% at 4.4σ) band +compared to that in the 2 − 4 keV band (0.9% at 2.5σ). Recently, +Gnarini et al. (2022) carried out detailed polarimetric simulations of +NS-LMXBs considering two different coronal geometries, viz. shell +and slab. The authors reported the expected PD and PA from such +sources considering different system inclinations and spectral states +(HSS or LHS). The inclination of the GX 9+9 system lies between +40◦ −60◦ (Iaria et al. 2020; Hertz & Wood 1988) and the source was +found to be in the HSS (Figures 2 and 3). The simulations suggest +that at these source inclinations, the PA in 2 − 8 keV is less than +90◦ for shell geometry and always stays greater than 100◦ for slab +geometry when the source is in HSS. Our estimate of PA ∼ 63◦ +supports the shell type corona geometry. The spherical shell could +be in the form of a SL above the neutron star surface. However, +we also note here that the predicted trend for shell geometry (PD +decreasing with increasing energy, see Gnarini et al. 2022) does not +agree with the present results. +A +similar +shell-type +geometry +was +also +proposed +by +Farinelli et al. (2023) for Cyg X-2 using polarization measure- +ments from IXPE, wherein the neutron star surface provides the +seed photons and the surface is covered by a spherical corona or +SL. The uncovered disc emits the thermal component. Motivated +by this, we carried out model-dependent studies by fitting the +IXPE data with two spectral models generally associated with +these two corona geometries (shell: model 1, slab: model 2). Both +these models give similar PD and PA as obtained from PCUBE. +Using model 1, we were also able to ascertain the polarization of +the individual spectral components and the results suggest that +the Comptonized component is polarized and the disc emission is +possibly unpolarized. Farinelli et al. (2023) obtained a similar result +for Cyg X-2. +In the case of Sco X-1 and Cyg X-2, the PA was found to coin- +cide with the system symmetry axis (Long et al. 2022; Farinelli et al. +2023). Unfortunately, GX 9+9 is radio faint (van den Eijnden et al. +2021) and a jet has not yet been detected from the source. Provided +the polarization is along the system symmetry axis, which is perpen- +dicular to the accretion disc, our results may indicate that the Comp- +tonized emission originates in the boundary layer or transition layer, +as in the case of Sco X-1. Hence, simulations for alternative geome- +tries such as transition layer or wedge-like corona can provide fur- +ther insights on the polarization data from bright NS-LMXBs such +as GX 9+9 . In addition, more polarimetric observations of LMXBs +are required to address the geometry of the accretion flow for differ- +ent classes of NS-LMXBs in different spectral states. +ACKNOWLEDGMENTS +RC, VKA and KMJ thank GH, SAG; DD, PDMSA and Director, +URSC for encouragement and continuous support to carry out this +research. +This research used data products provided by the IXPE Team +(MSFC, SSDC, INAF, and INFN) and distributed with additional +software tools by the High-Energy Astrophysics Science Archive +Research Center (HEASARC), at NASA Goddard Space Flight +Center (GSFC). This work has used the data from the LAXPC +Instruments developed at TIFR, Mumbai and the LAXPC POC at +TIFR is thanked for verifying and releasing the data via the ISSDC +data archive. This work has used the data from the Soft X-ray +Telescope (SXT) developed at TIFR, Mumbai, and the SXT POC at +TIFR is thanked for verifying & releasing the data and providing +the necessary software tools. +DATA AVAILABILITY +Data underlying this work are available at High Energy Astro- +physics Science Archive Research Center (HEASARC) facility, lo- +cated at NASA-Goddard Space Flight Center and AstroSat-ISSDC +website (http://astrobrowse.issdc.gov.in/astro_archive/archive). The +MAXI light curve used in this work is publicly available at +http://maxi.riken.jp/top/slist.html. +REFERENCES +Agrawal V. K., Misra R., 2009, MNRAS, 398, 1352 +Agrawal V. K., Sreekumar P., 2003, MNRAS, 346, 933 +Agrawal V. K., Nandi A., Katoch T., 2023, MNRAS, 518, 194 +Arnaud K. A., 1996, in Jacoby G. 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N., Magdziarz P., 1996, MNRAS, 283, 193 +van den Eijnden J., et al., 2021, MNRAS, 507, 3899 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–5 (2021) + diff --git a/rdFQT4oBgHgl3EQfuDap/content/tmp_files/load_file.txt b/rdFQT4oBgHgl3EQfuDap/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15fc7464eeda45df3173df685baecdf7b1bac08b --- /dev/null +++ b/rdFQT4oBgHgl3EQfuDap/content/tmp_files/load_file.txt @@ -0,0 +1,596 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf,len=595 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='13394v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='HE] 31 Jan 2023 MNRAS 000, 1–5 (2021) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 Spectro-polarimetric view of bright atoll source GX 9+9 using IXPE and AstroSat Rwitika Chatterjee1⋆, Vivek K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Agrawal1, Kiran M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Jayasurya1, Tilak Katoch2 1Space Astronomy Group, ISITE Campus, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Rao Satellite Center, ISRO, Bengaluru 560037, India 2Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We have carried out the first spectro-polarimetric study of the bright NS-LMXB GX 9+9 using IXPE and AstroSat observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We report a significant detection of polarization of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4% over the 2 − 8 keV energy band, with a polarization angle of 63◦ ± 7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The polarization is found to be energy-dependent, with a 3σ polarization degree consistent with null polarization in 2−4 keV, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2% in 4−8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Typical of the spectra seen in NS-LMXBs, we find that a combination of soft thermal emission from the accretion disc and Comptonized component from the optically thick corona produces a good fit to the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We also attempt to infer the individual polarization of these components, and obtain a 3σ upper limit of ∼ 11% on the polarization degree of the thermal component, and constrain that of the Comptonized component to ∼ 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We comment on the possible corona geometry of the system based on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Key words: accretion, accretion discs - polarization - X-rays: binaries - X-rays: individual: GX 9+9 1 INTRODUCTION Atoll and Z-sources are luminous low-mass X-ray binaries (LMXBs) where a neutron star accretes matter from a low-mass companion through Roche-lobe overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' These two types of LMXBs trace different patterns on their colour-colour diagram (CCD, Hasinger & van der Klis 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Atoll sources exhibit high variability over timescales ranging from milliseconds to years, and depending on their X-ray luminosity, may be found either in the high soft state (HSS) or low hard state (LHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' X-ray spectra of the NS-LMXBs are complex and re- quire multiple components to describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' A combination of multi-temperature disc blackbody and Comptonized emis- sion (Di Salvo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2000a, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Agrawal & Sreekumar 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Tarana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Agrawal & Misra 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2023) has been widely used to describe the X-ray spectra of the atoll and Z- sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Here, the softer component from the disc is directly ob- served, and the Comptonized emission may come from the boundary layer (BL) or spreading layer (SL) near the surface of the neutron star (‘eastern model’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' An alternative scenario is that the accretion disc is Comptonized and blackbody emission comes from the neu- tron star surface (‘western model’), and a combination of blackbody plus a Comptonized component is adopted for describing the spec- tra of these sources (Piraino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2000, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Di Salvo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2000b, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Barret & Olive 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' GX 9+9 is an NS-LMXBs, classified as an atoll source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' It was dis- covered by a sounding rocket experiment on 1967 July 7 (Bradt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Hertz & Wood (1988) reported a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='02 h orbital pe- riod using scanning mode of HEAO A-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Several models have been used to describe the spectrum of this source, such as power-law ⋆ E-mail: rwitika@ursc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='in with exponential cutoff (Church & Baluci´nska-Church 2001), black- body plus Comptonization model (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2006), and multicolor disc (MCD) plus Comptonized blackbody emission from the SL (Savolainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Göˇgü¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (2007) described the RXTE- PCA spectrum of this source by a double blackbody plus a broad iron line at 6 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Iaria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (2020) reported the presence of a blurred re- flection component using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='3 − 40 keV spectrum of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Spectral modeling of such sources helps to understand the nature of the coronal plasma (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' temperature, optical depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' However, these models are degenerate with respect to the geometry and lo- cation of corona, and polarization information, if available, can be used to constrain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Recent polarimetric studies of several NS- LMXBs such as Sco X-1 (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2022), Cyg X-2 (Farinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2023) and GS 1826 − 238 (Capitanio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2022) have revealed the capability of polarization studies in inferring the source geometry and emission properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In this letter, we present the first spectro- polarimetric study of the bright atoll source GX 9+9 using IXPE ob- servations and archival AstroSat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2 OBSERVATIONS AND DATA ANALYSIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1 AstroSat The source GX 9+9 was observed from 2020 July 25 to 2020 July 27 with AstroSat for a net exposure time of 56 ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' To understand the spectral nature of the source, we analyzed data from the Soft X-ray Telescope (SXT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='3 − 8 keV) and Large Area X-ray Proportional Counter (LAXPC, 3 − 60 keV), which were operated in photon- counting mode and event analysis mode respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' © 2021 The Authors 2 Rwitika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 500 600 700 800 Counts/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2 Soft−colour 0 5×104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 Hard−colour Time (s) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' LAXPC 3−25 keV count rate (top), soft-colour (middle) and hard- colour (bottom) during the AstroSat observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Bin time of 16 s has been used to generate the light curve and hardness plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2 B1 B2 B3 B4 Hard-Colour Soft-Colour Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' CCD of GX 9+9 , with different parts denoted by labels B1, B2, B3 and B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Each data points corresponds to 256 s of integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' HEASOFT FTOOL1 XSELECT was used to create the image, spec- tra and light curves from the Level-2 data of SXT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The SXT count rate during the observation was above the pileup limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Hence, we used an annular region excluding the central 2′, and with outer radius 12′ to create light curves and spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' ARF file was created using the task sxtARFModule2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' LAXPC light curves and spectra were generated by the latest version of the LAXPC analysis software LaxpcSoft3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The background spectra of the dark sky was provided by the instrument team and was used for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Figure 1 shows the variation of the LAXPC 3 − 25 keV count rate, soft-color and hard-colour as a function of time, over the AstroSat observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We defined soft-colour as the ratio of count rates in 5 − 7 keV and 3 − 5 keV, and hard colour as the ratio of count rates in 10 − 20 keV and 7 − 10 keV energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We also produced the CCD of the source, which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We note that the source is in the ‘banana’ state during the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We divided the CCD into 4 sections and denote them by B1, B2, B3 and B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We created spectra for the four sections and performed joint 1 http://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='nasa.' metadata={'source': 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+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='in/∼astrosat_laxpc/LaxpcSoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='html 56500 57500 58500 59500 MJD 100 150 200 250 300 350 Count rate (mcrab) Astrosat IXPE Jul 2013 Apr 2016 Jan 2019 Oct 2021 Date Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' MAXI/GSC 2−20 keV light curve of GX 9+9 from 2012 till present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The epochs of the AstroSat and IXPE observations are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The horizontal line indicates the mean count rate of 220 mcrab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' spectral fitting of each section using the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 − 7 keV spectra from SXT and 4 − 25 keV spectra from LAXPC4 using XSPEC5 (Arnaud 1996) v12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We found that a Comptonized emission component (Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 1996 nthcomp in XSPEC) plus a thermal compo- nent, either described by a single temperature blackbody or an MCD (bbodyrad and diskbb respectively), modified by ISM absorption (tbabs), give a satisfactory fit to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 − 25 keV spectra of the source6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The seed photons for Comptonization are assumed to follow a blackbody distribution (inp_type parameter set to 0 in nthcomp), when the thermal component comes from the accretion disc (here- after, model 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' However, multi-temperature disc blackbody is used for seed photons (inp_type = 1) when thermal component is de- scribed by bbodyrad (hereafter, model 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We note here that model 1 is reminiscent of the ‘eastern’ type model where thermal component from the disc is directly observed and the Comptonized emission comes from a shell-like corona around the neutron star/SL, whereas model 2 resembles a ‘western’-like scenario, with the corona as a slab or wedge above the accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2 IXPE The Imaging X-ray Polarimetry Explorer (IXPE, Weisskopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2022), launched on 2021 December 9, consists of three identical telescopes each consisting of a mirror module assembly with a polarization-sensitive imaging X-ray detector at the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' IXPE ob- served GX 9+9 on 2022 October 9 for about 92 ks of net exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In Figure 3, we show the long-term MAXI/GSC light curve of the source and note that the source is stable and remains in the HSS over the period of both AstroSat and IXPE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We analyzed the processed Level-2 data7 using IXPEOBSSIM soft- ware v30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 (Baldini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Source was defined as a circular region of radius 60" centered at the intensity centroid, and back- ground as an annular region with the same center and inner and outer radii as 180" and 240" respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Source and background events were filtered using XPSELECT task and various binning algo- rithms were applied using XPBIN task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The average count rate was 4 Beyond 25 keV, the source is weak and the background dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 5 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='gov/xanadu/xspec/ 6 We also need two edge components at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 keV and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 keV (instrumental features) to obtain a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 7 Publicly available on the HEASARC archive MNRAS 000, 1–5 (2021) Spectro-polarimetric view of GX 9+9 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Best-fit parameters of model 1 to SXT+LAXPC spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' NH: hy- drogen column density (1022 cm−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Γ: photon index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' τ: optical depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' kTin: inner disc temperature (keV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' kTe: plasma temperature (keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Ndbb and Nnth are the normalizations of the diskbb and nthcomp components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 90% uncer- tainties are quoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Parameters B1 B2 B3 B4 Average NH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 kTin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='04 Ndbb 345±75 373±127 221±62 108±18 405±64 Γ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='14 kTe 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='55±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='59 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='56±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='51 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='82 Seed kT a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='03 Nnth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 τb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='69 χ2/DOF 598/654 656/654 608/654 683/654 628/654 a Blackbody seed photons b Assuming spherical corona 13 cts/s with the background contribution ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The polarization parameters (polarization degree PD and polarization angle PA) were extracted in the energy bands 2 − 4 keV, 4 − 8 keV and 2 − 8 keV using the model-independent PCUBE algorithm (Kislat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' For calculating the uncertainties, PCUBE assumes that the PD and PA are independent whereas they are not actually so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Hence, the con- tours of the joint measurement of PD and PA represent the uncertain- ties more appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We also performed a spectro-polarimetric model-dependent fit (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Strohmayer 2017) of the data using XSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Source and background spectra corresponding to the Stokes parameters I, Q and U were extracted using algorithms PHA1, PHA1Q and PHA1U re- spectively, and the latest response files (v12) were used in spec- tral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The best-fit models obtained from the joint SXT and LAXPC fits (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1), modified by a constant polarization (polconst8 in XSPEC), was used in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 3 RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1 Spectral properties We show the best fit spectral parameters obtained by fitting the dif- ferent sections of the atoll track using model 1 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' As the source moves along the banana track from B1 to B4, the spectrum becomes harder, with the photon index decreasing from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The temperature at the inner radius of the accretion disc increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 keV to ∼ 1 keV, and the seed photon temperature ranges from 1 keV to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The temperature of the corona varies in the range 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2 keV to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The corona is optically thick, with τ in the range ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='9 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' It is evident from Figures 1 and 2 that soft-colour and hard-colour vary significantly and can trace a significant portion of the banana track on the timescale of a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We also note that the spectral properties show only subtle variations as the source moves along the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The PD and PA have been calculated using ∼ 90 ks of 8 Assumes a constant PD and PA over the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1 Photons cm−2 s−1 keV−1 1 10 2 5 20 −2 0 2 χ Energy (keV) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (Top) Unfolded average X-ray spectrum (SXT:black, LAXPC: red) of GX 9+9 , fitted with model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (Bottom) Residuals in units of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Polarimetric parameters obtained from PCUBE in different energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Values reported are for all three DUs combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Associated 1σ uncer- tainties are quoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Parameter 2 − 4 keV 4 − 8 keV 2 − 8 keV Q/I (%) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='38 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='39 U/I (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='39 PD (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='39 PA (deg) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 IXPE observations (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2) and it is difficult to identify the lo- cation of the source on the CCD during the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Hence, we also constructed an average spectrum across all the sections of the CCD (B1 to B4), which can be described by a combination of disc blackbody with kTin ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 keV and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6 keV corona (see Table 1 and Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We note that model 2 is able to fit the spectra equally well, and the average spectrum can also, equivalently, be described by a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 keV blackbody and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='8 keV disc (seed) photons Comptonized by a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7 keV corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2 Spectro-polarimetric properties The results obtained using PCUBE for the three identical detector units (DUs) combined, in 2 − 4 keV, 4 − 8 keV and 2 − 8 keV energy bands, are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The normalized stokes param- eters of each DU and all three DUs combined are also presented graphically in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In 2 − 4 keV, the measured PD is below the minimum detectable polarization (MDP) at the 99% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' However, in 4−8 keV as well as over the entire IXPE energy range of 2−8 keV, we report a significant detection of polarization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4σ and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2σ re- spectively), with PD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7%, and PA = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6◦ ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4◦ and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='6◦ ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7◦ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' To verify the robustness of the polarization parameters ob- tained with PCUBE, we also followed a model-dependent spectro- polarimetric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We simultaneously fitted the Stokes I, Q and U spectra of all three DUs with model 1 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1), multiplied by polconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Owing to statistics and limited energy band of IXPE, the parameters NH, kTin, Γ and kTe were frozen to the best fit values obtained by fitting the average spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We performed the spectral fits for the three energy bands as defined earlier, and the results are shown in Table 3 and Figures 6a and 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The obtained PD and PA values are consistent with those obtained via PCUBE, within uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The contours, although not entirely coincident, are in good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' MNRAS 000, 1–5 (2021) 4 Rwitika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' −6 −4 −2 0 2 4 6 Q/I (%) −6 −4 −2 0 2 4 6 U/I (%) 2 − 4 keV DU1 DU2 DU3 All DUs −6 −4 −2 0 2 4 6 Q/I (%) −6 −4 −2 0 2 4 6 U/I (%) 4 − 8 keV DU1 DU2 DU3 All DUs −6 −4 −2 0 2 4 6 Q/I (%) −6 −4 −2 0 2 4 6 U/I (%) 2 − 8 keV DU1 DU2 DU3 All DUs Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Normalised stokes parameters (Q/I and U/I) obtained using PCUBE in the energy bands 2 − 4 keV (left), 4 − 8 keV (middle) and 2 − 8 keV (right), for DU1 (orange), DU2 (blue), DU3 (green) and for the three DUs combined (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In each panel, the red star corresponds to null polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 0° 30° 60° 90° 120° 150° 180° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 N S E 2 − 4 keV 4 − 8 keV XSPEC pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% PCUBE pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% XSPEC pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% PCUBE pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% (a) 0° 30° 60° 90° 120° 150° 180° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='0 N S E XSPEC pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='45% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='73% PCUBE pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='45% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='73% (b) 0° 30° 60° 90° 120° 150° 180° 0 1 2 3 4 5 6 7 8 N S E diskbb pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='45% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='73% nthcomp pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='27% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='45% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='73% (c) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (a) Results of PCUBE (star, dashed lines) and XSPEC (triangle, solid lines) in the 2 − 4 keV (blue) and 4 − 8 keV energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 1σ contours are plotted (b) Results of PCUBE (star, dashed lines) and XSPEC (triangle, solid lines) over 2 − 8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 1σ (blue), 2σ (red) and 3σ (green) contours are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (c) Individual polconst for diskbb (triangle, blue contours) and star, green contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In all three panels, the IXPE spectrum has been fitted with the parameters of the average spectrum from AstroSat, fitted with model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Spectro-polarimetric parameters obtained by simultaneously fitting Stokes spectra in XSPEC in different energy bands using model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Results using single polconst as well as two polconst (one for each spectral com- ponent) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Uncertainties, upper limits quoted are at 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Parameter Single polconst Multiple polconst* 2 − 4 keV 4 − 8 keV 2 − 8 keV diskbb nthcomp PD (%) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='77 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='99 < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='96 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='01 PA (◦) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='1 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4 χ2/DOF 400/434 839/884 1281/1334 1274/1332 Fitted over 2 − 8 keV Motivated by the energy-dependence of the PD, we attempted to compute the PD corresponding to each spectral component, and hence we fitted the model tbabs*(polconst*diskbb+polconst*nthcomp) to the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The results are shown in Table 3 and Figure 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The polarization of diskbb could be constrained at only the 1σ level, with PD = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='49% and PA = 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' However, the polarization of nthcomp is well-constrained, with PD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='96% and PA = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4◦, roughly perpendicular to that of diskbb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' We also repeated the spectro-polarimetric fitting using model 2, with the spectral parameters frozen to the best-fit values correspond- ing to the average spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Using a single polconst to describe the emission, we obtained equivalent fits as with model 1, with sim- ilar PD and PA values in 2 − 4 keV, 4 − 8 keV and 2 − 8 keV energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' However, we were not able to constrain the polarization prop- erties of the individual model components (bbodyrad and nthcomp) at the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 4 DISCUSSION In this letter, we attempt to understand the geometry of the X-ray emitting region and radiative processes in the bright atoll X-ray bi- nary GX 9+9 using IXPE and AstroSat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' As in the case of sev- eral NS-LMXBs, it has been observed that X-ray spectral fits of this source with respect to different models are degenerate and produce equivalent fits to its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In particular, it is of interest to deter- mine the location and geometry of the Comptonizing corona, and po- larization information along with spectroscopy can potentially dis- tinguish between the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In the present work, we found that the spectrum of GX 9+9 can be described by an optically thick Comptonized emission plus a thermal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The thermal component can be represented by either a MCD (model 1) or blackbody (model 2), and both these approaches produce statistically good descriptions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' To break the spec- tral degeneracy in the NS-LMXBs, sensitive polarization measure- MNRAS 000, 1–5 (2021) Spectro-polarimetric view of GX 9+9 5 ments were long-awaited and thanks to the superior quality of polari- metric data from IXPE, it is now possible to attempt to distinguish between the different spectral models, and also provide an important probe of the geometry of the corona (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Farinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' An analysis of the IXPE data of GX 9+9 reveals that the 2−8 keV X-ray emission of the source is polarized, with PD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='7% and PA ∼ 63◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The polarization is higher in the 4−8 keV (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='2% at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='4σ) band compared to that in the 2 − 4 keV band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='9% at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='5σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Recently, Gnarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (2022) carried out detailed polarimetric simulations of NS-LMXBs considering two different coronal geometries, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' shell and slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The authors reported the expected PD and PA from such sources considering different system inclinations and spectral states (HSS or LHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The inclination of the GX 9+9 system lies between 40◦ −60◦ (Iaria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Hertz & Wood 1988) and the source was found to be in the HSS (Figures 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The simulations suggest that at these source inclinations, the PA in 2 − 8 keV is less than 90◦ for shell geometry and always stays greater than 100◦ for slab geometry when the source is in HSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Our estimate of PA ∼ 63◦ supports the shell type corona geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The spherical shell could be in the form of a SL above the neutron star surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' However, we also note here that the predicted trend for shell geometry (PD decreasing with increasing energy, see Gnarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2022) does not agree with the present results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' A similar shell-type geometry was also proposed by Farinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (2023) for Cyg X-2 using polarization measure- ments from IXPE, wherein the neutron star surface provides the seed photons and the surface is covered by a spherical corona or SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' The uncovered disc emits the thermal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Motivated by this, we carried out model-dependent studies by fitting the IXPE data with two spectral models generally associated with these two corona geometries (shell: model 1, slab: model 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Both these models give similar PD and PA as obtained from PCUBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Using model 1, we were also able to ascertain the polarization of the individual spectral components and the results suggest that the Comptonized component is polarized and the disc emission is possibly unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Farinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' (2023) obtained a similar result for Cyg X-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In the case of Sco X-1 and Cyg X-2, the PA was found to coin- cide with the system symmetry axis (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Farinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Unfortunately, GX 9+9 is radio faint (van den Eijnden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' 2021) and a jet has not yet been detected from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Provided the polarization is along the system symmetry axis, which is perpen- dicular to the accretion disc, our results may indicate that the Comp- tonized emission originates in the boundary layer or transition layer, as in the case of Sco X-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' Hence, simulations for alternative geome- tries such as transition layer or wedge-like corona can provide fur- ther insights on the polarization data from bright NS-LMXBs such as GX 9+9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' In addition, more polarimetric observations of LMXBs are required to address the geometry of the accretion flow for differ- ent classes of NS-LMXBs in different spectral states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' ACKNOWLEDGMENTS RC, VKA and KMJ thank GH, SAG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' DD, PDMSA and Director, URSC for encouragement and continuous support to carry out this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' This research used data products provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional software tools by the High-Energy Astrophysics Science Archive Research Center (HEASARC), at NASA Goddard Space Flight Center (GSFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' This work has used the data from the LAXPC Instruments developed at TIFR, Mumbai and the LAXPC POC at TIFR is thanked for verifying and releasing the data via the ISSDC data archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' This work has used the data from the Soft X-ray Telescope (SXT) developed at TIFR, Mumbai, and the SXT POC at TIFR is thanked for verifying & releasing the data and providing the necessary software tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content=' DATA AVAILABILITY Data underlying this work are available at High Energy Astro- physics Science Archive Research Center (HEASARC) facility, lo- cated at NASA-Goddard Space Flight Center and AstroSat-ISSDC website (http://astrobrowse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='issdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} +page_content='in/astro_archive/archive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFQT4oBgHgl3EQfuDap/content/2301.13394v1.pdf'} 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a/sNAzT4oBgHgl3EQfPPuE/content/tmp_files/2301.01180v1.pdf.txt b/sNAzT4oBgHgl3EQfPPuE/content/tmp_files/2301.01180v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1740587226a8268fe9098aef4fac9ba928403e77 --- /dev/null +++ b/sNAzT4oBgHgl3EQfPPuE/content/tmp_files/2301.01180v1.pdf.txt @@ -0,0 +1,1793 @@ +Astronomy & Astrophysics manuscript no. sandin +©ESO 2023 +January 4, 2023 +Three-component modelling of O-rich AGB star winds +I. Effects of drift using forsterite +C. Sandin1, L. Mattsson2, K. L. Chubb3, M. Ergon4, 1, and P. M. Weilbacher5 +1 Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden +e-mail: christer.sandin@astro.su.se +2 Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, SE-10691 Stockholm, Sweden +3 Centre for Exoplanet Science, University of St Andrews, North Haugh, St Andrews, KY16 9SS, United Kingdom +4 The Oskar Klein Centre, AlbaNova, SE-10691 Stockholm, Sweden +5 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany +Submitted January 3, 2023 +ABSTRACT +Stellar winds of cool and pulsating asymptotic giant branch (AGB) stars enrich the interstellar medium with large amounts of pro- +cessed elements and various types of dust. We present a first study on the influence of gas-to-dust drift on ab initio simulations of +stellar winds of M-type stars driven by radiation pressure on forsterite particles. Our study is based on our radiation hydrodynamic +model code T-800 that includes frequency-dependent radiative transfer, dust extinction based on Mie scattering, grain growth and +ablation, gas-to-dust drift using one mean grain size, a piston that simulates stellar pulsations, and an accurate high spatial resolution +numerical scheme. To enable this study, we calculated new gas opacities based on the exomol database, and we extended the model +code to handle the formation of minerals that may form in M-type stars. We discern effects of drift by comparing drift models to +our new and extant non-drift models. Compared to our recent results of C-rich stellar winds, our two new drift models based on an +oxygen-rich chemistry show drift velocities that are higher by about a factor ten, that is 310–360 km s−1. Our new drift model mass- +loss rates are 8–20 times lower than our own non-drift models, but compared to extant models that use the same stellar parameters, our +mass-loss rates are 10–420 times lower. Meanwhile, a comparison of other properties such as the expansion velocity and grain size +show similar values. Our results show that the inclusion of gas-to-drift is of fundamental importance in stellar wind models driven by +transparent grains such as forsterite. Assuming that the drift velocity is insignificant, properties such as the mass-loss rate may be off +from more realistic values by a factor one hundred and more. +Key words. hydrodynamics – radiative transfer – stars: atmospheres – stars: AGB and post-AGB – stars: mass-loss – stars: winds, +outflows +1. Introduction +Stellar winds rule the final and decisive stages of evolution of +low-to-intermediate mass stars when they ascend the asymptotic +giant branch (AGB). The dynamic AGB stage involves increas- +ing luminosities, low effective temperatures, and stellar pulsa- +tions. Dust formation begins at a couple of stellar radii where +temperatures are low enough to prevent the newly formed grains +from evaporating, and the new dust grains absorb or scatter the +radiation and in that way attain a momentum. The grains acceler- +ate outwards and collide with particles in the gas that are dragged +along as the particles drift through the same gas. Considering all +the needed physics, it is a physical problem of a kind to simulate +the resulting dust-driven wind where low expansion velocities +are about 10 km s−1 and high mass-loss rates vary from 10−8 up +to, in extreme cases, 10−4 M⊙ yr−1. +Depending on what element dominates, AGB stars are either +oxygen-rich (M-type stars) or carbon-rich (C-type stars). The di- +chotomy is reflected in stellar wind models where dust formation +in a carbon-rich chemistry is more simple where mostly amor- +phous carbon forms. Other types of dust and minerals do not +form in sufficient numbers to be influential. +Dust formation in an oxygen-rich chemistry is more com- +plex. Spectra of circumstellar envelopes of M-type AGB stars +show characteristic silicate features at 9.7 and 18 µm (see, e.g. +Woolf & Ney 1969; Low 1970; Molster et al. 2002; Dorschner +2010; Molster et al. 2010). These features indicate that silicon- +containing grains are a dominant component of the in M-type +AGB stars. Crystalline silicate dust with features at 11, 23, 28, +33, and 69 µm is also seen (Blommaert et al. 2014), but the crys- +tallinity does not appear to be correlated with the mass-loss rats +(Liu et al. 2017). Various minerals form depending on the avail- +ability of elements that are part of the different minerals, includ- +ing olivine, pyroxene, and iron (Gail & Sedlmayr 1999). Metallic +iron, moreover, appears to be a significant component in the cos- +mic dust budget owing to the large iron depletion seen in the in- +terstellar medium (Mattsson et al. 2019). Such grains can proba- +bly form in AGB atmospheres and their scattering cross-sections +are typically large, so if they form in sufficient number, they may +contribute to the driving of the wind. Gail & Sedlmayr (2014, +Article number, page 1 of 13 +arXiv:2301.01180v1 [astro-ph.SR] 3 Jan 2023 + +A&A proofs: manuscript no. sandin +hereafter, GS14) present a refined and many ways complete ap- +proach on how to implement mineral formation in both carbon +and oxygen-rich chemistries. +Höfner (2008, hereafter H08) presents first working models +(darwin) of stellar winds in oxygen-rich chemistry. She finds that +the dust scattering cross section of larger micron-sized iron-free +silicates particles provide a high enough radiative pressure to +drive a stellar wind. Bladh & Höfner (2012) and Bladh et al. +(2013) then argue, based mostly on parameterized models of +dust, that forsterite and enstatite are the most likely dust species +that drive the stellar wind; they also present photometric prop- +erties of models that agree well with observations. Bladh et al. +(2015) present a larger set of radiation hydrodynamic models +that include non-equilibrium dust formation. The authors con- +clude that they can calculate mass-loss rates as well as spectra, +which visual and near-IR diagnostics agree with observations. +Bladh et al. (2019, hereafter B19) present the most extensive +set of calculated M-type stellar wind models available this far. +Whilst the stellar wind models of H08 up to B19 show agreement +with observations, they are based on some assumptions that we +find interesting to explore in more detail. The authors emphasize +that they calculate high mass-loss rates and photometric proper- +ties that agree well with observations. They also point out that +there are few free parameters in their radiation hydrodynamic +models. In particular, the only such free parameter they mention +is the seed particle abundance. The authors, moreover, appear to +use sticking coefficients that are always set to unity (1) to form as +much dust as possible, instead of using extant lower empirically +based values. Additional assumptions include only modeling one +(or two) dust species at a time. +Physical arguments imply that effects of drift are stronger +in these winds than in carbon-rich environments (Mattsson & +Sandin 2021, hereafter MS21). Only one extant study addresses +effects of drift, whilst assuming very low drift velocities, lack- +ing any "evidence" for higher values (Tosi et al. 2022). We think +there is good reason to check the influence of drift on results +more carefully. As we show here, drift velocities turn out to be +dramatically higher in models of M-type stars than in models of +C-type stars. Correspondingly, we also find dramatically lower +mass-loss rates. We are – with our physically and numerically +extended models – unable to reproduce the higher valued dar- +win-based mass-loss rates that the authors base their results of +good agreement with observations on. Whilst more reliable ob- +servations of mass loss show higher mass-loss rates, it seems +something important is missing in the picture of understanding +the formation of stellar winds in M-type stars. +Extant ab initio stellar wind models that include drift are +all based on a carbon-rich chemistry. Sandin & Mattsson (2020, +hereafter Paper V) include frequency-dependent radiative trans- +fer and opacity tables of both the gas and the dust, and calculate +models at high spatial resolution: results indicate important dif- +ferences between drift models and position-coupled (PC) mod- +els. Mass-loss rates, expansion velocities, and yields of dust are +affected. An additional example of a carbon-rich model where +drift is found to be an important component to understand the +observations is presented in a study of grain alignment about +IRC+10◦216 (Andersson et al. 2022); this object shows a very +high mass-loss rate, where our model nevertheless show a drift +velocity that is twice as high as the expansion velocity. +We use our simulation code T-800 of Paper V and extend it +with the rates-based description of dust formation in an oxygen- +rich chemistry. Specifically, we here focus on a wind where only +forsterite is formed. To enable this study, we calculated new +gas opacity tables for solar metallicities based on the exomol +database (Tennyson et al. 2020), and also added free-free and +bound-free opacities calculated using the jekyll code (Ergon +et al. 2018; Ergon & Fransson 2022). We are thereby, for the +first time, able to study time-dependent models using high spa- +tial resolution in an oxygen-rich chemistry that include drift. +We first make semi-analytical predictions of the drift veloc- +ity in Sect. 2 to see what we can deduce based on simple phys- +ical arguments. Thereafter, we describe the physics features of +our physically enhanced models in Sect. 3. Presentations of the +modelling procedure and results follow in Sect. 4. We discuss +the influence of drift on our results in Sect. 5 and close the paper +with our conclusions in Sect. 6. +2. Semi-analytic predictions of drift +Before we engage in numerical and physical details of our up- +dated version of T-800, we look at a simplified treatment of the +oxygen-rich stellar-wind formation problem to estimate what the +associated drift velocities could be. We address the balancing +forces that give rise to the wind in Sect. 2.1. Thereafter we, anew, +look at the concept of complete momentum coupling in Sect. 2.2 +and conclude this analysis in Sect. 2.3. +2.1. Balancing dust extinction and radiation pressure +To estimate the drift velocity for a given set of stellar parameters, +we need to estimate the radiation pressure on the dust compo- +nent. Thus, we need to know the photon-to-dust grain momen- +tum transfer efficiency. Absorption and scattering of photons by +dust grains is modelled with the effective cross sections σabs, ν +and σsca, ν, where ν is the frequency. The efficiency of absorption +and scattering, or the combination of the two (extinction), is usu- +ally defined relative to the geometric cross section σ. For spher- +ical grains, σ = πa2, where a is the grain radius. The absorption +efficiency Qabs,ν = σabs, ν/σ is related to the extinction and scat- +tering efficiencies Qext,ν = σext, ν/σ and Qsca,ν = σsca, ν/σ, as +Qabs,ν = Qext,ν − Qsca,ν. +(1) +To calculate a correct radiation pressure, it is necessary to use the +absorption efficiency (we name this term Qabs,ν(pr) in Paper V), +Qrp, ν = Qext,ν − gsca, νQsca,ν = Qabs,ν − (gsca, ν − 1) Qsca,ν, +(2) +where gsca, ν = ⟨cos θ⟩ν is the average scattering angle. Mie the- +ory (Bohren & Huffman 1983) provides these efficiencies as well +as the average scattering angle. +In many extant works on AGB winds (e.g., Sandin & Höfner +2003; Mattsson et al. 2008, 2010), the radiation pressure is calcu- +lated assuming dust grains are small compared to the wavelength +of the incident radiation. This, so called, small-particle limit +(SPL) approximation leads to the simplification Qext,ν = Q′ +ν/a, +where Q′ +ν is a function of only the frequency (Wickramasinghe +1972). Figure 1 shows how Qrp, ν, as computed based on Mie +theory, compares to the corresponding SPL value of Qext,ν, using +optical constants of (Jäger et al. 2003). An important feature is +the [blue] “peacock feather” region where the Mie-theory based +radiation force on grains of radius a in an optically thin atmo- +sphere is (where the Eddington flux Hν(r) ≈ 0.25(R⋆/r)2), +frad,d = π +c +�R⋆ +r +�2 +nd (a) +� ∞ +0 +a2 Qrp, ν (a) Bν(Teff) dν, +(3) +where R⋆ is the stellar radius, nd the grain number density, and +Bν the Planck function. Grain sizes in the region of relevant val- +Article number, page 2 of 13 + +C. Sandin et al.: Three-component modelling of O-rich AGB star winds +6.5 +6.0 +5.5 +5.0 +4.5 +4.0 +3.5 +log(a) [cm] +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +log( ) [cm] +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +log(Qrp / QSPL +ext ) +Fig. 1. Ratio of the radiation pressure efficiency +Qrp, ν to the SPL extinction efficiency versus the +grain radius and wavelength, assuming spheri- +cal forsterite grains. +ues (0.1 <∼ a <∼ 0.5 µm) result in a radiation pressure about 300– +30 times lower in the spectral region near the typical flux peak +of M-type AGB stars (λ ≃ 1 µm) when the SPL is assumed. +2.2. Momentum coupling and equilibrium drift +The fraction of the momentum that is transferred from the ra- +diation field to the gas is referred to as the level of momentum +coupling. In Paper V, complete momentum coupling (CMC) was +defined as the case of force balance between radiation on the one +hand and drag and gravity on the other, although the amount of +momentum lost owing to the gravitational potential is negligible. +Equating radiation and drag force is also a common definition of +CMC. +Drift is a non-linear dynamic phenomenon, but simulations +(in particular those of Paper V) show that an equilibrium tends +to develop in most cases. Equilibrium drift can be defined as +the situation where the Lagrangian derivatives of gas and dust +velocity are equal, i.e., dv/dt = du/dt. This, in turn, means that +the equilibrium drift velocity ˚vD is constant with respect to time +and that ˚vD is governed by a simple algebraic equation instead of +a hard-to-solve partial differential equation. +Assuming equilibrium drift and CMC defined as above, the +drag force is fdrag = frad,d − fgrav,d. Defining the dimensionless +variable +S D = ˚vD +vζ +, +and +vζ = +� +ζTg = +� +128kB +9πµmH +Tg, +where vζ is a modified thermal velocity, kB is the Boltzmann con- +stant, µ the mean molecular weight, mH the mass of a hydrogen +atom, and Tg the gas temperature. We then have +S 2 +D = −1 +2 + +������� +1 +4 + +� frad, d − fgrav, d +fζ +�2������� +1 +2 +. +(4) +where fgrav,d ≈ ρd GM⋆r−2 is the point-mass approximation for +the gravitational force, ρd the dust density, G the Gravitational +constant, and fζ = πa2 nd ρg v2 +ζ can be seen as a thermal coupling +7.0 +6.5 +6.0 +5.5 +5.0 +4.5 +4.0 +3.5 +log(a) [cm] +2 +1 +0 +1 +2 +3 +log(vD) [km s +1] +L3.85T24 +L3.85T27 +log(M) = +8 [M +yr +1] +log(M) = +7 [M +yr +1] +log(M) = +6 [M +yr +1] +log(M) = +5 [M +yr +1] +Fig. 2. Equilibrium drift velocity ˚vD versus grain radius, assuming +CMC. Four lines show different wind densities (mass-loss rates) as ex- +plained in the legend. The grey bullet and square indicate the locations +of the two numerical simulations with drift presented in Sect. 4.3. +force between gas and dust; and here, ρg is the gas density. For +a given mass-loss rate ˙M, wind expansion velocity u∞, luminos- +ity L⋆, effective temperature Teff, and stellar mass M⋆, we can +now estimate the equilibrium drift velocity ˚vD using the above +equation in combination with the condition for mass conserva- +tion ˙M = 4π r2 ρgu∞ and L⋆ = 4π σSB R2 +⋆ T 4 +eff, where σSB is the +Stefan-Boltzmann constant. +Article number, page 3 of 13 + +A&A proofs: manuscript no. sandin +2.2.1. Predicted drift velocities +In Fig. 2 we show the expected equilibrium drift velocity ˚vD +versus grain radius a for M⋆ = 1 M⊙, log(L⋆) = 3.85 L⊙, +Teff = 2700 K and four different assumptions of mass-loss rates, +chosen to represent the range of mass-loss rates predicted by +models (B19) and observations (see, e.g., Uttenthaler et al. 2019, +and references therein). We note that there is ample room for +high drift velocities and that the maximum drift velocity of each +curve occurs for grain radii that are quite typical for M-type AGB +stars according to extant models (B19). +It is clear that a wind driven by radiation pressure on +forsterite grains leads to high drift velocities, even in case of +massive outflows. It has been argued that drift is negligible in +winds associated with very high mass-loss rates (Höfner & Olof- +sson 2018), which seems to be the case for carbon stars (see +Fig. 3a in Paper V); however, our more recent model of the stel- +lar wind in IRC+10◦216 says otherwise (see Fig. 8 in Andersson +et al. 2022). But given that M-type AGB stars have wind speeds +of u∞ ∼ 10 km s−1 and typical grain sizes a ≃ 0.5 µm, the drift +factor FD = 1 + vD/u∞ > 2 also for intense outflows. (FD > 2 +corresponds to a situation where the drift velocity vD exceeds the +gas expansion velocity u∞ and the dust mass-loss rate is typically +reduced by an order of magnitude or more.) We tentatively con- +clude that including drift in the modelling of M-type AGB stars +is of absolute fundamental importance, as PC models do not pro- +vide a correct result even in the high-mass-loss limit. As we shall +see, this conclusion is also confirmed by our detailed modelling, +described below. +2.3. Physical interpretation of the PC assumption +The PC assumption is incompatible with the idea that AGB +winds form by friction between radiatively accelerated dust +grains and gas particles. MS21 argue that there is no realistic +physical limiting case which leads to PC. Whilst this must still +be true, we shall here discuss the (unrealistic) limit where PC is +formally true. +Considering Eq. (4), we note that ( frad,d − fgrav,d)/ fζ ≪ 1 im- +plies S D = 0, i.e., no drift. If we ignore the case frad,d = fgrav,d, +this limit requires that fζ is very large and, in particular, much +larger than the net radiation force ˜frad,d = frad,d − fgrav,d. For this +to occur in the case given in the previous section (with frad,d ob- +tained from Eq. (3)), the modified thermal velocity vζ has to be +of the order 100 km s−1 unless ρg is several orders of magnitude +higher than expected in a realistic wind. Such a high vζ corre- +sponds to gas temperatures of the order 105 K, which is com- +pletely unrealistic. It is, in fact, fair to say that the physical in- +terpretation (or consequence) of the assumption of PC is quite +absurd. The semi-analytic predictions of drift velocity presented +here provide a solid theoretically founded reason for pursuing +detailed modelling of winds of M-type AGB stars with gas and +dust treated as dynamically decoupled phases. +3. Model features and improvements of T-800 +The model features of our radiation hydrodynamic model code +T-800 is described in Paper V. As in the C-rich chemistry, we +model three components in the O-rich chemistry described here: +the gas, the radiation field, and a dust component consisting of +forsterite (Fo) mineral grains. +In comparison to the moments method used to describe dust +formation in the C-rich chemistry, we replace the four dust mo- +ment equations (K0–K3) with one rate equation for the forma- +tion of each mineral k, and the carbon number density equa- +tion (Eq. (5) in Paper V) with corresponding equations for each +affected tracer element j. All physics of mineral formation in +oxygen-rich chemistry we require is developed and described by +Gail & Sedlmayr (1999) and GS14. The adjusted equations are +∂ +∂tnd,kNk + ∇ · �nd,kNkvk +� += +qk, +(5) +∂ +∂tn j + ∇ · +� +nju +� += +−ν j +kqk, +(6) +where t is the time, nd,k the seed particle density, Nk the number +of monomers, vk the mean dust particle velocity, qk the sum of +the source and sink terms owing to grain formation, nj the tracer +element atom number density, ν j +k the number of tracer element +atoms per monomer, and u the gas velocity. In this approach, +there is no description of nucleation. Instead, seed particles of +species are assumed to exist everywhere and +nd,k = ρg +ϵk +muµ, +(7) +where ρg is the gas density, ϵk the seed particle abundance, mu the +atomic mass constant, and µ = 1.26 the mean molecular weight. +3.1. The mineral rate equation +The rate equations describe the number of monomers Nk +throughout the model domain. The rate equation source term ow- +ing to grain growth, evaporation, and destruction is +qk = nd,k +������4πa2 +k +� +Jgr,k − Jev,k +� +− +1 +τksp,n +������ , +(8) +where ak is the particle radius, Jgr (Jev) the growth (evaporation) +rate, and 1/τsp,n the rate of non-thermal sputtering. The grain +radius is described using the grain volume V and the monomer +volume V1, +ak = +�3Vk +4π +� 1 +3 +, +Vk = NkV1,k, +V1,k = Akmu +ρm,k +, +(9) +where Ak is the molecular weight and ρm,k the mineral intrinsic +density. +The term that describes grain growth and evaporation is writ- +ten as (GS14, Eqs. 12.101, 12.102, and 12.108) +Jgr,k − Jev,k = ξk +p j +�2πmjkBTg +���������φk − 1 +ac +k +� +Tg +Td +��������� , +(10) +where ξk is the the drift-velocity-dependent sticking coeffi- +cient, p j and mj are the partial pressure and mass of the rates- +determining component, respectively. Moreover, kB is Boltz- +mann’s constant, Tg the gas temperature, φk the drift correction +factor, ac +k the reaction activity, and Td the dust temperature. +The sticking coefficient1 ξk is assumed to decrease when the +drift velocity becomes high in relation to the binding energy +Eb,k (Eq. (14) in Krüger & Sedlmayr 1997 and Eq. (13) in Sandin +& Höfner 2004, hereafter Paper III). +ξk = ξ(k) exp +��������− +������ +Akmu ˜w2 +k +8Eb,k +������ +3�������� , +(11) +1 Please note that inMS21 ξ was used to denote the grain-growth ve- +locity, which is a different, although not unrelated, quantity. +Article number, page 4 of 13 + +C. Sandin et al.: Three-component modelling of O-rich AGB star winds +where the velocity of dust grains relative to gas particles ˜wk is +(Eqs. 11 and 12 in Paper III) +˜wk = +������� +8kBTg +16πAkmu ++ +v2 +D,k +16 +������� +1 +2 +. +(12) +Here, vD,k = vk − u is the drift velocity. Moreover, the drift cor- +rection factor φk is (Eq. (12.19) in GS14) +φk = +�������1 + πAkmu +8kB +v2 +D,k +Tg +������� +1 +2 +. +(13) +We use the same expression for non-thermal sputtering (1/τk +sp,n) +as we do in Paper III. Although, here we account for collisions +with H2 molecules, in addition to H and He atoms. +3.2. Growth and evaporation of forsterite +We use two tracer elements: silicon and magnesium. There +are in this case eleven equations, instead of thirteen equations +when using the moments approach and a carbon-rich chemistry. +Forsterite grain growth takes place through collisions of seed +grains and extant grains with either SiO molecules or Mg atoms; +when addition of SiO (Mg) is the rate determining reaction step, +pj = pSiO and mj = mSiO (pj = pMg and mj = mMg). +The basic chemical reaction for the forsterite formation, as +well as its evaporation through chemical sputtering, is +2Mg + SiO + 3H2O ↔ Mg2SiO4(s) + 3H2 +(14) +and the (chemical sputtering) reaction activity ac +Fo is (see +Eqs. 12.60, 12.103, and 12.104 in GS14) +1 +ac +Fo += +p3 +H2 +pSiOp2 +Mgp3 +H2O +Kp (SiO) K3 +p (H2O) +Kp +�Mg2SiO4 +� K3p (H2), +(15) +where the four equilibrium constants Kp are calculated at the +dust temperature Td. +3.3. Partial pressures of atoms and molecules +The number densities of the molecules in the gas phase that are +part of the grain formation as well as the activities that deter- +mine when dust grains form are calculated in an equilibrium +chemistry of molecules with hydrogen, oxygen, carbon, nitro- +gen, aluminium, silicon, and sulfur, following the approach of +GS14 (chapter 10.3). The considered atoms and molecules are: +H, H2, O, OH, H2O, CO, CO2, CH4, N, N2, NH3, HCN, Al, +AlO, AlS, AlOH, AlO2H, Al2O, Al2O2, Si, SiO, SiO2, S, SO, +HS, H2S, SiS, and S2. Magnesium is assumed to be present as +free atoms. +All number densities and activities are calculated for the tem- +perature range 100 ≤ Tg ≤ 10 000 K. We use equilibrium con- +stants Kp – that are often referred to as dissociation constants +– of Sharp & Huebner (1990), GS14 (see their Table A.5), and +NIST JANAF.2. +4. Modelling procedure and results +We first briefly point at our modelling procedure in Sect. 4.1 and +then describe the physics setup and choice of model parameter +sets in Sect. 4.2. We present our results in Sect. 4.3. +2 Equilibrium-constant data of NIST/JANAF can be retrieved from +https://janaf.nist.gov/ +Table 1. References of used atom and molecule datasets as well as num- +ber of energy levels accounted for in each entry. +molecule +dataset +References +Energy levels +C +Kurucz +1 +999 +C2 +8states +2, 3 +44 189 +C2H2 +aCeTY +4 +5 160 803 +CH +MoLLIST +5, 6 +2526 +CH4 +YT34to10 +7, 8 +8 194 057 +CN +Trihybrid +9, 10, 11 +7703 +CO +Li2015 +12, 13 +6383 +CO2 +UCL-4000 +14 +3 562 798 +CS +JnK +15 +11497 +CrH +MoLLIST +6 +1646 +FeH +MoLLIST +16, 6 +3564 +H2 +RACPPK +17 +302 +H2O +POKAZATEL +18 +810 269 +H2S +AYT2 +19 +220 618 +HCl +HITRAN +20 +335 +HCN +Harris +21, 22 +168 110 +HF +Coxon-Hajig +23, 24, 13 +684 +LaO +BDL +25 +38 208 +MgH +XAB +26 +1303 +N +Kurucz +1 +283 +N2 +WCCRMT +27, 28, 29 +40 380 +NH3 +CoYuTe +30, 31 +5 095 730 +O +Kurucz +1 +201 +OH +MoLLIST +32, 33, 6 +1878 +SO2 +ExoAmes +34 +3 270 270 +SiO +SiOUVenIR +35 +174 250 +SiS +UCTY +36 +10 104 +TiO +Toto +37 +236 308 +VO +VOMYT +38 +638 958 +YO +SSYT +39 +79 440 +References. (1) Kurucz & Bell (1995); (2) Yurchenko et al. (2018b); +(3) McKemmish et al. (2020); (4) Chubb et al. (2020); (5) Masseron +et al. (2014); (6) Bernath (2020); (7) Yurchenko & Tennyson (2014); +(8) Yurchenko et al. (2017); (9) Brooke et al. (2014); (10) Syme & +McKemmish (2020); (11) Syme & McKemmish (2021); (12) Li et al. +(2015); (13) Somogyi et al. (2021); (14) Yurchenko et al. (2020); +(15) Paulose et al. (2015); (16) Dulick et al. (2003); (17) Roueff et al. +(2019); (18) Polyansky et al. (2018); (19) Azzam et al. (2016); (20) Gor- +don et al. (2017); (21) Harris et al. (2006); (22) Barber et al. (2014); +(23) Li et al. (2013); (24) Coxon & Hajigeorgiou (2015); (25) Bernath +et al. (2022); (26) Owens et al. (2022); (27) Western et al. (2018); +(28) Western (2017); (29) Shemansky (1969); (30) Al Derzi et al. +(2015); (31) Coles et al. (2019); (32) Brooke et al. (2016); (33) Yousefi +et al. (2018); (34) Underwood et al. (2016); (35) Yurchenko et al. +(2022); (36) Upadhyay et al. (2018); (37) McKemmish et al. (2019); +(38) McKemmish et al. (2016); (39) Smirnov et al. (2019) +4.1. Modelling procedure +We follow the modelling procedure described in Sect. 3.1 in +Paper V. Due to the low outflow velocity of the wind (u∞ <∼ +10 km s−1), we set the outer boundary here at rext +final = 20R⋆. We +use Nd = 840 grid points, which very nearly corresponds to the +grid point arrangement we achieve when using Nd = 1024 and +rext +final = 40R⋆. It appears to be sufficient to evolve the wind mod- +els for a time interval of about 100 P (stellar pulsation periods) +as the wind structures reach a state of equilibrium before that. +Article number, page 5 of 13 + +A&A proofs: manuscript no. sandin +4.2. Physics setup and selection of model parameters +We introduce effects of gas-to-dust drift using one mean dust ve- +locity. We compare the new drift models to PC (non-drift) mod- +els that are in all other ways are equivalent to the drift models. +We used solar abundances of Anders & Grevesse (1989), +with values for C and O of Grevesse & Sauval (1994). As do +B19, we set the pulsation period (P) using the P–L∗-relation of +Whitelock et al. (2009). To correct for too small bolometric vari- +ations (Gautschy-Loidl et al. 2004), B19 (see their Sect. 2.2) in- +troduce a free parameter fL that allows larger variations of the +luminosity at the inner boundary. Whilst we added the option to +T-800 to use a freely chosen value on fL, we use fL = 1 here +as we do not calculate model spectra and results of a PC model +using both a approaches are indistinguishable. +Next, we describe our approach to calculate gas opacities in +Sect. 4.2.1, dust properties in Sect. 4.2.2, and our selection of +model parameters in Sect. 4.2.3. +4.2.1. Gas opacities +In Paper V, we used tabulated gas opacities κν +� +ρg, Tg +� +that were +created for carbon-rich chemistries with the coma code (Aringer +2000; Aringer et al. 2009) for 319 wavenumbers in the interval +400 ≤ ˜ν ≤ 39 480 cm−1, 50 temperatures in the interval 1000 ≤ +Tg ≤ 10 000 K, and 24 densities in the interval −18 ≤ log10 ρg ≤ +−6 g cm−3. +Here, we calculated new bound-bound gas opacities based on +data of the exomol project (Tennyson et al. 2020).3 The calcula- +tions make use of data for the following 30 atoms and molecules: +C, C2, C2H2, CH, CH4, CN, CO, CO2, CS, CrH, FeH, H2, H2O, +H2S, HCl, HCN, HF, LaO, MgH, N, N2, NH3, O, OH, SO2, SiO, +SiS, TiO, VO, and YO, see Table 1. The number of energy lev- +els for each dataset is specified here, however the corresponding +number of transitions, or lines, is typically at least an order of +magnitude larger. For example, the ExoMol aCeTY C2H2 line +list has around 5.2 million energy levels and 4.3 billion tran- +sitions (Chubb et al. 2021). We used exocross (Yurchenko et al. +2018a) to calculate cross sections σl for each atom and molecule +l at 102 750 wavenumbers in the interval 100 ≤ ˜ν ≤ 41 200 cm−1, +105 temperatures in the interval 100 ≤ Tg ≤ 10 000 K, and 24 +gas densities in the interval −18 ≤ log10 ρg ≤ −6 g cm−3. The +cross sections are resampled to a coarse grid of a pre-defined set +of wavenumbers, where the resulting cross section is the aver- +age of the ten nearest cross sections on the finer grid. Currently, +we used 384 wavenumbers; this is an even multiple of the num- +ber of cores available on each node (2 × 64) on a current high- +performance cluster we used. Individual cross sections are there- +after converted to bound-bound opacities by multiplying with the +corresponding partial pressure pl as +κbb, l = +pl +kBTg +σl +ρg +� +cm2g−1� +. +(16) +We calculated partial pressures for the 27 molecules listed above +as well as the three individual atoms using the same approach as +in Sect. 3.3. +Bound-bound opacities become low at higher temperatures +(Tg >∼ 2000 K), where instead bound-free opacities κbf and free- +free opacities κff dominate. We calculated such opacities using +the jekyll code (Ergon et al. 2018; Ergon & Fransson 2022), +see Appendix A for more information. Free-free opacities κff,i +were calculated for each ion i separately using an expression that +3 https://www.exomol.com/ +Table 2. Dust parameters: forsterite +parameter +value +B19a +unit +Reference +ϵFo +10−15 +10−15 +1 +AFo +140.694 +140 +2, 3 +ρm,Fo +3.21 +3.27 +g cm−3 +2, 3 +ξFo +0.1 +1.0 +4 +Eb,Fo +3.5 +– +eV +5 +νSi +Fo, νMg +Fo +1, 2b +1, – +References. (1) B19; (2) Lide (1995); (3) GS14, Table 12.1; (4) GS14, +Sect. 12.7.1; (5) Barlow (1978), the “Silicate” entry in Table 4 +Notes.a The values we assume B19 use are specified by Höfner et al. +(2016). (b) For all elements j, but Mg and Si, ν j +Fo = 0. +0 +20 +40 +60 +vD [km s−1] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ξk +Fo +Fa +En +Fs +Crn +Irn +Car +Mos +Nin +Tg = 1000K +Fig. 3. The sticking coefficient ξk (Eq. (11)) versus the drift velocity vD +for nine different minerals, assuming a gas temperature Tg = 1000 K. +depends on provided electron and ion densities (e.g. Eq. (5.149) +in Hubeny & Mihalas 2015). +Finally, bound-bound and bound-free opacities of individual +atoms, molecules, and ions and free-free opacities of ions are +summed up to provide a total abundances-dependent gas opacity +for each pair of gas density and gas temperature, +κν +� +ρg, Tg +� += +� +l +κbb, l + +� +i +�κbf, i + κff, i +� . +Each set of abundances-specific opacities are saved in a bi- +nary file tabulated in wavenumber, density, and temperature. The +opacities are interpolated in density and temperature for each in- +dividual wavenumber in the radiative transfer calculations using +two-dimensional rational splines (Späth 1995). +We were at first kindly provided with the same opacity table +for solar metallicities that B19 use (Aringer, priv. comm.). Due to +unknown reasons, we were unsuccessful in using these data with +our new models. We discuss these extant opacity data and make +a simple comparison with our new opacities in Appendix B. +4.2.2. Dust properties +We list all forsterite-specific dust parameters we used in Table 2. +We illustrate how the sticking coefficient ξk varies with the drift +velocity for forsterite and eight other minerals in Fig. 3 (cf. Fig. 1 +in Paper III); the parameters of the eight additional minerals are +taken from the same set of references as for forsterite, noting that +Article number, page 6 of 13 + +C. Sandin et al.: Three-component modelling of O-rich AGB star winds +Table 3. Model parameters. Six columns specify the: model name, stel- +lar mass M⋆, stellar luminosity L⋆, effective temperature Teff, pulsation +amplitude ∆ up, and luminosity-dependent pulsation period P. +model +M⋆ +log (L⋆) +Teff +∆ up +P +[M⊙] +[L⊙] +[K] +[ km s−1] +[d] +L3.85T24 +1.00 +3.85 +2400 +3.0 +478 +L3.85T27 +1.00 +3.85 +2700 +4.0 +478 +the binding energies of several minerals are highly uncertain. +The figure shows that the sticking coefficient drops to 0 when +vD >∼ 20 km s−1 for larger mineral monomers such as forsterite +(Fo), fayalite (Fa), enstatite (En), and ferrosilite (Fs) and also +for carbon (Car). Higher velocities are possible with corundum +(Crn; vD ≈ 40 km s−1) as well as iron (Irn), moissanite (Mos), +and niningerite (Nin) where the cutoff drift velocity is about +60 km s−1. Hence, there is for forsterite no grain growth when +the drift velocity is higher than 20 km s−1. +All models were calculated using Mie scattering. Optical +constants (nν, kν) are taken from Jäger et al. (2003).4 To achieve +results comparable with B19, we used the seed particle abun- +dance ϵFo = 10−15 with all our calculations, as well as the same +sticking coefficient with our PC models, ξ(Fo)(PC) = 1.0. +4.2.3. Selection of model parameters +Our new model calculations of M-star objects are as demanding +as those of C stars in Paper V. Here we calculated two sets of +models to get a first impression of how the formation of stel- +lar winds of M stars work when the dust component is allowed +to drift relative to the gas. At first, we selected the proof-of- +concept-model B of H08, which she uses to successfully illus- +trate that dust-driven winds also form in M stars when Mie scat- +tering is used in the description of dust extinction instead of the +SPL approximation. Secondly, we selected a model of B19 one +with a lower effective temperature. The model parameters are +collected in Table 3. +4.3. Results +As in Paper V, we characterize wind models with a set of proper- +ties that are temporally averaged at the outer boundary. The mass +loss rate ⟨ ˙M⟩ and the terminal velocity ⟨u∞⟩ characterize the gas. +The degree of condensation of silicon ⟨ f Si +cond⟩ and magnesium +⟨ f Mg +cond⟩, the dust-to-gas mass-loss ratio ⟨ ˙Md/ ˙M⟩, the mean grain +radius ⟨rd⟩, and the terminal drift velocity ⟨vD,∞⟩ characterize the +dust. +The drift-velocity dependent [true] degree of condensation +of tracer element j accounting for all minerals k is calculated as +(cf. Eq. (12.81) in GS145 and Eq. (33) in Paper V) +f j +cond = +� +k +4πa3 +j +3 +1 +V1, j +1 +FD,k +ϵkν j +k +εj += 1 +εj +� +k +Nkϵkν j +k +FD,k +, +(17) +where FD,k = vk/u = 1 + vD,k/u is the drift factor and εj the +abundance of element j. In PC models, the modified degree of +4 The optical data can be retrieved from https://www.astro.uni- +jena.de/Laboratory/OCDB/amsilicates.html. +5 We ignore the seed particle radius a0,j used by GS14, noting that the +contribution of seed particles to the degree of condensation is miniscule +already at small particle radii a j that are barely larger than a0,j. +Table 4. Temporally averaged quantities at the outer boundary; see Section 4.3. From the left, the first three columns specify the model name (see Table 3), if PC or drift is used (P/D), and the +sticking coefficient (ξ(Fo)). Nine column pairs show the temporally averaged: mass loss rate ⟨ ˙M⟩, terminal velocity ⟨u∞⟩, terminal drift velocity ⟨vD,∞⟩, modified ⟨�f Si +cond⟩ and true ⟨ f Si +cond⟩ degree of +condensation of silicon, modified ⟨�f Mg +cond⟩ and true ⟨ f Mg +cond⟩ degree of condensation of magnesium, dust-to-gas mass-loss ratio ⟨δdgFD⟩, and dust radius ⟨aFo⟩. A relative fluctuation amplitude ˆr is +provided for each property; a subscript m indicates that the shown value was multiplied with a factor 103. The final columns show the outflow classification class: periodic (l×p), and quasi-periodic +(lq); l indicates the (multi-)periodicity of the gas/dust outflow in the unit of the piston period P. Rows of drift models are shown in boldface. +model +P/D ξ(Fo) +108 ⟨ ˙M⟩ +⟨u∞⟩ +⟨vD,∞⟩ +⟨�f Si +cond⟩ +⟨ f Si +cond⟩ +⟨�f Mg +cond⟩ +⟨f Mg +cond⟩ +104� +δdgFD +� +102⟨aFo⟩ +class +[M⊙ yr−1] +[ km s−1] +[ km s−1] +[µm] +ˆr +ˆr +ˆr +ˆr +ˆr +ˆr +ˆr +ˆr +ˆr +L3.85T24 +P +1.0 +174 +98 +15.1 +1.3 +0.282 0.088 +0.282 0.088 +0.526 0.16 +0.526 0.16 +10.1 +3.2 +55.0 +6.9 +3.1q +P +0.1 +26.7 +0.76 +5.07 46m +0.127 2.3m +0.127 2.3m +0.236 4.2m +0.236 4.2m +4.55 +80m +42.7 +0.26 +1q +B19 +P +1.0 +248 +10.0 +0.18 +0.18 +6.49 +48 +– +D +1.0 +1.4 +0.16 +13.3 +0.17 +311 19 +0.339 0.13 +13.7m +4.9m +0.632 0.24 +25.6m +9.1m +301 +120 +58.5 +6.8 +1q +D +0.1 +23.2 +0.31 +12.6 +0.24 +241 19 +0.303 0.17 +14.7m +7.6m +0.565 0.32 +27.4m +0.014 +226 +140 +55.4 +9.8 +q +L3.85T27 +P +1.0 +17.8 +0.62 +4.66 25m +0.124 2.6m +0.124 2.6m +0.231 4.9m +0.231 4.9m +4.44 +95m +42.4 +0.30 +1q +H08-B +P +1.0 +80 +7 +0.15 +0.15 +45 +– +B19 +P +1.0 +362 +9.2 +0.15 +0.15 +5.20 +45 +– +D +0.1 +0.86 +0.20 +11.9 +0.58 +363 43 +0.315 0.18 +9.87m 5.1m +0.587 0.33 +18.4m +9.5m +365 +230 +56.1 10 +2p +Article number, page 7 of 13 + +A&A proofs: manuscript no. sandin +condensation is (cf. Eq. (12.85) in GS14) +�f j +cond = 1 +εj +� +k +Nkϵkν j +k. +(18) +Moreover, the dust-to-mass mass-loss ratio is +˙Md +˙M = +� +k +ρd,k +ρg +v∞,k +u∞ += +� +k +δdg,kFD,k, +(19) +where δdg,k = ρd,k/ρg. +We calculated a relative fluctuation amplitude ˆr for each +property Q as ˆr = σs/Q, where σs is the (sample) standard devia- +tion of the property Q in the time interval that is used to measure +the same property. We show results of our model calculations in +Table 4. +5. Discussion +We analyze the new results of our PC models in Sect. 5.1 and +then compare PC models with our new drift models in Sect. 5.2. +5.1. Comparing results of the PC (non-drift) models +5.1.1. The lower temperature model L3.85T24 +Our PC models using a unity sticking coefficient (ξ(Fo) = 1.0) re- +veal a more massive wind than when ξ(Fo) = 0.1. The mass-loss +rate and expansion velocity are 6.5 and 3.0 times higher, respec- +tively. A lower ratio of 2.2 is seen in the degree of condensation +and the dust-to-gas density ratio. The average grain radius is only +30 % higher. The fluctuation amplitudes are 26–40 times higher +in all values, but the mass-loss rate, where it is 130 times higher. +Clearly, a unity sticking coefficient results in a much more vari- +able wind where all values but the grain radius are drastically +higher, but the increase is with the exception of the mass-loss +rate 30–200 % and not a factor 10. +In comparison to B19, and assuming a unity sticking coeffi- +cient, our values on the degrees of condensation and dust-to-gas +density ratio are 51–58 % higher, the grain radius 15 % higher +and the mass-loss rate 30% lower. These values compare fairly +well. +5.1.2. The proof-of-concept model L3.85T27 +H08 presents model L3.85T27 (“model B”) as a proof of the con- +cept that scattering on larger dust particles in place of absorption +allows formation of massive stellar winds in M stars. B19 calcu- +late a model with the same stellar parameters, also with 100 grid +points, but use a somewhat longer pulsation period (P = 478 d, +instead of P = 390 d) and also favor a lower pulsation ampli- +tude (∆ up = 2 km s−1). We used the same stellar parameters as +B19, where ∆ up = 4 km s−1. We also set the sticking coefficient +to unity, but we used a higher spatial resolution achieved with +ND = 840 grid points. We show the resulting values of both H08 +and B19 in Table 4 for easy reference. +Notably, our new values on the silicon degree of condensa- +tion and the mean grain radius agree well with these two studies; +our values are 83 % and 94 % of their values, respectively. Sim- +ilarly, our value on the dust-to-gas ratio is 85 % of the value of +B19. Regarding the mass-loss rate and expansion velocity, our +values are 25 % and 67 % (4.7 % and 51 %) of the values of H08 +(B19). Our mass-loss rate is, obviously, closer to the value of +H08, whilst the difference is much larger in comparison to B19. +5.2. Comparing results of the drift models +5.2.1. Effects of the sticking coefficient +The drift model shows a different result in comparison to the PC +model L3.85T24 – the values of the two drift models are similar +(they differ by 5.6–33 %), with the exception of the mass-loss +rate where the value of the unity sticking coefficient model is +6.0 % of the other model. Thus, the model with the lower sticking +coefficient results in a much higher mass loss. Plausibly, because +less efficiently formed dust grains can flow to regions where they +more effectively contribute to wind formation, before there is +enough dust to accelerate both the dust and gas outwards. The +fluctuation amplitudes are, moreover, all very similar (they differ +by 0–48 %). Notably, the very high drift velocity of 240 km s−1 +increases by 29% to 310 km s−1 in the unity model. The true de- +grees of condensation are a factor twenty lower than the modified +degrees of condensation owing to the high drift velocities. +We compare the drift model using ξ(Fo) = 0.1 with our PC +model where ξ(Fo) = 1.0. The mass-loss rate of the drift model is +86 % lower, the dust-to-gas density ratio 22 times higher, and the +true degrees of condensation 95 % lower than the corresponding +values of the PC model. The differences are much smaller in the +expansion velocity (17 % lower) and the grain radius (0.73 % +higher). The same comparison with the model of B19 gives a +90 % lower mass-loss rate, a 35 times higher dust-to-gas density +ratio, and a 92 % lower degree of condensation of silicon. Sim- +ilarly, differences are smaller in the expansion velocity (26 % +lower) and the grain radius (15 % higher). The huge increase in +the dust-to-gas density ratio of the drift model must be put in +context of the drastically lower mass-loss rate. Nevertheless, the +differences in comparison to the PC models of about a factor ten +in the mass-loss rate and a factor 35 in the dust-to-gas density +ratio are large and must not be ignored. +5.2.2. The proof-of-concept drift model L3.85T27 +The values of the drift model again differ from our PC model. +The mass-loss rate is here 95 % lower, the dust-to-gas density ra- +tio 82 times higher, and the degrees of condensation 92 % lower. +Furthermore, the expansion velocity is 160 % higher, and the +mean grain radius 32 % higher. Temporal fluctuations are higher +in the dusty properties, but the fluctuations are generally low. +Both the drift model and the PC model show periodic variations. +The mass-loss rate is 0.24 % of the value of B19, the dust-to-gas +density ratio 70 times higher, and the degree of condensation of +silicon 93 % lower. The expansion velocity is 29 % higher and +the grain radius 25 % higher. +We plot our PC and drift models versus the radius in Fig. 4. +The inefficient mass loss of the drift model is seen in the gas den- +sity that is a hundred times lower than in the PC model (Fig. 4b). +Despite the lower mass-loss rate, the expansion velocity of the +drift model is more than twice as high than in the PC model, see +Fig. 4a. The gas opacity κH increases somewhat towards higher +radii (Fig. 4j), whilst the Rosseland mean opacity κR decreases +to be a millionth and less of κH at higher radii; notably, the same +ratio is closer to a factor ten in the carbon-rich model shown in +Fig. 8j in Paper V. The Eddington factor of both models (Fig. 4g) +indicates that there is no need to solve the equation of radiative +transfer where R >∼ 10 R⋆, as the factor is very close to unity. +The drift velocity shown in Fig. 4c attains high values al- +ready near the star. The same high values cause an abrupt cut- +off in the dust formation rates at radii r <∼ 3 R⋆ (Fig. 4e). The +Article number, page 8 of 13 + +C. Sandin et al.: Three-component modelling of O-rich AGB star winds +(b) +−18 −16 −14 −12 +ρg +104×ρd +log ρ [g cm−3] +0 +10 +20 +30 +[AU] +(d) +0.050.100.15 0.20 +fSi +cond +fMg +cond +fcond +(f) +0.30 0.40 0.50 +rd [µm] +(h) +−22 −20 −18 −16 +χH +χ +χR +log χ [cm−1] +(j) +−6 +−4 +−2 +0 +κH +κR +log κ [cm2g−1] +(l) +0 +5 +10 +15 +Radius [R*] +−3.5 −3.0 −2.5 +log δdg +(a) +−10−5 0 +5 10 15 +ug +cs +ug [km s−1] +0 +10 +20 +30 +[AU] +(c) +0 +100 +200 +300 +vD [km s−1] +(e) +−10−5 0 +5 10 15 +τ−1 +gr +τ−1 +dc +J* +log τ−1 [s−1] +(g) +0.4 +0.6 +0.8 +1.0 +fEdd +(i) +0.5 1.0 1.5 2.0 2.5 +Tg +Tr +Td +T [103 °K] +(k) +0 +5 +10 +15 +Radius [R*] +0.6 +0.8 +1.0 +Td/Tr +(Tg/Tr)eq +T ratio +Fig. 4. Radial structure of a snapshot of setup L3.85T27 for the full modelled region. The drift model (where ξ(Fo) = 0.1; PC model, where +ξ(Fo) = 1.0) is shown with purple (orange) lines. From the top left, the 12 panels show: (a) gas velocity ug, sound speed cs; (b) gas density ρg, dust +density 104 × ρd (log); (c) drift velocity vD; (d) [true] degree of condensation of silicon f (Si) +cond and magnesium f (Mg) +cond ; (e) net growth rate τ−1 +gr , net +decay rate τ−1 +dc (log); (f) average grain radius rd; (g) Eddington factor fEdd; (h) extinction coefficient χH, Rosseland mean extinction coefficient χR; +(i) gas temperature Tg, radiative temperature Tr, and dust temperature Td; (j) gas opacity κH and Rosseland mean opacity κR (log); (k) temperature +ratios Td/Tr and (Tg/Tr)eq; and (l) dust-to-gas density ratio δdg. All properties are drawn versus the stellar radius R∗ (lower axis) and astronomical +units (AU; upper axis). Grey horizontal lines are guides. +[true] degree of condensation (Fig. 4d) and extinction coefficient +(Fig. 4h) are significantly lower in the drift model. +The dust-to-radiative temperature ratio in Fig. 4k shows a +value that is the inverse of the same ratio in a stellar wind of a +carbon star (cf. Fig. 8k in Paper V). The forsterite dust temper- +ature is always lower than the radiative temperature; it is about +half as high in the outer regions. In comparison, the amorphous +carbon dust temperature is always higher than the radiative tem- +perature; and it is up to about 50 % higher in the outer regions. +No other indicator illustrates the difference as clearly between +scattering and absorption dominated dust extinction. +5.2.3. Implications of allowing gas-to-dust drift +The differences between the two drift models and the corre- +sponding PC models are enormous. Our values on the mass-loss +rates of the PC models are 7.5 and 21 times higher than the cor- +responding drift model. These values grow to 11 and 420 times +when we instead compare with the corresponding values of B19. +And the mass-loss rate of the proof-of-concept model of H08 is +93 times higher than in our drift model. Besides different model +parameters and modeling approach, the differences are owing to +the scattering-dominated dust extinction of forsterite, which re- +sults in extraordinarily high drift velocities. Such high drift ve- +locities imply that any features in the diluted dust component +in observations will be minuscule. More reliable observations of +mass loss based on radio observations of CO, which make fewer +assumptions on the dust component, indicate that mass-loss rates +Article number, page 9 of 13 + +A&A proofs: manuscript no. sandin +are high – evidently something is needed to achieve these high +mass-loss rates also in simulations. +The question is what happens when additional dust species +are added to the simulations. Not all species are as transparent +as forsterite. It seems valuable to study changes in the radia- +tion field and wind driving mechanism where also enstatite is +added and species that include iron, such as fayalite, ferrosilite +and pure iron dust. Bladh & Höfner (2012) and also GS14 (see +Chapter 12) argue that these minerals form much farther out than +iron-free minerals, but it is still unknown what the result will be +in a multi-fluid model where temperature gradients of drift mod- +els may be much steeper in the inner wind forming region (see +Fig. 4i). Drift must not be ignored in stellar winds that are driven +by minerals such as forsterite where extinction is dominated by +scattering. +6. Conclusions +We have extended our frequency-dependent dust-driven high- +spatial-resolution wind model code T-800 of Paper V with de- +scriptions for mineral formation in oxygen-rich chemistry, as +laid out by GS14. We have also calculated new opacity tables +that are based on bound-bound cross sections of thirty [atoms +and] molecules of the exomol project and free-free and bound- +free opacities of the jekyll code. With our improved model code +and opacity data, we can choose molecular compositions and +wavelengths freely and model stellar winds of both C-type and +M-type stars that form various types of minerals. To our under- +standing, T-800 is the physically and numerically most detailed +dynamic stellar wind code there is, and it is the only one that can +accurately calculate effects of gas-to-dust drift in either type of +star. +We have calculated models to explore effects of drift in M- +type winds that are driven by forsterite particles. Extant studies +favor this species as a wind driver. We selected model parameters +of simulations that have shown high mass-loss rates. Our new PC +models show a fair comparison with extant results (of H08 and +B19); we cannot be more specific as details of those existing +simulations are unavailable in the literature. +Differences are much larger when we compare results of PC +models with drift models. Whilst changes in expansion veloc- +ities and grain sizes are modest, this is not so for the degree +of condensation, dust-to-gas density ratio, and mass-loss rates. +The drift velocity is 310–360 km s−1in the presented models; +these high values result in very low degrees of condensation. The +mass-loss rate is 7.5 times lower in one case and 21 times lower +in a second case. The second model is important as is H08 used +a model with the same parameters to prove the concept of stellar +wind formation in M-type stars. Drift is more important in M- +type stars than in C-type stars – the biggest difference is that mo- +mentum is transferred from the radiation field to the dust grains +through scattering on transparent grains instead of through ab- +sorption in opaque grains. +More studies are needed that explore the use simultaneous +formation of additional dust species to explain how observed +high mass-loss rates form. 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N., Tennyson, J., Syme, A.-M., et al. 2022, MNRAS, 510, 903 +Article number, page 11 of 13 + +A&A proofs: manuscript no. sandin +a) +−5 0 +5 10 15 +ug +cs +ug [km s−1] +2 +4 +6 +8 +10 +12 +[AU] +b) +−14 −12 −10 +ρg +104×ρd +log ρ [g cm−3] +c) +1 +2 +3 +4 +5 +Radius [R*] +1 +2 +3 +Tg +Tr +Td +T [103 °K] +Fig. B.1. Radial structure of a snapshot of setup L3.85T24 for the in- +ner modelled region. A PC model using the gas opacities of Aringer is +shown with orange lines. The 3 panels show: (a) gas velocity ug, sound +speed cs; (b) gas density ρg, dust density 104 × ρd (log); and (c) gas +temperature Tg, radiative temperature Tr, and dust temperature Td. All +properties are drawn versus the stellar radius R∗ (lower axis) and astro- +nomical units (AU; upper axis). Grey horizontal lines are guides. +Appendix A: Calculation of the bound-free opacity +Here is a brief description of the methods and data used in the +determination of the bound-free opacity, which is calculated us- +ing the jekyll code (Ergon et al. 2018; Ergon & Fransson 2022). +As in the calculations of our bound-bound opacities, we first de- +termine cross sections of bound-free level populations and then +calculate opacities based on the physical conditions. +The photo-ionization cross-sections for ground states are cal- +culated using analytic fits of Verner & Yakovlev (1995) and +Verner et al. (1996), and for the lowest excited states of He i, +C i, O i, Mg i, Mg ii, Si i, S i, and Ca ii, using data from TOPbase +of the Opacity Project6. For all other excited states, the photo- +ionization cross-sections are calculated using a hydrogenic ap- +proximation by Rybicki & Lightman (1979). +The bound-free opacity is then calculated based on LTE pop- +ulations of excited and ionized states for given values of density, +temperature, and composition. The atomic data (excitation and +ionization energies and statistical weights) are obtained from the +Atomic Spectra Database of NIST7 and the online tables by R. +Kurucz8. +Appendix B: On the role of the gas opacity data +Developing our new oxygen-rich chemistry models we calcu- +lated test wind models using the same gas opacity table that B19 +use (Aringer 2000; Aringer et al. 2009, the data are described +in). The first test models indicated a problem when starting the +stellar wind; some kind of noise appears in the gas temperature +6 http://cdsweb.u-strasbg.fr/topbase/topbase.html +7 https://www.nist.gov/pml/atomic-spectra-database +8 https://lweb.cfa.harvard.edu/amp/ampdata/kurucz23/sekur.html +of all models when dust is present and where Tg ≃ 2000 K. +We show an example of such noise in Fig. B.1, which presents +a snapshot of a stellar wind model where calculations have +just begun. The noise occurs in the temperature in the interval +1.9 <∼ r <∼ 2.2 R⋆. The same noise is [spatially] unresolved in +models using Nd = 100 grid points (not shown), and here, the +problem is much smaller. The problem becomes prohibitive in +drift models, which are more sensitive to variations of this kind. +The origin of the noise is unknown, but no terms in the radiation +hydrodynamic equations appear to be responsible considering +the noise always appears at the same gas temperature. There- +fore, we hypothesized that the noise originate in the gas opacity +data. +We calculated new gas opacity tables based on exomol data, +as we describe in Sect. 4.2.1. The extant and new opacity data +sets are compared in Fig. B.2, for the wavelength λ = 1µm. The +figures show large differences at all temperatures but the low- +est where the offset is about 0.7 dex. Differences are also larger +at lower densities. At higher temperatures and lower densities +our new bound-bound opacities drop faster to low values. The +same low values are not seen in the opacity data of Aringer et al., +whose opacity values at the higher temperatures T >∼ 4000 K are +up to 108 times higher than our values. In our new opacities, +data at these higher temperatures are completely dominated by +the free-free and bound-free components that show structure at +the lower densities (ρ <∼ 10 × 10−10 g cm−3). The opacity data +of Aringer et al. show values that appear to be more constant at +higher temperatures and lower densities; their data show much +less structure than our new opacities. +With our new opacities, the stellar wind calculations are not +hampered by the noise at T ≃ 2000 K described above. The ex- +tant opacity tables only allow is to speculate on why these data +give rise to noise in the temperature when we use them to at- +tempt to calculate a stellar wind. Our guess is that the problem is +connected to the much higher values on the opacities throughout +the parameter domains. +Article number, page 12 of 13 + +C. Sandin et al.: Three-component modelling of O-rich AGB star winds +Fig. B.2. The total gas opacity at the wavelength that is the closest to λ = 1µm for the data of Aringer et al. (left panel) and our new data +(right panel). The opacity (z-axis) is shown versus the temperature (x-axis) and the gas density (y-axis). The temperature and density ranges are +1000 ≤ T ≤ 10 000 K and −18 ≤ log ρ ≤ −6 g cm−3, respectively. The two panels use the same ranges on all three axes. +Article number, page 13 of 13 + diff --git a/sNAzT4oBgHgl3EQfPPuE/content/tmp_files/load_file.txt b/sNAzT4oBgHgl3EQfPPuE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..47894251674414955f1c063e4e11e3fa686ae97c --- /dev/null +++ b/sNAzT4oBgHgl3EQfPPuE/content/tmp_files/load_file.txt @@ -0,0 +1,1441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf,len=1440 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin ©ESO 2023 January 4, 2023 Three-component modelling of O-rich AGB star winds I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Effects of drift using forsterite C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin1, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Mattsson2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Chubb3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Ergon4, 1, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Weilbacher5 1 Department of Astronomy, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden e-mail: christer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='sandin@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='se 2 Nordita,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' KTH Royal Institute of Technology and Stockholm University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Hannes Alfvéns väg 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' SE-10691 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sweden 3 Centre for Exoplanet Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' University of St Andrews,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' North Haugh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' St Andrews,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' KY16 9SS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' United Kingdom 4 The Oskar Klein Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' AlbaNova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' SE-10691 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sweden 5 Leibniz-Institut für Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' An der Sternwarte 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 14482 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Germany Submitted January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2023 ABSTRACT Stellar winds of cool and pulsating asymptotic giant branch (AGB) stars enrich the interstellar medium with large amounts of pro- cessed elements and various types of dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We present a first study on the influence of gas-to-dust drift on ab initio simulations of stellar winds of M-type stars driven by radiation pressure on forsterite particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our study is based on our radiation hydrodynamic model code T-800 that includes frequency-dependent radiative transfer, dust extinction based on Mie scattering, grain growth and ablation, gas-to-dust drift using one mean grain size, a piston that simulates stellar pulsations, and an accurate high spatial resolution numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' To enable this study, we calculated new gas opacities based on the exomol database, and we extended the model code to handle the formation of minerals that may form in M-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We discern effects of drift by comparing drift models to our new and extant non-drift models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Compared to our recent results of C-rich stellar winds, our two new drift models based on an oxygen-rich chemistry show drift velocities that are higher by about a factor ten, that is 310–360 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our new drift model mass- loss rates are 8–20 times lower than our own non-drift models, but compared to extant models that use the same stellar parameters, our mass-loss rates are 10–420 times lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Meanwhile, a comparison of other properties such as the expansion velocity and grain size show similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our results show that the inclusion of gas-to-drift is of fundamental importance in stellar wind models driven by transparent grains such as forsterite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Assuming that the drift velocity is insignificant, properties such as the mass-loss rate may be off from more realistic values by a factor one hundred and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' hydrodynamics – radiative transfer – stars: atmospheres – stars: AGB and post-AGB – stars: mass-loss – stars: winds, outflows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Introduction Stellar winds rule the final and decisive stages of evolution of low-to-intermediate mass stars when they ascend the asymptotic giant branch (AGB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The dynamic AGB stage involves increas- ing luminosities, low effective temperatures, and stellar pulsa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Dust formation begins at a couple of stellar radii where temperatures are low enough to prevent the newly formed grains from evaporating, and the new dust grains absorb or scatter the radiation and in that way attain a momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The grains acceler- ate outwards and collide with particles in the gas that are dragged along as the particles drift through the same gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Considering all the needed physics, it is a physical problem of a kind to simulate the resulting dust-driven wind where low expansion velocities are about 10 km s−1 and high mass-loss rates vary from 10−8 up to, in extreme cases, 10−4 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Depending on what element dominates, AGB stars are either oxygen-rich (M-type stars) or carbon-rich (C-type stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The di- chotomy is reflected in stellar wind models where dust formation in a carbon-rich chemistry is more simple where mostly amor- phous carbon forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Other types of dust and minerals do not form in sufficient numbers to be influential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Dust formation in an oxygen-rich chemistry is more com- plex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Spectra of circumstellar envelopes of M-type AGB stars show characteristic silicate features at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='7 and 18 µm (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Woolf & Ney 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Low 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Molster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Dorschner 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Molster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' These features indicate that silicon- containing grains are a dominant component of the in M-type AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Crystalline silicate dust with features at 11, 23, 28, 33, and 69 µm is also seen (Blommaert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2014), but the crys- tallinity does not appear to be correlated with the mass-loss rats (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Various minerals form depending on the avail- ability of elements that are part of the different minerals, includ- ing olivine, pyroxene, and iron (Gail & Sedlmayr 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Metallic iron, moreover, appears to be a significant component in the cos- mic dust budget owing to the large iron depletion seen in the in- terstellar medium (Mattsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Such grains can proba- bly form in AGB atmospheres and their scattering cross-sections are typically large, so if they form in sufficient number, they may contribute to the driving of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Gail & Sedlmayr (2014, Article number, page 1 of 13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='01180v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='SR] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin hereafter, GS14) present a refined and many ways complete ap- proach on how to implement mineral formation in both carbon and oxygen-rich chemistries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Höfner (2008, hereafter H08) presents first working models (darwin) of stellar winds in oxygen-rich chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' She finds that the dust scattering cross section of larger micron-sized iron-free silicates particles provide a high enough radiative pressure to drive a stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Bladh & Höfner (2012) and Bladh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2013) then argue, based mostly on parameterized models of dust, that forsterite and enstatite are the most likely dust species that drive the stellar wind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' they also present photometric prop- erties of models that agree well with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Bladh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2015) present a larger set of radiation hydrodynamic models that include non-equilibrium dust formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The authors con- clude that they can calculate mass-loss rates as well as spectra, which visual and near-IR diagnostics agree with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Bladh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2019, hereafter B19) present the most extensive set of calculated M-type stellar wind models available this far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Whilst the stellar wind models of H08 up to B19 show agreement with observations, they are based on some assumptions that we find interesting to explore in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The authors emphasize that they calculate high mass-loss rates and photometric proper- ties that agree well with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' They also point out that there are few free parameters in their radiation hydrodynamic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In particular, the only such free parameter they mention is the seed particle abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The authors, moreover, appear to use sticking coefficients that are always set to unity (1) to form as much dust as possible, instead of using extant lower empirically based values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Additional assumptions include only modeling one (or two) dust species at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Physical arguments imply that effects of drift are stronger in these winds than in carbon-rich environments (Mattsson & Sandin 2021, hereafter MS21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Only one extant study addresses effects of drift, whilst assuming very low drift velocities, lack- ing any "evidence" for higher values (Tosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We think there is good reason to check the influence of drift on results more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' As we show here, drift velocities turn out to be dramatically higher in models of M-type stars than in models of C-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Correspondingly, we also find dramatically lower mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We are – with our physically and numerically extended models – unable to reproduce the higher valued dar- win-based mass-loss rates that the authors base their results of good agreement with observations on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Whilst more reliable ob- servations of mass loss show higher mass-loss rates, it seems something important is missing in the picture of understanding the formation of stellar winds in M-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Extant ab initio stellar wind models that include drift are all based on a carbon-rich chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin & Mattsson (2020, hereafter Paper V) include frequency-dependent radiative trans- fer and opacity tables of both the gas and the dust, and calculate models at high spatial resolution: results indicate important dif- ferences between drift models and position-coupled (PC) mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Mass-loss rates, expansion velocities, and yields of dust are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' An additional example of a carbon-rich model where drift is found to be an important component to understand the observations is presented in a study of grain alignment about IRC+10◦216 (Andersson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' this object shows a very high mass-loss rate, where our model nevertheless show a drift velocity that is twice as high as the expansion velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We use our simulation code T-800 of Paper V and extend it with the rates-based description of dust formation in an oxygen- rich chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Specifically, we here focus on a wind where only forsterite is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' To enable this study, we calculated new gas opacity tables for solar metallicities based on the exomol database (Tennyson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2020), and also added free-free and bound-free opacities calculated using the jekyll code (Ergon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Ergon & Fransson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We are thereby, for the first time, able to study time-dependent models using high spa- tial resolution in an oxygen-rich chemistry that include drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We first make semi-analytical predictions of the drift veloc- ity in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2 to see what we can deduce based on simple phys- ical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Thereafter, we describe the physics features of our physically enhanced models in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Presentations of the modelling procedure and results follow in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We discuss the influence of drift on our results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5 and close the paper with our conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Semi-analytic predictions of drift Before we engage in numerical and physical details of our up- dated version of T-800, we look at a simplified treatment of the oxygen-rich stellar-wind formation problem to estimate what the associated drift velocities could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We address the balancing forces that give rise to the wind in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Thereafter we, anew, look at the concept of complete momentum coupling in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2 and conclude this analysis in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Balancing dust extinction and radiation pressure To estimate the drift velocity for a given set of stellar parameters, we need to estimate the radiation pressure on the dust compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Thus, we need to know the photon-to-dust grain momen- tum transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Absorption and scattering of photons by dust grains is modelled with the effective cross sections σabs, ν and σsca, ν, where ν is the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The efficiency of absorption and scattering, or the combination of the two (extinction), is usu- ally defined relative to the geometric cross section σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' For spher- ical grains, σ = πa2, where a is the grain radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The absorption efficiency Qabs,ν = σabs, ν/σ is related to the extinction and scat- tering efficiencies Qext,ν = σext, ν/σ and Qsca,ν = σsca, ν/σ, as Qabs,ν = Qext,ν − Qsca,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (1) To calculate a correct radiation pressure, it is necessary to use the absorption efficiency (we name this term Qabs,ν(pr) in Paper V), Qrp, ν = Qext,ν − gsca, νQsca,ν = Qabs,ν − (gsca, ν − 1) Qsca,ν, (2) where gsca, ν = ⟨cos θ⟩ν is the average scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Mie the- ory (Bohren & Huffman 1983) provides these efficiencies as well as the average scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In many extant works on AGB winds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', Sandin & Höfner 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Mattsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2008, 2010), the radiation pressure is calcu- lated assuming dust grains are small compared to the wavelength of the incident radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' This, so called, small-particle limit (SPL) approximation leads to the simplification Qext,ν = Q′ ν/a, where Q′ ν is a function of only the frequency (Wickramasinghe 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Figure 1 shows how Qrp, ν, as computed based on Mie theory, compares to the corresponding SPL value of Qext,ν, using optical constants of (Jäger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' An important feature is the [blue] “peacock feather” region where the Mie-theory based radiation force on grains of radius a in an optically thin atmo- sphere is (where the Eddington flux Hν(r) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='25(R⋆/r)2), frad,d = π c �R⋆ r �2 nd (a) � ∞ 0 a2 Qrp, ν (a) Bν(Teff) dν, (3) where R⋆ is the stellar radius, nd the grain number density, and Bν the Planck function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Grain sizes in the region of relevant val- Article number, page 2 of 13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' : Three-component modelling of O-rich AGB star winds 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 log(a) [cm] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 log( ) [cm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 log(Qrp / QSPL ext ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Ratio of the radiation pressure efficiency Qrp, ν to the SPL extinction efficiency versus the grain radius and wavelength, assuming spheri- cal forsterite grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' ues (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 <∼ a <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 µm) result in a radiation pressure about 300– 30 times lower in the spectral region near the typical flux peak of M-type AGB stars (λ ≃ 1 µm) when the SPL is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Momentum coupling and equilibrium drift The fraction of the momentum that is transferred from the ra- diation field to the gas is referred to as the level of momentum coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In Paper V, complete momentum coupling (CMC) was defined as the case of force balance between radiation on the one hand and drag and gravity on the other, although the amount of momentum lost owing to the gravitational potential is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Equating radiation and drag force is also a common definition of CMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Drift is a non-linear dynamic phenomenon, but simulations (in particular those of Paper V) show that an equilibrium tends to develop in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Equilibrium drift can be defined as the situation where the Lagrangian derivatives of gas and dust velocity are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', dv/dt = du/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' This, in turn, means that the equilibrium drift velocity ˚vD is constant with respect to time and that ˚vD is governed by a simple algebraic equation instead of a hard-to-solve partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Assuming equilibrium drift and CMC defined as above, the drag force is fdrag = frad,d − fgrav,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Defining the dimensionless variable S D = ˚vD vζ , and vζ = � ζTg = � 128kB 9πµmH Tg, where vζ is a modified thermal velocity, kB is the Boltzmann con- stant, µ the mean molecular weight, mH the mass of a hydrogen atom, and Tg the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We then have S 2 D = −1 2 + ������� 1 4 + � frad, d − fgrav, d fζ �2������� 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (4) where fgrav,d ≈ ρd GM⋆r−2 is the point-mass approximation for the gravitational force, ρd the dust density, G the Gravitational constant, and fζ = πa2 nd ρg v2 ζ can be seen as a thermal coupling 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 log(a) [cm] 2 1 0 1 2 3 log(vD) [km s 1] L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T24 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T27 log(M) = 8 [M yr 1] log(M) = 7 [M yr 1] log(M) = 6 [M yr 1] log(M) = 5 [M yr 1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Equilibrium drift velocity ˚vD versus grain radius, assuming CMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Four lines show different wind densities (mass-loss rates) as ex- plained in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The grey bullet and square indicate the locations of the two numerical simulations with drift presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' force between gas and dust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' and here, ρg is the gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' For a given mass-loss rate ˙M, wind expansion velocity u∞, luminos- ity L⋆, effective temperature Teff, and stellar mass M⋆, we can now estimate the equilibrium drift velocity ˚vD using the above equation in combination with the condition for mass conserva- tion ˙M = 4π r2 ρgu∞ and L⋆ = 4π σSB R2 ⋆ T 4 eff, where σSB is the Stefan-Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Article number, page 3 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Predicted drift velocities In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2 we show the expected equilibrium drift velocity ˚vD versus grain radius a for M⋆ = 1 M⊙, log(L⋆) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85 L⊙, Teff = 2700 K and four different assumptions of mass-loss rates, chosen to represent the range of mass-loss rates predicted by models (B19) and observations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', Uttenthaler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2019, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We note that there is ample room for high drift velocities and that the maximum drift velocity of each curve occurs for grain radii that are quite typical for M-type AGB stars according to extant models (B19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' It is clear that a wind driven by radiation pressure on forsterite grains leads to high drift velocities, even in case of massive outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' It has been argued that drift is negligible in winds associated with very high mass-loss rates (Höfner & Olof- sson 2018), which seems to be the case for carbon stars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3a in Paper V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' however, our more recent model of the stel- lar wind in IRC+10◦216 says otherwise (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 8 in Andersson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' But given that M-type AGB stars have wind speeds of u∞ ∼ 10 km s−1 and typical grain sizes a ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 µm, the drift factor FD = 1 + vD/u∞ > 2 also for intense outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (FD > 2 corresponds to a situation where the drift velocity vD exceeds the gas expansion velocity u∞ and the dust mass-loss rate is typically reduced by an order of magnitude or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=') We tentatively con- clude that including drift in the modelling of M-type AGB stars is of absolute fundamental importance, as PC models do not pro- vide a correct result even in the high-mass-loss limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' As we shall see, this conclusion is also confirmed by our detailed modelling, described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Physical interpretation of the PC assumption The PC assumption is incompatible with the idea that AGB winds form by friction between radiatively accelerated dust grains and gas particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' MS21 argue that there is no realistic physical limiting case which leads to PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Whilst this must still be true, we shall here discuss the (unrealistic) limit where PC is formally true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (4), we note that ( frad,d − fgrav,d)/ fζ ≪ 1 im- plies S D = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', no drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' If we ignore the case frad,d = fgrav,d, this limit requires that fζ is very large and, in particular, much larger than the net radiation force ˜frad,d = frad,d − fgrav,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' For this to occur in the case given in the previous section (with frad,d ob- tained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (3)), the modified thermal velocity vζ has to be of the order 100 km s−1 unless ρg is several orders of magnitude higher than expected in a realistic wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Such a high vζ corre- sponds to gas temperatures of the order 105 K, which is com- pletely unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' It is, in fact, fair to say that the physical in- terpretation (or consequence) of the assumption of PC is quite absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The semi-analytic predictions of drift velocity presented here provide a solid theoretically founded reason for pursuing detailed modelling of winds of M-type AGB stars with gas and dust treated as dynamically decoupled phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Model features and improvements of T-800 The model features of our radiation hydrodynamic model code T-800 is described in Paper V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' As in the C-rich chemistry, we model three components in the O-rich chemistry described here: the gas, the radiation field, and a dust component consisting of forsterite (Fo) mineral grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In comparison to the moments method used to describe dust formation in the C-rich chemistry, we replace the four dust mo- ment equations (K0–K3) with one rate equation for the forma- tion of each mineral k, and the carbon number density equa- tion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (5) in Paper V) with corresponding equations for each affected tracer element j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' All physics of mineral formation in oxygen-rich chemistry we require is developed and described by Gail & Sedlmayr (1999) and GS14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The adjusted equations are ∂ ∂tnd,kNk + ∇ · �nd,kNkvk � = qk, (5) ∂ ∂tn j + ∇ · � nju � = −ν j kqk, (6) where t is the time, nd,k the seed particle density, Nk the number of monomers, vk the mean dust particle velocity, qk the sum of the source and sink terms owing to grain formation, nj the tracer element atom number density, ν j k the number of tracer element atoms per monomer, and u the gas velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In this approach, there is no description of nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Instead, seed particles of species are assumed to exist everywhere and nd,k = ρg ϵk muµ, (7) where ρg is the gas density, ϵk the seed particle abundance, mu the atomic mass constant, and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='26 the mean molecular weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mineral rate equation The rate equations describe the number of monomers Nk throughout the model domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The rate equation source term ow- ing to grain growth, evaporation, and destruction is qk = nd,k ������4πa2 k � Jgr,k − Jev,k � − 1 τksp,n ������ , (8) where ak is the particle radius, Jgr (Jev) the growth (evaporation) rate, and 1/τsp,n the rate of non-thermal sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The grain radius is described using the grain volume V and the monomer volume V1, ak = �3Vk 4π � 1 3 , Vk = NkV1,k, V1,k = Akmu ρm,k , (9) where Ak is the molecular weight and ρm,k the mineral intrinsic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The term that describes grain growth and evaporation is writ- ten as (GS14, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='101, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='102, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='108) Jgr,k − Jev,k = ξk p j �2πmjkBTg ���������φk − 1 ac k � Tg Td ��������� , (10) where ξk is the the drift-velocity-dependent sticking coeffi- cient, p j and mj are the partial pressure and mass of the rates- determining component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Moreover, kB is Boltz- mann’s constant, Tg the gas temperature, φk the drift correction factor, ac k the reaction activity, and Td the dust temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The sticking coefficient1 ξk is assumed to decrease when the drift velocity becomes high in relation to the binding energy Eb,k (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (14) in Krüger & Sedlmayr 1997 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (13) in Sandin & Höfner 2004, hereafter Paper III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' ξk = ξ(k) exp ��������− ������ Akmu ˜w2 k 8Eb,k ������ 3�������� , (11) 1 Please note that inMS21 ξ was used to denote the grain-growth ve- locity, which is a different, although not unrelated, quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Article number, page 4 of 13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' : Three-component modelling of O-rich AGB star winds where the velocity of dust grains relative to gas particles ˜wk is (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 11 and 12 in Paper III) ˜wk = ������� 8kBTg 16πAkmu + v2 D,k 16 ������� 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (12) Here, vD,k = vk − u is the drift velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Moreover, the drift cor- rection factor φk is (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='19) in GS14) φk = �������1 + πAkmu 8kB v2 D,k Tg ������� 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (13) We use the same expression for non-thermal sputtering (1/τk sp,n) as we do in Paper III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Although, here we account for collisions with H2 molecules, in addition to H and He atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Growth and evaporation of forsterite We use two tracer elements: silicon and magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' There are in this case eleven equations, instead of thirteen equations when using the moments approach and a carbon-rich chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Forsterite grain growth takes place through collisions of seed grains and extant grains with either SiO molecules or Mg atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' when addition of SiO (Mg) is the rate determining reaction step, pj = pSiO and mj = mSiO (pj = pMg and mj = mMg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The basic chemical reaction for the forsterite formation, as well as its evaporation through chemical sputtering, is 2Mg + SiO + 3H2O ↔ Mg2SiO4(s) + 3H2 (14) and the (chemical sputtering) reaction activity ac Fo is (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='60, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='103, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='104 in GS14) 1 ac Fo = p3 H2 pSiOp2 Mgp3 H2O Kp (SiO) K3 p (H2O) Kp �Mg2SiO4 � K3p (H2), (15) where the four equilibrium constants Kp are calculated at the dust temperature Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Partial pressures of atoms and molecules The number densities of the molecules in the gas phase that are part of the grain formation as well as the activities that deter- mine when dust grains form are calculated in an equilibrium chemistry of molecules with hydrogen, oxygen, carbon, nitro- gen, aluminium, silicon, and sulfur, following the approach of GS14 (chapter 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The considered atoms and molecules are: H, H2, O, OH, H2O, CO, CO2, CH4, N, N2, NH3, HCN, Al, AlO, AlS, AlOH, AlO2H, Al2O, Al2O2, Si, SiO, SiO2, S, SO, HS, H2S, SiS, and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Magnesium is assumed to be present as free atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' All number densities and activities are calculated for the tem- perature range 100 ≤ Tg ≤ 10 000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We use equilibrium con- stants Kp – that are often referred to as dissociation constants – of Sharp & Huebner (1990), GS14 (see their Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5), and NIST JANAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Modelling procedure and results We first briefly point at our modelling procedure in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 and then describe the physics setup and choice of model parameter sets in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We present our results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2 Equilibrium-constant data of NIST/JANAF can be retrieved from https://janaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='gov/ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' References of used atom and molecule datasets as well as num- ber of energy levels accounted for in each entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' molecule dataset References Energy levels C Kurucz 1 999 C2 8states 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3 44 189 C2H2 aCeTY 4 5 160 803 CH MoLLIST 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 6 2526 CH4 YT34to10 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 8 8 194 057 CN Trihybrid 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 11 7703 CO Li2015 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 13 6383 CO2 UCL-4000 14 3 562 798 CS JnK 15 11497 CrH MoLLIST 6 1646 FeH MoLLIST 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 6 3564 H2 RACPPK 17 302 H2O POKAZATEL 18 810 269 H2S AYT2 19 220 618 HCl HITRAN 20 335 HCN Harris 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 22 168 110 HF Coxon-Hajig 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 13 684 LaO BDL 25 38 208 MgH XAB 26 1303 N Kurucz 1 283 N2 WCCRMT 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 29 40 380 NH3 CoYuTe 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 31 5 095 730 O Kurucz 1 201 OH MoLLIST 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 6 1878 SO2 ExoAmes 34 3 270 270 SiO SiOUVenIR 35 174 250 SiS UCTY 36 10 104 TiO Toto 37 236 308 VO VOMYT 38 638 958 YO SSYT 39 79 440 References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (1) Kurucz & Bell (1995);' metadata={'source': 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(34) Underwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (35) Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (36) Upadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (37) McKemmish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (38) McKemmish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (39) Smirnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2019) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Modelling procedure We follow the modelling procedure described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 in Paper V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Due to the low outflow velocity of the wind (u∞ <∼ 10 km s−1), we set the outer boundary here at rext final = 20R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We use Nd = 840 grid points, which very nearly corresponds to the grid point arrangement we achieve when using Nd = 1024 and rext final = 40R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' It appears to be sufficient to evolve the wind mod- els for a time interval of about 100 P (stellar pulsation periods) as the wind structures reach a state of equilibrium before that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Article number, page 5 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Physics setup and selection of model parameters We introduce effects of gas-to-dust drift using one mean dust ve- locity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We compare the new drift models to PC (non-drift) mod- els that are in all other ways are equivalent to the drift models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We used solar abundances of Anders & Grevesse (1989), with values for C and O of Grevesse & Sauval (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' As do B19, we set the pulsation period (P) using the P–L∗-relation of Whitelock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' To correct for too small bolometric vari- ations (Gautschy-Loidl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2004), B19 (see their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2) in- troduce a free parameter fL that allows larger variations of the luminosity at the inner boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Whilst we added the option to T-800 to use a freely chosen value on fL, we use fL = 1 here as we do not calculate model spectra and results of a PC model using both a approaches are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Next, we describe our approach to calculate gas opacities in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1, dust properties in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2, and our selection of model parameters in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Gas opacities In Paper V, we used tabulated gas opacities κν � ρg, Tg � that were created for carbon-rich chemistries with the coma code (Aringer 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Aringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2009) for 319 wavenumbers in the interval 400 ≤ ˜ν ≤ 39 480 cm−1, 50 temperatures in the interval 1000 ≤ Tg ≤ 10 000 K, and 24 densities in the interval −18 ≤ log10 ρg ≤ −6 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Here, we calculated new bound-bound gas opacities based on data of the exomol project (Tennyson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3 The calcula- tions make use of data for the following 30 atoms and molecules: C, C2, C2H2, CH, CH4, CN, CO, CO2, CS, CrH, FeH, H2, H2O, H2S, HCl, HCN, HF, LaO, MgH, N, N2, NH3, O, OH, SO2, SiO, SiS, TiO, VO, and YO, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The number of energy lev- els for each dataset is specified here, however the corresponding number of transitions, or lines, is typically at least an order of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' For example, the ExoMol aCeTY C2H2 line list has around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2 million energy levels and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3 billion tran- sitions (Chubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We used exocross (Yurchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2018a) to calculate cross sections σl for each atom and molecule l at 102 750 wavenumbers in the interval 100 ≤ ˜ν ≤ 41 200 cm−1, 105 temperatures in the interval 100 ≤ Tg ≤ 10 000 K, and 24 gas densities in the interval −18 ≤ log10 ρg ≤ −6 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The cross sections are resampled to a coarse grid of a pre-defined set of wavenumbers, where the resulting cross section is the aver- age of the ten nearest cross sections on the finer grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Currently, we used 384 wavenumbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' this is an even multiple of the num- ber of cores available on each node (2 × 64) on a current high- performance cluster we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Individual cross sections are there- after converted to bound-bound opacities by multiplying with the corresponding partial pressure pl as κbb, l = pl kBTg σl ρg � cm2g−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (16) We calculated partial pressures for the 27 molecules listed above as well as the three individual atoms using the same approach as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Bound-bound opacities become low at higher temperatures (Tg >∼ 2000 K), where instead bound-free opacities κbf and free- free opacities κff dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We calculated such opacities using the jekyll code (Ergon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Ergon & Fransson 2022), see Appendix A for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Free-free opacities κff,i were calculated for each ion i separately using an expression that 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='exomol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='com/ Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Dust parameters: forsterite parameter value B19a unit Reference ϵFo 10−15 10−15 1 AFo 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='694 140 2, 3 ρm,Fo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='27 g cm−3 2, 3 ξFo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 4 Eb,Fo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 – eV 5 νSi Fo, νMg Fo 1, 2b 1, – References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (1) B19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2) Lide (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (3) GS14, Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (4) GS14, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (5) Barlow (1978), the “Silicate” entry in Table 4 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='a The values we assume B19 use are specified by Höfner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (b) For all elements j, but Mg and Si, ν j Fo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 0 20 40 60 vD [km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 ξk Fo Fa En Fs Crn Irn Car Mos Nin Tg = 1000K Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The sticking coefficient ξk (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (11)) versus the drift velocity vD for nine different minerals, assuming a gas temperature Tg = 1000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' depends on provided electron and ion densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='149) in Hubeny & Mihalas 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Finally, bound-bound and bound-free opacities of individual atoms, molecules, and ions and free-free opacities of ions are summed up to provide a total abundances-dependent gas opacity for each pair of gas density and gas temperature, κν � ρg, Tg � = � l κbb, l + � i �κbf, i + κff, i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Each set of abundances-specific opacities are saved in a bi- nary file tabulated in wavenumber, density, and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The opacities are interpolated in density and temperature for each in- dividual wavenumber in the radiative transfer calculations using two-dimensional rational splines (Späth 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We were at first kindly provided with the same opacity table for solar metallicities that B19 use (Aringer, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Due to unknown reasons, we were unsuccessful in using these data with our new models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We discuss these extant opacity data and make a simple comparison with our new opacities in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Dust properties We list all forsterite-specific dust parameters we used in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We illustrate how the sticking coefficient ξk varies with the drift velocity for forsterite and eight other minerals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 1 in Paper III);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' the parameters of the eight additional minerals are taken from the same set of references as for forsterite, noting that Article number, page 6 of 13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' : Three-component modelling of O-rich AGB star winds Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Six columns specify the: model name, stel- lar mass M⋆, stellar luminosity L⋆, effective temperature Teff, pulsation amplitude ∆ up, and luminosity-dependent pulsation period P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' model M⋆ log (L⋆) Teff ∆ up P [M⊙] [L⊙] [K] [ km s−1] [d] L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85 2400 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 478 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85 2700 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 478 the binding energies of several minerals are highly uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The figure shows that the sticking coefficient drops to 0 when vD >∼ 20 km s−1 for larger mineral monomers such as forsterite (Fo), fayalite (Fa), enstatite (En), and ferrosilite (Fs) and also for carbon (Car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Higher velocities are possible with corundum (Crn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' vD ≈ 40 km s−1) as well as iron (Irn), moissanite (Mos), and niningerite (Nin) where the cutoff drift velocity is about 60 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Hence, there is for forsterite no grain growth when the drift velocity is higher than 20 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' All models were calculated using Mie scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Optical constants (nν, kν) are taken from Jäger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='4 To achieve results comparable with B19, we used the seed particle abun- dance ϵFo = 10−15 with all our calculations, as well as the same sticking coefficient with our PC models, ξ(Fo)(PC) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Selection of model parameters Our new model calculations of M-star objects are as demanding as those of C stars in Paper V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Here we calculated two sets of models to get a first impression of how the formation of stel- lar winds of M stars work when the dust component is allowed to drift relative to the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' At first, we selected the proof-of- concept-model B of H08, which she uses to successfully illus- trate that dust-driven winds also form in M stars when Mie scat- tering is used in the description of dust extinction instead of the SPL approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Secondly, we selected a model of B19 one with a lower effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The model parameters are collected in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Results As in Paper V, we characterize wind models with a set of proper- ties that are temporally averaged at the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mass loss rate ⟨ ˙M⟩ and the terminal velocity ⟨u∞⟩ characterize the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The degree of condensation of silicon ⟨ f Si cond⟩ and magnesium ⟨ f Mg cond⟩, the dust-to-gas mass-loss ratio ⟨ ˙Md/ ˙M⟩, the mean grain radius ⟨rd⟩, and the terminal drift velocity ⟨vD,∞⟩ characterize the dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The drift-velocity dependent [true] degree of condensation of tracer element j accounting for all minerals k is calculated as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='81) in GS145 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (33) in Paper V) f j cond = � k 4πa3 j 3 1 V1, j 1 FD,k ϵkν j k εj = 1 εj � k Nkϵkν j k FD,k , (17) where FD,k = vk/u = 1 + vD,k/u is the drift factor and εj the abundance of element j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In PC models, the modified degree of 4 The optical data can be retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='uni- jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='de/Laboratory/OCDB/amsilicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5 We ignore the seed particle radius a0,j used by GS14, noting that the contribution of seed particles to the degree of condensation is miniscule already at small particle radii a j that are barely larger than a0,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Temporally averaged quantities at the outer boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' From the left, the first three columns specify the model name (see Table 3), if PC or drift is used (P/D), and the sticking coefficient (ξ(Fo)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Nine column pairs show the temporally averaged: mass loss rate ⟨ ˙M⟩, terminal velocity ⟨u∞⟩, terminal drift velocity ⟨vD,∞⟩, modified ⟨�f Si cond⟩ and true ⟨ f Si cond⟩ degree of condensation of silicon, modified ⟨�f Mg cond⟩ and true ⟨ f Mg cond⟩ degree of condensation of magnesium, dust-to-gas mass-loss ratio ⟨δdgFD⟩, and dust radius ⟨aFo⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' A relative fluctuation amplitude ˆr is provided for each property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' a subscript m indicates that the shown value was multiplied with a factor 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The final columns show the outflow classification class: periodic (l×p), and quasi-periodic (lq);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' l indicates the (multi-)periodicity of the gas/dust outflow in the unit of the piston period P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Rows of drift models are shown in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' model P/D ξ(Fo) 108 ⟨ ˙M⟩ ⟨u∞⟩ ⟨vD,∞⟩ ⟨�f Si cond⟩ ⟨ f Si cond⟩ ⟨�f Mg cond⟩ ⟨f Mg cond⟩ 104� δdgFD � 102⟨aFo⟩ class [M⊙ yr−1] [ km s−1] [ km s−1] [µm] ˆr ˆr ˆr ˆr ˆr ˆr ˆr ˆr ˆr L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T24 P 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='33 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='4m 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5m 365 230 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 10 2p Article number, page 7 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin condensation is (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85) in GS14) �f j cond = 1 εj � k Nkϵkν j k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (18) Moreover, the dust-to-mass mass-loss ratio is ˙Md ˙M = � k ρd,k ρg v∞,k u∞ = � k δdg,kFD,k, (19) where δdg,k = ρd,k/ρg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We calculated a relative fluctuation amplitude ˆr for each property Q as ˆr = σs/Q, where σs is the (sample) standard devia- tion of the property Q in the time interval that is used to measure the same property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We show results of our model calculations in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Discussion We analyze the new results of our PC models in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 and then compare PC models with our new drift models in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Comparing results of the PC (non-drift) models 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The lower temperature model L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T24 Our PC models using a unity sticking coefficient (ξ(Fo) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0) re- veal a more massive wind than when ξ(Fo) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mass-loss rate and expansion velocity are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 times higher, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' A lower ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2 is seen in the degree of condensation and the dust-to-gas density ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The average grain radius is only 30 % higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The fluctuation amplitudes are 26–40 times higher in all values, but the mass-loss rate, where it is 130 times higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Clearly, a unity sticking coefficient results in a much more vari- able wind where all values but the grain radius are drastically higher, but the increase is with the exception of the mass-loss rate 30–200 % and not a factor 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In comparison to B19, and assuming a unity sticking coeffi- cient, our values on the degrees of condensation and dust-to-gas density ratio are 51–58 % higher, the grain radius 15 % higher and the mass-loss rate 30% lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' These values compare fairly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The proof-of-concept model L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T27 H08 presents model L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T27 (“model B”) as a proof of the con- cept that scattering on larger dust particles in place of absorption allows formation of massive stellar winds in M stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' B19 calcu- late a model with the same stellar parameters, also with 100 grid points, but use a somewhat longer pulsation period (P = 478 d, instead of P = 390 d) and also favor a lower pulsation ampli- tude (∆ up = 2 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We used the same stellar parameters as B19, where ∆ up = 4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We also set the sticking coefficient to unity, but we used a higher spatial resolution achieved with ND = 840 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We show the resulting values of both H08 and B19 in Table 4 for easy reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Notably, our new values on the silicon degree of condensa- tion and the mean grain radius agree well with these two studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' our values are 83 % and 94 % of their values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sim- ilarly, our value on the dust-to-gas ratio is 85 % of the value of B19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Regarding the mass-loss rate and expansion velocity, our values are 25 % and 67 % (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='7 % and 51 %) of the values of H08 (B19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our mass-loss rate is, obviously, closer to the value of H08, whilst the difference is much larger in comparison to B19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Comparing results of the drift models 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Effects of the sticking coefficient The drift model shows a different result in comparison to the PC model L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T24 – the values of the two drift models are similar (they differ by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='6–33 %), with the exception of the mass-loss rate where the value of the unity sticking coefficient model is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 % of the other model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Thus, the model with the lower sticking coefficient results in a much higher mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Plausibly, because less efficiently formed dust grains can flow to regions where they more effectively contribute to wind formation, before there is enough dust to accelerate both the dust and gas outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The fluctuation amplitudes are, moreover, all very similar (they differ by 0–48 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Notably, the very high drift velocity of 240 km s−1 increases by 29% to 310 km s−1 in the unity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The true de- grees of condensation are a factor twenty lower than the modified degrees of condensation owing to the high drift velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We compare the drift model using ξ(Fo) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1 with our PC model where ξ(Fo) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mass-loss rate of the drift model is 86 % lower, the dust-to-gas density ratio 22 times higher, and the true degrees of condensation 95 % lower than the corresponding values of the PC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The differences are much smaller in the expansion velocity (17 % lower) and the grain radius (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='73 % higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The same comparison with the model of B19 gives a 90 % lower mass-loss rate, a 35 times higher dust-to-gas density ratio, and a 92 % lower degree of condensation of silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sim- ilarly, differences are smaller in the expansion velocity (26 % lower) and the grain radius (15 % higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The huge increase in the dust-to-gas density ratio of the drift model must be put in context of the drastically lower mass-loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Nevertheless, the differences in comparison to the PC models of about a factor ten in the mass-loss rate and a factor 35 in the dust-to-gas density ratio are large and must not be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The proof-of-concept drift model L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T27 The values of the drift model again differ from our PC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mass-loss rate is here 95 % lower, the dust-to-gas density ra- tio 82 times higher, and the degrees of condensation 92 % lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Furthermore, the expansion velocity is 160 % higher, and the mean grain radius 32 % higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Temporal fluctuations are higher in the dusty properties, but the fluctuations are generally low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Both the drift model and the PC model show periodic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mass-loss rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='24 % of the value of B19, the dust-to-gas density ratio 70 times higher, and the degree of condensation of silicon 93 % lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The expansion velocity is 29 % higher and the grain radius 25 % higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We plot our PC and drift models versus the radius in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The inefficient mass loss of the drift model is seen in the gas den- sity that is a hundred times lower than in the PC model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Despite the lower mass-loss rate, the expansion velocity of the drift model is more than twice as high than in the PC model, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The gas opacity κH increases somewhat towards higher radii (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4j), whilst the Rosseland mean opacity κR decreases to be a millionth and less of κH at higher radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' notably, the same ratio is closer to a factor ten in the carbon-rich model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 8j in Paper V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The Eddington factor of both models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4g) indicates that there is no need to solve the equation of radiative transfer where R >∼ 10 R⋆, as the factor is very close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The drift velocity shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4c attains high values al- ready near the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The same high values cause an abrupt cut- off in the dust formation rates at radii r <∼ 3 R⋆ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The Article number, page 8 of 13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' : Three-component modelling of O-rich AGB star winds (b) −18 −16 −14 −12 ρg 104×ρd log ρ [g cm−3] 0 10 20 30 [AU] (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='20 fSi cond fMg cond fcond (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='50 rd [µm] (h) −22 −20 −18 −16 χH χ χR log χ [cm−1] (j) −6 −4 −2 0 κH κR log κ [cm2g−1] (l) 0 5 10 15 Radius [R*] −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 log δdg (a) −10−5 0 5 10 15 ug cs ug [km s−1] 0 10 20 30 [AU] (c) 0 100 200 300 vD [km s−1] (e) −10−5 0 5 10 15 τ−1 gr τ−1 dc J* log τ−1 [s−1] (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 fEdd (i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 Tg Tr Td T [103 °K] (k) 0 5 10 15 Radius [R*] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0 Td/Tr (Tg/Tr)eq T ratio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Radial structure of a snapshot of setup L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T27 for the full modelled region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The drift model (where ξ(Fo) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' PC model, where ξ(Fo) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='0) is shown with purple (orange) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' From the top left, the 12 panels show: (a) gas velocity ug, sound speed cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (b) gas density ρg, dust density 104 × ρd (log);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (c) drift velocity vD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (d) [true] degree of condensation of silicon f (Si) cond and magnesium f (Mg) cond ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (e) net growth rate τ−1 gr , net decay rate τ−1 dc (log);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (f) average grain radius rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (g) Eddington factor fEdd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (h) extinction coefficient χH, Rosseland mean extinction coefficient χR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (i) gas temperature Tg, radiative temperature Tr, and dust temperature Td;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (j) gas opacity κH and Rosseland mean opacity κR (log);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (k) temperature ratios Td/Tr and (Tg/Tr)eq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' and (l) dust-to-gas density ratio δdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' All properties are drawn versus the stellar radius R∗ (lower axis) and astronomical units (AU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' upper axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Grey horizontal lines are guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' [true] degree of condensation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4d) and extinction coefficient (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4h) are significantly lower in the drift model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The dust-to-radiative temperature ratio in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4k shows a value that is the inverse of the same ratio in a stellar wind of a carbon star (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 8k in Paper V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The forsterite dust temper- ature is always lower than the radiative temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' it is about half as high in the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In comparison, the amorphous carbon dust temperature is always higher than the radiative tem- perature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' and it is up to about 50 % higher in the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' No other indicator illustrates the difference as clearly between scattering and absorption dominated dust extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Implications of allowing gas-to-dust drift The differences between the two drift models and the corre- sponding PC models are enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our values on the mass-loss rates of the PC models are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 and 21 times higher than the cor- responding drift model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' These values grow to 11 and 420 times when we instead compare with the corresponding values of B19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' And the mass-loss rate of the proof-of-concept model of H08 is 93 times higher than in our drift model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Besides different model parameters and modeling approach, the differences are owing to the scattering-dominated dust extinction of forsterite, which re- sults in extraordinarily high drift velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Such high drift ve- locities imply that any features in the diluted dust component in observations will be minuscule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' More reliable observations of mass loss based on radio observations of CO, which make fewer assumptions on the dust component, indicate that mass-loss rates Article number, page 9 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin are high – evidently something is needed to achieve these high mass-loss rates also in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The question is what happens when additional dust species are added to the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Not all species are as transparent as forsterite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' It seems valuable to study changes in the radia- tion field and wind driving mechanism where also enstatite is added and species that include iron, such as fayalite, ferrosilite and pure iron dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Bladh & Höfner (2012) and also GS14 (see Chapter 12) argue that these minerals form much farther out than iron-free minerals, but it is still unknown what the result will be in a multi-fluid model where temperature gradients of drift mod- els may be much steeper in the inner wind forming region (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Drift must not be ignored in stellar winds that are driven by minerals such as forsterite where extinction is dominated by scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Conclusions We have extended our frequency-dependent dust-driven high- spatial-resolution wind model code T-800 of Paper V with de- scriptions for mineral formation in oxygen-rich chemistry, as laid out by GS14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We have also calculated new opacity tables that are based on bound-bound cross sections of thirty [atoms and] molecules of the exomol project and free-free and bound- free opacities of the jekyll code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' With our improved model code and opacity data, we can choose molecular compositions and wavelengths freely and model stellar winds of both C-type and M-type stars that form various types of minerals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' To our under- standing, T-800 is the physically and numerically most detailed dynamic stellar wind code there is, and it is the only one that can accurately calculate effects of gas-to-dust drift in either type of star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We have calculated models to explore effects of drift in M- type winds that are driven by forsterite particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Extant studies favor this species as a wind driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We selected model parameters of simulations that have shown high mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our new PC models show a fair comparison with extant results (of H08 and B19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' we cannot be more specific as details of those existing simulations are unavailable in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Differences are much larger when we compare results of PC models with drift models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Whilst changes in expansion veloc- ities and grain sizes are modest, this is not so for the degree of condensation, dust-to-gas density ratio, and mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The drift velocity is 310–360 km s−1in the presented models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' these high values result in very low degrees of condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The mass-loss rate is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='5 times lower in one case and 21 times lower in a second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The second model is important as is H08 used a model with the same parameters to prove the concept of stellar wind formation in M-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Drift is more important in M- type stars than in C-type stars – the biggest difference is that mo- mentum is transferred from the radiation field to the dust grains through scattering on transparent grains instead of through ab- sorption in opaque grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' More studies are needed that explore the use simultaneous formation of additional dust species to explain how observed high mass-loss rates form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' And this cannot be done without drift – the resulting simulations is a multi-fluid problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' This article is a proof of concept of the influential effects of drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' acknowledges funding from STFC, under project number ST/V000861/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Parts of the computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC), partially funded by the Swedish Research Council through grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2018- 05973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We thank B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Aringer (Vienna and Padova) for kindly providing us with an opacity table that we could use to develop and test our new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 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+page_content=' 2014, MNRAS, 440, 1649 Yurchenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', Tennyson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', Syme, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2022, MNRAS, 510, 903 Article number, page 11 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' sandin a) −5 0 5 10 15 ug cs ug [km s−1] 2 4 6 8 10 12 [AU] b) −14 −12 −10 ρg 104×ρd log ρ [g cm−3] c) 1 2 3 4 5 Radius [R*] 1 2 3 Tg Tr Td T [103 °K] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Radial structure of a snapshot of setup L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='85T24 for the in- ner modelled region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' A PC model using the gas opacities of Aringer is shown with orange lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The 3 panels show: (a) gas velocity ug, sound speed cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (b) gas density ρg, dust density 104 × ρd (log);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' and (c) gas temperature Tg, radiative temperature Tr, and dust temperature Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' All properties are drawn versus the stellar radius R∗ (lower axis) and astro- nomical units (AU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' upper axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Grey horizontal lines are guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Appendix A: Calculation of the bound-free opacity Here is a brief description of the methods and data used in the determination of the bound-free opacity, which is calculated us- ing the jekyll code (Ergon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Ergon & Fransson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' As in the calculations of our bound-bound opacities, we first de- termine cross sections of bound-free level populations and then calculate opacities based on the physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The photo-ionization cross-sections for ground states are cal- culated using analytic fits of Verner & Yakovlev (1995) and Verner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (1996), and for the lowest excited states of He i, C i, O i, Mg i, Mg ii, Si i, S i, and Ca ii, using data from TOPbase of the Opacity Project6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' For all other excited states, the photo- ionization cross-sections are calculated using a hydrogenic ap- proximation by Rybicki & Lightman (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The bound-free opacity is then calculated based on LTE pop- ulations of excited and ionized states for given values of density, temperature, and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The atomic data (excitation and ionization energies and statistical weights) are obtained from the Atomic Spectra Database of NIST7 and the online tables by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Kurucz8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Appendix B: On the role of the gas opacity data Developing our new oxygen-rich chemistry models we calcu- lated test wind models using the same gas opacity table that B19 use (Aringer 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Aringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 2009, the data are described in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The first test models indicated a problem when starting the stellar wind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' some kind of noise appears in the gas temperature 6 http://cdsweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='fr/topbase/topbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='html 7 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='gov/pml/atomic-spectra-database 8 https://lweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='edu/amp/ampdata/kurucz23/sekur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='html of all models when dust is present and where Tg ≃ 2000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We show an example of such noise in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1, which presents a snapshot of a stellar wind model where calculations have just begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The noise occurs in the temperature in the interval 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='9 <∼ r <∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2 R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The same noise is [spatially] unresolved in models using Nd = 100 grid points (not shown), and here, the problem is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The problem becomes prohibitive in drift models, which are more sensitive to variations of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The origin of the noise is unknown, but no terms in the radiation hydrodynamic equations appear to be responsible considering the noise always appears at the same gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' There- fore, we hypothesized that the noise originate in the gas opacity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' We calculated new gas opacity tables based on exomol data, as we describe in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The extant and new opacity data sets are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2, for the wavelength λ = 1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The figures show large differences at all temperatures but the low- est where the offset is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='7 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Differences are also larger at lower densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' At higher temperatures and lower densities our new bound-bound opacities drop faster to low values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The same low values are not seen in the opacity data of Aringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=', whose opacity values at the higher temperatures T >∼ 4000 K are up to 108 times higher than our values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' In our new opacities, data at these higher temperatures are completely dominated by the free-free and bound-free components that show structure at the lower densities (ρ <∼ 10 × 10−10 g cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The opacity data of Aringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' show values that appear to be more constant at higher temperatures and lower densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' their data show much less structure than our new opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' With our new opacities, the stellar wind calculations are not hampered by the noise at T ≃ 2000 K described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The ex- tant opacity tables only allow is to speculate on why these data give rise to noise in the temperature when we use them to at- tempt to calculate a stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Our guess is that the problem is connected to the much higher values on the opacities throughout the parameter domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Article number, page 12 of 13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' : Three-component modelling of O-rich AGB star winds Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The total gas opacity at the wavelength that is the closest to λ = 1µm for the data of Aringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' (left panel) and our new data (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The opacity (z-axis) is shown versus the temperature (x-axis) and the gas density (y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The temperature and density ranges are 1000 ≤ T ≤ 10 000 K and −18 ≤ log ρ ≤ −6 g cm−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' The two panels use the same ranges on all three axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} +page_content=' Article number, page 13 of 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfPPuE/content/2301.01180v1.pdf'} diff --git a/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf b/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf new file mode 100644 index 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b/t9E5T4oBgHgl3EQfLQ4L/content/tmp_files/2301.05471v1.pdf.txt @@ -0,0 +1,584 @@ +arXiv:2301.05471v1 [hep-ph] 13 Jan 2023 +Are LHCb exotics Tc¯s0(2900)0, Tc¯s0(2900)++ and X0(2900) members +of an SUF(3) 6-plet? +V. Dmitraˇsinovi´c∗ +Institute of Physics, Belgrade University, Pregrevica 118, Zemun, +P.O.Box 57, 11080 Beograd, Serbia +(Dated: January 16, 2023) +Abstract +Manifestly exotic scalar resonance X0(2900)0 with minimal quark content of [¯c¯sud] was reported +by LHCb [1] in 2020. More recently LHCb reported [2, 3] discovery of manifestly exotic Tc¯s0(2900)0 +and Tc¯s0(2900)++ scalar states, degenerate with the X0(2900). We argue that these are three of +six members of the flavor SU(3)F symmetry 6-plet. We predict the partial widths of D(2900)∗0, +the crucial non-strange missing member of the tetraquark 6-plet, and discuss the optimal decay +channels for its detection. +Keywords: charmed mesons; scalars; tetraquarks +∗Electronic address: dmitrasin@ipb.ac.rs +1 + +Introduction Manifestly exotic scalar resonance X0(2900) in the D−K+ channel with mass +around 2900 MeV and minimal quark content [¯c¯sud] was reported in 2020 by LHCb [1]. Its +decay width is reported as Γ(X0(2900) → D−K+) = 57 ± 12 ± 4 MeV. More recently LHCb +reported [2, 3] a discovery of manifestly exotic Tc¯s0(2900)0 and Tc¯s0(2900)++ scalar states, +with decay widths Γ(Tc¯s0(2900)0 → D+ +s π−) = 119 ± 26 ± 13 MeV; and Γ(Tc¯s0(2900)++ → +D+ +s π+) = 137 ± 32 ± 17 MeV, while noticing its (manifest) degeneracy with the X0(2900), +[29], albeit without noticing that this degeneracy is one condition for their membership in +the flavor SUF(3) symmetry 6-plet. +In this Letter we argue that the Tc¯s0(2900)0, Tc¯s0(2900)++, X0(2900) are but three of six +degenerate members of an SUF(3) symmetry 6-plet, as suggested in [4, 5]. The remaining +members should be the (isovector) Tc¯s0(2900)+ and an isodoublet (D∗0(2900), D∗+(2900)) of +scalar hidden-strangeness cryptoexotics. All members of this 6-plet were predicted at the +same mass, even in the broken SU(3)F symmetry case, i.e., with ms ̸= mu/d. The agreement +of the predicted mass degeneracy with the observed masses Tc¯s0(2900)++ and X0(2900) is +impressive, but the different decay widths may yet provide a challenge, which we address +below. +We remind the reader of the 2005 prediction [5] of an SUF(3) symmetry 6-plet of mass- +degenerate tetraquarks with a bare mass of 2725 MeV, in a simple nonrelativistic constitent +quark model (NRCQM) with ’tHooft strong-hyperfine interaction. The mass 2725 MeV was +predicted by fitting model parameters so as to accommodate the then-new, but now defunct +[6, 7] D∗ +sJ(2632) SELEX state [8]. Thus, the 6-plet mass is now free to be refitted at 2900 +MeV. Taking the common mass of 6-plet as 2900 MeV, we calculate the branching ratios +using only model-independent features such as the SUF(3) symmetry and two-body phase +space. The predicted total width is consistent with the measured ones of Tc¯s0(2900)0,++, +but larger than the observed X0(2900) one, roughly by a factor of two. We briefly discuss a +number of open theoretical issues that may influence the widths. +Masses of the 6-plet are subject to certain SUF(3) flavour symmetry conditions, which +we derive below from the SU(3) flavor wave functions of single-charmed tetraquarks. The +SU(3) flavor multiplets of C=1 tetraquarks are given by the flavor SU(3) Clebsch-Gordan +series 3 ⊗ ¯3 ⊗ ¯3 = 3 ⊗ (3 ⊗ ¯6) = ¯3A ⊕ ¯3S ⊕ 6 ⊕ 15. +For corresponding SU(3) weight +diagrams see Fig. 1. There are two distinct flavour 3-plets in this Clebsch-Gordan series, +that are distinguished by their permutational symmetry, or antisymmetry with respect to +2 + +W +S +D +S +15 +6 +3 +Y +Y +Y +I3 +X +S +D +DS +FIG. 1: +Weight diagrams of SU(3) irreducible representations appearing in the single-charm +tetraquark Clebsch-Gordan series. +the interchange of the two quarks: 3S,A. The two tetraquark anti-triplets (¯3) are analogous +to c¯q mesons, which we call cryptoexotics; three members of the sextet (6) and many of the +15-plet do not have c¯q analogons, which makes them exotics. +The two antisymmetric charmed tetraquark flavour multiplets (3A-plet and 6-plet) have +the curious property that all of their members have the same mass[30], in the linear approx- +imation to SU(3) symmetry breaking, irrespective of their manifest strangeness [5], which is +straightforward to see from their flavor wave functions, +|T0 +c¯sq¯q ⊂ 6⟩ = +1 +√ +2 +|cu +� ¯d¯s − ¯s ¯d +� +⟩ +|T+ +c¯sq¯q ⊂ 6⟩ = +1 +√ +2|cd (¯s¯u − ¯u¯s)⟩ +|T++ +c¯sq¯q ⊂ 6⟩ = 1 +2|c +� +u(¯u¯s − ¯s¯u) + d( ¯d¯s − ¯s ¯d) +� +⟩ +|D0∗ ⊂ 6⟩ = 1 +2|c +� +s(¯u¯s − ¯s¯u) + d( ¯d¯u − ¯u ¯d) +� +⟩ +|D+∗ ⊂ 6⟩ = 1 +2|c +� +s( ¯d¯s − ¯s ¯d) + u( ¯d¯u − ¯u ¯d) +� +⟩ +|X +0 +0 ⊂ 6⟩ = +1 +√ +2|cs +� ¯d¯u − ¯u ¯d +� +⟩ . +(1) +In Eq. (1) it can be seen that even the two states (D0∗, D+∗) with zero net strangeness contain +an s¯s pair one half of the time, which effectively increases their masses by one strange- +3 + +up/down quark mass difference ms − mu/d. This property turned out in (surprisingly) good +agreement with the measured masses of the D+ +s0(2317) and D0(2308) mesons [9–11]. Three, +T0 +c¯sq¯q(2900), T++ +c¯sq¯q(2900) and X ++ +0 (2900), of the six presumed members of the 6-plet have been +discovered with degenerate masses, within combined uncertainties, so it (only) remains to be +seen if this degeneracy will equally hold true for the remaining strangeness-zero isodoublet +(D0∗, D+∗)? +This situation closely resembles that of the lowest-mass spin- 3 +2 baryon 10-plet before the +discovery of Ω−(1670). There, also, the SUF(3) symmetry breaking patterns led Gell-Mann +[12] to the (spectacular) prediction of the mass of the previously missing hyperon, Ω−(1670). +Of course, these tetraquark states have substantial decay widths which may influence +their “dressed” masses. Next, we examine the observed decay widths, so as to see if they +can be used to predict the widths of the as yet undiscovered states? +Decay widths LHCb announced [2, 3] the results of a search for exotic isovector c¯s states, +as two new resonances with masses of +T 0,++ +c¯s(q¯q)I=1(2900) : M = 2.908 ± 0.011 ± 0.020 GeV +and widths of +T 0,++ +c¯s(q¯q)I=1(2900) : Γ(Tc¯s → π±D+ +s ) = 136 ± 23 ± 11 MeV +In the D−K+ channel, on the other hand, there are two resonances [1], both described with +Breit-Wigner line shapes, the scalar (JP = 0+) one with parameters +X0(2900) : M = 2.866 ± 0.007 ± 0.002 GeV/c2; +Γ(X0 → K+D−) = 57 ± 12 ± 4 MeV; +which is roughly two times smaller than the width of T 0,++ +c¯s(q¯q)I=1(2900). Moreover, this is +substantially smaller than the expected total width (≥ 300 MeV) of such tetraquarks. Can +one understand these differences? +First, note that T ++ +c¯s(q¯q)I=1(2900) need not decay only into D+ +s π+, but may also decay into +D+K+, which has not been observed (as yet), with only a slightly smaller phase space. +Similarly, T 0 +c¯s(q¯q)I=1(2900) need not decay only into D+ +s π−, but may also decay into D0K0, +which has not been observed due to the neutrality of decay products. Finally, X0 → K+D− +4 + +is not the only allowed mode of decay - X0(2900) → D +0K +0 is also allowed. We must therefore +examine all these decay widths. +Two-body decay widths are given by +Γ(M → m1 + m2) = 1 +8π|M|2λ(M, m1, m2) +M2 +where |M|2 = f 2 +SU(3)|m|2 is the quantum mechanical decay amplitude squared, which factors +into f 2 +SU(3), the squared SU(3) “isoscalar factor” in the given specific flavor channel and the +flavor-independent decay amplitude squared |m|2, and +λ(M, m1, m2) = |p1| = |p2| += +� +(M2 − (m1 − m2)2) (M2 − (m1 + m2)2) +2M +(2) +is the phase space factor (also known as the Mandelstam function). The flavor-independent +decay amplitudes m are equal, for equal masses M, m1, m2, to linear approximation. +This circumstance allows us to calculate the ratios of partial widths as +Γ(T → D+ +s π+) +Γ(T → D+K−) = +f 2 +SU(3)(T, D+ +s , π−)λ(T, D+ +s , π−) +f 2 +SU(3)(T, D+, K−)λ(T, D+, K−). +provided we are given SU(3) isoscalar factors which are just the flavor SU(3) symmetry +off-diagonal matrix elements fSU(3)(Tc(2900) → f) = ⟨f|Tc(2900)⟩. +We use the 6-plet +tetraquark flavor wave functions, Eq. (1), and K− = s¯u, ¯K0 = s ¯d, π0 = +1 +√ +2(u¯u − d ¯d), +π+ = u ¯d, η8 = +1 +√ +6(u¯u + d ¯d − 2s¯s), η0 = +1 +√ +3(u¯u + d ¯d + s¯s), K+ = u¯s, K0 = d¯s, D0 = c¯u, +D+ = c ¯d, D+ +s = c¯s. +SU(3) isoscalar factors of observed tetraquarks, which may proceed in (at least) two different +channels, e.g. T0 +c¯sq¯q → π−D+ +s , T0 +c¯sq¯q → K0D0. Their SU(3) isoscalar factors are +f(T 0 +c¯sq¯q → π−D+ +s ) = 1/ +√ +2 +f(T 0 +c¯sq¯q → K0D0) = 1/ +√ +2 +The kaonic decay channel is difficult to detect due to two neutrals in the final state. +Similarly, the T ++ +c¯s(q¯q)I=1(2900) SU(3) isoscalar factors are f(T ++ +c¯sq¯q → π+D+ +s ) = 1/ +√ +2, +f(T ++ +c¯sq¯q → K+D+) = 1/ +√ +2 and finally f( ¯X0(2900) → D+K−) = 1/ +√ +2, f( ¯X0(2900) → +D0K0) = 1/ +√ +2. +5 + +TABLE I: Numerical values of SU(3) matrix elements ⟨f|D∗0(2900)⟩ for various final states ⟨f|. +Here η8 and η0 denote the eighth member of the octet and the SU(3) singlet, respectively. +⟨D+π−| ⟨D0π0| ⟨D0η8| ⟨D0η0| ⟨D+ +s K−| +|D∗0(2900)⟩ +1 +2 +−1 +2 +√ +2 +−1 +2 +√ +6 +1 +√ +3 +−1 +2 +√ +2 +SU(3) +isoscalar +factors +of +unobserved +tetraquarks +The +isodoublet +states +(D∗(2900)0, D∗(2900)+) decay into more than two flavor channels: D∗(2900)0 → D0π0, +D∗(2900)0 → D+π−, D∗(2900)0 → D0η, D∗(2900)0 → D+ +s K−, with SU(3) isoscalar +factors shown in Table I. Some of these are eminently observable, e.g., both in the pion +(D∗(2900)0 → D+π−) and in the kaon (D∗(2900)0 → D+ +s K−) channels, due to the charged +decay products. These channels should be prime candidates for an experimental search. +Decays of observed tetraquarks A quick calculation using Eq. (2) and the meson masses from +Particle Data Group (PDG) [22], shows that the differences in phase space range from 4% +to 28 %, which cannot account for the large differences, by a factor of two, in the widths +of Tc¯s and X0. That also shows that the unobserved neutral channel decay T0 +c¯sq¯q → K0D0 +carries approximately one half of the total width. Similarly, the T ++ +c¯s(q¯q)I=1(2900) → D+K+ +decay, which is observable in principle, but has not been reported as yet, carries about 50 +% of the total width, which is therefore at least 300 MeV, in agreement with expectations. +The X0(2900) → D0K0 decay, which is difficult to detect on account of the final products +neutrality, also carries about 50 % of the total width. Thus we see that the SU(3) isoscalar +factors and phase space considerations do not lead to the convergence of the two decay +widths T ++ +c¯sq¯q and X0(2900). Nevertheless, this disagreement should not discourage us too +much, as it is comparatively smaller than in the case of the spin- 3 +2 baryon 10-plet, see the +Comments below. +Predicted widths of unobserved states The absolute widths of the cryptoexotic D mesons +can be calculated from the (known) phase-space factors and the isoscalar factors and one +measured tetraquark decay width. +1) We predict the ratio of D∗(2900)0 → D+π− and D∗0(2900) → D+ +s K− partial decay +widths as +Γ(D∗(2900)0 → D+π−) +Γ(D∗0(2900) → D+ +s K−) = 2 × 1.28 = 2.56, +6 + +Using the Γ(Tc¯s0(2900)0 → π−D+ +s ) decay width to set the total width, we find +Γ(D∗(2900)0 → D+π−) ≃ 74 ± 19 MeV +and +Γ(D∗(2900)0 → D+ +s K−) ≃ 29 ± 7 MeV +Using the Γ(X0 → D+K−) decay width to set the total width, we find +Γ(D∗(2900)0 → D+π−) ≃ 25 ± 7 MeV +and +Γ(D∗(2900)0 → D+ +s K−) ≃ 17 ± 5 MeV, +which puts them within the realm of the measurable. +2) The decay widths of the charged member of the isodoublet D∗(2900)+: Γ(D∗(2900)+ → +D+π0), Γ(D∗(2900)+ → D0π+), Γ(D∗(2900)+ → D+η), Γ(D∗(2900)+ → D+ +s ¯K0) are com- +parable to those of the neutral state, but not easily measurable. Here, again, the trouble is +that one of the two decay products is always neutral, so we do not anticipate detection in +the foreseeable future. +3) The single-charge isovector tetraquark’s Tc¯s0(2900)+ widths Γ(Tc¯s0(2900)+ → π0D+ +s ), +Γ(Tc¯s0(2900)+ → K+D0) are comparable to those of Tc¯s0(2900)++ and Tc¯s0(2900)0. Again, +the main obstruction to an experimental search is that one of its two decay products is +always neutral, which makes it unlikely to be observed in the near run. +Comments The observed common decay width of Γ(T 0,++ +c¯s0 +(2900)) = 136 ± 34 MeV is (at +least) two times smaller than naively expected, however. The observed width is only one half +of the total width, however, the other half going into (unobserved) decays with at least one +neutral object in the final state, which ought to settle the issue of the total width. The same +holds for the observed and total widths of X0(2900), the latter still being approximately two +times too small. That discrepancy should not unsettle us as analogous discrepancies in the +lowest-mass spin- 3 +2 baryon 10-plet are comparatively larger. Nevertheless, the (resonance- +peak) Breit-Wigner masses of mass spin- 3 +2 baryons closely correspond with the bare masses +in the flavour SU(3) symmetry schemes [12, 17] and in the constituent quark model [24]. +Complicated production and decay mechanisms have been invoked for the calculation of each +hyperon’s individual width [22]. Something similar may be expected for scalar tetraquarks +[31], as well. +7 + +Conclusions In this Letter we suggested a simple test of the conjecture that the isotriplet +states T 0,++ +c¯s0 +(2900) and isosinglet ¯X0(2900) belong to an SU(3)F symmetry 6-plet. +An +isodoublet of non-strange cryptoexotic states, (D0 +c(¯uq¯q)I=1/2, D+ +c( ¯dq¯q)I=1/2) must exist with the +(same) mass around 2900 MeV. The neutral member D∗0(2900) of this isodoublet ought to +also decay into two charged particle channels, π−D+ and K−D+ +s , which should allow ready +detection. The observable partial widths Γ(D∗0(2900)) ought to be at most one half of the +corresponding partial widths of the isotriplet T 0 +c¯s0(2900), and/or of the isosinglet X0(2900). +This is a rather weak constraint, due to the factor two difference(s) between the decay widths +of T 0 +c¯s0(2900) and X0(2900). The prospective discovery of a neutral resonance D∗0(2900) at, +or near its predicted mass of 2900 MeV, would constitute an unassailable proof of its being +a member of a 6-plet, and therefore of the 6-plet as a whole, vagaries about decay widths +notwithstanding. +Several exotic tetraquarks have been discovered experimentally and discussed theoreti- +cally before the latest LHCb batch [1–3] - see e.g. the reviews [19, 20], yet there has not been +a single instance, to our knowledge, of an exotic discovery following a prediction. Terasaki +[27] predicted manifestly exotic (double-charged) isovector partners of the D+ +s (2317), which +were searched for by Belle [23] in a narrow strip (± 33 MeV) around 2317 MeV, and by +BaBar [21] up to 2600 MeV invariant mass, both without success. +Perhaps the only successful prediction of D0(2308) in 2004, Ref. [4], led to other specific +mass predictions, viz. that of an isotriplet of c¯sq¯q states, and of an isoscalar cs¯q¯q state, +all belonging to a flavor SU(3)F symmetry 6-plet, at 2724 MeV in [5]. After removing the +SELEX constraint, the 6-plet mass can now be raised to 2900 MeV. Refs. [4, 5, 15, 16] also +predicted existence of a new 15-plet of tetraquarks, but little can be said about their masses +with any certainty. +All of the exotics observed thus far have good isospin, yet their (prospective) SU(3) labels +are rarely, if ever discussed. Here we suggested, perhaps for the first time, a specific test +of an exotic charmed tetraquarks’ new SU(3) multiplet. It appears desirable to put this +suggestion to an experimental test. +Acknowledgments The author acknowledges informative correspondence with Drs. Yin- +rui Liu and Ma Ruiting of the LHCb Collaboration. +This research was funded by the +Serbian Ministry of Education Science and Technological Development, grant number 451- +8 + +03-68/2020-14/200024. +[1] R. Aaij et al. [LHCb], Phys. Rev. D 102, 112003 (2020) doi:10.1103/PhysRevD.102.112003 +[arXiv:2009.00026 [hep-ex]]. +[2] [LHCb], [arXiv:2212.02716 [hep-ex]]. +[3] [LHCb], [arXiv:2212.02717 [hep-ex]]. +[4] V. Dmitraˇsinovi´c, Phys. Rev. D 70, 096011 (2004) doi:10.1103/PhysRevD.70.096011 +[5] V. Dmitraˇsinovi´c, Phys. Rev. Lett. 94, 162002 (2005) doi:10.1103/PhysRevLett.94.162002 +[6] E. +S. +Swanson, +Phys. +Rept. +429, +243-305 +(2006) +doi:10.1016/j.physrep.2006.04.003 +[arXiv:hep-ph/0601110 [hep-ph]]. +[7] H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. 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Rev. 101 (1956), 1570-1579 doi:10.1103/PhysRev.101.1570 +[26] L. +Castillejo, +R. +H. +Dalitz +and +F. +J. +Dyson, +Phys. +Rev. +101 +(1956), +453-458 +doi:10.1103/PhysRev.101.453 +[27] K. +Terasaki, +Phys. +Rev. +D +68, +011501 +(2003) +doi:10.1103/PhysRevD.68.011501 +[arXiv:hep-ph/0305213 [hep-ph]]. +[28] H. X. Chen, W. Chen, R. R. Dong and N. Su, Chin. Phys. Lett. 37, no.10, 101201 (2020) +doi:10.1088/0256-307X/37/10/101201 [arXiv:2008.07516 [hep-ph]]. +[29] Indeed, LHCb [2] noted that “The obtained mass of the Tc¯s0 state is consistent with that of +another 0+ open-charm tetraquark, the X0(2900)([cs¯u ¯d]) state discovered in the D+K− final +state [19, 20], but their widths and flavor contents are different.” +[30] Note that this degeneracy is a function of the SU(3) symmetry breaking patterns, which, in +turn, depend on the the strong hyperfine interaction [13–15]. +[31] The fact that the NRCQM predictions are for bare states is important, as it means that there +are no decays (zero width) in that approximation. The physical exotics’ (bare and/or dressed) +masses lie far above their corresponding two-body thresholds, and therefore should be very +wide, with only a lower bound on the width, Γ > 350 MeV, due to the “fall-apart” nature +of the decay, much like the σ(500) and K∗ +0(700) = κ light-flavour scalar states, see Ref. [22]. +Opening up of decay channels (“unitarization” of the calculation) would change not only the +decay width, but also the dressed mass of the state. Resonant states could be dynamically +10 + +generated, see “Review of scalar mesons” in Ref. [22], i.e., they need not have a bare quark- +state “seed” (or a Castillejo-Dalitz-Dyson (CDD) pole [26]) at all. Indeed, the ∆ resonance was +predicted as a state in a meson-nucleon model, without quarks, and only later confirmed by +Chew and Low [25] in a “booststrap” kind of calculation. In the same vein, H.-X. Chen et al. +[28] have recently argued that “X0(2900) can be interpreted as the S-wave D−K+ molecular +state”. Even if the as yet unobserved states turn out according to predictions, we shall not +know their dynamical origin without further tests. +11 + diff --git a/t9E5T4oBgHgl3EQfLQ4L/content/tmp_files/load_file.txt b/t9E5T4oBgHgl3EQfLQ4L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e817e04746ae17b9ac0681fbb45c1b5da03f77e --- /dev/null +++ b/t9E5T4oBgHgl3EQfLQ4L/content/tmp_files/load_file.txt @@ -0,0 +1,391 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf,len=390 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='05471v1 [hep-ph] 13 Jan 2023 Are LHCb exotics Tc¯s0(2900)0, Tc¯s0(2900)++ and X0(2900) members of an SUF(3) 6-plet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Dmitraˇsinovi´c∗ Institute of Physics, Belgrade University, Pregrevica 118, Zemun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='Box 57, 11080 Beograd, Serbia (Dated: January 16, 2023) Abstract Manifestly exotic scalar resonance X0(2900)0 with minimal quark content of [¯c¯sud] was reported by LHCb [1] in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' More recently LHCb reported [2, 3] discovery of manifestly exotic Tc¯s0(2900)0 and Tc¯s0(2900)++ scalar states, degenerate with the X0(2900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' We argue that these are three of six members of the flavor SU(3)F symmetry 6-plet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' We predict the partial widths of D(2900)∗0, the crucial non-strange missing member of the tetraquark 6-plet, and discuss the optimal decay channels for its detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Keywords: charmed mesons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' scalars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' tetraquarks ∗Electronic address: dmitrasin@ipb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='rs 1 Introduction Manifestly exotic scalar resonance X0(2900) in the D−K+ channel with mass around 2900 MeV and minimal quark content [¯c¯sud] was reported in 2020 by LHCb [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Its decay width is reported as Γ(X0(2900) → D−K+) = 57 ± 12 ± 4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' More recently LHCb reported [2, 3] a discovery of manifestly exotic Tc¯s0(2900)0 and Tc¯s0(2900)++ scalar states, with decay widths Γ(Tc¯s0(2900)0 → D+ s π−) = 119 ± 26 ± 13 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' and Γ(Tc¯s0(2900)++ → D+ s π+) = 137 ± 32 ± 17 MeV, while noticing its (manifest) degeneracy with the X0(2900), [29], albeit without noticing that this degeneracy is one condition for their membership in the flavor SUF(3) symmetry 6-plet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' In this Letter we argue that the Tc¯s0(2900)0, Tc¯s0(2900)++, X0(2900) are but three of six degenerate members of an SUF(3) symmetry 6-plet, as suggested in [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The remaining members should be the (isovector) Tc¯s0(2900)+ and an isodoublet (D∗0(2900), D∗+(2900)) of scalar hidden-strangeness cryptoexotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' All members of this 6-plet were predicted at the same mass, even in the broken SU(3)F symmetry case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=', with ms ̸= mu/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The agreement of the predicted mass degeneracy with the observed masses Tc¯s0(2900)++ and X0(2900) is impressive, but the different decay widths may yet provide a challenge, which we address below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' We remind the reader of the 2005 prediction [5] of an SUF(3) symmetry 6-plet of mass- degenerate tetraquarks with a bare mass of 2725 MeV, in a simple nonrelativistic constitent quark model (NRCQM) with ’tHooft strong-hyperfine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The mass 2725 MeV was predicted by fitting model parameters so as to accommodate the then-new, but now defunct [6, 7] D∗ sJ(2632) SELEX state [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Thus, the 6-plet mass is now free to be refitted at 2900 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Taking the common mass of 6-plet as 2900 MeV, we calculate the branching ratios using only model-independent features such as the SUF(3) symmetry and two-body phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The predicted total width is consistent with the measured ones of Tc¯s0(2900)0,++, but larger than the observed X0(2900) one, roughly by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' We briefly discuss a number of open theoretical issues that may influence the widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Masses of the 6-plet are subject to certain SUF(3) flavour symmetry conditions, which we derive below from the SU(3) flavor wave functions of single-charmed tetraquarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The SU(3) flavor multiplets of C=1 tetraquarks are given by the flavor SU(3) Clebsch-Gordan series 3 ⊗ ¯3 ⊗ ¯3 = 3 ⊗ (3 ⊗ ¯6) = ¯3A ⊕ ¯3S ⊕ 6 ⊕ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' For corresponding SU(3) weight diagrams see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' There are two distinct flavour 3-plets in this Clebsch-Gordan series, that are distinguished by their permutational symmetry, or antisymmetry with respect to 2 W S D S 15 6 3 Y Y Y I3 X S D DS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 1: Weight diagrams of SU(3) irreducible representations appearing in the single-charm tetraquark Clebsch-Gordan series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' the interchange of the two quarks: 3S,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The two tetraquark anti-triplets (¯3) are analogous to c¯q mesons, which we call cryptoexotics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' three members of the sextet (6) and many of the 15-plet do not have c¯q analogons, which makes them exotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The two antisymmetric charmed tetraquark flavour multiplets (3A-plet and 6-plet) have the curious property that all of their members have the same mass[30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' in the linear approx- imation to SU(3) symmetry breaking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' irrespective of their manifest strangeness [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' which is straightforward to see from their flavor wave functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' |T0 c¯sq¯q ⊂ 6⟩ = 1 √ 2 |cu � ¯d¯s − ¯s ¯d � ⟩ |T+ c¯sq¯q ⊂ 6⟩ = 1 √ 2|cd (¯s¯u − ¯u¯s)⟩ |T++ c¯sq¯q ⊂ 6⟩ = 1 2|c � u(¯u¯s − ¯s¯u) + d( ¯d¯s − ¯s ¯d) � ⟩ |D0∗ ⊂ 6⟩ = 1 2|c � s(¯u¯s − ¯s¯u) + d( ¯d¯u − ¯u ¯d) � ⟩ |D+∗ ⊂ 6⟩ = 1 2|c � s( ¯d¯s − ¯s ¯d) + u( ¯d¯u − ¯u ¯d) � ⟩ |X 0 0 ⊂ 6⟩ = 1 √ 2|cs � ¯d¯u − ¯u ¯d � ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' (1) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' (1) it can be seen that even the two states (D0∗, D+∗) with zero net strangeness contain an s¯s pair one half of the time, which effectively increases their masses by one strange- 3 up/down quark mass difference ms − mu/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' This property turned out in (surprisingly) good agreement with the measured masses of the D+ s0(2317) and D0(2308) mesons [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Three, T0 c¯sq¯q(2900), T++ c¯sq¯q(2900) and X + 0 (2900), of the six presumed members of the 6-plet have been discovered with degenerate masses, within combined uncertainties, so it (only) remains to be seen if this degeneracy will equally hold true for the remaining strangeness-zero isodoublet (D0∗, D+∗)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' This situation closely resembles that of the lowest-mass spin- 3 2 baryon 10-plet before the discovery of Ω−(1670).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' There, also, the SUF(3) symmetry breaking patterns led Gell-Mann [12] to the (spectacular) prediction of the mass of the previously missing hyperon, Ω−(1670).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Of course, these tetraquark states have substantial decay widths which may influence their “dressed” masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Next, we examine the observed decay widths, so as to see if they can be used to predict the widths of the as yet undiscovered states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Decay widths LHCb announced [2, 3] the results of a search for exotic isovector c¯s states, as two new resonances with masses of T 0,++ c¯s(q¯q)I=1(2900) : M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='908 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='020 GeV and widths of T 0,++ c¯s(q¯q)I=1(2900) : Γ(Tc¯s → π±D+ s ) = 136 ± 23 ± 11 MeV In the D−K+ channel, on the other hand, there are two resonances [1], both described with Breit-Wigner line shapes, the scalar (JP = 0+) one with parameters X0(2900) : M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='002 GeV/c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Γ(X0 → K+D−) = 57 ± 12 ± 4 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' which is roughly two times smaller than the width of T 0,++ c¯s(q¯q)I=1(2900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Moreover, this is substantially smaller than the expected total width (≥ 300 MeV) of such tetraquarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Can one understand these differences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' First, note that T ++ c¯s(q¯q)I=1(2900) need not decay only into D+ s π+, but may also decay into D+K+, which has not been observed (as yet), with only a slightly smaller phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Similarly, T 0 c¯s(q¯q)I=1(2900) need not decay only into D+ s π−, but may also decay into D0K0, which has not been observed due to the neutrality of decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Finally, X0 → K+D− 4 is not the only allowed mode of decay - X0(2900) → D 0K 0 is also allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' We must therefore examine all these decay widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Two-body decay widths are given by Γ(M → m1 + m2) = 1 8π|M|2λ(M, m1, m2) M2 where |M|2 = f 2 SU(3)|m|2 is the quantum mechanical decay amplitude squared, which factors into f 2 SU(3), the squared SU(3) “isoscalar factor” in the given specific flavor channel and the flavor-independent decay amplitude squared |m|2, and λ(M, m1, m2) = |p1| = |p2| = � (M2 − (m1 − m2)2) (M2 − (m1 + m2)2) 2M (2) is the phase space factor (also known as the Mandelstam function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The flavor-independent decay amplitudes m are equal, for equal masses M, m1, m2, to linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' This circumstance allows us to calculate the ratios of partial widths as Γ(T → D+ s π+) Γ(T → D+K−) = f 2 SU(3)(T, D+ s , π−)λ(T, D+ s , π−) f 2 SU(3)(T, D+, K−)λ(T, D+, K−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' provided we are given SU(3) isoscalar factors which are just the flavor SU(3) symmetry off-diagonal matrix elements fSU(3)(Tc(2900) → f) = ⟨f|Tc(2900)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' We use the 6-plet tetraquark flavor wave functions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' (1), and K− = s¯u, ¯K0 = s ¯d, π0 = 1 √ 2(u¯u − d ¯d), π+ = u ¯d, η8 = 1 √ 6(u¯u + d ¯d − 2s¯s), η0 = 1 √ 3(u¯u + d ¯d + s¯s), K+ = u¯s, K0 = d¯s, D0 = c¯u, D+ = c ¯d, D+ s = c¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' SU(3) isoscalar factors of observed tetraquarks, which may proceed in (at least) two different channels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' T0 c¯sq¯q → π−D+ s , T0 c¯sq¯q → K0D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Their SU(3) isoscalar factors are f(T 0 c¯sq¯q → π−D+ s ) = 1/ √ 2 f(T 0 c¯sq¯q → K0D0) = 1/ √ 2 The kaonic decay channel is difficult to detect due to two neutrals in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Similarly, the T ++ c¯s(q¯q)I=1(2900) SU(3) isoscalar factors are f(T ++ c¯sq¯q → π+D+ s ) = 1/ √ 2, f(T ++ c¯sq¯q → K+D+) = 1/ √ 2 and finally f( ¯X0(2900) → D+K−) = 1/ √ 2, f( ¯X0(2900) → D0K0) = 1/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 5 TABLE I: Numerical values of SU(3) matrix elements ⟨f|D∗0(2900)⟩ for various final states ⟨f|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Here η8 and η0 denote the eighth member of the octet and the SU(3) singlet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' ⟨D+π−| ⟨D0π0| ⟨D0η8| ⟨D0η0| ⟨D+ s K−| |D∗0(2900)⟩ 1 2 −1 2 √ 2 −1 2 √ 6 1 √ 3 −1 2 √ 2 SU(3) isoscalar factors of unobserved tetraquarks The isodoublet states (D∗(2900)0, D∗(2900)+) decay into more than two flavor channels: D∗(2900)0 → D0π0, D∗(2900)0 → D+π−, D∗(2900)0 → D0η, D∗(2900)0 → D+ s K−, with SU(3) isoscalar factors shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Some of these are eminently observable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=', both in the pion (D∗(2900)0 → D+π−) and in the kaon (D∗(2900)0 → D+ s K−) channels, due to the charged decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' These channels should be prime candidates for an experimental search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Decays of observed tetraquarks A quick calculation using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' (2) and the meson masses from Particle Data Group (PDG) [22], shows that the differences in phase space range from 4% to 28 %, which cannot account for the large differences, by a factor of two, in the widths of Tc¯s and X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' That also shows that the unobserved neutral channel decay T0 c¯sq¯q → K0D0 carries approximately one half of the total width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Similarly, the T ++ c¯s(q¯q)I=1(2900) → D+K+ decay, which is observable in principle, but has not been reported as yet, carries about 50 % of the total width, which is therefore at least 300 MeV, in agreement with expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The X0(2900) → D0K0 decay, which is difficult to detect on account of the final products neutrality, also carries about 50 % of the total width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Thus we see that the SU(3) isoscalar factors and phase space considerations do not lead to the convergence of the two decay widths T ++ c¯sq¯q and X0(2900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Nevertheless, this disagreement should not discourage us too much, as it is comparatively smaller than in the case of the spin- 3 2 baryon 10-plet, see the Comments below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Predicted widths of unobserved states The absolute widths of the cryptoexotic D mesons can be calculated from the (known) phase-space factors and the isoscalar factors and one measured tetraquark decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 1) We predict the ratio of D∗(2900)0 → D+π− and D∗0(2900) → D+ s K− partial decay widths as Γ(D∗(2900)0 → D+π−) Γ(D∗0(2900) → D+ s K−) = 2 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='28 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='56, 6 Using the Γ(Tc¯s0(2900)0 → π−D+ s ) decay width to set the total width, we find Γ(D∗(2900)0 → D+π−) ≃ 74 ± 19 MeV and Γ(D∗(2900)0 → D+ s K−) ≃ 29 ± 7 MeV Using the Γ(X0 → D+K−) decay width to set the total width, we find Γ(D∗(2900)0 → D+π−) ≃ 25 ± 7 MeV and Γ(D∗(2900)0 → D+ s K−) ≃ 17 ± 5 MeV, which puts them within the realm of the measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 2) The decay widths of the charged member of the isodoublet D∗(2900)+: Γ(D∗(2900)+ → D+π0), Γ(D∗(2900)+ → D0π+), Γ(D∗(2900)+ → D+η), Γ(D∗(2900)+ → D+ s ¯K0) are com- parable to those of the neutral state, but not easily measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Here, again, the trouble is that one of the two decay products is always neutral, so we do not anticipate detection in the foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 3) The single-charge isovector tetraquark’s Tc¯s0(2900)+ widths Γ(Tc¯s0(2900)+ → π0D+ s ), Γ(Tc¯s0(2900)+ → K+D0) are comparable to those of Tc¯s0(2900)++ and Tc¯s0(2900)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Again, the main obstruction to an experimental search is that one of its two decay products is always neutral, which makes it unlikely to be observed in the near run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Comments The observed common decay width of Γ(T 0,++ c¯s0 (2900)) = 136 ± 34 MeV is (at least) two times smaller than naively expected, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The observed width is only one half of the total width, however, the other half going into (unobserved) decays with at least one neutral object in the final state, which ought to settle the issue of the total width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The same holds for the observed and total widths of X0(2900), the latter still being approximately two times too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' That discrepancy should not unsettle us as analogous discrepancies in the lowest-mass spin- 3 2 baryon 10-plet are comparatively larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Nevertheless, the (resonance- peak) Breit-Wigner masses of mass spin- 3 2 baryons closely correspond with the bare masses in the flavour SU(3) symmetry schemes [12, 17] and in the constituent quark model [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Complicated production and decay mechanisms have been invoked for the calculation of each hyperon’s individual width [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Something similar may be expected for scalar tetraquarks [31], as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 7 Conclusions In this Letter we suggested a simple test of the conjecture that the isotriplet states T 0,++ c¯s0 (2900) and isosinglet ¯X0(2900) belong to an SU(3)F symmetry 6-plet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' An isodoublet of non-strange cryptoexotic states, (D0 c(¯uq¯q)I=1/2, D+ c( ¯dq¯q)I=1/2) must exist with the (same) mass around 2900 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The neutral member D∗0(2900) of this isodoublet ought to also decay into two charged particle channels, π−D+ and K−D+ s , which should allow ready detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The observable partial widths Γ(D∗0(2900)) ought to be at most one half of the corresponding partial widths of the isotriplet T 0 c¯s0(2900), and/or of the isosinglet X0(2900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' This is a rather weak constraint, due to the factor two difference(s) between the decay widths of T 0 c¯s0(2900) and X0(2900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The prospective discovery of a neutral resonance D∗0(2900) at, or near its predicted mass of 2900 MeV, would constitute an unassailable proof of its being a member of a 6-plet, and therefore of the 6-plet as a whole, vagaries about decay widths notwithstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Several exotic tetraquarks have been discovered experimentally and discussed theoreti- cally before the latest LHCb batch [1–3] - see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' the reviews [19, 20], yet there has not been a single instance, to our knowledge, of an exotic discovery following a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Terasaki [27] predicted manifestly exotic (double-charged) isovector partners of the D+ s (2317), which were searched for by Belle [23] in a narrow strip (± 33 MeV) around 2317 MeV, and by BaBar [21] up to 2600 MeV invariant mass, both without success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Perhaps the only successful prediction of D0(2308) in 2004, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [4], led to other specific mass predictions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' that of an isotriplet of c¯sq¯q states, and of an isoscalar cs¯q¯q state, all belonging to a flavor SU(3)F symmetry 6-plet, at 2724 MeV in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' After removing the SELEX constraint, the 6-plet mass can now be raised to 2900 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [4, 5, 15, 16] also predicted existence of a new 15-plet of tetraquarks, but little can be said about their masses with any certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' All of the exotics observed thus far have good isospin, yet their (prospective) SU(3) labels are rarely, if ever discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Here we suggested, perhaps for the first time, a specific test of an exotic charmed tetraquarks’ new SU(3) multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' It appears desirable to put this suggestion to an experimental test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Acknowledgments The author acknowledges informative correspondence with Drs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Yin- rui Liu and Ma Ruiting of the LHCb Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' This research was funded by the 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+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Dong and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Su, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='10, 101201 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='1088/0256-307X/37/10/101201 [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='07516 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [29] Indeed, LHCb [2] noted that “The obtained mass of the Tc¯s0 state is consistent with that of another 0+ open-charm tetraquark, the X0(2900)([cs¯u ¯d]) state discovered in the D+K− final state [19, 20], but their widths and flavor contents are different.” [30] Note that this degeneracy is a function of the SU(3) symmetry breaking patterns, which, in turn, depend on the the strong hyperfine interaction [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [31] The fact that the NRCQM predictions are for bare states is important, as it means that there are no decays (zero width) in that approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' The physical exotics’ (bare and/or dressed) masses lie far above their corresponding two-body thresholds, and therefore should be very wide, with only a lower bound on the width, Γ > 350 MeV, due to the “fall-apart” nature of the decay, much like the σ(500) and K∗ 0(700) = κ light-flavour scalar states, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Opening up of decay channels (“unitarization” of the calculation) would change not only the decay width, but also the dressed mass of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Resonant states could be dynamically 10 generated, see “Review of scalar mesons” in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [22], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=', they need not have a bare quark- state “seed” (or a Castillejo-Dalitz-Dyson (CDD) pole [26]) at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Indeed, the ∆ resonance was predicted as a state in a meson-nucleon model, without quarks, and only later confirmed by Chew and Low [25] in a “booststrap” kind of calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' In the same vein, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' [28] have recently argued that “X0(2900) can be interpreted as the S-wave D−K+ molecular state”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' Even if the as yet unobserved states turn out according to predictions, we shall not know their dynamical origin without further tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E5T4oBgHgl3EQfLQ4L/content/2301.05471v1.pdf'} diff --git a/wNFRT4oBgHgl3EQfgTdV/content/tmp_files/2301.13579v1.pdf.txt b/wNFRT4oBgHgl3EQfgTdV/content/tmp_files/2301.13579v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..00fbc6598e04154b2c6ea46a274d341152190a44 --- /dev/null +++ b/wNFRT4oBgHgl3EQfgTdV/content/tmp_files/2301.13579v1.pdf.txt @@ -0,0 +1,1862 @@ +Twisted cohomology and likelihood ideals +Saiei-Jaeyeong Matsubara-Heo and Simon Telen +Abstract +A likelihood function on a smooth very affine variety gives rise to a twisted de Rham +complex. We show how its top cohomology vector space degenerates to the coordinate +ring of the critical points defined by the likelihood equations. We obtain a basis for +cohomology from a basis of this coordinate ring. We investigate the dual picture, where +twisted cycles correspond to critical points. We show how to expand a twisted cocycle +in terms of a basis, and apply our methods to Feynman integrals from physics. +1 +Introduction +Very affine varieties are closed subvarieties of an algebraic torus. They have applications +in algebraic statistics [13] and particle physics [20]. We study smooth such varieties given +by hypersurface complements in the algebraic torus. Fix ℓ Laurent polynomials f1, . . . , fℓ ∈ +C[x±1 +1 , . . . , x±1 +n ] in n variables. Localizing at the product f1 · · · fℓ gives the very affine variety +X = {x ∈ (C∗)n : fi(x) ̸= 0, for all i} = (C∗)n \ V (f1 · · · fℓ). +(1) +In statistics and physics applications, the functions fi arise from a likelihood function +L(x) = f −s xν = f −s1 +1 +· · · f −sℓ +ℓ +xν1 +1 · · · xνn +n , +(2) +encountered as the integrand of a generalized Euler integral [1, 26]. These are Bayesian inte- +grals in statistics, and Feynman integrals in physics. Outside these applications, generalized +Euler integrals are interesting objects in their own right. They represent hypergeometric +functions and solutions to GKZ systems [8, 17]. Computations with these integrals can be +done in a twisted cohomology vector space Hn(X, ω) associated to X and L [1]. This paper +establishes a crucial relation between Hn(X, ω) and an ideal in the coordinate ring of X, +called the likelihood ideal. It makes computations in Hn(X, ω) explicit, by showing how to +compute a basis and how to find coefficients in this basis. +We think of the exponents s, ν in (2) as complex parameters, so the likelihood L is multi- +valued. The logarithm of L is the log-likelihood function, whose derivatives are single valued +and well-defined on X. The complex critical points of the log-likelihood function are the +solutions of ω(x) = 0, where ω is the one-form +ω(x) = dlogL(x) = −s1 dlogf1 − · · · − sℓ dlogfℓ + ν1 dx1 +x1 ++ · · · + νn dxn +xn +. +1 +arXiv:2301.13579v1 [math.AG] 31 Jan 2023 + +Expanding ω = g1dx1+· · ·+gndxn in the basis dx1, . . . , dxn gives n equations g1 = · · · = gn = +0 on X. The gi generate an ideal I in the coordinate ring O(X) of X, called the likelihood +ideal. June Huh has showed that, for generic s, ν, the likelihood ideal defines |χ(X)| critical +points, with χ(X) the Euler characteristic [13]. This means that dimC O(X)/I = |χ(X)|. +The Euler characteristic also counts the dimension of the twisted cohomology Hn(X, ω) +of X [1]. We briefly recall the definition. The form ω is regular on X, in the algebraic sense. +We write ω ∈ Ω1(X). More generally, Ωk(X) denotes the regular k-forms on X. Our vector +space Hn(X, ω) is the n-th cohomology of the twisted de Rham complex 0 → Ω0(X) → +Ω1(X) → · · · → Ωn(X) → 0, where the differential is d + ω∧. In symbols: +Hn(X, ω) = Ωn(X) / (d + ω∧)(Ωn−1(X)). +Using an identification Ωn(X) ≃ O(X), we can write this alternatively as Hn(X, ω) = +O(X)/V , where V ⊂ O(X) is a vector space which is not an ideal. We will sometimes write +V (ω) to emphasize the dependence of V on the twist ω. +Our first aim is to relate bases of the |χ(X)|-dimensional vector spaces O(X)/I and +O(X)/V , defined by the likelihood ideal I and the image V of the twisted differential. In +the physics literature, a basis of O(X)/V (or, the corresponding set of Feynman integrals) +is called a set of master integrals [12]. It is an important computational problem to find +such a basis. Computing a basis for O(X)/I can be done by computing the critical points +numerically, which is a task of numerical nonlinear algebra [1, Section 5]. For an element +g ∈ O(X), let [g]I be its residue class in O(X)/I, and [g]V its residue class in O(X)/V . +Theorem 1.1. Let (s, ν) ∈ Cℓ+n be generic complex parameters in the sense of Assumption +1 and let {[β1]I, . . . , [βχ]I} be a basis for O(X)/I. The set {[β1]V (ω/δ), . . . , [βχ]V (ω/δ)} is a +basis for Hn(X, ω/δ) = O(X)/V (ω/δ), for almost all δ ∈ C \ {0}. +Under stronger assumptions (Remark 4.1) one can use δ = 1 in this theorem. However, +there are special choices of s, ν and [βi]I for which this does not work, see Example 4.2. +We also prove an analog of Theorem 1.1 over the field K = C(s, ν) of rational functions +in s and ν. We use the notation XK for our very affine variety, but now defined over K. The +cohomology module of the twisted de Rham complex is the K-vector space O(XK)/VK = +Hn(XK, ω). Here VK ⊂ O(XK) is the image of d + ω∧ in our twisted de Rham complex +(Ω•(XK), d+ω∧) over K. The likelihood ideal is here reinterpreted as an ideal IK in O(XK). +Theorem 1.2. The K-vector spaces O(XK)/VK and O(XK)/IK have dimension |χ(X)|. If +{β1, . . . , βχ} ⊂ O(XK) represents a constant basis of O(XK)/IK, in the sense of Definition +3, then {[β1]VK, . . . , [βχ]VK} is a K-basis of O(XK)/VK. +Next to finding bases of cohomology, we also address the following problem. Given a +basis [β1]VK, . . . , [βχ]VK of O(XK)/VK and an element [g] ∈ O(XK)/VK, find the coefficients +ci ∈ K of g in this basis: [g] = c1 [β1]VK + · · · + cχ [βχ]VK. We show that O(XK)/VK is +isomorphic to a quotient of a non-commutative ring of difference operators R by a left ideal +J ⊂ R. This is an isomorphism of left R-modules: O(XK)/VK ≃ R/J (Theorem 2.2). The +unknown coefficients ci ∈ K are found from a set of contiguity matrices for J. These are χ×χ +matrices over K which encode how the difference operators act on the basis elements [βi]VK. +2 + +We show how to compute these matrices and provide an implementation. Our algorithm, +inspired by border basis algorithms [22], exploits the fact that a basis for O(XK)/VK can be +computed a priori. For the physics application, it offers an alternative for Laporta’s algorithm +to systematically compute all integration by parts relations among Feynman integrals [14]. +The intuition for the connection between twisted cohomology and the likelihood ideal +comes from a degeneration. We introduce a new parameter δ, so that making δ move from +1 to 0 turns O(XK)/VK into O(XK)/IK. This degeneration also appears in [18], where it +was used to relate the cohomology intersection pairing to Grothendieck’s residue pairing. +It can be defined over C as well, and turns Theorems 1.1 and 1.2 into practice: bases of +cohomology turn into bases of the likelihood quotient. The cohomology intersection pairing +over K is characterized as a unique bilinear pairing compatible with the R-module structure +mentioned above (Theorem 4.1). Contiguity matrices for δ → 0 become pairwise commuting +K-linear maps representing multiplication modulo IK (Theorem 5.1). +The dual vector space of the twisted cohomology Hn(X, ω) is the twisted homology +Hn(X, −ω) = HomC(O(X)/V, C) [1, Section 2]. A preferred basis of Hn(X, −ω) in this +article consists of the Lefschetz thimbles [28, Section 3]. These are called Lagrangian cy- +cles in [2, Section 4.3]. There is one Lefschetz thimble Γj ⊂ X for each critical point x(j) +satisfying ω(x(j)) = 0, with the property that x(j) ∈ Γj. They represent the linear functionals +[g]V �−→ +� +Γj +g(x) · L(x) dx1 +x1 +∧ · · · ∧ dxn +xn +. +In Section 4, we will show that when δ → 0, these Lefschetz thimbles degenerate to the +evaluation functionals [g]I �−→ g(x(j)) on the likelihood quotient O(X)/I. +The paper is organized as follows. Section 2 recalls the twisted de Rham complex and +establishes the isomorphism O(XK)/VK ≃ R/J. Section 3 introduces the likelihood ideal +and sets up the degeneration which takes the twisted de Rham cohomology to the likelihood +quotient. It contains a proof of Theorem 1.2. In Section 4, we prove Theorem 1.1 via our +degeneration and a perfect pairing of cohomology. We also discuss a different perfect pairing +with twisted homology, whose degeneration turns Lefschetz thimbles into evaluation at crit- +ical points. Section 5 deals with our computational goals: computing bases for cohomology +and computing expansions in this basis. We implement our algorithms in Julia. The code +uses the packages HomotopyContinuation.jl [6] and Oscar.jl [23]. It is made available at +https://mathrepo.mis.mpg.de/TwistedCohomology. In Section 6, we apply our methods +to compute contiguity matrices in several examples, including some Feynman integrals. +2 +Twisted de Rham cohomology +Fix ℓ Laurent polynomials f1, . . . , fℓ ∈ C[x±1 +1 , . . . , x±1 +n ] and let XC = X be as in (1). As +alluded to in the introduction, we will need analogous schemes XA over different rings A. In +our setting, the ring A satisfies A = C or C[s1, . . . , sℓ, ν1, . . . , νn] ⊂ A. Our schemes XA are +XA = Spec A[x±1 +1 , . . . , x±1 +n ]f1···fℓ. +3 + +The A-module of regular k-forms on XA is denoted by +Ωk(XA) = +� +� +� +� +1≤j1<... 0, +dH(x(t)) +dt += 0. +(33) +For each critical point x(j), the Lefschetz thimbles Γ− +j and Γ+ +j are unions of trajectories: +Γ∓ +j = +� +x0 ∈ X : the trajectory x(t) with x(0) = x0 satisfies limt→±∞ x(t) = x(j)� +. +13 + +Figure 1: Lefschetz thimbles for the data from (34). +Both Γ− +j and Γ+ +j are n-dimensional real manifolds containing x(j). By (33) and Assumption +1, they do not contain any of the other critical points x(i), i ̸= j. When restricted to Γ− +j +(Γ+ +j ), G reaches a maximum (minimum) at x(j), and tends to −∞ (+∞) away from x(j). +Intuitively, this explains why the integrals (31) and (32) converge when Γ∓ = Γ∓ +j . +Example 4.1 (n = 1). Figure 1 visualizes (part of) the Lefschetz thimbles for the data +n = ℓ = 1, +f = 1 − x3, +s = 1/2 + 3 +√ +−1, +ν = 1/7 + 7 +√ +−1. +(34) +The Mathematica code used to generate this picture is available at https://mathrepo. +mis.mpg.de/TwistedCohomology. The Euler characteristic of X is χ(X) = −χ = −3. The +three critical points of log L(x) are the black dots in the picture. The thimbles Γ+ +j and Γ− +j +are shown in the same color (blue, orange or green), for j = 1, 2, 3. They are flow lines +of a vector field that is a scaled version of the gradient field of G(x) = Re(log f −sxν) [2, +§4.3.4], visualized in the background of the figure. The points flowing towards/away from +the critical point are Γ− +j /Γ+ +j . We plotted this using ContourPlot on the imaginary part +H(x), see (33). This is tricky because H is multi-valued. Figure 1 shows the level lines +{H(x) = H(x(j)) + k · π}, for a few integer values of k. +⋄ +To turn our Lefschetz thimbles into twisted cycles, we need to choose which branch of +our multi-valued likelihood function L to integrate in (31) and (32). We do this by selecting +a value L(x0)±1 at some x0 ∈ Γ± +j , and analytically continuing along x(t) for t ∈ R. +Lemma 4.1. Under Assumption 1, the Lefschetz thimbles [Γ± +j ] form a basis for Hn(X, ±ω). +Proof. Under Assumption 1, we have dimC Hn(X, ±ω) = χ. By the discussion preceding [2, +Theorem 4.7], the Lefschetz thimbles generate Hn(X, ±ω). This implies the Lemma. +One of the advantages of using Lefschetz thimbles as a basis for twisted homology is +that it gives an easy formula for the intersection pairing between the cohomology spaces +14 + +1→ +Fi +G +()rHn(X, ω) and Hn(X, −ω). This is a bilinear map ⟨·, ·⟩ch : Hn(X, ω) × Hn(X, −ω) → C with +⟨g+, g−⟩ω +ch = +χ +� +j=1 +⟨g+, Γ− +j ⟩ω +per · ⟨g−, Γ+ +j ⟩−ω +per. +(35) +This formula is a special instance of [18, Equation (2.1)] and [20, Equation (18)], which holds +when Lefschetz thimbles are used as bases for homology. Just like the period pairing, the +intersection pairing is perfect. It has gained recent interest in physics, as it turns out to +compute scattering amplitudes in some special cases [20]. It follows from the formula (35) +that the matrix Q(ω) of ⟨·, ·⟩ω +ch, using the (arbitrary) bases β+ +i , β− +i for cohomology as above, +is Q(ω) = P(ω) · P(−ω)⊤. We also point out the following shift relations: +⟨f −1 +i +g+, fig−⟩ω(s,ν) +ch += ⟨g+, g−⟩ω(s+ei,ν) +ch +, +⟨xjg+, x−1 +j g−⟩ω(s,ν) +ch += ⟨g+, g−⟩ +ω(s,ν+ej) +ch +, +(36) +which will be useful later. Here ω(s, ν) emphasizes the dependence of ω on s, ν. +4.3 +Intersection pairing over K +While twisted homology is only defined over C, previous sections have used purely algebraic +descriptions of Hn(XA, ω) over different rings A. This subsection discusses the cohomology +intersection pairing ⟨·, ·⟩ω +ch over the field K = C(s, ν). Our first result says that the function +(s, ν) �→ ⟨g+, g−⟩ω(s,ν) +ch +belongs to K. +Proposition 4.1. For g± ∈ Hn(X, ±ω), the function (s, ν) �→ ⟨g+, g−⟩ω(s,ν) +ch +is rational. +Proof. Recall that Serre’s duality pairing is a composition +H0( ¯X, Ωn +log(kD)) × Hn( ¯X, O ¯ +X(−(k + 1)D)) +∪ +−→ Hn( ¯X, Ωn +¯ +X) +tr +−→ C +(37) +of the cup product ∪ and the trace map tr [11, Chapter 3, §7]. By the degeneration of the +spectral sequence (8) in Lemma 2.1, we obtain the following representations of Hn(X, ±ω): +Hn(X, ω) = +H0( ¯X, Ωn +log(kD)) +im +� +∇ω : H0( ¯X, Ωn−1 +log (kD)) → H0( ¯X, Ωn +log(kD)) +�, +Hn(X, −ω) = ker +� +∇−ω : Hn( ¯X, O ¯ +X(−(k + 1)D)) → Hn( ¯X, Ω1 +log(−(k + 1)D)) +� +. +(38) +Via (38), Serre duality (37) induces a bilinear pairing Hn(X, ω) × Hn(X, −ω) → C , which +is identical to ⟨·, ·⟩ω +ch [18, Eq. (2.3)]. This construction works over K = C(s, ν), as Serre +duality holds for any projective scheme defined over a field. The K-valued pairing +⟨·, ·⟩ω +ch,K : Hn(XK, ω) × Hn(XK, −ω) → K +(39) +obtained from Serre duality specializes to ⟨·, ·⟩ω +ch for s, ν generic as in Definition 1. +15 + +The cohomology intersection pairing ⟨·, ·⟩ω +ch,K over K is distinguished from other perfect +pairings by its compatibility with the R-action on twisted cohomology groups, where R is +the ring of difference operators from (11). The rest of this subsection makes this statement +precise. We define another action •− of R on O(XK), slightly different from (12): +σsi •− g(s, ν) = fi · g(s + ei, ν), +σνj •− g(s, ν) = x−1 +j +· g(s, ν + ej). +(40) +As in the discussion around (12), the action (40) induces an R-action on Hn(XK, −ω). The +ring R also acts on K via the shifts σsi • a(s, ν) = a(s+ei, ν) and σνj • a(s, ν) = a(s, ν +ej). +A K-bilinear pairing ⟨·, ·⟩ : Hn(XK, ω) × Hn(XK, −ω) → K is compatible with R if +⟨σ • [g+], σ •− [g−]⟩ = σ • ⟨[g+], [g−]⟩ +(41) +holds for any [g±] ∈ Hn(XK, ±ω) and σ = σsi, σνj. Here is a K-version of (36). +Proposition 4.2. The cohomology intersection pairing (39) is compatible with R. +Proof. Let us write [ξ±(s, ν)] ∈ Hn(XK, ±ω) for the cohomology classes in the right-hand +side of (38). Then, σsi •[ξ+(s, ν)] (resp. σsi •−[ξ−(s, ν)]) is represented by a cohomology class +[f −1 +i +ξ+(s + ei, ν)] ∈ H0( ¯XK, Ωn +log(kD − divfi)) (resp. [fiξ−(s + ei, ν)] ∈ Hn( ¯XK, O ¯ +XK(−(k + +1)D + divfi))). Here, divfi denotes the divisor of fi viewed as a rational function on ¯XK. +We obtain a sequence of identities +⟨σsi • [ξ+(s, ν)], σsi •− [ξ−(s, ν)]⟩ω +ch,K = tr([f −1 +i +ξ+(s + ei, ν)] ∪ [fiξ−(s + ei, ν)]) += tr([ξ+(s + ei, ν)] ∪ [ξ−(s + ei, ν)]) += σsi • tr([ξ+(s, ν)] ∪ [ξ−(s, ν)]) += σsi • ⟨[ξ+(s, ν)], [ξ−(s, ν)]⟩ω +ch,K. +Theorem 4.1. Up to a non-zero scalar multiplication by C, ⟨·, ·⟩ω +ch,K is the unique perfect +K-bilinear pairing ⟨·, ·⟩ : Hn(X, ω) × Hn(X, −ω) → K compatible with R. +Our proof of Theorem 4.1 uses the following Lemma. +Lemma 4.2. Let N be a left R-module that is finite dimensional over K. If N is a simple +R-module, then the dimension of EndR(N) over C is 1. +Proof. Let ϕ ∈ EndR(N). Writing ¯K for the algebraic closure of K, the action of R on +K extends to that on ¯K. Thus, we may regard ϕ as an element of End ¯ +K⊗KR( ¯N) where +we set ¯N := +¯K ⊗K N. +We first prove that any eigenvalue of ϕ is in C. +Let us take +an eigenvector v ∈ N of ϕ. It is straightforward to see that σi +s1v is an eigenvector with +eigenvalue σi +s1α. Therefore, there exists an integer i so that σi +s1α = α. Similarly, we can +prove that α is periodic for all s1, . . . , sℓ, ν1, . . . , νn. Since such a function α ∈ ¯K must be a +constant function, it must belong to C. Now, suppose that dimC EndR(N) ≥ 2 and take a +morphism ϕ ∈ EndR(N) linearly independent from idN over C. For any eigenvalue α ∈ C of +ϕ, α · idN − ϕ has a non-trivial kernel, which is a non-trivial R-submodule of N. This is a +contradiction. +16 + +Proof of Theorem 4.1. By Kashiwara’s equivalence [5, Chapter VI, Theorem 7.13], the lo- +cal cohomology group M defined by (14) is a simple Dℓ+n,C-module. It follows from [16, +Theorem 1.2.1] that Hn(XK, ω) is a simple R-module. +Any pair of K-bilinear pairings +⟨·, ·⟩, ⟨·, ·⟩′ : Hn(XK, ω)×Hn(XK, −ω) → K compatible with R gives rise to an R-morphism +Hn(XK, ω) → Hn(XK, ω). Now, the theorem follows from Lemma 4.2. +4.4 +Degeneration of pairings +We now turn to perfect pairings for the likelihood quotient O(X)/I. The dual vector space +(O(X)/I)∨ consists of all linear functionals on O(X) which vanish on the likelihood ideal I. +The evaluation pairing ⟨·, ·⟩ev : O(X)/I × (O(X)/I)∨ → C is given by +⟨g, v⟩ev = v(g). +Here g is short for [g]I, and v ∈ (O(X)/I)∨ is such that v(I) = 0. A canonical basis of +(O(X)/I)∨ is v1, . . . , vχ, where vj(g) = g(x(j))/√ηj represents evaluation at the j-th critical +point. This is normalized by √ηj = +� +Hess(x(j)). Together with a basis [β1]I, . . . , [βχ]I of +the likelihood quotient, the evaluation pairing has a matrix representation Eij = ⟨βi, vj⟩ev = +βi(x(j))/√ηj. Since the evaluation pairing is perfect, this matrix is invertible. +In analogy with Q(ω) = P(ω)·P(−ω)⊤, we can also consider the bilinear map represented +by the matrix G = E · E⊤. This is the Grothendieck residue pairing ⟨·, ·⟩res, given by +O(X)/I × O(X)/I → C, +with +⟨g, h⟩res = +χ +� +j=1 +⟨g, vj⟩ev · ⟨h, vj⟩ev = +χ +� +j=1 +g(x(j))h(x(j)) +ηj +. +We are now ready to bring in our deformation parameter δ. In Section 3, we did this by +replacing ∇ω with ∇δ +ω. As we have seen in the proof of Theorem 1.2, this is equivalent to +replacing ω with ω/δ. Here is what this looks like for our period pairings: +⟨g±, Γ∓⟩±ω/δ +per += +� +Γ∓ g±(x) · L(x)± 1 +δ dx1 +x1 +∧ · · · ∧ dxn +xn +. +We view this as a function of δ. +If Γ− +j is a Lefschetz thimble, log L(x)± 1 +δ has constant +imaginary part, and its real part reaches a maximum at x(j), see (33). When δ → 0, this +maximum value at x(j) grows, and the contribution of the rest of the integration contour is +more and more suppressed. This is the intuition behind the Proposition 4.3, which roughly +says that for δ → 0, integration over the Lefschetz thimble turns into evaluation at x(j). +Proposition 4.3. Let Γ∓ +j be the Lefschetz thimbles associated to the j-th critical point x(j) +of log L(x). Under Assumption 1, we have the following formulae as δ → 0: +⟨g+, Γ− +j ⟩ω/δ +per = (−2πδ) +n +2 · e +1 +δ log L(x(j)) · ⟨g+, vj⟩ev · (1 + O(δ)), +(42) +⟨g−, Γ+ +j ⟩−ω/δ +per += (−2πδ) +n +2 ( +√ +−1)−n · e− 1 +δ log L(x(j)) · ⟨g−, vj⟩ev · (1 + O(δ)). +(43) +Similarly, for the cohomology intersection pairing, we have +⟨g+, g−⟩ω/δ +ch += (2π +√ +−1δ)n · ⟨g+, g−⟩res · (1 + O(δ)). +(44) +17 + +Proof. The formulae (42)-(43) follow from stationary phase approximation [10, Chapter I]. +Equation (44) follows from (35) and (42)-(43). It appears in [18, Theorem 2.4]. +Propositions 4.1 and 4.3 lead to a proof of Theorem 1.1: +Proof of Theorem 1.1. Let [β1]I, . . . , [βχ]I be a basis for the likelihood quotient O(X)/I, and +let P(±ω/δ)ij = ⟨βi, Γ∓ +j ⟩±ω/δ +per +be the period pairing matrices. Proposition 4.3 implies +Q(ω/δ) = P(ω/δ) · P(ω/δ)⊤ = (2π +√ +−1δ)n · G · (1 + O(δ)). +Since the [βi]I are a basis, the matrix G is invertible, and hence also Q(ω/δ) is invertible +for δ → 0. Since the entries of Q(ω/δ) are rational functions of δ, see Proposition 4.1, this +implies that the classes of the βi form a basis for Hn(X, ±ω/δ), for almost all δ ∈ C. +Preferably, we would like to use δ = 1 in Theorem 1.1. Unfortunately, for this, genericity +of s, ν in the sense of Assumption 1 is not enough. Here is an example. +Example 4.2. Let n = 1 and f(x) = 1 − x. The Euler characteristic of X is −1. Genericity +in Definition 1 means ν, −s, s − ν /∈ Z. These three linear forms correspond to the three +boundary points {0}, {1}, {∞} in the compactification X ⊂ P1. In H1(X, ω), we have +�x + 1 +x2 +� +V (ω) += +��2ν − s − 1 +ν − 1 +�� +V (ω) +. +(45) +If ν = 1/4, s = −1/2, (45) implies that this is not a basis for H1(X, ω). However, [ x+1 +x2 ]I is +a basis of O(X)/I. Still, [ x+1 +x2 ]V (ω/δ) ∈ H1(X, ω/δ) is a basis for generic δ. +⋄ +Remark 4.1. We can set δ = 1 if we make a stronger genericity assumption on s, ν. In +addition to Assumption 1, we assume that det Q(ω(s, ν)) is neither 0 nor ∞ at s, ν. Here +Q(ω(s, ν)) ∈ Kχ×χ is the matrix of rational functions in s, ν which represents the cohomology +intersection pairing for the functions β± +i = βi from Theorem 1.1, see Proposition 4.1. Its +determinant is a nonzero rational function, because Q(ω(s/δ, ν/δ)) is given by (44). +5 +Bases for cohomology and contiguity matrices +This section is about computation. First, we show how to compute a basis for Hn(XK, ω). +By Theorem 1.2, it suffices to compute a subset of O(XK) which represents a constant basis +of O(XK)/IK in the sense of Definition 3. Our strategy relies on numerical computation. It +is based on some heuristics. However, in practice, it is highly reliable and effective. +We start by plugging in generic complex values of s and ν in the likelihood function +L(x) from (2). We then solve ω = dlogL(x) = 0 numerically, using the homotopy contin- +uation technique explained in [1, Section 5]. This reliably computes all χ = (−1)n · χ(X) +complex critical points, even for large Euler characteristics. See [26] for an example with +χ = 3628800. A list of regular functions β1, . . . , βχ ∈ O(X) gives a basis of O(X)/I if and +only if the evaluation pairing from Section 4.4 gives an invertible χ×χ-matrix Eij = ⟨βi, vj⟩ev. +Algorithm 1 exploits this observation. It takes the likelihood equations as input, as well as +18 + +a list G ⊂ O(X) of constant regular functions. The output is a subset of G that is maximal +independent in O(XK)/IK, i.e. it has the largest possible cardinality such that its elements +are K-linearly independent mod IK. One strategy to generate G is as follows. For fixed +d ≥ 1, we set Gd = {f −axb}|a|+|b|≤d, with |a| = a1 + · · · + aℓ, ai ≥ 0, and similarly for b. +If the list returned by Algorithm 1 for G = Gd contains m < χ elements, then Gd does not +contain a basis. In that case, we increase d and repeat. We note that O(XK)/IK is spanned +by the union �∞ +d=0 Gd. Both the computation of the critical points and the rank tests in this +algorithm are numerical, but this works well in practice. +Algorithm 1 Compute a maximal independent subset of constant functions mod IK +Input: ω(s, ν) = dlogL(x), G = {g1, . . . , gN} ⊂ O(X), g1 /∈ IK +Output: {β1, . . . , βm} ⊂ G such that [β1]IK, . . . , [βm]IK is maximal independent +ω ← ω(s∗, ν∗) for generic complex s∗, ν∗ +{x(1), . . . , x(χ)} ← solutions of ω(x) = 0 +E ← the row vector (g1(x(j)))j=1,...,χ +β1 ← g1, ℓ ← 2, k ← 2 +while rank(E) < χ and k ≤ N do +E′ ← append the row (gk(x(j)))j=1,...,χ to E +if rank E′ > rank E then +E ← E′, βℓ ← gk, ℓ ← ℓ + 1 +end if +k ← k + 1 +end while +return {β1, . . . , βℓ−1} +Next, our goal is to compute contiguity matrices for a given basis [β1]VK, . . . , [βχ]VK. These +are χ × χ matrices with entries in K encoding how the difference operators σsi, σνj ∈ R act +on the basis elements. For instance, the contiguity matrix Cs1 satisfies +σs1 • +� +� +� +[β1]VK +... +[βχ]VK +� +� +� = +� +� +� +σs1 • [β1]VK +... +σs1 • [βχ]VK +� +� +� = Cs1(s, ν) · +� +� +� +[β1]VK +... +[βχ]VK +� +� +� . +Notice that, although the difference operators σsi, σνj are pairwise commuting, the contiguity +matrices are not. This is easily seen in an example: +σsiσνj•[β]VK = σsi•Cνj(s, ν)·[β]VK = Cνj(s+ei, ν)·(σsi•[β]VK) = Cνj(s+ei, ν)·Csi(s, ν)·[β]VK. +Here the second equality applies (11). Expanding this in the opposite order shows that +Cνj(s + ei, ν) · Csi(s, ν) = Csi(s, ν + ej) · Cνj(s, ν). +(46) +More generally, for a, b ≥ 0, contiguity matrices can be used to compute σa +sσb +ν • [β]VK via +σa +sσb +ν • [β]VK = Cνn · · · · · Cν1 · Csℓ · · · · · Cs1 · [β]VK, +(47) +19 + +where the calligraphic C’s denote the following ordered products of matrices: +Cνj = +bj +� +q=1 +Cνj +� +s+a, ν + +� +k 0.05) +difference was observed in PAW prepared in different flasks. The percentage decrease in pH +of PAW of flasks (flask-I to flask-V) compared to control after 4-hours of plasma-water +exposure time varied between 55.2% to 62.7%, respectively. The observed trend of decrease +in pH of PAW with increasing plasma-water exposure time is also supported by previously +reported work of Ten Bosch et al.(5), Punith et al.(21), Sajib et al.(32), Jin et al.(30), Xiang et +al.(6), and Subramanian et al.(15), etc. + +The dissolution of various oxidizing species (H2O2, dissolved O3, OH, ONOOˉ, NO2ˉ +ions, and NO3ˉ, etc.) in PAW increases the oxidizing potential (ORP) of PAW(6, 12, 16, 28, + +29, 37). The variation in ORP of PAW with increasing plasma-water exposure time is shown +in figure 4 (b). Increasing plasma-water exposure time substantially (p < 0.05) increased the +ORP of PAW kept in flasks I to V. The ORP of PAW kept in the flasks (I to V) varies between +550 mV to 605 mV which was 139.1% and 163.0% higher compared to control after 4-hours +of plasma-water exposure. The past reported work of Xiang et al.(6) also confirmed an increase +in oxidizing tendency (ORP) of PAW with increasing plasma treatment time. + +The generated dissolved radicals/species in PAW increased the total dissolved solids +(TDS) and electrical conductivity (EC) of PAW. The inorganic ions such as NO2ˉ ions and +NO3ˉ ions etc. present in PAW makes the PAW electrically conducting. Figure 4 (c, d) reveals +increasing plasma treatment time with water increases the TDS and EC of PAW. This signifies +a continuous increase in the concentration of dissolved conducting species in PAW with longer +plasma-water exposure. For 4-hours of plasma treatment time, the highest value of TDS and +EC observed in PAW were given as 600 ppm and 1610 µS cm-1, respectively. The increase in +EC with increasing plasma-water exposure time was also shown by Xiang et al.(6), Jin et +al.(30), Subramanian et al.(15), and, Than et al.(18), respectively. + + +Figure 4. Physicochemical properties of plasma activated water prepared using multiple plasma +device batch setup (refer figure 1 (b)). The statistically significant difference (p < 0.05) among +the different properties of PAW groups with time (mean ± standard deviation) (µ ± σ) is shown +by different lowercase letters. +3.3 RONS concentration in PAW prepared in the batch process +The formation of RONS in PAW is shown in equations (10-21) of appendix Table A1. Some +of the stable RONS (NO3ˉ ions, NO2ˉ ions, H2O2, and dissolved O3) present in PAW were +identified using semi-quantitative estimation and their actual concentrations were measured +using UV-visible spectroscopy(37). Figure 5 showed the measured RONS concentration in +PAW. Initially, in the control (plasma-water exposure time 0 hours), no dissolved RONS were +present in PAW (figure 5). As the plasma exposure to water was initiated, the formation of +RONS in water starts occurring. Increasing plasma-water exposure time increases the NO3ˉ +0 +0.5 +1 +2 +4 +2 +3 +4 +5 +6 +7 +d'''' +c'''' +b'''' +ab'''' +a'''' +d''' +c''' +d'' +c'' +b''' +b''' +a''' +ab'' +b'' +a'' +d' +c' +b' +b' +a' +e +c +b +ab +pH +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +a +0 +0.5 +1 +2 +4 +200 +300 +400 +500 +600 +700 +d'''' +cd'''' +c'''' +b'''' +a'''' +d''' +cd''' +c''' +b''' +a''' +e'' +d'' +c'' +b'' +a'' +e' +d' +c' +b' +a' +d +cd +c +b +ORP (mV) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +a +0 +0.5 +1 +2 +4 +0 +100 +200 +300 +400 +500 +600 +700 +e'''' +d'''' +c'''' +b'''' +a'''' +e''' +d''' +c''' +b''' +a''' +e'' +d'' +c'' +b'' +a'' +e' +d' +c' +b' +a' +e +d +c +b +a +TDS (ppm) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +0 +0.5 +1 +2 +4 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +e'''' +d'''' +c'''' +b'''' +a'''' +e''' +d''' +c''' +b''' +a''' +e'' +d'' +c'' +b'' +a'' +e' +c' +b' +a' +a' +e +d +c +b +a +(d) +(c) +(b) +EC (µS/cm) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +(a) + +ions, NO2ˉ ions, and H2O2 concentration in PAW (figure 5 (a-c)). These RONS concentration +variations with time are also supported by work reported by Subramanian et al.(15), Xiang et +al.(6), Jin et al.(30), Sajib et al.(32), Lukes et al.(27), Punith et al.(21), and, Than et al.(18). +However, the concentration of dissolved O3 present in PAW initially increased and then +decreased with increasing plasma-water exposure time. This showed the dissolved O3 present +in PAW reacts with other RONS (equations (11, 13) of appendix Table A1) that results in +suppression of its concentration (figure 5 (d)). Moreover, the NO2ˉ ions and H2O2 also reacted +with each other and other RONS present in PAW by following equations (19) of appendix +Table A1 (27). However, NO2ˉ ions and H2O2 did not show a decrease in their concentration +with time. In addition, the observed maximum concentration of NO2ˉ ions and H2O2 in PAW +were 10.7 mg L-1 and 6.7 mg L-1. This was substantially low compared to the maximum +concentration of NO3ˉ ions which was 176.0 mg L-1. This showed the stability of NO3ˉ ions +(equations (11, 16, 20, 21) of appendix Table A1) over other RONS in PAW. This was due to +other RONS (NO2ˉ ions, dissolved O3, H2O2) reacted with each other to form more stable NO3ˉ +ions. As a result, the NO3ˉ ions concentration in PAW was substantially higher compared to +NO2ˉ ions, H2O2, and dissolved O3, respectively. The substantially higher concentration of +NO3ˉ ions in PAW relative to rest RONS is also shown in past literature by various +researchers(5, 6, 15, 28, 29, 32, 33, 37). + +In the batch production process, a 4-hour plasma-water exposure gave PAW of ORP up +to 605 mV and pH up to 2.5. The high ORP and low pH of PAW showed antimicrobial efficacy +of PAW. Since the high oxidizing PAW has the efficacy to oxidize the pathogenic microbes +and low pH supports this antimicrobial efficacy of PAW(6). Hence, the prepared PAW could +be used in applications like food (fruits, vegetables, daily products, and meat products +including seafood) preservations, surface disinfection including surgical equipment, dentistry, +and pathogens (bacteria, fungi, viruses, and pests, etc.) inactivation, etc.(1-13) + + +In conclusion, the batch process could be used to prepare PAW of high reactivity that +could be used in numerous applications. A prototype (scale-up) of the present batch model +could be used in industries to prepare PAW of high reactivity with a larger volume. + +Figure 5. Reactive oxygen-nitrogen species concentration ((a) NO3ˉ ions, (b) NO2ˉ ions, (c) +H2O2, and (d) Dissolved O3) in plasma activated water prepared using multiple plasma device +batch setup (refer figure 1 (b)). The statistically significant difference (p < 0.05) among the +different properties of PAW groups with time (mean ± standard deviation) µ ± σ is shown by +different lowercase letters. +3.4 Properties of PAW prepared in continuous process +Figure 6 showed the properties of PAW when prepared using continuous processes. Similar to +the batch process, the properties of PAW showed a similar trend with increasing plasma-water +exposure time. However, one exception was observed in the form of dissolved O3 +0 +0.5 +1 +2 +4 +0 +50 +100 +150 +200 +a'' +e'''' +d'''' +c'''' +b'''' +a'''' +e''' +d''' +c''' +b''' +a''' +e'' +d'' +c'' +b'' +a'' +e' +d' +c' +b' +a' +e +d +c +b +NO3 +- (mg L +-1) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +a +0 +0.5 +1 +2 +4 +0 +2 +4 +6 +8 +10 +12 +e'''' +d'''' +c'''' +b'''' +a'''' +e''' +d''' +c''' +b''' +a''' +e'' +d'' +c'' +b'' +a'' +e' +d' +c' +b' +a' +e +d +c +b +a +NO2 +- (mg L +-1) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +0 +0.5 +1 +2 +4 +0 +1 +2 +3 +4 +5 +6 +7 +e'''' +d'''' +c'''' +b'''' +a'''' +d''' +d''' +c''' +b''' +a''' +e'' +e' +d'' +c'' +b'' +d' +c' +b' +a' +e +d +c +b +a +H2O2 (mg L +-1) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +0 +0.5 +1 +2 +4 +0 +1 +2 +3 +4 +5 +6 +7 +c'''' +b'''' +b'''' +b'''' +a'''' +d''' +c''' +c''' +b''' +a''' +d'' +c'' +c'' +b'' +a'' +d' +c' +c' +b' +a' +e +b +b +b +a +(d) +(c) +(b) +Dissolved O3 (mg L +-1) +Plasma water exposure time (hours) + Flask - I + Flask - II + Flask - III + Flask - IV + Flask - V +(a) + +concentration. In which, the dissolved O3 concentration increases with increasing plasma-water +exposure time. This was due to the low reactivity (lower oxidizing tendency and higher pH) of +PAW in continuous processes which limits the reaction of dissolved O3 with other RONS in +PAW. Therefore, the decrease in dissolved O3 concentration was not observed. + +The chosen plasma-water exposure time was 10 hours for a continuous process which +was significantly higher compared to the batch process (4 hours). Since, the volume of water +for the continuous process was 20 liters that was larger compared to the batch process (2 liters). +As the multiple plasma device was used for both the batch and continuous process. Therefore, +a higher plasma-water exposure time was used in continuous processes compared to batch +processes. + +For a plasma-water exposure time of 10 hours, the percentage decrease in the pH of +PAW was 53.7% compared to control (water with no plasma treatment, t = 0 hours). +Simultaneously, the increase in ORP of PAW was 108.7%. Moreover, the TDS and EC of PAW +after a plasma-water exposure time of 10 hours reached the value of 340 ppm and 850 µS cm- +1, respectively. The measured maximum concentration of NO3ˉ ions, NO2ˉ ions, H2O2, and +dissolved O3 after 10 hours of plasma-water exposure were given as 93.5 mg L-1, 7.5 mg L-1, +4.7 mg L-1, and 4.7 mg L-1, respectively. + +As the reactivity (oxidizing tendency) and dissolved RONS concentration in PAW +prepared in a continuous process were lower than the batch process. Hence, this low reactive +PAW has applications in seed germination and plant growth and as a nitrogen source in +agriculture applications, etc.(17, 18, 21, 22, 53) Since, as discussed, the high reactive PAW has +applications in microbial inactivation. Hence, seeds and plants’ exposure to high reactive PAW +may damage the seeds and plants themselves (53). Therefore, control over the PAW properties +becomes extremely important while preparing PAW for different applications. + + +Figure 6. Physicochemical properties ((a) pH, (b) oxidation-reduction potential (ORP), (c) total +dissolved solids (TDS), and (d) Electrical conductivity (EC)) and reactive-oxygen nitrogen +species concentration ((e) NO3ˉ ions, (f) NO2ˉ ions, (g) H2O2, and (h) Dissolved O3) in plasma +activated water prepared using multiple plasma device continuous setup (refer figure 1 (c)). +The statistically significant difference (p < 0.05) among the different properties of PAW groups +with time (mean ± standard deviation) µ ± σ is shown by different lowercase letters. +4. Conclusion +The present work discussed a multiple plasma device setup to produce plasma activated water. +This multiple plasma device produces PAW in batch and continuous manner. Both the batch +and continuous PAW setup are designed in such a way as to optimize the dissolution of gases +reactive species in water. Moreover, the reactive species carried by effluent gases after plasma- +water exposure are trapped in water in both the batch and continuous PAW setup. Hence, it +reduced the environmental pollutants such as NOx and gases O3, etc. which comes out effluent +gases after plasma-water exposure. In the batch process, a high reactive PAW is produced with +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +3 +4 +5 +6 +7 +f +ef +e +de +de +d +cd +c +b +b +pH +Plasma water exposure time (hours) +a +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +225 +300 +375 +450 +525 +f +f +e +e +d +cd +bc +b +ab ab +a +ORP (mV) +Plasma water exposure time (hours) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +0 +100 +200 +300 +400 +k +j +i +h +g +f +e +d +c +b +a +TDS (ppm) +Plasma water exposure time (hours) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +0 +200 +400 +600 +800 +1000 +j +i +h +g +f +f +e +d +c +b +a +EC (µS cm +-1) +Plasma water exposure time (hours) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +0 +25 +50 +75 +100 +k +j +i +h +g +f +e +d +c +b +NO3 +- (mg L +-1) +Plasma water exposure time (hours) +a +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +0 +2 +4 +6 +8 +i +h +g +g +f +e +d +c +b +a +a +NO2 +- (mg L +-1) +Plasma water exposure time (hours) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +0 +1 +2 +3 +4 +5 +g +f +e +d +d +c +c +b +ab +ab +a +H2O2 (mg L +-1) +Plasma water exposure time (hours) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 +0 +1 +2 +3 +4 +5 +g +f +e +d +d +c +c +b +b +ab +a +(h) +(g) +(f) +(e) +(d) +(c) +(b) +Dissolved O3 (mg L +-1) +Plasma water exposure time (hours) +(a) + +a total volume of 2 liters. In the continuous process, a relatively lower reactive PAW is +produced with a total volume of 20 liters. The high reactive PAW has enhanced +physicochemical properties and RONS concentration. Hence, it has applications in pathogen +inactivation, food preservation, cancer cell inactivation, and surface disinfection, etc. +Moreover, less reactive PAW has moderate physicochemical properties and lower RONS +concentration compared to a high reactive PAW. Therefore, it has applications in seeds +germination, plant growth, and as a nitrogen source for numerous agriculture and aquaculture +applications, etc. +Acknowledgments +This work was supported by the Department of Atomic Energy (Government of India) graduate +fellowship scheme (DGFS). The authors sincerely thank O. R. Kaila, and Mr. Nimish for +providing constant support and useful suggestions during this work. +Data availability statement +The data that support the findings of this study are available upon reasonable request from the +authors. +Conflict of interests +The authors declare that there are no conflicts of interests. +Authors’ contributions +All authors contributed to the study conception and design. Material preparation, data +collection, and analysis were performed by Vikas Rathore. Design and development of power +supply was done by Chirayu Patil. The first draft of the manuscript was written by Vikas +Rathore, and all authors commented on previous versions of the manuscript. All authors read +and approved the final manuscript. + +ORCID iDs +Vikas Rathore https://orcid.org/0000-0001-6480-5009 +Annexure +Table A1: Reactions of formation of various reactive oxygen-nitrogen species in PAW through +plasma-water interaction. +Reaction +Rate constant (k) +Eq. no. +𝑁2 (𝑔) + 𝑒− → 2𝑁 (𝑔) + 𝑒− +6.3 × 10-6 Te-1.6 e-9.8/Te +cm3 s-1 +2 +𝐻2𝑂 (𝑔) + 𝑒− → 𝑂𝐻 (𝑔) + 𝐻 (𝑔) + 𝑒− +2.6 × 10-12 cm3 s-1 +3 +𝐻2𝑂 (𝑔) + 𝑁2(𝐴) (𝑔) → 𝑂𝐻 (𝑔) + 𝐻 (𝑔) + 𝑁2 (𝑔) +4.2 × 10-11 cm3 s-1 +4 +𝑁 (𝑔) + 𝑂2 (𝑔) → 𝑁𝑂 (𝑔) + 𝑂 (𝑔) +8.5 × 10-17 cm3 s-1 +5 +𝑂(𝑔) + 𝑁𝑂(𝑔) + 𝑀(𝑔) → 𝑁𝑂2(𝑔) + 𝑀(𝑔) +9.0 × 10-32 cm6 s-1 +6 +𝑂 (𝑔) + 𝑂2 (𝑔) → 𝑂3 (𝑔) +1.7 × 10-12 cm3 s-1 +7 +𝑂𝐻 (𝑔) + 𝑂𝐻 (𝑔) → 𝐻2𝑂2 (𝑔) +2.6 × 10-11 cm3 s-1 +8 +𝑂𝐻 (𝑔) + 𝑂3 (𝑔) → 𝐻𝑂2 (𝑔) + 𝑂2 (𝑔) +1.9 × 10-12 cm3 s-1 +9 +𝑂𝐻 (𝑎𝑞. ) + 𝑁𝑂 (𝑎𝑞.) → 𝑵𝑶𝟐 +− (𝑎𝑞.) + 𝐻+(𝑎𝑞.) +1.0 × 1010 M-1 s-1 +10 +𝑶𝟑 (𝑎𝑞. ) + 𝑵𝑶𝟐 +− (𝑎𝑞.) → 𝑂2 (𝑎𝑞.) + 𝑵𝑶𝟑 +− (𝑎𝑞.) +2.5 × 105 M-1 s-1 +11 +𝑂𝐻 (𝑎𝑞. ) + 𝑂𝐻 (𝑎𝑞.) → 𝑯𝟐𝑶𝟐 (𝑎𝑞.) +5.0 × 109 M-1 s-1 +12 +𝑂𝐻 (𝑎𝑞.) + 𝑶𝟑 (𝑎𝑞.) → 𝑂2 (𝑎𝑞. ) + 𝐻𝑂2 (𝑎𝑞.) +1.0 × 108 M-1 s-1 +13 +𝐻𝑂2 (𝑎𝑞. ) + 𝐻𝑂2 (𝑎𝑞.) → 𝑂2 (𝑎𝑞. ) + 𝑯𝟐𝑶𝟐 (𝑎𝑞. ) +1.0 × 106 M-1 s-1 +14 +𝑁𝑂2 (𝑎𝑞.) + 𝑁𝑂 (𝑎𝑞. ) + 𝐻2𝑂 (𝑎𝑞.) → 𝟐𝑵𝑶𝟐 +− (𝑎𝑞.) + 2𝐻+ (𝑎𝑞.) +2.0 × 108 M-1 s-1 +15 +2𝑁𝑂2 (𝑎𝑞. ) + 𝐻2𝑂 (𝑎𝑞.) → 𝑵𝑶𝟑 +− (𝑎𝑞.) + 𝑵𝑶𝟐 +− (𝑎𝑞. ) + 2𝐻+ (𝑎𝑞.) +0.5 × 108 M-1 s-1 +16 +𝑯𝟐𝑶𝟐 (𝑎𝑞. ) + 𝑂𝐻 (𝑎𝑞.) → 𝐻2𝑂 (𝑎𝑞.) + 𝑂2 +− (𝑎𝑞.) + 𝐻+ (𝑎𝑞.) +2.7 × 107 M-1 s-1 +17 +𝐻2𝑂 (𝑎𝑞.) + 𝐻𝑂2 (𝑎𝑞. ) + 𝑂2 +− (𝑎𝑞.) +→ 𝑂2 (𝑎𝑞.) + 𝑯𝟐𝑶𝟐 (𝑎𝑞.) + 𝐻𝑂− (𝑎𝑞.) +9.7 × 107 M-1 s-1 +18 +𝑵𝑶𝟐 +− (𝑎𝑞. ) + 𝑯𝟐𝑶𝟐 (𝑎𝑞. ) + 𝐻+ (𝑎𝑞.) → 𝑂𝑁𝑂𝑂𝐻 (𝑎𝑞. ) + 𝐻2𝑂 (𝑎𝑞. ) +1.1 × 103 M-1 s-1 +19 + +𝑵𝑶𝟐 +− (𝑎𝑞.) + 𝑂𝐻 (𝑎𝑞. ) + 𝐻+ (𝑎𝑞.) → 𝑵𝑶𝟑 +− (𝑎𝑞. ) + 2𝐻+ (𝑎𝑞.) +5.3 × 109 M-1 s-1 +20 +𝑂𝑁𝑂𝑂𝐻 (𝑎𝑞. ) → 𝑵𝑶𝟑 +− (𝑎𝑞.) + 𝐻+ (𝑎𝑞.) +0.9 s-1 +21 + +References +1. +Rathore V, Patel D, Butani S, Nema SK. 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Long-lived species in plasma- +activated water generated by an AC multi-needle-to-water discharge: effects of gas flow on +chemical reactions. Journal of Physics D: Applied Physics. 2020;54(6):065201. + +50. +Lim JS, Kim RH, Hong YJ, Lamichhane P, Adhikari BC, Choi J, et al. Interactions +between atmospheric pressure plasma jet and deionized water surface. Results in Physics. +2020;19:103569. +51. +Sergeichev KF, Lukina NA, Sarimov RM, Smirnov IG, Simakin AV, Dorokhov AS, et +al. Physicochemical Properties of Pure Water Treated by Pure Argon Plasma Jet Generated by +Microwave Discharge in Opened Atmosphere. Frontiers in Physics. 2021:596. +52. +Ghimire B, Szili EJ, Patenall BL, Lamichhane P, Gaur N, Robson AJ, et al. +Enhancement of hydrogen peroxide production from an atmospheric pressure argon plasma jet +and implications to the antibacterial activity of plasma activated water. Plasma Sources Science +Technology. 2021;30(3):035009. +53. +Sivachandiran L, Khacef A. Enhanced seed germination and plant growth by +atmospheric pressure cold air plasma: combined effect of seed and water treatment. RSC +advances. 2017;7(4):1822-32. + + diff --git a/ytE4T4oBgHgl3EQfYgyr/content/tmp_files/load_file.txt b/ytE4T4oBgHgl3EQfYgyr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3707549b3fdf28d562ae6b69b1dbbebb381efa2 --- /dev/null +++ b/ytE4T4oBgHgl3EQfYgyr/content/tmp_files/load_file.txt @@ -0,0 +1,973 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf,len=972 +page_content='Title: Continuous production for large quantity plasma activated water using multiple plasma device setup Authors Vikas Rathore1,2*, Chirayu Patil1, and Sudhir Kumar Nema1,2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Atmospheric Plasma Division, Institute for Plasma Research (IPR), Gandhinagar, Gujarat 382428, India 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India Email: vikas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='rathore@ipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='in Abstract In the present work, a batch and continuous production of plasma-activated water (PAW) is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' To produce PAW in a batch and continuous manner a multiple plasma device setup is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The multiple plasma device consists of a series of plasma devices that are powered simultaneously to produce PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This multiple plasma device is powered by indigenously developed high-voltage high-frequency power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The air plasma generated in this multiple plasma device setup is electrically characterized and the produced radicals/species are identified using optical emission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The post-discharge effluent gases left after plasma-water exposure carries some environmental pollutants (NOx and O3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The batch and continuous PAW production setup utilizes effluent (pollutants) gases in production of large volume PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, it substantially reduces the concentration of these pollutants in effluent gases which are released in environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The batch process produces high reactive PAW with less volume (2 liters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, in a continuous process, a high volume (20 liters) with low reactivity of PAW is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The high reactive PAW and low reactive PAW are used for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Inactivation of microbes (bacteria, fungi, viruses, and pests), food preservation, selective killing of cells, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' is carried out using high reactive PAW whereas low reactive PAW has applications in seeds germination, plant growth, and as a nitrogen source for agriculture and aquaculture applications, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In addition, the batch and continuous PAW production setup designs are scalable, therefore, can be used in industries for PAW production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Keywords: plasma activated water, multiple plasma device, batch and continuous process, reactive oxygen-nitrogen species, plasma characterization (electrical and optical emission) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Introduction The plasma activated water (PAW) technology is a step toward sustainable development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' It is an eco-friendly and economically viable technology that utilizes plasma and water as preliminary ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' PAW has applications in microbial inactivation (bacteria, fungi, viruses, algae, and pests, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' )(1-6), food preservation (fruits, vegetables, dairy products, and meat products including seafood, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' )(7-12), dental and medical equipment’s surface disinfection(13), pesticide removal(14), selective killing of cancer cells(15), virus vaccine preparation(16), seeds germination(17), plant growth(18), plasma acid(19), and as a source of nitrogen fertilizer(20), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (21, 22) Hence use for disinfectants (chemicals), preservatives, fertilizer, and medicine ingredients, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' that cause various environmental and health issues can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Since, they can cause soil and water pollution that causes various diseases in plants and animals (land and water) including humans(23-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, PAW has the potential to be used as a replacement of disinfectants (chemicals), preservatives, fertilizer, and medicine ingredients, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, PAW technology has the potential to be used in sustainable development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The mentioned applications of PAW are possible because of the presence of numerous reactive oxygen-nitrogen species (RONS) in PAW(1, 2, 4, 7, 13, 15-18, 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Some of the RONS present in PAW are given as NO3ˉ ions, NO2ˉ ions, H2O2, OH radical, ONOOˉ ions, and dissolved O3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (4, 7, 17, 19, 27-31) The presence of reactive oxygen species (ROS) such as H2O2, OH radical, ONOOˉ ions, and dissolved O3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' plays a significant role in applications like microbial inactivation, food preservation, and selective killing of cancer cells, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (1, 2, 4, 6, 7) Moreover, a higher concentration of reactive nitrogen species (RNS) like NO3ˉ ions and NO2ˉ ions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' exists in the form of nitric and nitrous acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' As a result, pH of PAW decreased significantly which also favors applications like microbial inactivation and food preservation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (1, 2, 7, 9) Moreover, low to moderate concentration of RONS present in PAW is used in applications like seed germination and plant growth(12, 17, 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The PAW treatment improves the wettability properties of seeds by removing the waxing from their surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, low to moderate ROS concentration acts as a signalling molecule that enhances plant growth(17, 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In addition, the RNS present in PAW can also be used as a nitrogen source for applications in agriculture and aquaculture(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, the concentration of RONS in PAW decides its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The present work also emphasizes the production of PAW with low/moderate and high RONS concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The agriculture and aquaculture applications of PAW required a large quantity of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, applications like microbial inactivation and food preservation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' required a relatively low volume of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Also, limited research has been conducted which emphasizes the production of PAW in large quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (30) reported mass production of PAW using a cylindrical dielectric barrier discharge reactor with very high plasma discharge power (8 kW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Also, they used a cooling assembly to cool down electrodes which created additional running costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' They showed the variation in pH, electrical conductivity (EC), and NO3ˉ ions concentration in PAW only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, Ĉech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (3) use hydrodynamic cavitation plasma jet (HCPJ) for mass production of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' However, Ĉech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (3) do not report the change in physicochemical properties and RONS concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, commenting on the reactivity of plasma activated water becomes difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Past literature of numerous authors mainly reported the production of small volume PAW using cold plasma with low plasma discharge power (1W to 40 W)(5, 10, 13-15, 21, 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, production of high volume of PAW using low plasma discharge power still need to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The insubstantial literature on high volume PAW production using cold plasma with low plasma discharge power and its properties analysis become the motivation of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, the present work emphasizes the production of high volume of PAW using cold plasma with low plasma discharge power (~ 35 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Also, tries to optimize the dissolution of generated plasma species/radicals in water by trapping gases reactive species/radicals carried by effluents gases after PAW production(33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The post-discharge effluent gas after plasma- water exposure carries environmental pollutants such as NOx and gases O3 etc (34-36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, the reduction of these gases’ pollutants concentration in effluent gases becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, trapping of these gases using PAW further enhances the physicochemical properties and RONS concentration in it(33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' As a result, maximum utilization of discharge gases species is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The present work discussed scalable, in terms of volume of plasma activated water setup using multiple plasma devices powered by an indigenously developed power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' These setups include batch production of PAW and continuous production of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In batch production, a comparatively low volume (2 liters) of PAW with high reactivity and RONS concentration tries to be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In continuous production, a high volume (20 liters) of PAW with moderate reactivity and RONS concentration tries to be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The importance of high and low/moderate reactivity PAW in applications is already discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The voltage- current waveform is used to characterize the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The generated air plasma species are diagnosed by using optical emission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The PAW characterization is performed by measuring the physicochemical properties of PAW and RONS concentration in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Material and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='1 Optical emission and electrical diagnosis of multiple plasma device setup Figure 1 (a) showed the schematic of multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A series of five plasma devices were used in multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The plasma devices are connected in parallel with the power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The plasma device is a cylindrical co-axial assembly in which quartz cone was used as dielectric, mesh as a ground electrode, and copper pipe as power electrode(28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The quartz double cone has a two-sided B24 male socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A flexible copper wire mesh was wrapped around the cone to make a ground electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The power electrode was made using a 16 mm outer diameter copper pipe of with a thickness of 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A diamond knurling of pitch size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 mm was introduced which disturbed the uniform electric field and increased the localized electric field at the sharp knurled edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' More detail about plasma device can be found in our past reported work (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A self-made high voltage high frequency (HVHF) and low current power supply was used to power multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The voltage drops across the multiple plasma device setup was measured using a high voltage 1000x probe (Tektronix P6015A) and a 4-channel, 100 MHz, 2 GS s-1 sampling rate digital oscilloscope (Tektronix TDS2014C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The total current and transported charge in multiple plasma device setup were measured using a voltage probe (Tektronix TPP0201) and an oscilloscope as shown in figure 1 (a)(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This probe measured the voltage drop across a 3-ohm non-inductive resistor and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='2 µF non-polar capacitor connected in series with the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The air plasma emission spectrum in the afterglow region produced in the plasma device was recorded using optical fiber and spectrometer (Model EPP2000-UV from StellarNet Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') in the wavelength range of 200-600 nm as shown in figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The captured light from the radiative decay of atoms and molecules was transmitted using optical fiber to the charge- coupled device (CCD) detector of the spectrometer with a detector integration time of 500 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The spectrometer has 1200 lines mm-1 grating groove density, 60 mm effective focal length, and a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 nm of spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The calibration of the spectrometer was performed using an emission line of wavelength 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='6 nm emitted from a mercury vapour lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='2 Batch process for PAW production The schematic of batch production of plasma activated water is shown in figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The multiple plasma device setup was loosely fit over the flat bottom flasks (B24 socket fitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' That substantially reduces the discharge gas leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Each flat bottom flask (500 capacity) carries 400 ml (5-flasks, total water volume of 2 liters) of ultrapure milli-Q water (control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Air was fed from the top using an air pump at a constant flow rate of 5 l min-1 using an air rotameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The plasma-water exposure time for batch production varies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 hours to 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The properties (RONS and physicochemical properties) of PAW was monitored periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The discharge gases effluent after plasma-water exposure from flask-I feed to flask-II, so the undissolved gases reactive in flask-I get dissolved in flask-II and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The series trapping of reactive species present in effluent gases enhances the physicochemical properties and dissolved RONS concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Furthermore, it reduces the risk of release of environmental pollutants species carried by post-discharge effluent gases such as NOx and O3 to the open atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A schematic of the series trapping assembly of gases reactive species in water is shown in figure 1 (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='3 Continuous process for PAW production The schematic of the continuous production of plasma activated water is shown in figure 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In a continuous process, water was recirculated between the flask during plasma-water exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The total volume of water taken in the continuous process was 20 liters kept in a closed water tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A water pump was used to feed the water from the tank to the flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The level of water in flasks is controlled using inlet and outlet control values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' After the water activation cycle is over, the activated water returns to the water tank and feedback again to the flask as shown in figure 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Furthermore, periodic monitoring of PAW was performed to study the change in its properties for up to 10 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The remaining operating condition was kept constant similar to the batch process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (a) Schematic of electrical and optical emission characterization of air plasma produced using multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (b) Schematic of batch production of plasma activated water using multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (c) Schematic of continuous production of plasma activated water using multiple plasma devices setup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='4 Schematic of power supply Figure 2 showed the schematic of the power supply used in the batch and continuous production of plasma activated water using multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The input to the power supply was single-phase (1-ϕ) 50 Hz 220 V alternative current (AC) power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The input AC is converted to direct current (DC) using a full-wave diode bridge rectification circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The filter circuit Dscilloscope Emission Spectra Spectrometer WaterTankconverts the rectified DC to pure DC as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The DC voltage was lies between 0 to 300 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The received DC were converted to high frequency pulsating quasi sine wave output voltage by designing a high-frequency inverter circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The inverter circuit was developed using a full-bridge Pulse Width Modulator (PWM) topology with a high switching frequency oscillator (OSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The high-frequency pulsating quasi sine wave output was connected to the primary side of the high voltage high frequency (HVHF) transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The HVHF transformer transforms 0-300 V to 0-10 kV output voltage with tuneable output frequency of 0-30 kHz (38- 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The high voltage (HV) output coming from the secondary side of the transformer followed by a filter was used as input to multiple plasma device setup to produce plasma activated water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Schematic of power supply used for PAW production using multiple plasma device setup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 Measurement of properties of PAW The physicochemical properties namely pH, oxidation reduction potential (ORP), total dissolved solids (TDS), and electrical conductivity (EC) of PAW were determined using a Hanna Instruments pH meter (HI98121), Hanna Instruments ORP meter (ORP-200), HM Digital TDS meter (AP1), and HM Digital EC meter (COM-360).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 个The reactive oxygen nitrogen species (RONS) concentrations namely NO3ˉ ions, NO2ˉ ions, H2O2, and dissolved O3 concentration in PAW were initially determined semi- quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The semi-quantitative estimation of RONS was performed using strips test and colorimetry test kits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The strips used for the estimation of NO2ˉ ions and H2O2 concentration were given as QUANTOFIX Nitrite and QUANTOFIX Peroxide 25 (MACHEREY-NAGEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The colorimetry test kits used to determine NO3ˉ ions and dissolved O3 were given as VISOCOLOR alpha Nitrate (MACHEREY-NAGEL) and Dissolved Oxygen Chemical Test Kit - HI3810 (Hanna Instruments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The UV-visible spectroscopy (SHIMADZU UV-2600)(37) were used to measure the RONS concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The NO3ˉ ions concentration in PAW was determined by measuring the absorbance of PAW at 220 nm (Deuterium lamp) and using a standard calibration curve of NO3ˉ ions (molar attenuation coefficient at wavelength 220 nm is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0602 (mg L-1)-1 cm-1)(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The NO2ˉ ions present in PAW (acidic region) when reacts with the reaction mixture sulfanilamide and N-(1-naphthyl)ethylenediamine dihydrochloride give radish purple azo dye and showed peak absorbance at 540 nm (Tungsten lamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The standard curve of NO2ˉ ions (molar attenuation coefficient at wavelength 540 nm is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0009 (µg L-1)-1 cm-1) was used to determine the unknown concentration of NO2ˉ ions(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Similarly, H2O2 is present in PAW (acidic region) when reacts with titanium (IV) ions to form peroxotitanium (pertitanic acid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' That formed a yellow color complex which showed maximum absorbance at 407 nm (Tungsten lamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The unknown H2O2 concentration in PAW was determined using a standard H2O2 curve having a molar attenuation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='4857 mM-1 cm-1(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, the NO2ˉ ion presence in PAW interferes with H2O2 determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Since, the NO2ˉ ions react with H2O2 and suppress its concentration beyond the detection limit(27, 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, to inhibit the NO2ˉ ions and H2O2 reaction, azide ions (N3ˉ) were added to PAW as soon as it was prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The N3ˉ ions react with NO2ˉ ions and for N2 gas, as a result, NO2ˉ ions interference in H2O2 concentration determine can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The indigo colorimetric method was used to determine the dissolved O3 concentration in PAW (37, 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The expression to calculate the unknown dissolved O3 in PAW is given as: 𝑚𝑔 𝑙 𝑜𝑓 𝑂3 = 100×∆𝐴 𝑓 ×𝑏 × 𝑣 (1) ΔA – difference in absorbance of PAW and control at 600 nm (Tungsten lamp) f – sensitivity factor (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='42) v – volume of sample in ml b – optical path length of cell (cm) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='6 Data Analysis All the experiments shown in present study were repeated at least three times (n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The data collected were expressed as means ± standard deviation (µ ± σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The statistically significant difference with a p-value (null hypothesis significance test) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05 between the observed results were calculated using one-way analysis of variance (ANOVA) followed by a post-hoc test (Tukey’s Honest Significant Difference (HSD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Results and Discussions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='1 Characterization of air plasma and identification of plasma species The voltage-current waveform of air plasma is shown in figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The current shown in figure 3 (a) is a combination of continuous alternating current and discontinuous discharge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The discharge current is the current associated with generated radicals and species in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' These charged particles carry discharge current in each rising and falling current half-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 3 (a) shows the discharge current is in the form of a combination of several filamentary current peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, this discharge is known as filamentary micro discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, the air plasma produced in plasma devices was DBD filamentary micro-discharge in nature(42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The energy and power consumed in multiple plasma device setup were given as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='74 mJ and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='8 W calculated using the charge voltage Lissajous figure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The emission spectrum of the air plasma afterglow region (the optical fiber capturing light photons placed 10 mm away from plasma discharge region in which mainly neutral molecules and atoms are present) is shown in figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The air plasma showed strong emission band peaks of nitrogen gas (N2) second positive system (SPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The observed vibrational band peaks in afterglow air plasma showed the transition of N2 (C 3Πu) higher state to N2 (B 3Πg) lower state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In addition, a weak intensity N2+ first negative system (FNS) was also observed in air plasma afterglow regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The observed vibrational band peaks of N2+ FNS showed the transition from N2+ (B 2Σu+) higher state to N2+ (X 2Σg+) lower state(43, 44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The radiative decay of N2 and N2+ higher energy state to lower energy state results in the formation of the mentioned electronic vibrational transition shown in figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (a) Voltage-current characterization of multiple plasma device setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (b) Emission spectrum of air plasma produced in plasma device 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='2 Physicochemical properties of PAW prepared in batch process 100 50 Current (mA) N*(B*-X,) 100 2-*3The charge radicals and species formed due to air discharge in plasma devices when exposed to water formed various stable and unstable reactive radicals and species in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The high energy identified species observed in the air plasma afterglow region participate in various gases reactions which formed species like N, H, eˉ, NO, NO2, OH, H2O2, HO2, H2O, O, O2, and O3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' are shown in equations (2-9) of appendix Table A1 (27, 28, 31, 45-48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' These generated species get dissolved in water and form various reactive oxygen- nitrogen species (RONS) in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The formation of various RONS in PAW is shown in equations (10-21) of appendix Table A1 (27, 28, 31, 45-52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The presence of these RONS in PAW responsible for the physicochemical changes observed in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 4 showed the variation in physicochemical properties of PAW with plasma treatment time kept in the flasks – I to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The occurrence of NO2ˉ and NO3ˉ ions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' in PAW exist in form of nitrous and nitric acid due to which the decrease in pH of PAW observed (1, 2, 5, 30, 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Increasing plasma- water treatment time significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05) decreases the pH of PAW as shown in figure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The decreasing pH signifies the increasing concentration of acidic species present in PAW with increasing plasma-water exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In addition, no statistically significant (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05) difference was observed in PAW prepared in different flasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The percentage decrease in pH of PAW of flasks (flask-I to flask-V) compared to control after 4-hours of plasma-water exposure time varied between 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='2% to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The observed trend of decrease in pH of PAW with increasing plasma-water exposure time is also supported by previously reported work of Ten Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (5), Punith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (21), Sajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (32), Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (30), Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (6), and Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (15), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The dissolution of various oxidizing species (H2O2, dissolved O3, OH, ONOOˉ, NO2ˉ ions, and NO3ˉ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') in PAW increases the oxidizing potential (ORP) of PAW(6, 12, 16, 28, 29, 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The variation in ORP of PAW with increasing plasma-water exposure time is shown in figure 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Increasing plasma-water exposure time substantially (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05) increased the ORP of PAW kept in flasks I to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The ORP of PAW kept in the flasks (I to V) varies between 550 mV to 605 mV which was 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='1% and 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0% higher compared to control after 4-hours of plasma-water exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The past reported work of Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (6) also confirmed an increase in oxidizing tendency (ORP) of PAW with increasing plasma treatment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The generated dissolved radicals/species in PAW increased the total dissolved solids (TDS) and electrical conductivity (EC) of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The inorganic ions such as NO2ˉ ions and NO3ˉ ions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' present in PAW makes the PAW electrically conducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 4 (c, d) reveals increasing plasma treatment time with water increases the TDS and EC of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This signifies a continuous increase in the concentration of dissolved conducting species in PAW with longer plasma-water exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' For 4-hours of plasma treatment time, the highest value of TDS and EC observed in PAW were given as 600 ppm and 1610 µS cm-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The increase in EC with increasing plasma-water exposure time was also shown by Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (6), Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (30), Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (15), and, Than et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (18), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Physicochemical properties of plasma activated water prepared using multiple plasma device batch setup (refer figure 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The statistically significant difference (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05) among the different properties of PAW groups with time (mean ± standard deviation) (µ ± σ) is shown by different lowercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='3 RONS concentration in PAW prepared in the batch process The formation of RONS in PAW is shown in equations (10-21) of appendix Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Some of the stable RONS (NO3ˉ ions, NO2ˉ ions, H2O2, and dissolved O3) present in PAW were identified using semi-quantitative estimation and their actual concentrations were measured using UV-visible spectroscopy(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 5 showed the measured RONS concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Initially, in the control (plasma-water exposure time 0 hours), no dissolved RONS were present in PAW (figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' As the plasma exposure to water was initiated, the formation of RONS in water starts occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Increasing plasma-water exposure time increases the NO3ˉ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 2 3 4 5 6 7 d'''' c'''' b'''' ab'''' a'''' d''' c''' d'' c'' b''' b''' a''' ab'' b'' a'' d' c' b' b' a' e c b ab pH Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V a 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 200 300 400 500 600 700 d'''' cd'''' c'''' b'''' a'''' d''' cd''' c''' b''' a''' e'' d'' c'' b'' a'' e' d' c' b' a' d cd c b ORP (mV) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V a 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 0 100 200 300 400 500 600 700 e'''' d'''' c'''' b'''' a'''' e''' d''' c''' b''' a''' e'' d'' c'' b'' a'' e' d' c' b' a' e d c b a TDS (ppm) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 0 250 500 750 1000 1250 1500 1750 e'''' d'''' c'''' b'''' a'''' e''' d''' c''' b''' a''' e'' d'' c'' b'' a'' e' c' b' a' a' e d c b a (d) (c) (b) EC (µS/cm) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V (a) ions, NO2ˉ ions, and H2O2 concentration in PAW (figure 5 (a-c))." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' These RONS concentration variations with time are also supported by work reported by Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (15), Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (6), Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (30), Sajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (32), Lukes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (27), Punith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (21), and, Than et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' However, the concentration of dissolved O3 present in PAW initially increased and then decreased with increasing plasma-water exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This showed the dissolved O3 present in PAW reacts with other RONS (equations (11, 13) of appendix Table A1) that results in suppression of its concentration (figure 5 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, the NO2ˉ ions and H2O2 also reacted with each other and other RONS present in PAW by following equations (19) of appendix Table A1 (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' However, NO2ˉ ions and H2O2 did not show a decrease in their concentration with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In addition, the observed maximum concentration of NO2ˉ ions and H2O2 in PAW were 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 mg L-1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 mg L-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This was substantially low compared to the maximum concentration of NO3ˉ ions which was 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 mg L-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This showed the stability of NO3ˉ ions (equations (11, 16, 20, 21) of appendix Table A1) over other RONS in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This was due to other RONS (NO2ˉ ions, dissolved O3, H2O2) reacted with each other to form more stable NO3ˉ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' As a result, the NO3ˉ ions concentration in PAW was substantially higher compared to NO2ˉ ions, H2O2, and dissolved O3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The substantially higher concentration of NO3ˉ ions in PAW relative to rest RONS is also shown in past literature by various researchers(5, 6, 15, 28, 29, 32, 33, 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In the batch production process, a 4-hour plasma-water exposure gave PAW of ORP up to 605 mV and pH up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The high ORP and low pH of PAW showed antimicrobial efficacy of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Since the high oxidizing PAW has the efficacy to oxidize the pathogenic microbes and low pH supports this antimicrobial efficacy of PAW(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, the prepared PAW could be used in applications like food (fruits, vegetables, daily products, and meat products including seafood) preservations, surface disinfection including surgical equipment, dentistry, and pathogens (bacteria, fungi, viruses, and pests, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') inactivation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (1-13) In conclusion, the batch process could be used to prepare PAW of high reactivity that could be used in numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' A prototype (scale-up) of the present batch model could be used in industries to prepare PAW of high reactivity with a larger volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Reactive oxygen-nitrogen species concentration ((a) NO3ˉ ions, (b) NO2ˉ ions, (c) H2O2, and (d) Dissolved O3) in plasma activated water prepared using multiple plasma device batch setup (refer figure 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The statistically significant difference (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05) among the different properties of PAW groups with time (mean ± standard deviation) µ ± σ is shown by different lowercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='4 Properties of PAW prepared in continuous process Figure 6 showed the properties of PAW when prepared using continuous processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Similar to the batch process, the properties of PAW showed a similar trend with increasing plasma-water exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' However, one exception was observed in the form of dissolved O3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 0 50 100 150 200 a'' e'''' d'''' c'''' b'''' a'''' e''' d''' c''' b''' a''' e'' d'' c'' b'' a'' e' d' c' b' a' e d c b NO3 (mg L 1) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V a 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 0 2 4 6 8 10 12 e'''' d'''' c'''' b'''' a'''' e''' d''' c''' b''' a''' e'' d'' c'' b'' a'' e' d' c' b' a' e d c b a NO2 (mg L 1) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 0 1 2 3 4 5 6 7 e'''' d'''' c'''' b'''' a'''' d''' d''' c''' b''' a''' e'' e' d'' c'' b'' d' c' b' a' e d c b a H2O2 (mg L 1) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content="5 1 2 4 0 1 2 3 4 5 6 7 c'''' b'''' b'''' b'''' a'''' d''' c''' c''' b''' a''' d'' c'' c'' b'' a'' d' c' c' b' a' e b b b a (d) (c) (b) Dissolved O3 (mg L 1) Plasma water exposure time (hours) Flask - I Flask - II Flask - III Flask - IV Flask - V (a) concentration." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In which, the dissolved O3 concentration increases with increasing plasma-water exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This was due to the low reactivity (lower oxidizing tendency and higher pH) of PAW in continuous processes which limits the reaction of dissolved O3 with other RONS in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, the decrease in dissolved O3 concentration was not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The chosen plasma-water exposure time was 10 hours for a continuous process which was significantly higher compared to the batch process (4 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Since, the volume of water for the continuous process was 20 liters that was larger compared to the batch process (2 liters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' As the multiple plasma device was used for both the batch and continuous process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, a higher plasma-water exposure time was used in continuous processes compared to batch processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' For a plasma-water exposure time of 10 hours, the percentage decrease in the pH of PAW was 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7% compared to control (water with no plasma treatment, t = 0 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Simultaneously, the increase in ORP of PAW was 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, the TDS and EC of PAW after a plasma-water exposure time of 10 hours reached the value of 340 ppm and 850 µS cm- 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The measured maximum concentration of NO3ˉ ions, NO2ˉ ions, H2O2, and dissolved O3 after 10 hours of plasma-water exposure were given as 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 mg L-1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 mg L-1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 mg L-1, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 mg L-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' As the reactivity (oxidizing tendency) and dissolved RONS concentration in PAW prepared in a continuous process were lower than the batch process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, this low reactive PAW has applications in seed germination and plant growth and as a nitrogen source in agriculture applications, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' (17, 18, 21, 22, 53) Since, as discussed, the high reactive PAW has applications in microbial inactivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, seeds and plants’ exposure to high reactive PAW may damage the seeds and plants themselves (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, control over the PAW properties becomes extremely important while preparing PAW for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Physicochemical properties ((a) pH, (b) oxidation-reduction potential (ORP), (c) total dissolved solids (TDS), and (d) Electrical conductivity (EC)) and reactive-oxygen nitrogen species concentration ((e) NO3ˉ ions, (f) NO2ˉ ions, (g) H2O2, and (h) Dissolved O3) in plasma activated water prepared using multiple plasma device continuous setup (refer figure 1 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The statistically significant difference (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='05) among the different properties of PAW groups with time (mean ± standard deviation) µ ± σ is shown by different lowercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Conclusion The present work discussed a multiple plasma device setup to produce plasma activated water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' This multiple plasma device produces PAW in batch and continuous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Both the batch and continuous PAW setup are designed in such a way as to optimize the dissolution of gases reactive species in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, the reactive species carried by effluent gases after plasma- water exposure are trapped in water in both the batch and continuous PAW setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, it reduced the environmental pollutants such as NOx and gases O3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' which comes out effluent gases after plasma-water exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' In the batch process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' a high reactive PAW is produced with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='4 ' 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+page_content=' In the continuous process, a relatively lower reactive PAW is produced with a total volume of 20 liters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The high reactive PAW has enhanced physicochemical properties and RONS concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Hence, it has applications in pathogen inactivation, food preservation, cancer cell inactivation, and surface disinfection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Moreover, less reactive PAW has moderate physicochemical properties and lower RONS concentration compared to a high reactive PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Therefore, it has applications in seeds germination, plant growth, and as a nitrogen source for numerous agriculture and aquaculture applications, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Acknowledgments This work was supported by the Department of Atomic Energy (Government of India) graduate fellowship scheme (DGFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The authors sincerely thank O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Kaila, and Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Nimish for providing constant support and useful suggestions during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Data availability statement The data that support the findings of this study are available upon reasonable request from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Conflict of interests The authors declare that there are no conflicts of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Authors’ contributions All authors contributed to the study conception and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Material preparation, data collection, and analysis were performed by Vikas Rathore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Design and development of power supply was done by Chirayu Patil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' The first draft of the manuscript was written by Vikas Rathore, and all authors commented on previous versions of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ORCID iDs Vikas Rathore https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='org/0000-0001-6480-5009 Annexure Table A1: Reactions of formation of various reactive oxygen-nitrogen species in PAW through plasma-water interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Reaction Rate constant (k) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 𝑁2 (𝑔) + 𝑒− → 2𝑁 (𝑔) + 𝑒− 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='3 × 10-6 Te-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='6 e-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='8/Te cm3 s-1 2 𝐻2𝑂 (𝑔) + 𝑒− → 𝑂𝐻 (𝑔) + 𝐻 (𝑔) + 𝑒− 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='6 × 10-12 cm3 s-1 3 𝐻2𝑂 (𝑔) + 𝑁2(𝐴) (𝑔) → 𝑂𝐻 (𝑔) + 𝐻 (𝑔) + 𝑁2 (𝑔) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='2 × 10-11 cm3 s-1 4 𝑁 (𝑔) + 𝑂2 (𝑔) → 𝑁𝑂 (𝑔) + 𝑂 (𝑔) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 × 10-17 cm3 s-1 5 𝑂(𝑔) + 𝑁𝑂(𝑔) + 𝑀(𝑔) → 𝑁𝑂2(𝑔) + 𝑀(𝑔) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 × 10-32 cm6 s-1 6 𝑂 (𝑔) + 𝑂2 (𝑔) → 𝑂3 (𝑔) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 × 10-12 cm3 s-1 7 𝑂𝐻 (𝑔) + 𝑂𝐻 (𝑔) → 𝐻2𝑂2 (𝑔) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='6 × 10-11 cm3 s-1 8 𝑂𝐻 (𝑔) + 𝑂3 (𝑔) → 𝐻𝑂2 (𝑔) + 𝑂2 (𝑔) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='9 × 10-12 cm3 s-1 9 𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑁𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑵𝑶𝟐 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝐻+(𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 × 1010 M-1 s-1 10 𝑶𝟑 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑵𝑶𝟐 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑵𝑶𝟑 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 × 105 M-1 s-1 11 𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑯𝟐𝑶𝟐 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 × 109 M-1 s-1 12 𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑶𝟑 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 × 108 M-1 s-1 13 𝐻𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑯𝟐𝑶𝟐 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 × 106 M-1 s-1 14 𝑁𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑁𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻2𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝟐𝑵𝑶𝟐 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 2𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='0 × 108 M-1 s-1 15 2𝑁𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻2𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑵𝑶𝟑 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑵𝑶𝟐 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 2𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='5 × 108 M-1 s-1 16 𝑯𝟐𝑶𝟐 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝐻2𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑂2 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 × 107 M-1 s-1 17 𝐻2𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝐻𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑂2 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑂2 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑯𝟐𝑶𝟐 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝐻𝑂− (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='7 × 107 M-1 s-1 18 𝑵𝑶𝟐 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝑯𝟐𝑶𝟐 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑂𝑁𝑂𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻2𝑂 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='1 × 103 M-1 s-1 19 𝑵𝑶𝟐 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') → 𝑵𝑶𝟑 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) + 2𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='3 × 109 M-1 s-1 20 𝑂𝑁𝑂𝑂𝐻 (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' ) → 𝑵𝑶𝟑 − (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') + 𝐻+ (𝑎𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='9 s-1 21 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Rathore V, Patel D, Butani S, Nema SK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Investigation of physicochemical properties of plasma activated water and its bactericidal efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Plasma Chemistry Plasma Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='41(3):871-902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Rathore V, Patel D, Shah N, Butani S, Pansuriya H, Nema 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Effect of plasma-activated water on the microbial decontamination and food quality of thin sheets of bean curd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Applied Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content='9(20):4223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE4T4oBgHgl3EQfYgyr/content/2301.05050v1.pdf'} +page_content=' Kang C, Xiang Q, Zhao D, Wang W, Niu L, Bai 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