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+A Global Optimization Algorithm for K-Center
+Clustering of One Billion Samples
+Jiayang Ren1, Ningning You2, Kaixun Hua1, Chaojie Ji3, Yankai Cao1
+1Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada,
+[email protected], [email protected], [email protected]
+2Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China,
+[email protected]
+3Department of Mathematics, University of British Columbia, Vancouver, BC, Canada,
+[email protected]
+This paper presents a practical global optimization algorithm for the K-center clustering problem, which
+aims to select K samples as the cluster centers to minimize the maximum within-cluster distance. This
+algorithm is based on a reduced-space branch and bound scheme and guarantees convergence to the global
+optimum in a finite number of steps by only branching on the regions of centers. To improve efficiency,
+we have designed a two-stage decomposable lower bound, the solution of which can be derived in a closed
+form. In addition, we also propose several acceleration techniques to narrow down the region of centers,
+including bounds tightening, sample reduction, and parallelization. Extensive studies on synthetic and real-
+world datasets have demonstrated that our algorithm can solve the K-center problems to global optimal
+within 4 hours for ten million samples in the serial mode and one billion samples in the parallel
+mode. Moreover, compared with the state-of-the-art heuristic methods, the global optimum obtained by our
+algorithm can averagely reduce the objective function by 25.8% on all the synthetic and real-world datasets.
+Key words : global optimization; K-center clustering; branch and bound; two-stage decomposition; bounds
+tightening
+1. Introduction
+Cluster analysis is a task to group similar samples into the same cluster while separating less
+similar samples into different clusters. It is a fundamental unsupervised machine learning task that
+explores the character of datasets without the need to annotate cluster classes. Clustering plays
+a vital role in various fields, such as data summarization (Kleindessner et al. 2019, Hesabi et al.
+1
+arXiv:2301.00061v1  [math.OC]  30 Dec 2022
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+2
+2015), customer grouping (Aggarwal et al. 2004), facility location determination (Hansen et al.
+2009), and etc.
+There are several typical cluster models, including connectivity-based models, centroid-based
+models, distribution-based models, density-based models, etc. This work focuses on one of the fun-
+damental centroid-based clustering models called the K-center problem. The goal of the K-center
+problem is to minimize the maximum within-cluster distance
+(Kaufman and Rousseeuw 2009).
+Specifically, given a dataset with S samples and the desired number of clusters K, the K-center
+problem aims to select K samples from the dataset as centers and to minimize the maximum
+distance from other samples to its closest center. The K-center problem is a combinatorial opti-
+mization problem that has been widely studied in theoretical computer science (Lim et al. 2005).
+Moreover, it has been intensively explored as a symmetric and uncapacitated case of the p-center
+facility location problem in operations research and management science (Garcia-Diaz et al. 2019),
+where the number of facilities corresponds to the variable k in a standard K-center problem.
+Formally, provided a K, the objective function of K-center problem can be formulated as follows:
+min
+µ∈X max
+s∈S min
+k∈K ||xs − µk||2
+2
+(1)
+where X = {x1,...,xS} is the dataset with S samples and A attributes, in which xs = [xs,1,...,xs,A] ∈
+RA is the sth sample and xs,a is the ath attribute of ith sample, s ∈ S := {1,··· ,S} is the index set
+of samples. As to the variables related to clusters, k ∈ K := {1,··· ,K} is the index set of clusters,
+µ := {µ1,··· ,µK} represents the center set of clusters, µk = [µk
+1,...,µk
+A] ∈ RA is the center of kth
+cluster. Here, µ are the variables to be determined in this problem. We use µ ∈ X to denote the
+“centers on samples” constraint in which each cluster’s center is restricted to the existing samples.
+1.1. Literature Review
+The K-center problem has been shown to be NP-hard (Gonzalez 1985), which means that it is
+unlikely to find an optimal solution in polynomial time unless P = NP (Garey and Johnson 1979).
+As a remedy, heuristic algorithms, which aim to find a good but not necessarily optimal solution,
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+3
+are often used to solve the K-center problem on large-scale datasets. The study of exact algorithms,
+which provide an optimal solution but may hardly be terminated in an acceptable time, is restricted
+to small-scale datasets due to this poor scalability on larger datasets.
+Regarding heuristic algorithms, there are several 2-approximation algorithms that provide a
+theoretical guarantee of their distance from the optimal solution for the K-center problem, but do
+not provide a guarantee on their running time (Plesn´ık 1987, Gonzalez 1985, Dyer and Frieze 1985,
+Hochbaum and Shmoys 1985, Cook et al. 1995). Among these 2-approximation algorithms, Furthest
+Point First (FPF) algorithm proposed by Gonzalez (1985) is known to be the fastest in practice
+(Miheliˇc and Robic 2005). It works by starting with a randomly selected center and then adding
+points that are farthest from the existing centers to the center set. Despite their solution quality
+guarantee, these 2-approximation algorithms may not always provide close-to-optimal solutions in
+practice (Garcia-Diaz et al. 2019). Another kind of heuristic methods with a polynomial running
+time but a weaker solution quality guarantee is also intensively studied in the literature (Miheliˇc
+and Robic 2005, Garcia-Diaz et al. 2017). Besides heuristic methods, there are also metaheuristic
+methods that do not have a polynomial running time or a solution quality guarantee, but have
+been shown to provide near-optimal solutions in some cases (Mladenovi´c et al. 2003, Pullan 2008,
+Davidovi´c et al. 2011). In sum, none of these algorithms can deterministically guarantee a global
+optimal solution for the K-center problem.
+In contrast to the numerous heuristic algorithms, the study of exact algorithms, which provide
+the optimal solution but no solution time guarantee, is still struggling with small-scale problems
+(e.g., thousands of samples). Early exact works are inspired by the relationship between K-center
+and set-covering problems (Minieka 1970). Daskin (2000) transferred the K-center problem to a
+maximal covering problem, in which the number of covered samples by K centers is maximized.
+Then, they proposed an iterative binary search scheme to accelerate the solving procedure. Ilhan
+and Pinar (2001) considered iteratively setting a maximum distance and validating if it can cover
+all the samples. Elloumi et al. (2004) designed a new integer linear programming formulation of
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+4
+the K-center problem, then solved this new formulation by leveraging the binary search scheme
+and linear programming relaxation. These algorithms have been shown to provide practical results
+on small-scale datasets with up to 1,817 samples.
+Another research direction models the K-center problem as a Mixed Integer Programming (MIP)
+formulation, allowing for the use of the branch and bound technique to find an optimal solution.
+However, the vanilla implementations of the branch and bound technique are confined to small-scale
+datasets with fewer than 250 samples (Brusco and Stahl 2005). Hence, constraint programming is
+introduced to address the larger scale K-center problems. Dao et al. (2013) designed two sets of
+variables describing the cluster centers and sample belongings, then updated the solution through
+constraint propagation and branching. They further reduced the sets of variables and proposed a
+more general framework in Duong et al. (2017). By involving constraint programming, their works
+can solve the datasets with up to 5,000 samples.
+Recently, researchers have explored iterative techniques to solve the K-center problem on large
+datasets by breaking it down into smaller subproblems, such as iterative sampling (Aloise and
+Contardo 2018) and row generation (Contardo et al. 2019). In Aloise and Contardo (2018), a
+sampling-based algorithm was proposed that alternates between an exact procedure on a small
+subset of the data and a heuristic procedure to test the optimality of the current solution. This
+algorithm is capable to solve a dataset containing 581,012 samples within 4 hours. However, a report
+about the optimality gap is absent, which is an important measure of solution quality. According to
+that computing the covering set for a subset of all samples is cheaper than all (Chen and Handler
+1987, Chen and Chen 2009), the same research group proposed a row generation algorithm that
+relies on computing a much smaller sub-matrix (Contardo et al. 2019). This approach is able to
+solve a dataset with 1 million samples to a 6% gap in 9 hours. However, neither of these methods
+provides a finite-step convergence guarantee, which results in that they may not always converge
+to an arbitrarily small gap within a finite number of steps. Therefore, these methods can lead to a
+nontrivial optimality gap, especially for large datasets.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+5
+1.2. Main contributions
+Recently, Cao and Zavala (2019) proposed a reduced-space spatial branch and bound (BB) scheme
+for two-stage stochastic nonlinear programs. Hua et al. (2021) adopted this reduced-space BB
+scheme and Lagrangian decomposition to solve the K-means clustering problem with a global
+optimal guarantee. They solve the large-scale K-means problems up to 210,000 samples to 2.6%
+optimality gap within 4 hours. However, these works can not be directly applied to the K-center
+problem. The challenge is that the K-center problem minimizes the maximum within-cluster dis-
+tance instead of the average within-cluster distance. Therefore, utilizing the Lagrangian decom-
+position method to compute the lower bound is impossible. Moreover, because of the “centers on
+samples” constraint in the K-center problem, the direct application of Hua’s algorithm will lead
+to infeasible solutions.
+To address these challenges, we propose a tailored reduced-space branch and bound algorithm
+for the K-center problem. We also design several bounds tightening (BT) and sample reduction
+methods to accelerate the BB procedure. Our algorithm is unique in that it only branches on the
+region of centers, which allows us to guarantee convergence to the global optimum within a finite
+number of steps. In contrast, traditional branch and bound algorithms must branch on all integer
+variables, which can become computationally infeasible for large-scale problems. By focusing on
+the limited region of centers, our algorithm is capable to solve even large-scale K-center problems.
+Specifically, the main contributions of this paper are as follows:
+• We propose an exact global optimization algorithm based on a tailored reduced-space branch
+and bound scheme for the K-center problem. To increase efficiency, we develop a two-stage decom-
+posable lower bounding method with a closed-form solution, eliminating the need for using any
+MIP solver in the optimization process. Moreover, the convergence of our algorithm to the global
+optimum is guaranteed by branching only on the region of centers.
+• We demonstrate that the assignment of clusters can be determined for many samples without
+knowing the optimal solution. Based on this characteristic, we propose several bounds tightening
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+6
+and sample reduction techniques to further reduce the search space and accelerate the solving
+procedure. Moreover, we also implement a sample-level parallelization strategy to fully utilize
+computational resources.
+• An open-source Julia implementation of the algorithm is provided. Extensive studies on 5
+synthetic and 33 real-world datasets have demonstrated that we can obtain the global solution for
+datasets with up to 1 billion samples and 12 features, a feat that has not been achieved so far.
+Especially, compared with the heuristic methods, the global optimum obtained by our algorithm
+can averagely reduce the objective function by 25.8% on all the synthetic and real-world datasets.
+This paper is an expanded version of our proceeding publication (Shi et al. 2022) that includes one
+new acceleration technique called sample reduction and a parallel implementation. These improve-
+ments have significantly increased the scale of the optimally solvable K-center problem from 14
+million samples to 1 billion. In this version, we provide more detailed proof of the global opti-
+mum convergence of our algorithm. In addition, we have designed more comprehensive numerical
+experiments on a broader range of datasets and parameters.
+1.3. Outline
+This paper is organized as follows: Section 2 introduces a two-stage formulation and a Mixed Integer
+Nonlinear Programming (MINLP) formulation for the K-center problem. Section 3 presents the
+details of the reduced-space branch and bound algorithm, including the lower bound, upper bound
+methods, and convergence analysis. Section 4 discusses the accelerating techniques for our BB
+algorithm, including bounds tightening, sample reduction, and parallel implementation techniques.
+Section 5 presents the detailed proof of convergence to the global optimum in the finite steps.
+Section 6 gives extensive numerical results compared with other algorithms. Finally, Section 7
+concludes the paper.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+7
+2. K-center Formulation
+2.1. Two-stage Formulation
+To introduce the lower bounding method in the branch and bound scheme, we first propose a
+two-stage optimization form of the K-center Problem 1. The first-stage problem is as follows:
+z =
+min
+µ∈X∩M0 max
+s∈S Qs(µ).
+(2)
+where the center set µ is the so-called first-stage variable, Qs(µ) is the optimal value of the second-
+stage optimization problem:
+Qs(µ) = min
+k∈K ||xs − µk||2
+2
+(3)
+We denote a closed set M0 := {µ | µ ≤ µ ≤ ¯µ} as the region of centers, where µ is the lower bound of
+centers and ¯µ is the upper bound, i.e., µk
+a = min
+s∈S Xs,a, ¯µk
+a = max
+s∈S Xs,a, ∀k ∈ K, a ∈ {1,··· ,A}. Here,
+the constraint µ ∈ M0 is introduced to simplify the discussion of the BB scheme. Since M0 can be
+inferred directly from data, it will not affect the optimal solution of Problem 1. Constraint µ ∈
+X ∩ M0 means the center of each cluster is selected from the samples belonging to the intersection
+set of the corresponding region M0 and the dataset X
+2.2. MINLP Formulation
+To introduce the bounds tightening and sample reduction methods, we propose a MINLP formu-
+lation of the K-center Problem 1:
+min
+µ,d,b,λ d∗
+(4a)
+s.t. dk
+s ≥ ||xs − µk||2
+2
+(4b)
+− N1(1 − bk
+s) ≤ d∗
+s − dk
+s ≤ 0
+(4c)
+d∗ ≥ d∗
+s
+(4d)
+�
+k∈K
+bk
+s = 1
+(4e)
+bk
+s ∈ {0,1}
+(4f)
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+8
+− N2(1 − λk
+s) ≤ xs − µk ≤ N2(1 − λk
+s)
+(4g)
+�
+s∈S
+λk
+s = 1
+(4h)
+λk
+s ∈ {0,1}
+(4i)
+bk
+s ≥ λk
+s
+(4j)
+s ∈ S,k ∈ K
+(4k)
+where dk
+s represents the distance between sample xs and center µk, d∗
+s denotes the distance between
+xs and the center of its cluster, N1 and N2 are both arbitrary large values. bk
+s and λk
+s are two binary
+variables. bk
+s is equal to 1 if sample xs belongs to the Kth cluster, and 0 otherwise. λk
+s is equal to
+1 if xs is the center of the Kth cluster µk, and 0 otherwise.
+Constraint 4c is a big M formulation and ensures that d∗
+s = dk
+s if bk
+s = 1 and d∗
+s ≤ dk
+s otherwise.
+Constraint 4e guarantees that sample xs belongs to one cluster. We also adopt Constraint 4g, 4h
+and 4j to represent the “centers on samples” constraints, µ ∈ X. Specifically, Constraint 4g uses a
+big M formula to make sure that µk = xs if λk
+s = 1 and Constraint 4h confirms that each center can
+only be selected on one sample. Constraint 4j ensures that if xs is the center of the Kth cluster,
+then it is assigned to the Kth cluster. It should be noted that the global optimizer CPLEX also
+relies on this formulation to solve the K-center problem.
+3. Tailored Reduced-space Branch and Bound Scheme
+This section introduces a tailored reduced-space branch and bound algorithm for the K-center
+problem with lower and upper bounding methods.
+3.1. Lower Bounds
+In this section, we adopt the two-stage formulation and derive a closed-form solution to obtain the
+lower bound of the K-center Problem 1.
+At each node in the BB procedure, we deal with a subset of M0, which is denoted as M, and
+solve the following problem concerning M:
+z(M) = min
+µ∈X∩M max
+s∈S Qs(µ)
+(5)
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+9
+This problem can be equivalently reformulated as the following problem by duplicating µ across
+samples and enforcing them to be equal:
+min
+µs∈X∩M max
+s∈S Qs(µs)
+(6a)
+s.t.
+µs = µs+1,s ∈ {1,··· ,S − 1}
+(6b)
+We call constraints 6b the non-anticipativity constraints. By removing the “centers on samples”
+constraint µ ∈ X and the non-anticipativity constraints 6b, we attain a lower bound formulation
+as follow:
+β(M) := min
+µs∈M max
+s∈S Qs(µs).
+(7)
+With constraints relaxed, the feasible region of Problem 7 is a superset of Problem 6’s feasible
+region. Therefore, it is obvious that β(M) ≤ z(M).
+In Problem 7, since µ of each sample is independent, it is obvious that:
+β(M) = max
+s∈S min
+µs∈M Qs(µs).
+(8)
+Clearly, problem 8 can be decomposed into S subproblems with β(M) = max
+s∈S βs(M):
+βs(M) = min
+µ∈M Qs(µ).
+(9)
+Denote the region of kth cluster’s center as M k := {µk : µk ≤ µk ≤ ¯µk} where µk and ¯µk are the
+lower and upper bound of µk respectively. Since Qs(µ) = min
+k∈K ||xs − µk||2
+2, we have
+βs(M) = min
+k∈K min
+µk∈Mk ||xs − µk||2
+2,
+(10)
+which can be further decomposed into K subsubproblems with βs(M)=min
+k∈K βk
+s (M k):
+βk
+s (M k) = min
+µk∈Mk ||xs − µk||2
+2.
+(11)
+The analytical solution to Problem 11 is: µk
+a
+∗ = mid{µk
+a, xs,a, ¯µk
+a},∀a ∈ {1,··· ,A}. Consequently,
+the closed-form solution to Problem 7 can be easily computed by the max-min operation on all the
+samples.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+10
+3.2. Upper Bounds
+At each node in the BB procedure, the upper bounds of Problem 5 can be obtained by fixing the
+centers at a candidate feasible solution ˆµ ∈ X ∩ M. In this way, we can compute the upper bound
+base on the following equation:
+α(M) = max
+s∈S min
+k∈K ||xs − ˆµk||2
+2
+(12)
+Since ˆµ is a feasible solution, we have z(M) ≤ α(M), ∀ˆµ ∈ X ∩ M. In our implementation, we
+use two methods to obtain the candidate feasible solutions. At the root node, we use a heuristic
+method called Farthest First Traversal (Gonzalez 1985) to obtain a candidate solution ˆµ ∈ X ∩M0.
+Using this method, we randomly pick an initial point and select each following point as far as
+possible from the previously selected points. Algorithm 2 describes the details of the farthest first
+traversal, where d(xs,T) represents the minimum distance from sample xs to any sample in set T.
+We use FFT(M0) to denote the upper bound obtained using this approach. At a child node with
+center region M, for each cluster, we select the data sample closest to the middle point of M k as
+ˆµk, and obtain the corresponding upper bound α(M).
+3.3. Branching
+Our algorithm only needs to branch on the region of centers, M := {µ : µ ≤ µ ≤ ¯µ}, to guarantee
+convergence, which would be theoretically discussed in Section 5, o. Since the desired number of
+clusters is K and the number of attributes is A, the number of possible branching variables is K ×A.
+The selection of branching variables and values will dramatically influence the BB procedure’s
+efficiency. In our implementation, we select the max-range variable at each node as the branching
+variable and the midpoint of this variable as the branching value.
+3.4. Branch and Bound Scheme
+The detailed reduced-space branch and bound algorithm for the K-center Problem 1 are given in
+the Algorithm 1. In the algorithm, We use relint(.) to denote the relative interior of a set. We can
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+11
+also establish the convergence of the branch-and-bound scheme in Algorithm 1. The BB procedure
+can generate a monotonically non-ascending sequence {αi} and a monotonically non-descending
+sequence {βi}. We can show that they both converge to z in a finite number of steps.
+Theorem 1. Algorithm 1 is convergent to the global optimal solution after a finite step L, with
+βL = z = αL, by only branching on the region of centers.
+Since the following acceleration techniques also influence the global convergence in Section 4. We
+present the detailed proof of Theorem 1 in Section 5 after introducing the acceleration techniques.
+Algorithm 1 Branch and Bound Scheme
+Initialization
+Initialize the iteration index i ← 0;
+Set M ← {M0}, and tolerance ϵ > 0;
+Compute initial lower and upper bounds βi = β(M0), αi =
+FFT(M0) // Alg. 2 ;
+Select K farthest initial seeds // Sec.4.1.1;
+while M ̸= ∅ do
+Node Selection
+Select a set M satisfying β(M) = βi from M and delete it
+from M;
+Update i ← i + 1;
+Bounds Tightening
+Cluster Assignment // Alg. 3;
+Bounds Tightening // Alg. 4;
+Obtain the tightened node ˆ
+M;
+If i % isr = 0, Sample Reduction // Alg. 5;
+if ∃|X ∩ M k| > 1,k ∈ K then
+Branching
+Find two subsets M1
+and M2
+s.t. relint(M1) ∩
+relint(M2) = ∅ and M1 ∪ M2 = M;
+Update M ← M∪{Mi}, if X ∩M k
+i ̸= ∅,∀k ∈ K,i ∈ 1,2;
+end if
+Bounding
+Compute upper and lower bound α(M1), β(M1), α(M2),
+β(M2);
+Let βi ← min{β(M ′) | M ′ ∈ M};
+Let αi ← min{αi−1,α(M1),α(M2)};
+Remove all M ′ from M if β(M ′) ≥ αi;
+If βi − αi ≤ ϵ, STOP;
+end while
+Algorithm 2 Farthest First Traversal
+Initialization
+Randomly pick s ∈ S;
+Denote T as the set of K points selected by farthest first
+traversal;
+Set T ← {xs};
+while |T| < K do
+Compute xs ∈ arg max
+xs∈X d(xs,T) to find xs which is the
+farthest away from set T;
+T ← T ∪ {xs};
+end while
+Algorithm 3 Cluster Assignment
+Center Based Assignment
+for sample xs ∈ X do
+if bk
+s == 0,∀k ∈ K then
+if βk
+s (M k) > α,∀k ∈ K \ {k′} then
+xs is assigned to cluster k′ with bk′
+s = 1;
+end if
+end if
+end for
+Sample Based Assignment
+if All clusters have at least one sample assigned then
+for sample xs ∈ X do
+if ∀k ∈ K \ {k′}, ∃ xj assigned to kth cluster, ||xs −
+xj||2
+2 > 4α then
+xs is assigned to cluster k′ with bk′
+s = 1.
+end if
+end for
+end if
+Algorithm 4 Bounds Tightening
+Given the current center region M and upper bound α
+for Cluster k ∈ K do
+Obtain the assigned sample set J k using Alg.3;
+Compute the ball-based or box-boxed area of each
+assigned sample, Bα(xj) or Rα(xj);
+Tighten the center region by M k ∩Bα(xj) or M k ∩Rα(xj)
+, ∀j ∈ J k;
+Further tighten according to the “centers on samples”
+constraint;
+end for
+Algorithm 5 Sample Reduction
+Initialize the index set of redundant samples as R ← S
+for all BB nodes do
+Obtain the index set of redundant samples for lower
+bounds, RLB, according to the criterion in Sec. 4.2.1;
+Obtain the index set of redundant samples for upper
+bounds, RUB, according to the criterion in Sec. 4.2.2;
+Update the redundant index set, R ← R ∩ RLB ∩ RUB;
+end for
+Delete samples in the redundant set R from the current
+dataset.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+12
+4. Acceleration Techniques
+Although the lower bound introduced in Section 3.1 is enough to guarantee convergence, it might
+not be very tight, leading to tremendous iterations. Therefore, we propose several acceleration
+techniques to reduce the search space and speed up the BB procedure. Since Algorithm 1 only
+branches on the region of centers M := {µ : µ ≤ µ ≤ ¯µ}, we focus on reducing the region of centers
+to accelerate the solution process while not excluding the optimal solution of the original K-center
+problem.
+4.1. Bounds Tightening Techniques
+In each node, the assignment of many samples (i.e., which cluster the sample is assigned to) can be
+pre-determined by the geometrical relationship of samples and regions of centers. This information
+can be further used to reduce the region of µ.
+4.1.1. Cluster Assignment
+The task of cluster assignment is to pre-determine some values
+of bk
+s in the MINLP Formulation 4 at each BB node before finding the global optimal solution.
+We first demonstrate the relations between samples and centers. Denote α as the upper bound
+obtained using methods described in Section 3.2. Then based on Objective 4a and Constraint 4d,
+we have d∗
+s ≤ d∗ ≤ α. From Constraint 4b and 4c, we can conclude that if bk
+s = 1, then ||xs −µk||2
+2 ≤
+d∗
+s ≤ α. Therefore, we can derive Lemma 1:
+Lemma 1. If sample xs is in the kth cluster, then ||xs − µk||2
+2 ≤ α, where α is an upper bound of
+the K-center problem.
+Besides the relation between samples and centers, cluster assignments may also be determined
+from the distance of two samples. Suppose sample xi and xj belong to the kth cluster, then from
+Lemma 1 we have ||xi − µk||2
+2 ≤ α and ||xj − µk||2
+2 ≤ α. Thus ||xi − xj||2
+2 = ||xi − µk + µk − xj||2
+2 ≤
+(||xi − µk||2 + ||µk − xj||2)2 ≤ 4α. Therefore, we have Lemma 2:
+Lemma 2. If two samples xi and xj are in the same cluster, then ||xi − xj||2
+2 ≤ 4α where α is an
+upper bound of the K-center problem.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+13
+We propose three methods for pre-assigning samples based on these two Lemmas:
+K Farthest Initial Seeds: From Lemma 2, if ||xi − xj||2
+2 > 4α, then xi and xj are not in the
+same cluster. At the root node, if we can find K samples with the distance between any two of
+these samples xi and xj satisfying ||xi − xj||2
+2 > 4α, then we can conclude that these K samples
+must belong to K distinct clusters. Figure 1 shows an example of this property, in which three
+samples are pre-assigned to 3 distinct clusters. We call these K points initial seeds. To find the
+initial seeds, every two samples must be as far as possible. Therefore, in our implementation, we use
+the heuristic Farthest First Traversal (FFT) (Algorithm 2) to obtain K farthest points. For about
+half of the case studies shown in Section 6, we can obtain the initial seeds using FFT. However, for
+other cases, initial seeds can not be obtained using FFT, or the initial seeds may not even exist.
+Center-Based Assignment: From Lemma 1, if ||xs − µk||2
+2 > α, then xs does not belong to
+kth cluster, which is bk
+s = 0. Consequently, if we can determine that bk
+s = 0,∀k ∈ K \ {k′}, then
+bk′
+s = 1. However, the value of µ here is unknown before obtaining the optimal solution. One
+observation is that if the BB node with region M contains the optimal solution, then we have
+βk
+s (M k) = min
+µk∈Mk ||xs − µk||2
+2 ≤ ||xs − µk||2
+2. Therefore, if βk
+s (M k) > α, sample xs is not in the kth
+cluster and bk
+s = 0. In summary, for sample xs, if ∀k ∈ K \ {k′}, βk
+s (M k) > α, then xs is assigned to
+cluster k′ with bk′
+s = 1. Figure 2 illustrates an example in two-dimensional space with three clusters.
+This center-based method can be adopted at every node of the BB scheme. Since βk
+s (M k) is
+already obtained when computing the lower bound in Section 4.2.1, there is no additional compu-
+tational cost. Nevertheless, we do not need to apply this method at the root node since M 1
+0 = ··· =
+M K
+0 . As the BB scheme continues branching on the regions of centers, M k becomes more and more
+different from others. Then more samples can be pre-assigned using this center-based method.
+Sample-Based Assignment: Besides utilizing centers to pre-assign samples, assigned samples
+can also help pre-assign other samples. From Lemma 2, if ||xi −xj||2
+2 > 4α, then xi and xj are not in
+the same cluster. If xj belongs to kth cluster, then obviously xi cannot be assigned to kthe cluster
+and bk
+i = 0. With this relationship, if all the other K − 1 clusters are excluded, xi will be assigned
+to the remaining cluster. Figure 3 shows an example of the sample-based assignment.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+14
+There is a prerequisite to using this sample-based method. For each cluster, there must be at least
+one sample already assigned to the cluster. Based on this prerequisite, sample-based assignment is
+utilized only after at least one sample is pre-assigned for each cluster.
+4.1.2. Bounds Tightening
+In this subsection, we adopt the Bounds Tightening (BT) tech-
+nique and the cluster assignment information to reduce the region of µ.
+Ball-based Bounds Tightening: For a sample j, Bα(xj)={x| ||x − xj||2
+2 ≤ α} represents the
+ball with center xj and radius √α. By using cluster assignment methods in Section 4.1.1, assuming
+that sample j belongs to kth cluster is already known, by Lemma 1, then µk ∈ Bα(xj) holds. We
+use J k to denote the index of all samples assigned to kth cluster, i.e., J k = {j ∈ S | bk
+j = 1},
+then µk ∈ Bα(xj),∀j ∈ J k. Besides this, we also know that µk ∈ X ∩ M k. Denote Sk
++ as the index
+set of samples satisfying all these constraints, Sk
++(M) := {s ∈ S |xs ∈ X ∩ M k,xs ∈ Bα(xj),∀j ∈
+J k}. In this way, we can obtain a tightened box containing all feasible solutions of kth center,
+ˆ
+M k={µk|ˆµk ≤ µk ≤ ˆ¯µk}, with the bounds of ath attribute in kth center to be ˆµk
+a=
+min
+s∈Sk
++(M)xk
+s,a and
+ˆ¯µk
+s= max
+s∈Sk
++(M)xk
+s,a. Figure 4 gives an example of bounds tightening using this method. One challenge
+of this ball-based bounds tightening method is that it needs to compute the distance of xs and xj
+for all s ∈ S and j ∈ J k. If we know the assignments of the majority of the samples, we need to
+do at most S2 times of distance calculation. Note that we only need to do S ∗ K times of distance
+calculation to compute a lower bound. To reduce the computational time, we set a threshold on
+the maximum number of balls (default: 50) utilized to tighten bounds in our implementation.
+Box-based Bounds Tightening: Another strategy to reduce the computation burden is based
+on the relaxation of Bα(xj). For any ball Bα(xj), the closed set Rα(xj) = {x | xj − √α ≤ x ≤
+xj + √α} is the smallest box containing Bα(xj). Then we have µk ∈ Rα(xj),∀j ∈ J k. Since Rα(xj)
+and M k are all boxes, we can easily compute the tighten bounds ˆ
+M k=�
+j∈J k Rα(xj) ∩ M k. Figure
+5 gives an example of box-based bounds tightening using this method. Obviously, the bounds
+generated in Figure 4 is much tighter, while the method in Figure 5 is much faster. Consequently,
+if |J k| is small for all clusters, the ball-based bounds tightening method gives more satisfactory
+results. While if |J k| is large for any k, box-based bounds tightening provides a cheaper alternative.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+15
+4.1.3. Symmetry Breaking
+Another way to get tighter bounds is based on symmetry-
+breaking constraints. We add the constraints µ1
+1 ≤ µ2
+1 ≤ ··· ≤ µK
+1 in the BB algorithm 1, in which
+µk
+a denotes ath attribute of kth center. Note that symmetry-breaking constraints and FFT-based
+initial seeds in Section 4.1.1 both break symmetry by providing a certain order for the clusters, so
+they cannot be combined. Our implementation uses symmetric breaking only when initial seeds are
+not found from FFT at the root node. It should be noted that we also add this symmetry-breaking
+constraints when using CPLEX to solve the MINLP formulation 4 of the K-center problem.
+4.2. Sample Reduction
+Some samples may become redundant during the lower and upper bounding procedure without
+contributing to the bound improvements. If these samples are proven to be redundant in all the
+current and future branch nodes, we can conclude they will not influence the bounding results
+anymore, resulting in sample reduction.
+4.2.1. Redundant samples in lower bounding
+Denote β as the current best lower bound
+obtained using methods described in Section 3.1. According to Equation 8, lower bound β(M) is
+the maximum value of each sample’s optimal value, βs(M). Based on this observation, we further
+define the best maximum distance of sample s to the center region of µ as
+αs(M) = min
+k∈K max
+µk∈Mk ||xs − µk||2
+2,
+(13)
+It is obvious that βs(M) ≤ αs(M). If αs(M) < β, we have βs(M) < β, which means sample s is
+not the sample corresponding to maximum within-cluster distance. Hence, we can conclude that
+sample s is a redundant sample in lower bounding for this BB node. Moreover, ∀M ′ ⊂ M, we
+have βs(M ′) ≤ αs(M ′) ≤ αs(M). According to the shrinking nature of center region M and the
+non-descending nature of lower bound β, if αs(M) < β is true in a BB node, sample s will remain
+redundant in all the child nodes of this branch node. It should be noted that αs(M) can be
+calculated using an analytical solution similar to βs(M), which is µk
+a = µk
+a if |µk
+a −xs,a| > |¯µk
+a −xs,a|,
+otherwise ¯µk
+a.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+16
+4.2.2. Redundant samples in upper bounding
+Obviously, a sample xj cannot be the
+center for kth cluster if it does not belong to M k. Moreover, according to Lemma 1, if a sample xj
+is the center for cluster K, ||xi −xj||2
+2 ≤ α must hold for all the samples xi assigned to this cluster.
+Hence, a sample xj also cannot be the center for kth cluster, if there exists a sample xi assigned to
+kth cluster satisfying ||xi −xj||2
+2 > α. If sample xj cannot be centers for any cluster, we denote this
+sample xj as a redundant sample for upper bounding. Since the non-ascending nature of upper
+bound α, if sample s is redundant for upper bounding in a branch node, it will remain redundant
+in all the child nodes of this branch node. It should be noted that the calculations in this method
+are identical to Sample-Based Assignment in Section 4.1.1 with no extra calculations introduced
+in this method.
+4.2.3. Sample reduction
+If a sample s is redundant in lower bounding, it implies that sample
+s is not the “worst-case sample” corresponding to the maximum within-cluster distance. If a sample
+s is redundant in upper bounding, then it means that sample s cannot be a center for any cluster. If
+the sample s is redundant in both lower bounding and upper bounding, then removing this sample
+will not affect the solution of this BB node and all its child BB nodes. Algorithm 5 describes the
+procedure of sample reduction: first, obtain the redundant samples for lower and upper bounding
+in each branch node; then, we can delete the samples that are redundant for both lower and upper
+bounding in all the branch nodes. In our implementation, this sample reduction method is executed
+for every isr iterations.
+4.2.4. Effects on computation
+Sample reduction can reduce the number of samples that
+need to be explored by deleting redundant samples every isr iterations, as described in Algorithm
+5. It can also accelerate the calculation of lower bounds and bounds tightening at each iteration.
+For the lower bounding method in Section 3.1, we only need to solve the second-stage problems for
+non-redundant samples that have been validated by the lower-bounding criterion in Section 4.2.1.
+Additionally, once a sample is deemed redundant for lower bounding in a particular node, it will
+remain redundant in all child nodes of that node. This means that we do not need to solve the
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+17
+second-stage problem for this sample in the current node or any of its child nodes. For the bounds
+tightening methods in Section 4.1.2, we only need to calculate the bounds based on non-redundant
+samples that have been validated by the upper-bounding criterion in Section 4.2.2. Similarly, if
+a sample is redundant for upper bounding in a node, it will remain redundant in all child nodes
+of that node, and can be eliminated from the bounds tightening calculations in the current node
+and its child nodes. In this way, sample reduction can not only delete redundant samples at every
+isr iterations, but also eliminate redundant information in the current node and its child nodes,
+thereby accelerating the overall calculation.
+4.3. Parallelization
+We also provide a parallel implementation of the whole algorithm to accelerate the solving process.
+Since our algorithm is primarily executed at the sample level, like βs(M) in the lower bounding, we
+can parallelize the algorithm by distributing the dataset to each process equally, then calculating
+on each process with the local dataset and communicating the results as needed. The detailed
+parallelization framework is shown in Figure 6. Here, the green modules represent the parallel
+operations at each process, and the blue modules represent serial reduction operations. This par-
+allelization framework is realized utilizing Message-Passing Interface (MPI) and MPI.jl by (Byrne
+et al. 2021).
+5. Convergence Analysis
+As stated in Theorem 1, the branch-and-bound scheme for the K-center problem in Algorithm 1
+converges to the global optimal solution after a finite step. In this section, we present the proof of
+this theorem.
+Specifically, the branch-and-bound scheme in Algorithm 1 branches on the region of centers, µ,
+and generates a rooted tree with the search space M0 at the root node. For the child node at qth
+level and lqth iteration, we denote the search space as Mlq. The search space of its child node is
+denoted as Mlq+1 satisfying Mlq+1 ⊂ Mlq. We denote the decreasing sequence from the root node
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+18
+with M0 to the child node with Mlq as {Mlq}. The search space of kth cluster center at Mlq is
+denoted as M k
+lq. Along the branch-and-bound process, we can obtain a monotonically non-ascending
+upper bound sequence {αi} and a monotonically non-descending lower bound sequence {βi}.
+In the following convergence analysis, we adapt the fundamental conclusions from (Horst and
+Tuy 2013) to our algorithm. It should be noted that the convergence of the K-center problem
+here is stronger than the convergence analysis in (Cao and Zavala 2019) for two-stage nonlinear
+optimization problems or the convergence proof in (Hua et al. 2021) for K-means clustering prob-
+lem. Both Cao and Zavala (2019) and Hua et al. (2021) guarantee the convergence in the sense of
+lim
+i→∞αi = lim
+i→∞βi = z. They can only produce a global ϵ-optimal solution in a finite number of steps.
+While for the K-center problem, the algorithm can obtain an exact optimal solution (e.g., ϵ = 0)
+in a finite number of steps.
+Definition 1. (Definition IV.3 (Horst and Tuy 2013)) A bounding operation is called finitely
+consistent if, at every step, any unfathomed partition element can be further refined and if any
+decreasing sequence {Mlq} successively refined partition elements is finite.
+Lemma 3. The bounding operation in Algorithm 1 is finitely consistent.
+Proof. Firstly, we prove that any unfathomed partition element Mlq can be further refined. Any
+unfathomed Mlq satisfies two conditions: (1) ∃|X ∩ M k
+lq| > 1,k ∈ K, and (2) αl − β(Mlq) > ϵ,ϵ > 0.
+Obviously, there exists at least one partition to be further refined.
+We then prove any decreasing sequences {Mlq} successively refined partition elements are finite.
+Assuming by contradiction that a sequence {Mlq} is infinite. In our algorithm, since we branch
+on the first-stage variable µ corresponding to the diameter of M, this subdivision is exhaustive.
+Therefore, we have lim
+q→∞δ(Mlq) = 0 and {Mlq} converge to one point ¯µ at each cluster, where δ(Mlq)
+is the the diameter of set Mlq.
+If this point ¯µ ∈ X, there exists a ball around ¯µ, denoted as Br(¯µ) = {µ | ||µ − ¯µ|| ≤ r}, fulfilling
+|X ∩Br(¯µ)| = 1. There exists a level q0 that Mlq ⊂ Br(¯µ),∀q ≥ q0. At this lq0th iteration, according
+to the terminal conditions |X ∩ M k
+lq| = 1,∀k ∈ K, the partition elements Mlq0 will not be branched
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+19
+anymore. Because the dataset X is finite, we have the sequence {Mlq} is finite in this case. If ¯µ ̸⊂ X,
+there is a ball around ¯µ, denoted as Br(¯µ) = {µ | ||µ − ¯µ|| ≤ r}, satisfying |X ∩ Br(¯µ)| = 0. There
+exists a level q0 that Mlq ⊂ Br(¯µ),∀q ≥ q0. At this lq0th iteration, Mlq0 will be deleted according to
+the terminal conditions. Consequently, the sequence {Mlq} is also finite in this case. In conclusion,
+it is impossible to exist a sequence {Mlq} that is infinite.
+Theorem 2. (Theorem IV.1 (Horst and Tuy 2013)) In a BB procedure, suppose that the bounding
+operation is finitely consistent. Then the procedure terminates after finitely many steps.
+Lemma 4. Algorithm 1 terminates after finitely many steps.
+Proof. From Lemma 3, the bounding operation in Algorithm 1 is finitely consistent. According to
+Theorem 2, we have Algorithm 1 terminates after finitely many steps
+Finally, we prove that the BB scheme for the K-center problem is convergent:
+Theorem 1. Algorithm 1 is convergent to the global optimal solution after a finite step L, with
+βL = z = αL, by only branching on the space of µ.
+Proof. From Lemma 4, Algorithm 1 terminates after finite steps. The algorithm terminates with
+two situations. The first situations is |βl − αl| ≤ ϵ,ϵ ≥ 0. When ϵ is set to be 0, we have βl = z = αl.
+The second situation is the branch node set M = ∅. A branch node with M is deleted from M
+and not further partitioned if it satisfies β(M) > αl or |X ∩ M k| = 1,∀k ∈ K. In the first case, it
+is obvious that this branch node does not contain the global optimal solution µ∗. Therefore, the
+branch node with M ′ containing the optimal solution µ∗ is not further partitioned because the
+second case |X ∩ M ′k| = 1,∀k ∈ K. After bounds tightening according to the “centers on samples”
+constraint, the tightened node M ′ = {µ∗}. Obviously for this tightened node, we have βl = β(M ′) =
+z = α(M ′) = αl. In this way, we have proved Theorem 1.
+6. Numerical Results
+In this section, we report the detailed implementation of our algorithm and the numerical results
+on synthetic and real-world datasets.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+20
+6.1. Implementation Details
+We denote our tailored reduced-space branch and bound algorithm 1 with and without acceleration
+techniques as BB+CF+BT and BB+CF correspondingly. All our algorithms are implemented in Julia,
+and the parallel version is realized using Message Passing Interface through the MPI.jl module.
+We compare the performance of our algorithm with the state-of-art global optimizer CPLEX 20.1.0
+(Cplex 2020) and the heuristic algorithm, Farthest First Traversal (FFT) as shown in Algorithm
+2. The initial points severely influence the results of FFT. Therefore, we execute FFT for 100 trails
+with randomly selected initial points and report the best results. As for CPLEX, we use the
+MINLP formulation 4 with the symmetry-breaking constraints to solve the K-center problem.
+We executed all experiments on the high-performance computing cluster Niagara in the Digital
+Research Alliance of Canada. Each computing node of the Niagara cluster has 40 Intel “Skylake”
+cores and 188 GiB of RAM. For the global optimizer CPLEX and our algorithms, a time limit of
+4 hours is set to compare the performance fairly and avoid unacceptable computational costs.
+For our algorithms, there is also an optimality gap limit of 0.1%. The source code is available at
+https://github.com/YankaiGroup/global_kcenter_extended.
+In order to evaluate the performance extensively, we execute all the algorithms on both syn-
+thetic and real-world datasets. The synthetic datasets are generated using Distributions.jl and
+Random.jl modules in Julia. We generate the synthetic datasets with 3 Gaussian clusters, 2
+attributes, and varying numbers of samples. As for the real-world datasets, we use 30 datasets
+from the UCI Machine Learning Repository (Dua and Graff 2017), datasets Pr2392 from (Padberg
+and Rinaldi 1991), Hemi from (Wang et al. 2022) and Taxi from (Schneider 2015). The number
+of samples ranges from 150 to 1,120,841,769. The number of attributes ranges from 2 to 68. The
+detailed characteristics of datasets can be found in the following result tables.
+We report four criteria in the following result tables to compare the performance of algorithms:
+upper bound (UB), optimality gap (Gap), the number of solved BB nodes (Nodes), and the run
+time (Time). UB is the best objective value of the K-center Problem 1. Gap represents the relative
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+21
+difference between the best lower bound (LB) and UB. It is defined as Gap = UB−LB
+UB
+× 100%.
+The optimality gap is a unique property of the deterministic global optimization algorithm. The
+heuristic algorithm (FFT) does not have this property. Nodes and Time are the iteration number
+and the run time of the BB scheme from the beginning to the termination.
+6.2. Serial Results on Synthetic Datasets
+Table 1 reports the serial results of synthetic datasets with different numbers of samples and
+different desired clusters (K = 3,5,10). Compared with the heuristic method FFT, our algorithm
+BB+LD+BT can reduce UB by 29.4% average on these synthetic datasets. These results validate the
+conclusion from Garcia-Diaz et al. (2019) that these 2-approximation heuristic algorithms perform
+poorly in practice despite the solution quality guarantee.
+As for the comparison of global optimizers, the direct usage of CPLEX on Problem 4 could not
+converge to a small optimality gap≤ 0.1% within 4 hours on all the synthetic datasets. BB+LD with-
+out acceleration techniques can obtain the small optimality gap≤ 0.1% within 4 hours on synthetic
+datasets smaller than 42,000 samples with desired clusters K = 3. The algorithm BB+LD+BT can
+obtain the best upper bounds and reach a satisfactory gap≤ 0.1% in most experiments within 4
+hours. Moreover, compared with BB+LD, BB+LD+BT needs fewer nodes and less run time to obtain
+the same optimality gap. For example, for the Syn-1200 dataset with K = 3, BB+LD need 1,155,375
+nodes and 3609 seconds to reach a gap≤ 0.1%, while BB+LD+BT only needs 23 nodes and 13.5 sec-
+onds. These comparisons between BB+LD and BB+LD+BT demonstrate the acceleration techniques in
+Section 4 can significantly reduce the search space and accelerate the BB procedure.
+6.3. Serial Results on Real-world datasets
+Table 2, Table 3, and Table 4 show the serial results on real-world datasets with different sample
+numbers and desired cluster numbers (K = 3,5,10). In these tables, we highlight the best results
+among these algorithms with the optimality gap≤ 0.1%. These real-world results are consistent
+with the results of synthetic datasets.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+22
+The best solutions generated by the heuristic method (FFT) can be far from optimal in these
+tables, even for very small datasets. For example, for IRIS dataset, FFT obtains UB of 3.66 while
+our algorithm and CPLEX give a UB of 2.04 with ≤ 0.1% gap. Compared with FFT, our algorithm
+BB+CF+BT can averagely reduce the UB by 22.2% on these real-world datasets and 25.8% on all the
+synthetic and real-world datasets. Even for experiments terminated with large gaps, in most cases,
+BB+CF+BT can obtain a smaller UB than FFT.
+For small datasets, our algorithms BB+CF and BB+CF+BT can obtain the same UB as CPLEX.
+However, CPLEX needs significantly more run time and nodes than our algorithms. For all datasets
+with more than 740 samples, CPLEX cannot even give an optimality gap≤ 50% within 4 hours. On
+the contrary, BB+CF+BT can obtain the best UB and a satisfactory gap≤ 0.1% for most datasets.
+The comparisons of the two versions of our algorithms BB+CF and BB+CF+BT demonstrate that
+the acceleration techniques in Section 4 can significantly reduce the computational time and the
+number of BB nodes to solve the problems. Remarkably, with these acceleration techniques, we
+can even solve several datasets in the root node (Nodes=1), e.g., the datasets iris, HF, and SGC.
+Besides, BB+CF+BT results with superscript 1 in these tables mean we can assign K farthest initial
+seeds through FFT at the root node as described in Section 4.1.1. We can obtain the initial seeds
+for about half of the datasets when K = 3. Moreover, the number of nodes is much smaller for the
+datasets with initial seeds than the datasets without initial seeds. This phenomenon indicates the
+initial seeds are essential for cluster assignment and bounds tightening since we need at least one
+assigned sample at each cluster to execute the sample-based assignment.
+For most of the datasets with millions of samples and K = 3 in Table 4, BB+CF+BT can converge
+to a small gap≤ 0.1% and provide the best optimal solution after 4 hours of running. To the best
+of our knowledge, it is the first time that the K-center problem is solved under a relatively small
+gap≤ 0.1% within 4 hours on datasets over 14 million samples in the serial mode.
+As a drawback, our algorithm BB+LD+BT still struggles to obtain a small optimality gap when the
+desired number of clusters is larger than 3. However, it should be noted the state-of-art global opti-
+mizer CPLEX cannot even solve any datasets to gap≤ 50% when K > 3. On the contrary, BB+LD+BT
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+23
+can obtain gap≤ 0.1% on most datasets with less than 5 million samples and K = 5. Moreover, for
+the cases when our algorithm BB+LD+BT cannot obtain a small optimality gap, it still gives the best
+UB among all the algorithms in these experiments.
+6.4. Parallel Results on Huge-scale Real-world datasets
+To fully utilize the computational ability of high-performance clusters, we implement our algorithm
+BB+CF+BT in a parallel manner as shown in Section 4.3. Here, we test the parallel algorithm on
+datasets that couldn’t obtain a small gap≤ 0.1% for K = 3 within 4 hours in the serial mode,
+including two datasets with ten million samples, HIGGS and BigCross. Moreover, we also extend
+the experiments to a billion-scale dataset called Taxi. This billion-scale dataset contains over 1.1
+billion individual taxi trips with 12 attributes in New York City from January 2009 through June
+2015. We preprocess the Taxi dataset according to the analysis by Schneider (2015) to remove
+outliers and missing values in the dataset. As an outcome shown in Table 5, the parallel version
+of BB+CF+BT can reach a small optimality gap≤ 0.1% and a better UB on the datasets BigCross
+and Taxi within 4 hours. For the dataset HIGGS, the parallel version achieves a smaller UB and
+gap compared to the heuristic method and the serial version. As far as we know, this is the first
+time that the K-center problem is solved under a relatively small gap≤ 0.1% within 4 hours on the
+billion-scale dataset.
+7. Conclusion
+We propose a global optimization algorithm for the K-center problem using a tailored reduced
+space branch and bound scheme. In this algorithm, we only need to branch on the region of cluster
+centers to guarantee convergence to the global optimal solution in a finite step.
+We give a two-stage decomposable formulation and an MINLP formulation of the K-center
+problem. With this two-stage formulation, we develop a lower bound with closed-form solutions by
+relaxing the non-anticipativity constraints and the “centers on sample” constraints. As an outcome,
+the proposed bounding methods are extremely computationally efficient with no needs to solve any
+optimization sub-problems using any optimizers.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+24
+Along with the BB procedure, we introduce several acceleration techniques based on the MINLP
+formulation, including bounds tightening, and sample reduction. Numerical experiments show these
+acceleration techniques can significantly reduce the search space and accelerate the solving pro-
+cedure. Moreover, we also give a parallel implementation of our algorithm to fully utilize the
+computational power of modern high performance clusters.
+Extensive numerical experiments have been conducted on synthetic and real-world datasets.
+These results exhibit the efficiency of our algorithm: we can solve the real-world datasets with up
+to ten million samples in the serial mode and one billion samples in the parallel mode to a
+small optimality gap (≤0.1%) within 4 hours.
+Finally, we also declare that our algorithm is promised to extend to deal with certain constrained
+versions of K-center problems. For example, the capacitated restricted version, absolute and vertex
+restricted version (Calik 2013). We are interested in developing these variants in future work.
+Acknowledgments
+The authors acknowledge funding from the discovery program of the Natural Science and Engineering
+Research Council of Canada under grant RGPIN-2019-05499 and the computing resources provided by SciNet
+(www.scinethpc.ca) and Digital Research Alliance of Canada (www.alliancecan.ca). Jiayang Ren acknowl-
+edges the financial support from the China Scholarship Council.
+References
+Aggarwal CC, Wolf JL, Yu PSl (2004) Method for targeted advertising on the web based on accumulated
+self-learning data, clustering users and semantic node graph techniques. US Patent 6,714,975.
+Aloise D, Contardo C (2018) A sampling-based exact algorithm for the solution of the minimax diameter
+clustering problem. Journal of Global Optimization 71(3):613–630.
+Brusco MJ, Stahl S (2005) Branch-and-Bound Applications in Combinatorial Data Analysis (New York:
+Springer).
+Byrne S, Wilcox LC, Churavy V (2021) MPI.jl: Julia bindings for the Message Passing Interface. Proceedings
+of the JuliaCon Conferences 1(1):68.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+25
+Calik H (2013) Exact Solution Methodologies for the P-Center Problem under Single and Multiple Allocation
+Strategies. Theses, Bilkent University.
+Cao Y, Zavala VM (2019) A scalable global optimization algorithm for stochastic nonlinear programs. Journal
+of Global Optimization 75(2):393–416.
+Chen D, Chen R (2009) New relaxation-based algorithms for the optimal solution of the continuous and
+discrete p-center problems. Computers & Operations Research 36(5):1646–1655.
+Chen R, Handler GY (1987) Relaxation method for the solution of the minimax location-allocation problem
+in euclidean space. Naval Research Logistics (NRL) 34(6):775–788.
+Contardo C, Iori M, Kramer R (2019) A scalable exact algorithm for the vertex p-center problem. Computers
+& Operations Research 103:211–220.
+Cook W, Lovasz L, Seymour P, eds. (1995) Combinatorial Optimization. DIMACS Series in Discrete Mathe-
+matics and Theoretical Computer Science (Providence, Rhode Island: American Mathematical Society).
+Cplex II (2020) V20.1.0: User’s Manual for CPLEX. International Business Machines Corporation .
+Dao TBH, Duong KC, Vrain C (2013) A declarative framework for constrained clustering. Machine Learning
+and Knowledge Discovery in Databases, volume 8190, 419–434.
+Daskin MS (2000) A new approach to solving the vertex p-center problem to optimality: Algorithm and
+computational results. Communications of the Operations Research Society of Japan 45(9):428–436.
+Davidovi´c T, Ramljak D, ˇSelmi´c M, Teodorovi´c D (2011) Bee colony optimization for the p-center problem.
+Computers and Operations Research 38(10):1367–1376.
+Dua D, Graff C (2017) UCI machine learning repository. URL http://archive.ics.uci.edu/ml.
+Duong KC, Vrain C, et al. (2017) Constrained clustering by constraint programming. Artificial Intelligence
+244:70–94.
+Dyer ME, Frieze AM (1985) A simple heuristic for the p-centre problem. Operations Research Letters
+3(6):285–288.
+Elloumi S, Labb´e M, Pochet Y (2004) A new formulation and resolution method for the p-center problem.
+INFORMS Journal on Computing 16(1):84–94.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+26
+Garcia-Diaz J, Menchaca-Mendez R, Menchaca-Mendez R, Pomares Hern´andez S, P´erez-Sansalvador JC,
+Lakouari N (2019) Approximation algorithms for the vertex K-Center problem: survey and experimental
+evaluation. IEEE Access 7:109228–109245.
+Garcia-Diaz J, Sanchez-Hernandez J, Menchaca-Mendez R, Menchaca-Mendez R (2017) When a worse
+approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center
+problem. Journal of Heuristics 23(5):349–366.
+Garey M, Johnson D (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness (New
+York: W. H. Freeman).
+Gonzalez TF (1985) Clustering to minimize the maximum intercluster distance. Theoretical Computer Sci-
+ence 38:293–306.
+Hansen P, Brimberg J, Uroˇsevi´c D, Mladenovi´c N (2009) Solving large p-median clustering problems by
+primal–dual variable neighborhood search. Data Mining and Knowledge Discovery 19(3):351–375.
+Hesabi ZR, Tari Z, Goscinski A, Fahad A, Khalil I, Queiroz C (2015) Data summarization techniques for
+big data—a survey. Handbook on Data Centers, 1109–1152.
+Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the K-Center problem. Mathematics of
+Operations Research 10(2):180–184.
+Horst R, Tuy H (2013) Global optimization: Deterministic approaches (Springer Science & Business Media).
+Hua K, Shi M, Cao Y (2021) A scalable deterministic global optimization algorithm for clustering problems.
+International Conference on Machine Learning, 4391–4401.
+Ilhan T, Pinar MC (2001) An efficient exact algorithm for the vertex p-center problem. Preprint.[Online].
+Available: http://www.ie.bilkent.edu.tr/mustafap/pubs .
+Kaufman L, Rousseeuw PJ (2009) Finding Groups in Data: an Introduction to Cluster Analysis (John Wiley
+& Sons).
+Kleindessner M, Awasthi P, Morgenstern J (2019) Fair k-center clustering for data summarization. Interna-
+tional Conference on Machine Learning, 3448–3457.
+Lim A, Rodrigues B, Wang F, Xu Z (2005) K-Center problems with minimum coverage. Theoretical Computer
+Science 332(1):1–17.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+27
+Miheliˇc J, Robic B (2005) Solving the k-center problem efficiently with a dominating set algorithm. Journal
+of computing and information technology 13:225–234.
+Minieka E (1970) The m-Center Problem. SIAM Review 12(01).
+Mladenovi´c N, Labb´e M, Hansen P (2003) Solving the p-Center problem with Tabu search and variable
+neighborhood Search. Networks 42(1):48–64.
+Padberg M, Rinaldi G (1991) A branch-and-cut algorithm for the resolution of large-scale symmetric traveling
+salesman problems. SIAM review 33(1):60–100.
+Plesn´ık J (1987) A heuristic for the p-center problems in graphs. Discrete Applied Mathematics 17(3):263–
+268.
+Pullan W (2008) A memetic genetic algorithm for the vertex p-center problem. Evolutionary Computation
+16(3):417–436.
+Schneider T (2015) Analyzing 1.1 billion NYC taxi and uber trips with a vengeance. URL https://
+toddwschneider.com/posts/analyzing-1-1-billion-nyc-taxi-and-uber-trips-with-a-vengeance/.
+Shi M, Hua K, Ren J, Cao Y (2022) Global optimization of K-Center clustering. Proceedings of the 39th
+International Conference on Machine Learning, 19956–19966.
+Wang E, Cai G, Ballachay R, Cao Y, Trajano HL (2022) Predicting xylose yield in prehydrolysis of hard-
+woods: a machine learning approach. Frontiers in Chemical Engineering 84.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+28
+Table 1
+Serial results on synthetic datasets
+Dataset
+Sam
+ple
+Dimen
+sion
+Method
+K=3
+K=5
+K=10
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+Syn-300
+3.0E+2
+2
+FFT
+69.68
+-
+-
+-
+43.33
+-
+-
+-
+21.88
+-
+-
+-
+CPLEX
+61.75
+2.9E+4
+≤0.1
+29
+37.14
+2.3E+7
+19.4
+4h
+16.06
+1.2E+7
+100.0
+4h
+BB+CF
+61.75
+5.5E+4
+≤0.1
+46
+37.14
+2.3E+6
+16.2
+4h
+15.64
+1.7E+6
+100.0
+4h
+BB+CF+BT
+61.75
+17
+≤0.1
+13
+37.14
+1,764
+≤0.1
+15
+12.31 2.0E+4 ≤0.1
+38
+Syn-1200
+1.2E+3
+2
+FFT
+93.34
+-
+-
+-
+58.46
+-
+-
+-
+30.49
+-
+-
+-
+CPLEX
+84.81
+5.8E+6
+1.6
+4h
+34.29
+3.5E+6
+7.8
+4h
+89.32
+8.1E+5
+100.0
+4h
+BB+CF
+84.81
+1.2E+6
+≤0.1
+3,609
+34.29
+1.4E+6
+12.5
+4h
+21.81
+1.0E+6
+100.0
+4h
+BB+CF+BT
+84.81
+23
+≤0.11
+14
+34.29
+411
+≤0.1
+15
+14.51 3.0E+4 ≤0.1
+148
+Syn-2100
+2.1E+3
+2
+FFT
+106.50
+-
+-
+-
+72.70
+-
+-
+-
+36.04
+-
+-
+-
+CPLEX
+95.10
+3.0E+6
+0.2
+4h
+49.32
+1.3E+6
+100.0
+4h
+193.26
+3.4E+5
+100.0
+4h
+BB+CF
+95.10
+1.5E+6
+≤0.1
+11,606
+42.58
+1.0E+6
+20.8
+4h
+25.78
+5.3E+5
+100.0
+4h
+BB+CF+BT
+95.10
+17
+≤0.11
+13
+42.58
+455
+≤0.1
+16
+17.65 8.9E+4 ≤0.1
+725
+Syn-42000
+4.2E+4
+2
+FFT
+161.98
+-
+-
+-
+96.12
+-
+-
+-
+47.21
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+142.33
+1.7E+5
+6.7
+4h
+63.40
+1.0E+5
+28.1
+4h
+44.24
+5.4E+4
+100.0
+4h
+BB+CF+BT 142.33
+103
+≤0.1
+21
+62.77 5.0E+3 ≤0.1
+363
+28.29
+5.8E+4
+36.1
+4h
+Syn-210000 2.1E+5
+2
+FFT
+175.81
+-
+-
+-
+120.78
+-
+-
+-
+66.79
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+168.57
+4.4E+4
+7.0
+4h
+77.02
+2.5E+4
+43.8
+4h
+53.73
+1.4E+4
+100.0
+4h
+BB+CF+BT 168.57
+5
+≤0.11
+21
+71.88 2.4E+3 ≤0.1 1,118
+44.48
+1.2E+4
+72.2
+4h
+1 Can assign K initial seeds through FFT at the root node. BB+CF+BT results without this superscript means can not assign initial seeds.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+29
+Table 2
+Serial results on small-scale datasets (S ≤ 1,000)
+Dataset
+Sam
+ple
+Dimen
+sion
+Method
+K=3
+K=5
+K=10
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+iris
+150
+4
+FFT
+2.65
+-
+-
+-
+1.80
+-
+-
+-
+0.95
+-
+-
+-
+CPLEX
+2.04
+1.2E+5
+≤0.1
+46
+1.54
+2.8E+6
+60.0
+4h
+1.21
+1.4E+7
+100.0
+4h
+BB+CF
+2.04
+1.3E+4
+≤0.1
+17
+1.20
+3.1E+6
+≤0.1
+5,472
+0.74
+2.2E+6
+100.0
+4h
+BB+CF+BT
+2.04
+1
+≤0.11
+12
+1.20
+409
+≤0.1
+14
+0.66
+9.6E+5
+25.8
+4h
+seeds
+210
+7
+FFT
+13.17
+-
+-
+-
+9.01
+-
+-
+-
+4.48
+-
+-
+-
+CPLEX
+10.44
+1.2E+6
+≤0.1
+542
+11.61
+2.5E+6
+96.1
+4h
+21.48
+5.6E+5
+100.0
+4h
+BB+CF
+10.44
+7.2E+3
+≤0.1
+17
+7.22
+2.7E+6
+8.3
+4h
+3.51
+1.5E+6
+100.0
+4h
+BB+CF+BT
+10.44
+21
+≤0.11
+13
+7.22
+1,444
+≤0.1
+15
+2.92
+2.1E+5 ≤0.1
+569
+glass
+214
+9
+FFT
+27.52
+-
+-
+-
+22.28
+-
+-
+-
+11.73
+-
+-
+-
+CPLEX
+Out of memory
+Out of memory
+Out of memory
+BB+CF
+27.52
+5.6E+3
+≤0.1
+15
+16.44
+9.7E+5
+≤0.1
+1,522
+10.64
+1.4E+6
+100.0
+4h
+BB+CF+BT
+27.52
+191
+≤0.1
+13
+16.44
+4.4E+3
+≤0.1
+17
+7.95
+1.7E+6 ≤0.1 9,180
+BM
+249
+6
+FFT
+1.52E+04
+-
+-
+-
+1.12E+04
+-
+-
+-
+5.33E+03
+-
+-
+-
+CPLEX
+No feasible solution
+1.48E+04
+8.8E+6
+100.0
+4h
+1.63E+04
+2.4E+6
+100.0
+4h
+BB+CF
+1.05E+04
+1.4E+4
+≤0.1
+22
+6.32E+03
+2.2E+6
+12.0
+4h
+5.01E+03
+1.4E+6
+100.0
+4h
+BB+CF+BT 1.05E+04
+63
+≤0.11
+13
+6.32E+03 1.8E+4
+≤0.1
+29
+4.98E+03
+6.7E+5
+97.9
+4h
+UK
+258
+5
+FFT
+0.70
+-
+-
+-
+0.57
+-
+-
+-
+0.42
+-
+-
+-
+CPLEX
+Out of memory
+Out of memory
+Out of memory
+BB+CF
+0.53
+3.2E+5
+≤0.1
+258
+0.43
+1.5E+6
+43.9
+4h
+0.33
+1.4E+6
+100.0
+4h
+BB+CF+BT
+0.53
+1.6E+4
+≤0.1
+23
+0.43
+8.9E+5
+26.9
+4h
+0.31
+6.1E+5
+97.3
+4h
+HF
+299
+12
+FFT
+2.69E+10
+-
+-
+-
+1.17E+10
+-
+-
+-
+1.68E+09
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+1.72E+10
+339
+≤0.1
+10
+1.02E+10
+2.1E+4
+≤0.1
+44
+1.52E+09
+3.4E+6
+100.0
+4h
+BB+CF+BT 1.72E+10
+1
+≤0.11
+12
+1.02E+10
+557
+≤0.1
+14
+1.44E+09
+1.2E+6
+53.2
+4h
+Who
+440
+8
+FFT
+4.58E+09
+-
+-
+-
+3.18E+09
+-
+-
+-
+9.81E+08
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+3.49E+09
+3.4E+3
+≤0.1
+15
+2.11E+09
+1.7E+5
+≤0.1
+341
+9.27E+08
+1.5E+6
+100.0
+4h
+BB+CF+BT 3.49E+09
+375
+≤0.1
+14
+2.11E+09 2.3E+3
+≤0.1
+16
+8.21E+08
+8.4E+5
+62.0
+4h
+HCV
+602
+12
+FFT
+1.75E+05
+-
+-
+-
+8.38E+04
+-
+-
+-
+3.03E+04
+-
+-
+-
+CPLEX
+1.41E+05
+9.5E+5
+≤0.1
+3,720
+8.73E+04
+5.1E+5
+100.0
+4h
+4.47E+04
+3.8E+5
+100.0
+4h
+BB+CF
+1.41E+05
+291
+≤0.1
+10
+6.37E+04
+2.2E+4
+≤0.1
+76
+2.36E+04
+1.4E+6
+100.0
+4h
+BB+CF+BT 1.41E+05
+39
+≤0.1
+13
+6.37E+04
+583
+≤0.1
+15
+2.16E+04 7.6E+5 ≤0.1 6,300
+Abs
+740
+21
+FFT
+1.94E+04
+-
+-
+-
+1.19E+04
+-
+-
+-
+7.81E+03
+-
+-
+-
+CPLEX
+1.72E+04
+9.3E+5
+52.0
+4h
+3.02E+04
+1.7E+4
+100.0
+4h
+No feasible solution
+BB+CF
+1.39E+04
+3.3E+4
+≤0.1
+153
+9.93E+03
+1.5E+6
+18.9
+4h
+6.15E+03
+9.8E+5
+100.0
+4h
+BB+CF+BT 1.39E+04
+611
+≤0.1
+15
+9.92E+03 5.0E+4
+≤0.1
+178
+6.37E+03
+4.6E+5
+98.3
+4h
+TR
+980
+10
+FFT
+7.32
+-
+-
+-
+6.42
+-
+-
+-
+4.41
+-
+-
+-
+CPLEX
+8.32
+1.6E+6
+54.5
+4h
+7.82
+2.0E+5
+100.0
+4h
+8.70
+3.3E+4
+100.0
+4h
+BB+CF
+5.94
+7.4E+5
+≤0.1
+2,953
+4.49
+1.3E+6
+47.7
+4h
+3.69
+9.8E+5
+100.0
+4h
+BB+CF+BT
+5.94
+3.3E+4
+≤0.1
+83
+4.49
+1.0E+6
+24.6
+4h
+3.73
+5.0E+5
+99.9
+4h
+SGC
+1,000
+21
+FFT
+1.33E+07
+-
+-
+-
+4.08E+06
+-
+-
+-
+9.50E+05
+-
+-
+-
+CPLEX
+9.45E+06
+5.0E+4
+100.0
+4h
+1.56E+08
+10
+100.0
+4h
+No feasible solution
+BB+CF
+9.45E+06
+411
+≤0.1
+12
+3.91E+06
+2.8E+4
+≤0.1
+185
+9.50E+05
+9.6E+5
+100.0
+4h
+BB+CF+BT 9.45E+06
+1
+≤0.11
+12
+3.91E+06
+1
+≤0.11
+12
+9.50E+05
+5.8E+5
+100.0
+4h
+1 Can assign K initial seeds through FFT at the root node. BB+CF+BT results without this superscript means can not assign initial seeds.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+30
+Table 3
+Serial results on large-scale datasets (1,000<S<1,000,000)
+Dataset
+Sam
+ple
+Dimen
+sion
+Method
+K=3
+K=5
+K=10
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+hemi
+1,955
+7
+FFT
+1.06E+05
+-
+-
+-
+3.31E+04
+-
+-
+-
+1.42E+04
+-
+-
+-
+CPLEX
+4.08E+05
+1.4E+5
+100.0
+4h
+4.08E+05
+1.2E+5
+100.0
+4h
+4.08E+05
+5
+100.0
+4h
+BB+CF
+4.08E+05
+1.4E+5
+50.0
+4h
+2.18E+04
+6.8E+5
+≤0.1
+4,212
+1.42E+04
+7.1E+5 100.0
+4h
+BB+CF+BT 6.49E+04
+11
+≤0.11
+14
+2.18E+04
+158
+≤0.1
+14
+7.20E+03 8.2E+3 ≤0.1
+89
+pr2392
+2,392
+2
+FFT
+3.75E+07
+-
+-
+-
+2.11E+07
+-
+-
+-
+1.02E+07
+-
+-
+-
+CPLEX
+6.51E+07
+2.1E+5
+100.0
+4h
+5.66E+07
+3.6E+4
+100.0
+4h
+4.23E+07
+4.8E+4 100.0
+4h
+BB+CF
+2.93E+07
+5.9E+4
+≤0.1
+297
+1.52E+07
+8.3E+5
+20.9
+4h
+1.02E+07
+6.0E+5 100.0
+4h
+BB+CF+BT 2.93E+07
+207
+≤0.1
+15
+1.46E+07 6.6E+3 ≤0.1
+45
+8.70E+06
+4.2E+5
+59.6
+4h
+TRR
+5,454
+24
+FFT
+101.55
+-
+-
+-
+95.26
+-
+-
+-
+86.87
+-
+-
+-
+CPLEX
+166.61
+1
+100.0
+4h
+No feasible solution
+No feasible solution
+BB+CF
+89.78
+3.6E+5
+73.1
+4h
+85.07
+2.7E+5
+100.0
+4h
+78.53
+1.7E+5 100.0
+4h
+BB+CF+BT
+88.30
+2.8E+5
+64.2
+4h
+84.80
+2.0E+5
+100.0
+4h
+77.37
+1.3E+5 100.0
+4h
+AC
+7,195
+22
+FFT
+3.59
+-
+-
+-
+2.79
+-
+-
+-
+2.28
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+2.75
+3.1E+5
+42.6
+4h
+2.26
+2.3E+5
+72.0
+4h
+2.14
+1.4E+5 100.0
+4h
+BB+CF+BT
+2.78
+2.7E+5
+38.9
+4h
+2.26
+1.8E+5
+70.2
+4h
+1.90
+1.2E+5 100.0
+4h
+rds cnt
+10,000
+4
+FFT
+1.93E+04
+-
+-
+-
+5.93E+03
+-
+-
+-
+1.44E+03
+-
+-
+-
+CPLEX
+6.86E+04
+3.2E+4
+100.0
+4h
+No feasible solution
+No feasible solution
+BB+CF
+1.39E+04
+639
+≤0.1
+25
+4.90E+03
+1.6E+5
+≤0.1
+6,048
+1.44E+03
+1.8E+5 100.0
+4h
+BB+CF+BT 1.39E+04
+1
+≤0.11
+12
+4.90E+03
+107
+≤0.11
+16
+1.44E+03
+1.8E+5 100.0
+4h
+HTRU2
+17,898
+8
+FFT
+7.11E+04
+-
+-
+-
+3.36E+04
+-
+-
+-
+1.37E+04
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+5.24E+04
+1.1E+4
+≤0.1
+627
+2.12E+04
+2.1E+5
+14.4
+4h
+1.37E+04
+9.6E+4 100.0
+4h
+BB+CF+BT 5.24E+04
+25
+≤0.11
+15
+2.09E+04 5.1E+3 ≤0.1
+282
+1.37E+04
+7.9E+4
+99.5
+4h
+GT
+36,733
+11
+FFT
+4.57E+03
+-
+-
+-
+4.00E+03
+-
+-
+-
+2.59E+03
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+3.07E+03
+1.4E+5
+27.6
+4h
+2.83E+03
+8.2E+4
+62.2
+4h
+2.35E+03
+4.7E+4 100.0
+4h
+BB+CF+BT 2.98E+03 2.1E+4 ≤0.1 2,053 2.81E+03
+6.3E+4
+60.9
+4h
+2.29E+03
+4.0E+4 100.0
+4h
+rds
+50,000
+3
+FFT
+0.11
+-
+-
+-
+0.06
+-
+-
+-
+0.03
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+0.08
+1.3E+5
+4.8
+4h
+0.05
+8.5E+4
+26.2
+4h
+0.03
+4.4E+4 100.0
+4h
+BB+CF+BT
+0.08
+719
+≤0.1
+16
+0.05
+3.6E+4 ≤0.1 3,429
+0.02
+4.1E+4 100.0
+4h
+KEGG
+53,413
+23
+FFT
+6.20E+06
+-
+-
+-
+1.70E+06
+-
+-
+-
+2.13E+05
+-
+-
+-
+CPLEX
+Out of memory
+No feasible solution
+No feasible solution
+BB+CF
+4.98E+06
+87
+≤0.1
+41
+7.58E+05
+5.8E+3
+≤0.1
+2,416
+2.13E+05
+2.3E+4 100.0
+4h
+BB+CF+BT 4.98E+06
+1
+≤0.11
+13
+7.58E+05
+1
+≤0.11
+14
+2.04E+05
+3.0E+4 100.0
+4h
+rng agr
+199,843
+7
+FFT
+4.68E+10
+-
+-
+-
+1.61E+10
+-
+-
+-
+7.47E+09
+-
+-
+-
+CPLEX
+Out of memory
+No feasible solution
+No feasible solution
+BB+CF
+3.16E+10
+3.7E+4
+4.8
+4h
+1.37E+10
+2.0E+4
+35.4
+4h
+7.47E+09
+1.2E+4 100.0
+4h
+BB+CF+BT 3.14E+10 2.3E+3 ≤0.11
+239
+1.20E+10 2.0E+4 ≤0.11 3,330 7.02E+09
+9.1E+3 100.0
+4h
+urbanGB 360,177
+2
+FFT
+7.63
+-
+-
+-
+5.62
+-
+-
+-
+2.81
+-
+-
+-
+CPLEX
+Out of memory
+No feasible solution
+No feasible solution
+BB+CF
+5.48
+1.6E+4
+≤0.1 10,713
+4.48
+1.5E+4
+59.1
+4h
+2.81
+7.7E+3 100.0
+4h
+BB+CF+BT
+5.48
+171
+≤0.11
+66
+3.86
+3.0E+3 ≤0.1 2,710
+2.60
+6.3E+3
+92.6
+4h
+spnet3D 434,876
+3
+FFT
+822.03
+-
+-
+-
+256.87
+-
+-
+-
+68.19
+-
+-
+-
+CPLEX
+Out of memory
+No feasible solution
+No feasible solution
+BB+CF
+569.91
+2.2E+4
+0.3
+4h
+216.25
+1.3E+4
+16.8
+4h
+68.19
+6.2E+3 100.0
+4h
+BB+CF+BT
+569.80
+85
+≤0.11
+28
+205.89
+3.5E+3 ≤0.11
+661
+68.19
+4.6E+3 100.0
+4h
+1 Can assign K initial seeds through FFT at the root node. BB+CF+BT results without this superscript means can not assign initial seeds.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+31
+Table 4
+Serial results on datasets with millions of samples
+Dataset
+Sam
+ple
+Dimen
+sion
+Method
+K=3
+K=5
+K=10
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+USC1990 2,458,285
+68
+FFT
+2.04E+11
+-
+-
+-
+7.47E+10
+-
+-
+-
+1.87E+10
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+1.69E+11
+916
+3.6
+4h
+7.47E+10
+352
+100.0
+4h
+1.87E+10
+168
+100.0
+4h
+BB+CF+BT 1.68E+11
+1
+≤0.11
+277
+6.05E+10
+256
+≤0.11 1,781
+1.87E+10
+396
+61.0
+4h
+Gas
+methane
+4,178,504
+18
+FFT
+1.31E+08
+-
+-
+-
+1.17E+08
+-
+-
+-
+6.95E+07
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+1.04E+08 1.2E+3
+31.1
+4h
+8.82E+07
+488
+100.0
+4h
+6.95E+07
+244
+100.0
+4h
+BB+CF+BT 1.02E+08
+65
+≤0.11 1,272 7.21E+07
+807
+19.9
+4h
+6.95E+07
+410
+100.0
+4h
+Gas
+CO
+4,208,261
+18
+FFT
+8.83E+08
+-
+-
+-
+5.17E+08
+-
+-
+-
+2.05E+08
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+5.66E+08 1.1E+3
+12.8
+4h
+4.09E+08
+449
+100.0
+4h
+2.05E+08
+241
+100.0
+4h
+BB+CF+BT 5.46E+08
+66
+≤0.1 1,053 2.80E+08
+670
+≤0.1
+9,612
+2.05E+08
+398
+100.0
+4h
+kddcup
+4,898,431
+38
+FFT
+4.71E+17
+-
+-
+-
+9.71E+16
+-
+-
+-
+5.96E+14
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+2.25E+17
+63
+≤0.1
+2,461
+4.73E+16
+229
+100.0
+4h
+5.96E+14
+124
+100.0
+4h
+BB+CF+BT 2.25E+17
+37
+≤0.1
+958
+4.73E+16
+417
+≤0.1 10,116 2.58E+14
+1
+≤0.11 586
+HIGGS
+11,000,000
+29
+FFT
+368.35
+-
+-
+-
+320.91
+-
+-
+-
+198.71
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+247.03
+368
+67.9
+4h
+249.45
+210
+100.0
+4h
+198.71
+98
+100.0
+4h
+BB+CF+BT
+237.91
+290
+65.5
+4h
+235.68
+185
+100.0
+4h
+198.71
+100
+100.0
+4h
+BigCross 11,620,300
+56
+FFT
+1.43E+07
+-
+-
+-
+7.54E+06
+-
+-
+-
+4.24E+06
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+1.09E+07
+148
+32.9
+4h
+7.54E+06
+122
+100.0
+4h
+4.24E+06
+66
+100.0
+4h
+BB+CF+BT
+9.97E+06
+211
+19.7
+4h
+7.54E+06
+135
+100.0
+4h
+4.24E+06
+66
+100.0
+4h
+Phones
+acceler
+ometer
+13,062,475
+6
+FFT
+2.04E+28
+-
+-
+-
+1.09E+28
+-
+-
+-
+3.89E+26
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+1.46E+28
+51
+≤0.1
+2,038
+6.17E+27
+303
+100.0
+4h
+3.89E+26
+148
+100.0
+4h
+BB+CF+BT 1.46E+28
+-
+≤0.11
+309
+6.17E+27
+354
+100.0
+4h
+3.89E+26
+154
+100.0
+4h
+Phones
+gyro
+scope
+13,932,632
+6
+FFT
+1.51E+28
+-
+-
+-
+1.09E+28
+-
+-
+-
+2.61E+26
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+1.46E+28
+51
+≤0.1
+2,195
+6.18E+27
+286
+100.0
+4h
+2.61E+26
+136
+100.0
+4h
+BB+CF+BT 1.46E+28
+1
+≤0.11
+294
+6.18E+27
+330
+100.0
+4h
+2.61E+26
+140
+100.0
+4h
+AADP
+14,057,567
+3
+FFT
+3.82E+03
+-
+-
+-
+2.98E+03
+-
+-
+-
+1.90E+03
+-
+-
+-
+CPLEX
+No feasible solution
+No feasible solution
+No feasible solution
+BB+CF
+2.66E+03
+602
+35.9
+4h
+2.49E+03
+324
+100.0
+4h
+1.90E+03
+147
+100.0
+4h
+BB+CF+BT 2.55E+03
+196
+≤0.1 4,321 2.46E+03
+290
+98.9
+4h
+1.90E+03
+145
+100.0
+4h
+1 Can assign K initial seeds through FFT at the root node. BB+CF+BT results without this superscript means can not assign initial seeds.
+Table 5
+Parallel results of BB+CF+BT (K = 3)
+Dataset
+Sample
+Dimension
+Method
+UB
+Nodes
+Gap
+(%)
+Time
+(s)
+HIGGS
+11,000,000
+29
+Heuristic
+368.35
+-
+-
+-
+Serial
+237.91
+290
+65.5
+4h
+Parallel
+(400 cores)
+227.91
+12,576
+30.2
+4h
+Bigcross
+11,620,300
+56
+Heuristic
+1.43E+07
+-
+-
+-
+Serial
+9.97E+06
+211
+19.7
+4h
+Parallel
+(400 cores)
+9.38E+06
+10,071
+≤0.1
+6,444
+Taxi
+1,120,841,769
+12
+Heuristic
+3.09E+04
+-
+-
+-
+Parallel
+(2000 cores)
+1.62E+04
+1,063
+≤0.1
+5,705
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+32
+× x1
+× x2
+×
+x3
+||x1 − x2||2
+2 > 4α
+||x2 − x3||2
+2 > 4α
+||x3 − x1||2
+2 > 4α
+Figure 1
+Initial seeds with 3 clusters. In this example, ||x1 − x2||2
+2 > 4α, ||x2 − x3||2
+2 > 4α and ||x3 − x1||2
+2 > 4α.
+Therefore, we can arbitrarily assign x1,x2,x3 to 3 distinct clusters.
+× xs
+M 1
+M 2
+M 3
+β2
+s(M 2) > α
+β3
+s(M 3) > α
+Figure 2
+Center-based assignment with 3 clusters. In this example, β2
+s(M 2) > α (b2
+s = 0) and β3
+s(M 3) > α (b3
+s = 0).
+Therefore, we assign xs to the first cluster (b1
+s = 1).
+× x1
+× x2
+×x3
+× xs
+M 1
+M 2
+M 3
+||xs − x1||2
+2 > 4α
+||xs − x2||2
+2 > 4α
+Figure 3
+Sample-based assignment with 3 clusters. Assume we already know that x1,x2,x3 belong to cluster 1,2
+and 3, respectively. xs is the sample to be determined. In this example, ||xs − x1||2
+2 > 4α (b1
+s = 0) and
+||xs − x2||2
+2 > 4α (b2
+s = 0). Therefore, xs is assigned to cluster 3 (b3
+s = 1).
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+33
+×
+×
+×
+×
+×
+×
+×
+×
+×
+×xi
+×
+xj
+M k
+×
+×
+Bα(xi)
+Bα(xj)
+√α
+√α
+Figure 4
+Ball-based bounds tightening in two-dimensional space. In this example, suppose it is determined that
+two points xi and xj belong to the Kth cluster. We first compute the index set of samples within all
+balls and original box, Sk
++(M) := {s ∈ S |xs ∈ X ∩M k ∩Bα(xi)∩Bα(xj)}. We then generate the smallest
+box containing these samples in Sk
++(M). The red rectangle is the tightened bounds we obtain.
+×
+×
+×
+×
+×
+×
+×
+×
+×
+×xi
+×
+xj
+M k
+×
+×
+Bα(xi)
+Bα(xj)
+Rα(xi)
+Rα(xj)
+√α
+√α
+Figure 5
+Box-based bounds tightening in two-dimensional space. In this example, we first generate two boxes
+with Rα(xi) := {x| xi −√α ≤ x ≤ xi +√α} and Rα(xj) = {x| xj −√α ≤ x ≤ xj +√α}. We then create a
+tighten bounds with ˆ
+M k=Rα(xi) ∩ Rα(xj) ∩ M k. The red rectangle is the tightened bounds we obtain.
+
+Ren et al.: Global Optimization for K-Center of One Billion Samples
+34
+Dataset 
+Subset 
+Subset 
+Subset 
+Subset 
+. . .
+Tightened space of centers: 
+Bound
+Tightening 
+Bound
+Tightening 
+Bound
+Tightening 
+Bound
+Tightening 
+. . .
+LB & UB
+Bounding 
+LB & UB
+Bounding 
+LB & UB
+Bounding 
+LB & UB
+Bounding 
+. . .
+Lower bounds: 
+,
+Upper bounds: 
+Sample
+reduction 
+Sample
+reduction 
+Sample
+reduction 
+Sample
+reduction 
+. . .
+Index set of redundant samples: 
+ 
+Update dataset according to the redundant index set
+Gather from each process
+Gather from each process
+Gather from each process
+Spread to each proccess equally
+Parallel
+(Map)
+Serial
+(Reduce)
+Figure 6
+Parallelization of the reduced-space branch and bound scheme
+

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+This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics
+1
+New Exact Betchov-like Relation for the
+Helicity Flux in Homogeneous Turbulence
+Damiano Capocci1†, Perry L. Johnson2, Sean Oughton3,
+Luca Biferale1 and Moritz Linkmann4‡
+1Department of Physics and INFN, University of Rome Tor Vergata, Rome, Italy
+2Department of Mechanical and Aerospace Engineering, University of California, Irvine, USA
+3Department of Mathematics, University of Waikato, Hamilton, New Zealand
+4School of Mathematics and Maxwell Institute for Mathematical Sciences,
+University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom
+(Received xx; revised xx; accepted xx)
+In homogeneous and isotropic turbulence, the relative contributions of different physical
+mechanisms to the energy cascade can be quantified by an exact decomposition of the
+energy flux (P. Johnson, Phys. Rev. Lett., 124, 104501 (2020), J. Fluid Mech. 922,
+A3(2021)). We extend the formalism to the transfer of kinetic helicity across scales,
+important in the presence of large-scale mirror breaking mechanisms, to identify physical
+processes resulting in helicity transfer and quantify their contributions to the mean flux
+in the inertial range. All subfluxes transfer helicity from large to small scales. About 50%
+of the mean flux is due to the scale-local vortex flattening and vortex twisting. We derive
+a new exact relation between these effects, similar to the Betchov relation for the energy
+flux, revealing that the mean contribution of the former is three times larger than that
+of the latter. Multi-scale effects account for the remaining 50% of the mean flux, with
+approximate equipartition between multi-scale vortex flattening, twisting and entangling.
+1. Introduction
+The kinetic helicity, defined as the L2-inner product of velocity u and vorticity ω, has
+dynamical, topological, geometrical, and statistical interpretations in turbulence. It is
+a dynamical and topological inviscid invariant, where the latter refers to its connection
+with the linking number of infinitesimal vortex lines (Moffatt 1969). Geometrically, it
+quantifies the alignment of velocity and vorticity in a volume-averaged sense. Within
+a statistical approach to turbulence, helicity is the correlation between velocity and
+vorticity. In a rotationally invariant ensemble, it is connected to the breaking of the
+symmetry under inversion of all axes. Inspired by its relevance to turbulence in atmo-
+spheric flows (Lilly 1986), dynamical and statistical effects connected with helicity have
+been studied in the atmospheric boundary layer (Deusebio & Lindborg 2014) and in
+rotating turbulence (Mininni & Pouquet 2010a,b), and more generally in homogeneous
+and isotropic turbulence (Chen et al. 2003a,b; Gledzer & Chkhetiani 2015; Kessar et al.
+2015; Sahoo et al. 2015; Stepanov et al. 2015; Alexakis 2017; Sahoo et al. 2017; Milanese
+et al. 2021; Yan et al. 2020), as well as shear flows (Yan et al. 2020; Yu et al. 2022) and
+in laboratory experiments (Scheeler et al. 2017).
+The level of helicity in a turbulent flow affects turbulent statistics and dynamics, and
+is thus of relevance from a fundamental theory perspective as well as for subgrid-scale
+† Email address for correspondence: [email protected]
+‡ Email address for correspondence: [email protected]
+arXiv:2301.04193v1  [physics.flu-dyn]  10 Jan 2023
+
+2
+D. Capocci, P. Johnson, S. Oughton, L. Biferale, M. Linkmann
+(SGS) modelling. As an alignment of velocity and vorticity weakens the nonlinearity of
+the Navier–Stokes equations, high levels of helicity have been connected with a depletion
+of the kinetic energy flux across scales by an analysis of the coupling between helical
+Fourier modes (Kraichnan 1973), and with regions of low dissipation (Moffatt 2014).
+These effects can be quantified by upper bound theory applied to helical forcing and
+direct numerical simulation — the energy flux of turbulence sustained by fully helical
+forcing is about 30% lower than in the non-helical case (Linkmann 2018).
+Helicity affects turbulence not only globally, that is, in terms of mean energy fluxes,
+but also on a scale-by-scale level. As a solenoidal vector field, the velocity field u can
+be decomposed into positively and negatively helical components u± (Herring 1974;
+Constantin & Majda 1988; Waleffe 1992), u(x, t) = u+(x, t) + u−(x, t), where u±
+are obtained by projecting the Fourier coefficients ˆu(k, t) onto basis vectors which are
+eigenfunctions of the curl operator in Fourier space. That is, ˆu±(k, t) = u±(k, t)h±(k) ,
+where ik × kh±(k) = ±h±(k) and u±(k, t) = ˆu(k, t) · h±(k). The energy flux can
+then be decomposed into different triadic couplings between positively and negatively
+helical velocity-field fluctuations (Waleffe 1992). Interestingly, interactions among helical
+Fourier modes of like-signed helicity leads to an inverse energy transfer across scales in
+the inertial range (Waleffe 1992; Biferale et al. 2012, 2013; Sahoo et al. 2015), while
+interactions of oppositely-signed helical modes transfer energy from large to small scales
+(Waleffe 1992; Alexakis 2017; Alexakis & Biferale 2018). For turbulent flows of electrically
+conducting fluids such as liquid metals or plasmas in the fluid approximation, helicity
+alters the evolution of both velocity and magnetic-field fluctuations profoundly. Here,
+small-scale kinetic helicity facilitates the formation of large-scale coherent magnetic
+structures through the large-scale dynamo (Steenbeck et al. 1966; Brandenburg 2001;
+Brandenburg & Subramanian 2005; Tobias et al. 2013; Linkmann et al. 2016, 2017).
+The cascade of kinetic helicity itself is predicted to be direct, that is, it proceeds from
+large to small scales (Brissaud et al. 1973; Waleffe 1992), and scale-local (Eyink 2005). It
+results, as discussed by Eyink (2006) in the context of a multi-scale gradient expansion,
+from a twisting of small-scale vortices into a local alignment with the small-scale velocity
+fluctuations by large-scale differential vorticity (‘screw’). However, being sign-indefinite,
+numerical results on helicity fluxes can be difficult to interpret as a loss of positive helicity
+at a given scale may be viewed as a gain of negative helicity at the same scale.
+In the context of SGS modelling, the effect helicity has on a turbulent flow is usually
+taken into account though additional diffusive model terms (Yokoi & Yoshizawa 1993;
+Li et al. 2006; Baerenzung et al. 2008; Inagaki et al. 2017). However, a combination of
+a-priori and a-posteriori analyses of different SGS models for isotropic helical turbulence
+found the effect of the additional diffusive model terms to be small and that a classical
+Smagorinsky model best represents the resolved-scale dynamics (Li et al. 2006). Similarly,
+based on analytical and numerical results, Linkmann (2018) suggests an adjustment of
+the Smagorinsky constant to account for high levels of helicity. So far, SGS analyses of
+helical turbulence have mainly been concerned with energy transfers.
+Here, we focus on the helicity flux across scales in statistically stationary homogeneous
+and isotropic turbulence, with large-scale forcing breaking mirror symmetry. For the
+energy flux, the Betchov (1956) relation states that the mean contribution from vortex
+stretching to the energy cascade is triple that due to strain self-amplification. Carbone
+& Wilczek (2022) recently showed that there are no further kinematic relations for the
+energy flux in statistically stationary homogeneous and isotropic turbulence with zero net
+helicity. However, we prove here that a new exact kinematic Betchov-type relation exists
+for the mean helicity flux. Furthermore, we also present an exact decomposition of the
+helicity flux in analogy to that of the kinetic energy flux derived by Johnson (2020, 2021),
+
+Helicity fluxes in homogeneous turbulence
+3
+whereby the relative contributions of physical mechanisms, such as vortex stretching and
+strain self-amplification, to the energy cascade can be quantified in terms of the overall
+contribution and their scale-locality. The aim is to identify physical mechanisms that
+transfer kinetic helicity across scales and to quantify their relative contributions to the
+mean helicity flux and its fluctuations, which may be useful for the construction of SGS
+models when resolving the helicity cascade is of interest.
+2. Exact decomposition of the kinetic helicity flux
+To derive the aforementioned exact decomposition of the helicity flux and relations
+between the resulting subfluxes, we begin with the three-dimensional (3D) incompressible
+Navier–Stokes equations, here written in component form
+∂tui + ∂j (uiuj) = −∂jpδij + 2ν∂jSij + fi ,
+(2.1)
+∂juj = 0 ,
+(2.2)
+where u = (u1, u2, u3) is the velocity field, p the pressure divided by the constant density,
+ν the kinematic viscosity, Sij the rate-of-strain tensor, and f = (f1, f2, f3) an external
+solenoidal force that may be present. To define the helicity flux across scales, we introduce
+a filtering operation to separate large- and small-scale dynamics (e.g., Germano 1992).
+Specifically, for a generic function φ, the filtered version at scale ℓ is φ
+ℓ = Gℓ ∗ φ , where
+Gℓ is a filter kernel with filter width ℓ and the asterisk denotes the convolution operation.
+Applying the filter to the Navier–Stokes equations (2.1)–(2.2) results in
+∂tuℓ
+i + ∂j
+�
+uℓ
+iuℓ
+j + pℓδij − 2νS
+ℓ
+ij + τ ℓ
+ij
+�
+= f
+ℓ
+i ,
+(2.3)
+where τ ℓ
+ij = τ ℓ(ui, uj) = uiujℓ − uℓ
+iuℓ
+j is the SGS stress tensor. Here, we follow the
+notation of Germano (1992) in defining the generalised second moment for any two fields
+as τ ℓ(a, b) = ab
+ℓ − aℓb
+ℓ. We also require the filtered vorticity equation
+∂tωℓ
+i + ∂j
+�
+ωℓ
+iuℓ
+j − uℓ
+iωℓ
+j − ν∂jωℓ
+i
+�
+− gℓ
+i = −∂j
+�
+ϵimn∂mτ ℓ
+nj
+�
+,
+(2.4)
+where g = ∇ × f. The large-scale helicity density, Hℓ = uℓ
+iωℓ
+i, then evolves according to
+∂tHℓ + ∂j
+�
+Hℓuℓ
+j + (pℓ − 1
+2uℓ
+iuℓ
+i)ωℓ
+j − ν∂jHℓ�
++ 2ν(∂juℓ
+i)(∂jωℓ
+i) − ωℓ
+if
+ℓ
+i − uℓ
+igℓ
+i
+= −∂j
+�
+2ωℓ
+iτ ℓ
+ij + ϵijkuℓ
+i∂mτ ℓ
+km
+�
++ 2τ ℓ
+ij∂jωℓ
+i
+(2.5)
+The last term in this equation is the helicity flux
+ΠH,ℓ = −2τ ℓ
+ij∂jωℓ
+i ,
+(2.6)
+and is the central focus herein. It has an alternative form (Yan et al. 2020),
+˜ΠH,ℓ = −τ ℓ
+ij∂jωℓ
+i −
+�
+τ ℓ(ωi, uj) − τ ℓ(ui, ωj)
+�
+∂juℓ
+i ,
+(2.7)
+and it can be shown that the RHSs of (2.6) and (2.7) differ by an expression that can
+be written as a divergence and therefore vanishes after averaging spatially, at least for
+statistically homogeneous turbulence (Yan et al. 2020). This implies ⟨ΠH,ℓ⟩ = ⟨ ˜ΠH,ℓ⟩.
+Eyink (2006) links the first term in (2.7) — which is proportional to ΠH,ℓ — to vortex
+twisting and Yan et al. (2020) attribute the second term to vortex stretching. In what
+follows we discuss an exact decomposition of ΠH,ℓ, and show that both effects can be
+identified therein. We also use ΠH,ℓ for our numerical evaluations (cf. Chen et al. 2003a;
+Eyink 2006).
+
+4
+D. Capocci, P. Johnson, S. Oughton, L. Biferale, M. Linkmann
+2.1. Gaussian filter relations for the helicity flux
+So far all expressions are exact and filter-independent. To derive exact decompositions
+of the helicity flux in both representations, we now focus on Gaussian filters. For that
+case, Johnson (2020, 2021) showed that the subgrid-scale stresses can be obtained as the
+solution of a forced diffusion equation with ℓ2 being the time-like variable, resulting in
+τ ℓ
+ij = τ ℓ(ui, uj) = ℓ2A
+ℓ
+ikA
+ℓ
+jk +
+� ℓ2
+0
+dθ τ φ
+�
+A
+√
+θ
+ik , A
+√
+θ
+kj
+�
+,
+(2.8)
+where φ(θ) =
+√
+ℓ2 − θ, and Aij = ∂jui are the velocity-field gradients. Since the SGS
+stress tensor τ ℓ
+ij is symmetric, for the first form of the helicity flux we obtain in analogy
+to the energy flux
+ΠH,ℓ = −2τ ℓ
+ijS
+ℓ
+ω,ij,
+(2.9)
+where Sω is the symmetric component of the vorticity gradient tensor, with components
+Sω,ij = (∂jωi + ∂iωj)/2. Employing (2.8) this yields
+ΠH,ℓ = −2ℓ2S
+ℓ
+ω,ijA
+ℓ
+ikA
+ℓ
+jk − 2
+� ℓ2
+0
+dθ S
+ℓ
+ω,ijτ φ
+�
+A
+√
+θ
+ik , A
+√
+θ
+kj
+�
+.
+(2.10)
+The first term involves a product of gradient tensors filtered at the same scale, ℓ; hence
+we refer to it as being single-scale, and denote it ΠH,ℓ
+s
+. In mean, it coincides with the
+nonlinear LES model for the SGS-stresses (Eyink 2006). In contrast, the second term
+encodes the correlation between resolved-scale vorticity-field gradients and (summed)
+velocity-field gradients at each scale smaller than ℓ, so that we refer to it as multi-scale.
+Splitting the velocity gradient tensors into symmetric and anti-symmetric parts, that
+is, into the rate-of-strain tensor S = (A + At)/2 and vorticity tensor Ω = (A − At)/2,
+where At is the transpose of A, the helicity flux can be decomposed into six subfluxes
+ΠH,ℓ = Πℓ
+s,SS + Πℓ
+s,ΩΩ + Πℓ
+s,SΩ + Πℓ
+m,SS + Πℓ
+m,ΩΩ + Πℓ
+m,SΩ,
+(2.11)
+where the single-scale terms are
+ΠH,ℓ
+s,SS = −2ℓ2S
+ℓ
+ω,ijS
+ℓ
+ikS
+ℓ
+jk = −2ℓ2tr
+�
+(S
+ℓ
+ω)tS
+ℓ(S
+ℓ)t�
+,
+(2.12)
+ΠH,ℓ
+s,ΩΩ = −2ℓ2S
+ℓ
+ω,ijΩ
+ℓ
+ikΩ
+ℓ
+jk = −2ℓ2tr
+�
+(S
+ℓ
+ω)tΩ
+ℓ(Ω
+ℓ)t�
+,
+(2.13)
+ΠH,ℓ
+s,SΩ = −2ℓ2S
+ℓ
+ω,ij
+�
+S
+ℓ
+ikΩ
+ℓ
+jk − Ω
+ℓ
+ikS
+ℓ
+jk
+�
+= −4ℓ2tr
+�
+(S
+ℓ
+ω)tS
+ℓ(Ω
+ℓ)t�
+,
+(2.14)
+and tr {·} denotes the trace. Similarly, the multi-scale terms are
+ΠH,ℓ
+m,SS = −2
+� ℓ2
+0
+dθ S
+ℓ
+ω,ijτ φ
+�
+S
+√
+θ
+ik , S
+√
+θ
+kj
+�
+,
+(2.15)
+ΠH,ℓ
+m,ΩΩ =
+2
+� ℓ2
+0
+dθ S
+ℓ
+ω,ijτ φ
+�
+Ω
+√
+θ
+ik , Ω
+√
+θ
+kj
+�
+,
+(2.16)
+ΠH,ℓ
+m,SΩ = −2
+� ℓ2
+0
+dθ S
+ℓ
+ω,ij
+�
+τ φ
+�
+S
+√
+θ
+ik , Ω
+√
+θ
+jk
+�
++ τ φ
+�
+Ω
+√
+θ
+ik , S
+√
+θ
+jk
+��
+= −4
+� ℓ2
+0
+dθ S
+ℓ
+ω,ijτ φ
+�
+S
+√
+θ
+ik , Ω
+√
+θ
+jk
+�
+.
+(2.17)
+We recall that ⟨ΠH,ℓ
+s,ΩΩ⟩, the spatial average of the contribution to the helicity flux due
+
+Helicity fluxes in homogeneous turbulence
+5
+to coupling of resolved-scale vorticity strain with resolved-scale vorticity, vanishes
+⟨ΠH,ℓ
+s,ΩΩ⟩ = −ℓ2
+4
+��
+∂jωℓ
+i + ∂iωℓ
+j
+�
+ωℓ
+iωℓ
+j
+�
+= −ℓ2
+4
+�
+∂j(ωℓ
+iωℓ
+iωℓ
+j)
+�
+= 0 ,
+(2.18)
+due to periodic boundary conditions and the divergence-free nature of the vorticity field,
+as previously discussed by Eyink (2006) in the context of a multi-scale gradient expansion
+of the SGS stress tensor.
+The physics encoded in these transfer terms may be understood in terms of three
+effects: (i) “vortex flattening” – compression and stretching of a vortex tube into a
+vortex sheet by large-scale straining motion, with the principal axes of the vorticity
+deformation tensor Sω aligning with that of the strain-rate tensor at smaller scale,
+see (2.12) and (2.15); (ii) “vortex twisting” – a twisting of small-scale vortex tubes
+by large-scale differential vorticity into thinner tubes consisting of helical vortex lines,
+and subsequent small-scale alignment between the resulting vorticity vectors and the
+extensile stress generated thereby (Eyink 2006), see (2.14) and (2.17); and (iii) “vortex
+entangling” – twisting of entangled vortex lines, see (2.13) and (2.16). Interpreting
+helicity as the correlation between velocity and vorticity, a change in this correlation
+(or alignment) across scales occurs by vorticity deformation through straining motions
+or differential vorticity. This results in decorrelation at large scales and an increase in
+small-scale correlation.
+2.2. An exact Betchov-type relation for the helicity flux
+In homogeneous turbulence, the Betchov (1956) relation is an exact expression con-
+necting the contributions associated with vortex stretching and strain self-amplification
+to the mean energy flux across scales. Here we show that there is an analogous exact
+expression relating two (single scale) mean helicity subfluxes: 3⟨ΠH,ℓ
+s,SS⟩ = ⟨ΠH,ℓ
+s,SΩ⟩. These
+subfluxes are associated with vortex flattening, ⟨ΠH,ℓ
+s,SS⟩, and vortex twisting, ⟨ΠH,ℓ
+s,SΩ⟩.
+Written in terms of the definitions given in (2.12) and (2.14), this expression reads
+3
+�
+tr
+�
+S
+ℓ
+ωS
+ℓS
+ℓ��
+= 2
+�
+tr
+�
+S
+ℓ
+ωΩ
+ℓS
+ℓ��
+.
+(2.19)
+The main steps in a proof of this are now summarised. Following an argument analogous
+to that used in proving the Betchov (1956) relation for the energy flux, and using tensor
+symmetry properties and (2.18), one obtains (Eyink 2006)
+�
+tr
+�
+S
+ℓ
+ωS
+ℓS
+ℓ��
+= −
+�
+tr
+�
+Ω
+ℓ
+ω(S
+ℓΩ
+ℓ + Ω
+ℓS
+ℓ)
+��
+= −2
+�
+tr
+�
+Ω
+ℓ
+ωΩ
+ℓS
+ℓ��
+,
+(2.20)
+where Ωω is the antisymmetric part of the vorticity gradient tensor. This yields
+1
+2
+�
+tr
+�
+∇ωℓ �
+∇uℓ�t ��
+∇uℓ + ∇uℓ�t���
+=
+�
+tr
+�3
+2 S
+ℓ
+ωS
+ℓS
+ℓ − S
+ℓ
+ωΩ
+ℓS
+ℓ��
+.
+(2.21)
+Thus, showing that the lefthand side (LHS) of this expression vanishes will prove the
+Betchov relation for the helicity flux, (2.19). To do so, we express the LHS of eq. (2.21)
+using the chain rule and in index notation
+�
+∂jωℓ
+i∂juℓ
+kS
+ℓ
+ki
+�
+=
+�
+∂j
+�
+ωℓ
+i∂juℓ
+kS
+ℓ
+ki
+��
+−
+�
+ωℓ
+i∂j∂juℓ
+kS
+ℓ
+ki
+�
+−
+�
+ωℓ
+iS
+ℓ
+kj∂jS
+ℓ
+ki
+�
+−
+�
+ωℓ
+iΩ
+ℓ
+kj∂jS
+ℓ
+ki
+�
+.
+(2.22)
+The first term on the RHS of this expression vanishes making use of periodic boundary
+conditions. Using incompressibility and integration by parts it can be shown that the
+
+6
+D. Capocci, P. Johnson, S. Oughton, L. Biferale, M. Linkmann
+N
+E
+ν
+ε
+εH
+L
+τ
+Reλ η/10−3 kmax kmaxη ∆t/τ
+#
+1024 7.26 0.001 3.33 5.02 1.12 0.50 327
+4.20
+340
+1.43
+0.60
+39
+Table 1. Simulation parameters and key observables, where N is the number of collocation
+points in each coordinate, E the (mean) total kinetic energy, ν the kinematic viscosity, ε the mean
+energy dissipation rate, εH the mean helicity dissipation rate, L = (3π/4E)
+� kmax
+0
+dk E(k)/k
+the integral scale, τ = L/
+�
+2E/3 the large-eddy turnover time, Reλ the Taylor-scale Reynolds
+number, η = (ν3/ε)1/4 the Kolmogorov microscale, kmax the largest wave number after
+de-aliasing, ∆t the sampling interval which is calculated from the length of the averaging
+interval divided by the number of equispaced snapshots, and # the number of snapshots. The
+data corresponds to run 22 of
+Sahoo et al. (2017). It is available for download using the
+SMART-Turb portal http://smart-turb.roma2.infn.it.
+last term also vanishes. The two remaining terms cancel out, which is shown by similar
+arguments and using the properties of the Levi-Civita tensor. This completes the proof.
+The mean single-scale terms also arise as the first-order contribution in a multi-scale
+expansion of the SGS stress tensor (Eyink 2006), where (2.20) is used to deduce that
+the full vorticity gradient, not only either its symmetric or antisymmetric component, is
+involved in the helicity flux across scales. In consequence, (2.19) and (2.20) assert that
+the mean transfers involving the symmetric or the antisymmetric parts of the vorticity
+gradient can be related to one another, and thus the single-scale contribution to the mean
+helicity flux can be written as
+�
+ΠH,ℓ
+s
+�
+= −8ℓ2 �
+tr
+�
+S
+ℓ
+ωS
+ℓS
+ℓ��
+= −16
+3 ℓ2 �
+tr
+�
+S
+ℓ
+ωΩ
+ℓS
+ℓ��
+.
+(2.23)
+3. Numerical details and data
+Data has been generated by direct numerical simulation of the incompressible 3D
+Navier–Stokes equations (2.1) and (2.2) on a triply periodic domain of size Lbox =
+2π in each direction, where the forcing f is a random Gaussian process with zero
+mean, fully helical f = f +, and active in the wavenumber band k ∈ [0.5, 2.4]. The
+spatial discretisation is implemented through the standard, fully dealiased pseudospectral
+method with 1024 collocation points in each direction. Further details and mean values
+of key observables are summarised in table 1.
+Figure 1(a) presents the time series of the total kinetic energy per unit volume, E(t).
+Time-averaged kinetic energy spectra of positively and negatively helical fluctuations,
+E±(k) = ⟨ 1
+2
+�
+k⩽|k|<k+1 |ˆu±(k)|2⟩ and the total energy spectrum E(k) = E+(k)+E−(k),
+are shown in in Kolmogorov-compensated form in Fig. 1(b). As can be seen by comparison
+of E+(k) and E−(k), the large-scale velocity-field fluctuations are dominantly positively
+helical, which is a consequence of the forcing. Decreasing in scale, we observe that
+negatively helical fluctuations increase in amplitude, and approximate equipartition
+between E+(k) and E−(k) is reached for k ⩾ 20. That is, a helically forced turbulent
+flow, where mirror-symmetry is broken at and close to the forcing scale, restores mirror-
+symmetry at smaller scales through nonlinear interactions (Chen et al. 2003a; Deusebio
+& Lindborg 2014; Kessar et al. 2015).
+
+Helicity fluxes in homogeneous turbulence
+7
+0
+5
+10
+15
+20
+25
+t/τ
+−0.4
+−0.2
+0.0
+0.2
+0.4
+E(t)/E − 1
+(a)
+10−2
+10−1
+100
+kη
+10−2
+10−1
+100
+k5/3E(k)/ε2/3
+E(k)
+E+(k)
+E−(k)
+(b)
+Figure 1. (a) Time evolution of the total energy normalised by its mean value, E. Time
+is given in units of large-eddy turnover time τ. The red dots correspond to the sampled
+velocity-field configurations. (b) Time-averaged energy spectra in Kolmogorov-compensated
+form. The grey-shaded area indicates the forcing range. The dashed line indicates a Kolmogorov
+constant CK ≈ 1.6.
+10−2
+10−1
+100
+k η
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+⟨ΠH,ℓ⟩/εH
+⟨ΠH,ℓ⟩
+⟨ΠH,ℓ
+s,SΩ⟩
+⟨ΠH,ℓ
+s,SS⟩
+⟨ΠH,ℓ
+s,ΩΩ⟩
+⟨ΠH,ℓ
+m,SS⟩
+⟨ΠH,ℓ
+m,ΩΩ⟩
+⟨ΠH,ℓ
+m,SΩ⟩
+3·⟨ΠH,ℓ
+s,SS⟩
+Figure 2. Decomposed helicity fluxes normalised with the mean helicity dissipation rate εH.
+Filled markers corresponds to single-scale contributions while empty symbols are related to
+multi-scale contributions. The error bars indicate one standard error. The subflux ⟨ΠH,ℓ
+s,SΩ⟩ has
+been superposed with 3⟨ΠH,ℓ
+s,SS⟩ in order to highlight the Betchov-type relation (2.19).
+4. Numerical results for mean subfluxes and fluctuations
+Figure 2 shows the total helicity flux and all subfluxes, normalised by the total helicity
+dissipation rate εH. As can be seen in the figure, the term ⟨ΠH,ℓ
+s,ΩΩ⟩ is identically zero,
+which must be the case according to (2.18). Moreover, the helicity Betchov relation
+(2.19) derived here is satisfied as it must be – the terms ⟨ΠH,ℓ
+s,SΩ⟩ and 3 ⟨ΠH,ℓ
+s,SS⟩ are
+visually indistinguishable, with a relative error between them of order 10−6 (not shown).
+A few further observations can be made from the data. The non-vanishing multi-scale
+terms, ⟨ΠH
+m,SΩ⟩, ⟨ΠH
+m,SS⟩ and ⟨ΠH
+m,ΩΩ⟩ are comparable in magnitude across all scales.
+They are approximately scale-independent in the interval 10−2 ⩽ kη ⩽ 10−1, with each
+accounting for about 15−20% of the total helicity flux in this range of scales. Even though
+clear plateaux are not present for the two non-vanishing single-scale terms, ⟨ΠH
+s,SΩ⟩ and
+⟨ΠH
+s,SS⟩, one could tentatively extrapolate that at higher Re, about 30% of the mean
+flux originates from scale-local vortex twisting and 10% from vortex flattening. That is,
+the multi-scale contributions amount to 50%-60% and the scale-local contributions to
+40-50% of the total helicity flux across scales, at least for this particular simulation.
+
+8
+D. Capocci, P. Johnson, S. Oughton, L. Biferale, M. Linkmann
+−100 −75 −50 −25
+0
+25
+50
+75
+100
+ΠH,ℓ
+X /σX
+10−1
+10−3
+10−5
+10−7
+10−9
+σX · P(ΠH,ℓ
+X )
+kη = 8.4×10−2
+(a)
+ΠH,ℓ
+s,ΩΩ
+ΠH,ℓ
+s
+ΠH,ℓ
+s,SS
+ΠH,ℓ
+s,SΩ
+−100 −75 −50 −25
+0
+25
+50
+75
+100
+ΠH,ℓ
+X /σX
+10−1
+10−3
+10−5
+10−7
+10−9
+σX · P(ΠH,ℓ
+X )
+kη = 8.4×10−2
+(b)
+ΠH,ℓ
+m,SS
+ΠH,ℓ
+m
+ΠH,ℓ
+m,SΩ
+ΠH,ℓ
+m,ΩΩ
+Figure 3. Standardised PDFs of helicity subfluxes ΠH,ℓ
+X , where X refers to the subflux identifier,
+for (a) single-scale and (b) multi-scale contributions; σX denotes the standard deviation of each
+respective term.
+Having discussed the mean subfluxes, we now consider the fluctuations of each subflux
+term, in order to quantify the level of fluctuations in each term and the presence and
+magnitude of helicity backscatter. Figure 3 presents standardised probability density
+functions (PDFs) of all helicity subfluxes at k = π/ℓ = 20, which is in the inertial range.
+These PDFs are fairly symmetric, much more so than for the kinetic energy fluxes, have
+wide tails, and are strongly non-Gaussian. Single- and multi-scale terms all have strong
+fluctuations of about 75 standard deviations. Interestingly, the subflux term ΠH,ℓ
+s,ΩΩ, which
+necessarily vanishes in mean (see (2.18)), has the strongest fluctuations (i.e., is the most
+intermittent). PDFs for all the other subfluxes are comparable. The symmetry is more
+pronounced in the single-scale rather than the multi-scale terms, as can be seen by
+comparison of the left and right panels of fig. 3. As all averaged fluxes (except ⟨ΠH,ℓ
+s,ΩΩ⟩
+which is zero) transfer positive helicity from large to small scales, symmetry in the PDFs
+indicates strong backscatter of positive helicity, or forward scatter of negative helicity.
+The PDFs become even broader with decreasing filter scale (not shown). A comparison
+between the PDFs of ΠH,ℓ and the alternate description based on SGS stresses related
+to vortex stretching, ˜ΠH,ℓ, has been carried out by Yan et al. (2020), indicating more
+intense backscatter in the latter compared to the former. Adding or removing a total
+gradient can strongly reduce the negative tail of the SGS energy transfer (Vela-Mart´ın
+2022), and the same may apply to the helicity flux.
+5. Conclusions
+We have derived an exact decomposition of the helicity flux across scales in terms
+of interactions between vorticity gradients and velocity gradients, and in terms of their
+scale locality. Decomposing all gradient tensors into symmetric and anti-symmetric parts
+allows for a discussion and quantification of different physical mechanisms that constitute
+the helicity cascade. Simulation results indicate that all subfluxes transfer helicity from
+large to small scales, albeit with strong backscatter. In the inertial range, about 50% of
+the total mean helicity flux is due to the action of two scale-local processes: (i) vortex
+flattening and (ii) vortex twisting. We have also shown that these two effects are related
+in mean through a newly derived exact (Betchov-type) relation, which implies that the
+contribution of the former is exactly three times larger than that of the latter. Multi-scale
+effects account for the remaining 50%, with approximate equipartition between multi-
+scale versions of the two aforementioned effects and multi-scale vortex entangling. Thus,
+it seems likely that, in LES contexts, accurate modeling of the helicity cascade should not
+neglect the multi-scale contributions. Although our numerical quantification of the fluxes
+is obtained using data from a single simulation with an inertial range of limited length,
+
+Helicity fluxes in homogeneous turbulence
+9
+we conjecture that the results obtained are robust in the sense that we expect them to
+hold for flows with larger Reynolds numbers. Similar flux decompositions can be derived
+for magnetohydrodynamics. We will report results of these investigations elsewhere in
+due course.
+Computational resources were provided through Scottish Academic Access on
+Cirrus (www.cirrus.ac.uk), and the UK Turbulence Consortium on ARCHER2
+(www.archer2.ac.uk). This work received funding from the European Research Council
+(ERC) under the European Union’s Horizon 2020 research and innovation programme
+(grant agreement No 882340) and from the Priority Programme SPP 1881 “Turbulent
+Superstructures” of the Deutsche Forschungsgemeinschaft (DFG, Li3694/1).
+Competing interests: the authors declare none.
+REFERENCES
+Alexakis, A. 2017 Helically Decomposed Turbulence. J. Fluid Mech. 812, 752–770.
+Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Physics Reports
+767-769, 1–101.
+Baerenzung, J., Politano, H., Ponty, Y. & Pouquet, A. 2008 Spectral modeling of
+turbulent flows and the role of helicity. Phys. Rev. E. 77, 046303.
+Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence.
+J. Fluid Mech. 1, 497.
+Biferale, L., Musacchio, S. & Toschi, F. 2012 Inverse energy cascade in three-dimensional
+isotropic turbulence. Phys. Rev. Lett. 108, 164501.
+Biferale, L., Musacchio, S. & Toschi, F. 2013 Split Energy-Helicity cascades in three
+dimensional Homogeneous and Isotropic Turbulence. J. Fluid Mech. 730, 309–327.
+Brandenburg, A. 2001 The inverse cascade and nonlinear alpha-effect in simulations of
+isotropic helical magnetohydrodynamic turbulence. Astrophys. J. 550, 824–840.
+Brandenburg, A. & Subramanian, K. 2005 Astrophysical magnetic fields and nonlinear
+dynamo theory. Phys. Reports 417, 1–209.
+Brissaud, A., Frisch, U., L´eorat, J., Lesieur, M. & Mazure, A. 1973 Helicity cascades
+in fully developed isotropic turbulence. Phys. Fluids 16, 1366–1367.
+Carbone, M. & Wilczek, M. 2022 Only two Betchov homogeneity constraints exist for
+isotropic turbulence. J. Fluid Mech. 948, R2.
+Chen, Q., Chen, S. & Eyink, G. L. 2003a The joint cascade of energy and helicity in three-
+dimensional turbulence. Phys. Fluids 15, 361–374.
+Chen, Q., Chen, S., Eyink, G. L. & Holm, D. D. 2003b Intermittency in the joint cascade
+of energy and helicity. Phys. Rev. Lett. 90, 214503.
+Constantin, P. & Majda, A. 1988 The Beltrami spectrum for incompressible flows. Commun.
+Math. Phys. 115, 435–456.
+Deusebio, E. & Lindborg, E. 2014 Helicity in the Ekman boundary layer. J. Fluid Mech.
+755, 654–671.
+Eyink, G. L. 2005 Locality of turbulent cascades. Physica D 207, 91–116.
+Eyink, G. L. 2006 Multi-scale gradient expansion of the turbulent stress tensor. J. Fluid Mech.
+549, 159–190.
+Germano, M. 1992 Turbulence — the filtering approach. J. Fluid Mech. 238, 325–336.
+Gledzer, E. B. & Chkhetiani, O. G. 2015 Inverse energy cascade in developed turbulence
+at the breaking of the symmetry of helical modes. JETP Letters 102, 465–472.
+Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859.
+Inagaki, K., Yokoi, N. & Hamba, F. 2017 Mechanism of mean flow generation in rotating
+turbulence through inhomogeneous helicity. Phys. Rev. Fluids 2, 114605.
+Johnson, P. L. 2020 Energy transfer from large to small scales in turbulence by multiscale
+nonlinear strain and vorticity interactions. Phys. Rev. Lett. 124, 104501.
+Johnson, P. L. 2021 On the role of vorticity stretching and strain self-amplification in the
+turbulence energy cascade. J. Fluid Mech. 922, A3.
+
+10
+D. Capocci, P. Johnson, S. Oughton, L. Biferale, M. Linkmann
+Kessar, M., Plunian, F., Stepanov, R. & Balarac, G. 2015 Non-Kolmogorov cascade of
+helicity-driven turbulence. Phys. Rev. E 92, 031004(R).
+Kraichnan, R. 1973 Helical turbulence and absolute equilibrium. J. Fluid Mech. 59, 745–752.
+Li, Y., Meneveau, C., Chen, S. & Eyink, G. L. 2006 Subgrid-scale modeling of helicity and
+energy dissipation in helical turbulence. Phys. Rev. E. 74, 026310.
+Lilly, D. K. 1986 The structure, energetics, and propagation of rotating convective storms.
+Part II: Helicity and storm stabilization. J. Atmos. Sci. 43, 126.
+Linkmann, M. 2018 Effects of helicity on dissipation in homogeneous box turbulence. J. Fluid
+Mech. 856, 79–102.
+Linkmann, M. F., Berera, A., McKay, M. E. & J¨ager, J. 2016 Helical mode interactions
+and spectral energy transfer in magnetohydrodynamic turbulence. J. Fluid Mech. 791,
+61–96.
+Linkmann, M. F., Sahoo, G., McKay, M. E., Berera, A. & Biferale, L. 2017 Effects
+of Magnetic and Kinetic Helicities on the Growth of Magnetic Fields in Laminar and
+Turbulent Flows by Helical Fourier Decomposition. Astrophys. J. 836, 26.
+Milanese, L. M., Loureiro, N. F. & Boldyrev, S. 2021 Dynamic Phase Alignment in
+Navier-Stokes Turbulence. Phys. Rev. Lett. 127, 274501.
+Mininni, P. D. & Pouquet, A. G. 2010a Rotating helical turbulence. I. Global evolution and
+spectral behavior. Phys. Fluids 22, 035105.
+Mininni, P. D. & Pouquet, A. G. 2010b Rotating helical turbulence. II. Intermittency, scale
+invariance, and structures. Phys. Fluids 22, 035106.
+Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35,
+117–129.
+Moffatt, H. K. 2014 Helicity and singular structures in fluid dynamics. Proc. Natl. Acad. Sci.
+111 (10), 3663–3670.
+Sahoo, G., Alexakis, A. & Biferale, L. 2017 Discontinuous transition from direct to inverse
+cascade in three-dimensional turbulence. Phys. Rev. Lett. 118, 164501.
+Sahoo, G., Bonaccorso, F. & Biferale, L. 2015 Role of helicity for large- and small-scale
+turbulent fluctuations. Phys. Rev. E 92, 051002.
+Scheeler, M. W., van Rees, W. M., Kedia, H., Kleckner, D. & Irvine, W. T. M. 2017
+Complete measurement of helicity and its dynamics in vortix tubes. Science 357, 487–491.
+Steenbeck, M., Krause, F. & R¨adler, K.-H. 1966 Berechnung der mittleren Lorentz-
+Feldst¨arke v x B f¨ur ein elektrisch leitendes Medium in turbulenter, durch Coriolis-Kr¨afte
+beeinflußter Bewegung. Z. Naturforsch. A 21, 369.
+Stepanov, R., Golbraikh, E., Frick, P. & Shestakov, A. 2015 Hindered energy cascade
+in highly helical isotropic turbulence. Phys. Rev. Lett. 115, 234501.
+Tobias, S. M., Cattaneo, F. & Boldyrev, S. 2013 MHD Dynamos and Turbulence. In Ten
+Chapters in Turbulence. Cambridge University Press.
+Vela-Mart´ın, A. 2022 Subgrid-scale models of isotropic turbulence need not produce energy
+backscatter. J. Fluid Mech. 937, A14.
+Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A
+4, 350–363.
+Yan, Z., Li, X., Yu, C., Wang, J. & Chen, S. 2020 Dual channels of helicity cascade in
+turbulent flows. J. Fluid Mech. 894, R2.
+Yokoi, N. & Yoshizawa, A. 1993 Statistical analysis of the effects of helicity in inhomogeneous
+turbulence. Phys. Fluids A 5, 464.
+Yu, C, Hu, R., Yan, Z. & Li, X. 2022 Helicity distributions and transfer in turbulent channel
+flows with streamwise rotation. J. Fluid Mech. 940, A18.
+

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+Experimental verification of the temperature coefficient of
+resistivity
+Robert D. Polak1, Michael R. Harris1, Kiet A. Nguyen1, and Anthony Kearns1
+1Department of Physics, Loyola University Chicago, Chicago, IL 60660, USA
+We have created an experimental procedure for determining the temperature coefficient of resis-
+tivity, αR, for introductory physics laboratories. As in the procedure from Henry [1], this method
+examines the relationship between temperature and resistivity to establish αR within 10% of the
+accepted value.
+Electrical resistivity, ρ, varies with temperature according to:
+ρ = ρo(1 + αR(T − To))
+(1)
+where ρo is the resistivity for a given temperature To, T is the temperature of the material, and αR
+is the temperature coefficient of resistivity. For a wire of length, L, and cross-sectional area, A, the
+resistance of a wire, R, is defined accordingly as
+R = ρ L
+A .
+(2)
+While resistance will increase as a product of both increased length and resistivity, the increase in
+length provides a negligible increase in resistance. This is evident from observing that the thermal
+coefficient of resistivity is approximately two orders of magnitude larger than the coefficient of
+thermal expansion. As such, the change in resistivity is primarily responsible for the increase in
+resistance. Hence, R will vary similarly with
+R = Ro(1 + αR(T − To))
+(3)
+where Ro is the resistance of the wire at a temperature To.
+By applying a current through the wire, its temperature will also vary as a result of Joule heating
+and the resistance of the wire can be measured based on a given current, I, and difference in voltage,
+∆V , by
+R = ∆V
+I
+.
+(4)
+Hence, by measuring the resistance as a function of temperature, we can determine αR by plotting
+R vs. T − To and performing a linear fit using Eq. (3).
+To perform the experiment, we created a closed circuit (see Fig. 1) in which a carbon steel
+wire [2] is suspended under tension above a surface, as in most stringed instruments. Two digital
+1
+arXiv:2301.04639v1  [physics.ed-ph]  21 Nov 2022
+
+multimeters were used to record the voltage across and current through a 0.016 inch (0.406 mm)
+diameter, 40-cm long wire. Temperature measurements were taken using liquid crystal thermometers
+[3] placed in thermal contact with the wire by fastening them to the wire with an adhesive backing.
+Two thermometers were used, one ranging from 14−31oC and the other from 32−49oC, to provide
+an overall temperature range of 14−49oC. For the most accurate temperature readings, we found it
+is essential to avoid all contact with the thermometers during the experiment. To collect the data,
+we applied different currents to the wire, ranging from 0.2A to 1.0A. We used a BK Precision 1787B
+power supply that allows for digital control of the current. We found that having initial steps of
+0.2A and later reduced to 0.1A created consistent temperature changes in the wire of 2 − 3oC. The
+experiment proved much more difficult to complete using an analog power supply because of the
+difficulty in creating the precise changes in current needed to have a well formed data set. After
+allowing around 30 seconds for the system to reach thermal equilibrium, the recorded temperature of
+each trial was given as the uppermost visible temperature reading of the liquid crystal thermometer,
+as seen in Fig. 2. We recorded the temperature of, current through and voltage across the wire, and
+calculated its resistance using Eq. (4).
+By graphing the resistance of the wire as a function of T −To, where To is the temperature of the
+wire with the lowest current applied, we can then apply a linear fit to the data with the y-intercept
+yielding Ro and slope giving RoαR, according to Eq. (3). Figure 4 shows example experimental
+results with the fit giving Ro = 0.744Ω and αR = 0.0039K−1. This is within 5% of the accepted
+value of αR = 0.0041K−1 [4]. Repeated experiments found these results to be reproducible with αR
+consistently measured within 10% of the accepted value.
+To get the best results, we found that the resistance of the wire should be at least 0.3Ω to allow
+for accurate resistance measurements. Furthermore, the wires need to be thick enough to support
+the thermometers.
+As such, we achieved the best results when using steel as opposed to other
+materials with lower resistivity and tensile strength, such as copper and aluminum.
+This experiment can be completed in less than 2 hours and uses equipment that is commonly
+present in a typical introductory physics lab, with the exception of relatively low cost supplies such
+as music wire and liquid crystal thermometers. It also reinforces key ideas from introductory physics
+such as conservation of energy, where electrical energy becomes thermal energy, Ohm’s Law, and
+the temperature dependence of resistance.
+References
+1. D. Henry, “Resistance of a wire as a function of temperature”, The Physics Teacher 33, 96-97
+(1995) https://doi.org/10.1119/1.2344149
+2. Precision Brand Music Wire (0.016 inch diameter); UPC. No. 21016
+3. TelaTemp reversible LCT strip model 416-2 (14 − 31oC) and 416-3 (32 − 49oC)
+4. S. Yafei , N. Dongjie, and S. Jing, 4th IEEE Conference on Industrial Electronics and Applications,
+368-372 (2009).
+2
+
+Figure 1: The experimental setup (bottom) with the wire suspended above the box, and a circuit
+diagram (top). The circuit consists of two multimeters set up to measure current and voltage and a
+digital power supply. The liquid crystal thermometers are affixed to the top of the wire.
+Figure 2: An example of a temperature reading corresponding to 37◦C (98◦F), according to the
+procedure of reading the uppermost visible indication of the liquid crystal thermometer (the yellow
+bar spanning 37◦C).
+3
+
+L.C.Thermometers
+Power
+V
+Supply
+A
+20cm
+15
+40
+25
+30
+10
+35
+TonsKala
+NEVA10440
+102
+39
+100
+38
+98
+37
+5
+35
+34
+33Figure 3: Plot of resistance against the associated change in temperature for currents from 0.2 A to
+1.0 A. A linear fit of the data yields αR = 0.0039K−1.
+4
+
+0.80 -
+0.79-
+0.78
+R
+0.77
+0.76 -
+0.75
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+T- To (K)

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+page_content='Experimental verification of the temperature coefficient of  resistivity  Robert D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Polak1, Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Harris1, Kiet A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Nguyen1, and Anthony Kearns1  1Department of Physics, Loyola University Chicago, Chicago, IL 60660, USA  We have created an experimental procedure for determining the temperature coefficient of resis-  tivity, αR, for introductory physics laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' As in the procedure from Henry [1], this method  examines the relationship between temperature and resistivity to establish αR within 10% of the  accepted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  Electrical resistivity, ρ, varies with temperature according to:  ρ = ρo(1 + αR(T − To))  (1)  where ρo is the resistivity for a given temperature To, T is the temperature of the material, and αR  is the temperature coefficient of resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' For a wire of length, L, and cross-sectional area, A, the  resistance of a wire, R, is defined accordingly as  R = ρ L  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  (2)  While resistance will increase as a product of both increased length and resistivity, the increase in  length provides a negligible increase in resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' This is evident from observing that the thermal  coefficient of resistivity is approximately two orders of magnitude larger than the coefficient of  thermal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' As such, the change in resistivity is primarily responsible for the increase in  resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Hence, R will vary similarly with  R = Ro(1 + αR(T − To))  (3)  where Ro is the resistance of the wire at a temperature To.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  By applying a current through the wire, its temperature will also vary as a result of Joule heating  and the resistance of the wire can be measured based on a given current, I, and difference in voltage,  ∆V , by  R = ∆V  I  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  (4)  Hence, by measuring the resistance as a function of temperature, we can determine αR by plotting  R vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' T − To and performing a linear fit using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  To perform the experiment, we created a closed circuit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' 1) in which a carbon steel  wire [2] is suspended under tension above a surface, as in most stringed instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Two digital  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='04639v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='ed-ph]  21 Nov 2022  multimeters were used to record the voltage across and current through a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='016 inch (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='406 mm)  diameter, 40-cm long wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Temperature measurements were taken using liquid crystal thermometers  [3] placed in thermal contact with the wire by fastening them to the wire with an adhesive backing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  Two thermometers were used, one ranging from 14−31oC and the other from 32−49oC, to provide  an overall temperature range of 14−49oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' For the most accurate temperature readings, we found it  is essential to avoid all contact with the thermometers during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' To collect the data,  we applied different currents to the wire, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='2A to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' We used a BK Precision 1787B  power supply that allows for digital control of the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' We found that having initial steps of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='2A and later reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='1A created consistent temperature changes in the wire of 2 − 3oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' The  experiment proved much more difficult to complete using an analog power supply because of the  difficulty in creating the precise changes in current needed to have a well formed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' After  allowing around 30 seconds for the system to reach thermal equilibrium, the recorded temperature of  each trial was given as the uppermost visible temperature reading of the liquid crystal thermometer,  as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' We recorded the temperature of, current through and voltage across the wire, and  calculated its resistance using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  By graphing the resistance of the wire as a function of T −To, where To is the temperature of the  wire with the lowest current applied, we can then apply a linear fit to the data with the y-intercept  yielding Ro and slope giving RoαR, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Figure 4 shows example experimental  results with the fit giving Ro = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='744Ω and αR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='0039K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' This is within 5% of the accepted  value of αR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='0041K−1 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Repeated experiments found these results to be reproducible with αR  consistently measured within 10% of the accepted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  To get the best results, we found that the resistance of the wire should be at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='3Ω to allow  for accurate resistance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Furthermore, the wires need to be thick enough to support  the thermometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  As such, we achieved the best results when using steel as opposed to other  materials with lower resistivity and tensile strength, such as copper and aluminum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  This experiment can be completed in less than 2 hours and uses equipment that is commonly  present in a typical introductory physics lab, with the exception of relatively low cost supplies such  as music wire and liquid crystal thermometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' It also reinforces key ideas from introductory physics  such as conservation of energy, where electrical energy becomes thermal energy, Ohm’s Law, and  the temperature dependence of resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Henry, “Resistance of a wire as a function of temperature”, The Physics Teacher 33, 96-97  (1995) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='1119/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='2344149  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Precision Brand Music Wire (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='016 inch diameter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' UPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' 21016  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' TelaTemp reversible LCT strip model 416-2 (14 − 31oC) and 416-3 (32 − 49oC)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Yafei , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Dongjie, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' Jing, 4th IEEE Conference on Industrial Electronics and Applications,  368-372 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  2  Figure 1: The experimental setup (bottom) with the wire suspended above the box, and a circuit  diagram (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' The circuit consists of two multimeters set up to measure current and voltage and a  digital power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' The liquid crystal thermometers are affixed to the top of the wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  Figure 2: An example of a temperature reading corresponding to 37◦C (98◦F), according to the  procedure of reading the uppermost visible indication of the liquid crystal thermometer (the yellow  bar spanning 37◦C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='  3  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='Thermometers  Power  V  Supply  A  20cm  15  40  25  30  10  35  TonsKala  NEVA10440  102  39  100  38  98  37  5  35  34  33Figure 3: Plot of resistance against the associated change in temperature for currents from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='2 A to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content='0 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
+page_content=' A linear fit of the data yields αR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfpArI/content/2301.04639v1.pdf'}
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+An Efficient Drifters Deployment Strategy to
+Evaluate Water Current Velocity Fields
+Murad Tukan, Eli Biton, Roee Diamant, Senior Member, IEEE
+Abstract
+Water current prediction is essential for understanding ecosystems, and to shed light on the role
+of the ocean in the global climate context. Solutions vary from physical modeling, and long-term
+observations, to short-term measurements. In this paper, we consider a common approach for water
+current prediction that uses Lagrangian floaters for water current prediction by interpolating the trajectory
+of the elements to reflect the velocity field. Here, an important aspect that has not been addressed before
+is where to initially deploy the drifting elements such that the acquired velocity field would efficiently
+represent the water current. To that end, we use a clustering approach that relies on a physical model of
+the velocity field. Our method segments the modeled map and determines the deployment locations as
+those that will lead the floaters to ’visit’ the center of the different segments. This way, we validate that
+the area covered by the floaters will capture the in-homogeneously in the velocity field. Exploration over
+a dataset of velocity field maps that span over a year demonstrates the applicability of our approach,
+and shows a considerable improvement over the common approach of uniformly randomly choosing the
+initial deployment sites. Finally, our implementation code can be found in [1].
+Index Terms
+Water currents, Positioning, Data processing, Coresets, Lagrangian floaters.
+M. Tukan(corresponding author [email protected]) and R. Diamant are with the Department of Marine Technologies,
+University of Haifa, 3498838 Haifa, Israel.
+E. Biton is with the Dept. of physical oceanography, Israel Oceanographic and Limnological Research, 3109701 Haifa, Israel.
+This work was sponsored in part by the MOST-BMBF German-Israeli Cooperation in Marine Sciences 2018-2020 (Grant #
+3-16573), by the MOST action for Agriculture, Environment, and Water for the year 2019 (Grant # 3-16728), and by a grant
+from the University of Haifa’s Data Science Research Center. This work has been submitted to the IEEE for possible publication.
+Copyright may be transferred without notice, after which this version may no longer be accessible.
+arXiv:2301.04216v1  [cs.LG]  10 Jan 2023
+
+I. INTRODUCTION
+Knowledge and information about the ocean’s flow is highly applicable to scientific purposes
+such as climate change, global heat distribution, air-sea interactions, eddy formations, convection,
+tides, biological productivity, to name a few. Prediction of the water current is also required for
+operational needs such as marine conservation, search and rescue, the fishing industry, navigation,
+the development of marine infrastructure, tracking oil-spill distribution, tsunami warnings, and
+renewable energy. The study of the complex oceanic flow field variability - characterized by
+a wide range of spatial-temporal scales of processes - demands the combination of physical
+models and observations. In particular, the latter can be used to calibrate or to validate the
+model’s parameters, as well as to serve as a database for statistical evaluation.
+Existing systems for directly measuring the water current (WC) mostly involve current profiling
+at a fixed deployment position, such as acoustic Doppler current profiles (e.g., ADCPs), or cover
+extensive areas, but only of the sea surface (e.g., HF radar and satellite elevation data). The data
+collected is used as an input for analytical and numerical models in a data assimilation fashion [2],
+[3], [4]. These models rely on local environmental information such as temperature, wind velocity,
+bathythermy and bathytermic, and need to be calibrated [5], [6] Another solution is to use
+Lagrangian floaters to in-situ evaluate the velocity field for data assimilation. Recently, [7] has
+developed a methodology to estimate the 3D flow field, based on the tracking of the trajectories of
+surface or submerged floaters. The method has been successfully implemented to restore/complete
+data gaps in a flow field. Other works have shown that the accuracy of the estimated flow field
+depends not only on the number of floats, but also on the locations of their initial deployments [8].
+As such, it is necessary to develop sophisticated methods for optimal dispersion of the floaters.
+Clearly, this optimal setup should be related to the flow’s characteristics.
+In this work, we develop a scheme how to plan ahead the initial positioning of Lagrangian
+floaters, such as to improve flow field reconstruction [7], [9].
+Our scheme is useful as a perpetration for in-situ calibration of a given water current flow field
+model. We analyze a given flow field map that is generated by a physical model to plan where
+to initially deploy a fixed number of floats. After executing our deployment planning scheme,
+the floats are released at the proposed locations and move freely with the water current for a
+
+fixed time frame, while their locations is tracked by e.g., acoustic positioning [10]. After this
+in-situ operation, the trajectories of the floats are used to generate a flow field using e.g., [7] or
+to validate or calibrate the given physical model. In the above described process, as illustrated
+good coverage of the flow field by the floats is required. This is illustrated in the two examples in
+Fig. 1. Here, we include a map of the WC velocity field’s magnitude and heading by the length
+and direction of the dark arrows. We observe homogeneous areas in the WC by small variations
+of the arrows and areas of a complex WC structure by a large diversity in the size and direction
+of the arrows. A group of 3 floats marked in orange lines are deployed in homogeneous sections
+of the flow field and thus do not capture the complexity of the water current, whereas a group
+of floats marked in green lines is well distributed to pass through all diverse sections in the
+flow field map. Clearly, the difference between the two groups is by their initial position. Our
+approach applies machine learning tools over the given flow field map. Particularly, inspired by
+computational geometry, we turn to coresets for the task of planning the initial location of the
+floaters. Informally speaking, given input data, a coreset is a weighted subset of the original data
+that approximates the original data in some provable sense with respect to a (usually infinite)
+set of queries or models and an objective loss/cost function [11]. We use coresets to find those
+“regions of interest” where the floaters should visit in order to most efficiently represent the WC
+structure. The method is tested on a finite resolution circulation model results that span over a
+year. Results show that using our algorithm, the floats better capture the complex structure of
+the WC, compared to the case of randomly deploying the floaters.
+To the best of our knowledge, our approach is the first to consider the problem of optimal de-
+ployment of floats for the task of WC prediction and is the first to use coresets for oceanographic
+applications. Our contribution is threefold:
+(i) A coreset-based solution for WC segmentation. A novel partitioning of the WC velocity
+field into segments of homogenous WC.
+(ii) A clustering scheme for the segmentation of the WC’s flow field. Primarily, reducing the
+problem of WC clustering into an instance of sets clustering.
+(iii) A sub-optimal solution to determine the deployment location of floaters for WC’s estimation.
+A graph theory-based approach to plan the deployment location of floaters such that the
+
+Fig. 1: The setting of a real-time sea experiment. This map was generated using SHYFEM [12]
+(System of Hydrodynamic Finite Element Modules) by setting different variables, e.g.,
+bathymetry, wind velocity, wind direction, etc. The direction of any arrow describes the direction
+of the WC at that discrete position p, while the length of the arrow represents the speed of the
+WC at p.
+floaters explore the different clusters of the WC.
+The paper is organized as follows. Related work is discussed in Section II. System setup
+and preliminaries are given in Section III. In Section IV, we discuss the methodology of our
+proposed approach. Section V presents the numerical and experimental results, and conclusions
+are drawn in Section VI.
+
+30
+25
+20
+15
+10
+5
+0
+0
+5
+10
+15
+20
+25
+30II. RELATED WORK
+Constructing WC flow fields based on Lagrangian particle trajectories has been gaining increas-
+ing attention over recent years [13], [14]. The main difference between the available approaches
+lies on the formalization of the relationships between the floaters’ trajectories. In [15], a model
+for the flow field is used to find such a connection. Another solution is offered in [7], where the
+relations between drifter trajectories are calculated by statistical models. In [16], a cooperative
+solution is adopted to recover the flow field by formulating the integration error. Specifically, the
+motion-integration errors of multiple autonomous underwater vehicles (AUVs) in a 2D flow are
+obtained. The relation between the flow model and motion-integration errors is then formulated
+as a system of nonlinear equations followed by an iterative algorithm that is designed to estimate
+the flow field. While the above techniques for data assimilation are able to merge measurements
+with a model to estimate the WC, as shown in [17], the results are sensitive to the initial
+deployment of the drifters. More specifically, an overly-close deployment would not capture the
+spatial dependency of the velocity field, while a too-far deployment, even below the Rossby
+radius of deformation, may break the assumed correlation between the sensors’ drifting velocity.
+A key challenge in determining the floaters’ deployment locations is to spread the sensors
+across diverse sections of the explored area. One option is to allow maneuvering such that floaters
+can escape areas of homogeneous velocity field [18]. Another option is to direct the initial floater
+positions along the out-flowing branch of Lagrangian boundaries for better relative dispersion
+of floaters [19]. In contrast, in [20] a sequential protocol was employed, where floaters are
+deployed one after the other, and their deployment locations are based on the trajectory obtained
+by the previously deployed floaters. First, the flow field is estimated using Gaussian processes
+(GP) [21], followed by trajectory estimation using OpenDrift [22]. The estimated trajectories are
+then ranked to find the next deployment location with the aim of obtaining a longer, unexplored,
+trajectory. The process then repeats until all floaters are deployed. While this method holds
+potential for exploring non-homogeneous patches in the velocity field, its optimality can only
+be reached when a large number of floats are in use. Further, the method makes perhaps a too
+hard assumption that the WC is stable throughout a long enough observation window to deploy
+and recover the floats one by one.
+
+While the models above are comparative to our work, they either assumed that (i) the moving
+agents have the ability to maneuver their own path, (ii) a large number of floaters is available
+to ensure good quality, (iii) or the velocity field is stable throughout a long enough observation
+window. These assumptions may be too hard in practical cases where the WC is time-varying,
+and when the number of floaters is limited. For these cases, we present an alternative solution.
+III. SYSTEM MODEL
+A. Setup details
+Consider a set of K submerged floaters X = {1, · · · , K}, each of which drifts with the
+WC for a time frame of T seconds. During their operations, the floaters’ locations are known
+- either through a self-navigation process or using acoustic localization [23]. The analysis is
+performed over a given time instance, 0 < t < T, that can be configured according to the
+expected time it takes a float to cover the given area for exploration. In this time frame, each
+of the floaters in X measures the velocity of the WC. This can be done directly, using sensors
+like Doppler velocity loggers; indirectly, using the time-varying position of the floaters; or by
+simple periodic surfacing to obtain GPS fixes as performed for the Argo floaters [24]. Assuming
+for simplicity, that at time instance t only one of the floaters measures the WC, we construct
+vector p(t) = [px(t), py(t), pz(t), t] for the x and y UTM coordinates of the floater, its depth
+pz(t) in meters, and the observation’s time instance, t, respectively. Similarly, we obtain vector
+v(t) = [vx(t), vy(t), vz(t)] representing the WC’s speed at the x, y, and z directions, respectively.
+We consider two scenarios: 1) the floaters are recovered after T seconds and the prediction
+of the WC’s velocity field is performed offline, and 2) the prediction is performed online based
+on past WC velocity observations, in which case the operation must involve communicating
+between the floaters. The first scenario mostly applies to the validation of a WC model, while
+the second can assist in the path planning of a submerged vessel. In this work, we are interested
+in determining the initial deployment position, p(0), of the floaters such that the WC is best
+predicted. To this end, we rely on our previously developed technique [7] as a utility metric to
+evaluate the WC’s velocity field from the floaters’ trajectories.
+
+B. Assumptions
+We make the following assumptions. First, the floaters are assumed to be Lagrangian, such
+that their motion is completely attributed to the WC. We also assume the existence of a WC
+model that provides velocity predictions in a two-dimensional plane for the explored area. The
+spatial resolution of the given model is fixed, and the accuracy of our approach is directly related
+to the model’s accuracy.
+The interesting case that we are aiming for is a non-homogeneous WC with patches of
+homogeneity, such that a number of floaters are needed in order to well explore the WC’s
+velocity field. These patches are assumed to change slowly in space, such that their borders
+are smooth and form convex sets. An example of such a velocity field is presented in Fig. 3a
+with arrows representing the magnitude and direction of the WC in a single cell in space. To
+simulate the floaters’ motion within the modeled WC, the trajectories of the floaters can follow
+these arrows. The example in Fig. 4 shows such motion for a set of two floaters. We observe
+differences in the trajectory of the simulated floaters, which reflects the non-homogeneity of the
+WC.
+C. Preliminaries
+a) Notations: For integers n and d ≥ 2, we denote by [n] the set {1, · · · , n}, by Rn×d the
+union over every n × d real matrix, and by Id ∈ Rd×d the identity matrix. A matrix A ∈ Rd×d
+is said to be (i) an orthogonal if and only if ATA = AAT = Id, or (ii) a positive definite matrix
+if and only if for every column vector x ̸= 0d, xTAx > 0. For every set A ⊆ Rd, we denote by
+|A| the number of elements of A. Finally, throughout the paper, vectors are addressed as column
+vectors.
+1) Volume approximation: In what follows, we define what is known as the Löwner ellipsoid,
+a tool that will aid us in obtaining an ε-coreset in the context of volume approximation.
+Definition 1 (Theorem III, [25]). Let L ⊆ Rd be a set of points. Let c ∈ Rd be a vector, and let
+G ∈ Rd×d be a positive definite matrix. We say that ellipsoid E =
+�
+x ∈ Rd���(x − c)T G (x − c) ≤ 1
+�
+,
+is an MVEE (short for the Minimum Volume Enclosing Ellipsoid) of L if
+1
+d (E − c) + c ⊆ Conv (L) ⊆ E,
+(1)
+
+where E − c denotes the set {x − c|x ∈ E}, 1
+dE denotes the set
+� 1
+dx
+��x ∈ E
+�
+, and Conv (L)
+denotes the convex hull of L.
+Definition 2 (Similar to that of [26]). Let ε > 0 be an approximation factor, and let X ⊆ Rd
+be a set of points. The set S ⊆ X is defined to be an ε-coreset for the MVEE of X, if
+Vol (MVEE (X)) ≤ (1 + ε) Vol (MVEE (S)) ,
+(2)
+where Vol (A) denotes the volume of A and MVEE (A) denotes the MVEE of A, for any A ⊆ Rd.
+2) Clustering of sets: To determine the initial location of the floaters, we first need to perform
+clustering to group together sets of similar WC. Thus, the following definitions will be used for
+the task of clustering.
+Definition 3 (Variant of Definition 2.3,[27]). Let m, n, d be a triplet of positive integers. An
+m-set P is a set of m distinct points in Rd. An (m, n)-set is a set P =
+�
+P
+��P ⊆ Rd, |P| = m
+�
+such that |P| = n.
+The following defines a distance between an m-set and a set of k centers.
+Definition 4 (Variant of Definition 2.1, [27]). Let
+�
+Rd, D
+�
+be a metric space, where D : P
+�
+Rd�
+×
+P
+�
+Rd�
+→ [0, ∞) be a function that maps every two subsets P, C ⊆ Rd to
+D (P, C) =
+min
+p∈P,x∈C ∥p − x∥2
+2 .
+(3)
+For an integer k ≥ 1, define Xk =
+�
+C ⊆ Rd��|C| = k
+�
+.
+The following defines a coreset for the sets clustering problem [27].
+Definition 5 ([27]). Let n, m, k be a triplet of positive integers, P be an (n, m)-set as in Defi-
+nition 3, and let ε, δ > 0 denote the approximation error and probability of failure, respectively.
+(S, v) is an ε-coreset, where S ⊆ P and v : S → [0, ∞) is a weight function, if for every
+C ⊆ Xk
+�����
+�
+P∈P
+D (P, C) −
+�
+Q∈S
+v(Q)D (Q, C)
+����� ≤ ε
+�
+P∈P
+D (P, C) ,
+(4)
+occurs with probability at least 1 − δ.
+
+3) Prediction of WC: In our work, we use the scheme in [7] as cost function for predicting the
+WC. The prediction is based on calculating a function that links the positions and velocities of
+the floaters. This function can be linear, in which case the calculation is performed by a weighted
+least squares; or non-linear, in which case the estimation involves support vector regression with
+a non-linear kernel function. Once the relation between the floaters’ positions and their velocity
+is established, the WC’s velocity at any given location (within the area explored by the floaters)
+is evaluated by operating the resulting function over the given location.
+IV. METHODOLOGY
+Fig. 2: A flow chart illustrating our approach for determining the floaters’ best deployment
+position.
+
+Iterative water current segmentation
+Sample point
+ε-approximation of
+from ε-grid
+water current
+Velocity map
+Is there
+anymore
+Active coreset
+Yes
+sampled
+based ellipsoid
+uncovered
+bounding
+point?
+Floater initial positioning
+Water current clustering
+Dissect each
+Apply a
+Either
+variant of k-
+segment into
+set of sets
+means
+Graph based
+Heuristically
+approachA. Key idea
+Recall that we are interested in a solution that, given a WC flow field map, how to setup
+the deployment of a fixed number of Lagrangian floaters such as to best explore the flow field
+in-situ. The problem of setting the floaters’ initial deployment is treated here as maximizing the
+information gained by the floaters with respect to the actual WC velocity’s field. Such a problem
+can be reduced to the robot coverage path planning problem [28], which aims to provide full
+coverage of an explored area while also minimizing the number of repeated visits. In the context
+of our problem, a variant of the coverage path planning problem is used: Given K robots without
+the ability to control their movement, and a state space where each state moves the robot to
+a different state, the goal is to cover as many states as possible while moving through already
+discovered states as little as possible. This problem can be shown to be NP-hard [29]. However,
+in our case, the space is not continuous, but rather discrete and bounded by the resolution of the
+given model. Such a setting simplifies the problem and makes it polynomial in nature (rather
+than exponential).
+The steps of our algorithm are illustrated in the block diagram in Fig. 2, and a toy example
+is illustrated in Fig. 4. Given a map M of M × N velocity vectors forming a snippet of
+a WC’s flow field (see example in Fig. 3a), the algorithm first partitions M into segments
+of homogeneous patches. This is translated into first applying an ε-grid on the map. That is,
+dissecting the map into a set of (M/ε) × (N/ε) cells (see Fig. 4a). Then, from each cell we
+sample one representative. For each sampled point, we next check whether the point is covered
+by a segment in which case we proceed to the next sampled point. Otherwise, we find the
+smallest ellipsoid in volume that encloses a homogeneous patch, including the sampled point.
+We then obtain an εβ-approximation towards the enclosed patch of the WC in the obtained
+ellipsoid from the previous step, as illustrated in Fig. 4b. The above steps are repeated over the
+set of sampled points until all points are covered.
+As an approach for segmenting the flow field map, ˆ
+M, a clustering scheme is applied. Each
+segment is dissected into an (n, 3)-set (see Definition 3), where n =
+� size of segment
+3
+�
+, such that
+each point in the (n, 3)-set is composed of its coordinates on the map M and the velocity vector
+that is present at those coordinates. We then normalize the velocity vector such that its norm
+
+is equal to the norm of its corresponding coordinates at M. In the last stage of clustering, we
+generate a coreset for the sets clustering problem on the merged set of sets M (see Definition 5),
+followed by a variant of the k-means algorithm, where k here is equal to the number of floaters.
+The result is a set of k centers that defines a clustering on �
+M as shown in Fig. 4c. Finally,
+based on the clustered �
+M, we determine the deployment position of the K floaters using two
+main techniques (i) heuristics: where the placement locations are chosen as the farthest point in
+the opposite direction of the dominating direction of each cluster or (ii) graph-based: following
+the longest path formed in the flow filed map.
+We handle the task of clustering by coresets. Coresets are a weighted subset of the input
+data. They were first introduced in computational geometry as a means to reduce the size of
+large datasets. Throughout recent years, coresets have been extended and developed for various
+optimization problems from different fields. One key component associated with coresets is that
+they aim to encapsulate the hidden structure in the data that the optimization problem at hand
+entails. Other approaches such as matrix sketches [30] or submodular maximization [31] can
+also be used for clustering. The advantage of coresets is that it is a subset of the data where
+the coreset guarantee is satisfied for any query, e.g., any k centers in the context of k-means
+clustering. Such coresets are referred as “strong coresets” in the literature [32].
+B. Our Floater Deployment Scheme
+We formulate our problem as follows. Let S denote the space of all possible placements.
+For every x ∈ X, let f(x) ∈ S, denote the position of floater x. Assume that each p ∈ S is
+associated with a loss function φ : S → S that maps p to some state q ∈ S. Finally, let P (S)
+denote the power set of S, π : S → P(S) denote the path of visited states given the initial state
+for a floater, ℓ(p(f(x)) denote the length of the path associated with the floater x, and S4
+i=1 be
+a set of orthants of S such that for each i, j ∈ [4], Si ⊆ S and Si ∩ Sj = ∅ with i ̸= j. The
+
+(a) Map of velocities
+(b) Zoomed area with respect to our toy map.
+Fig. 3: A toy example of a flow field map. The color bar denotes the magnitude of the velocity
+vectors. Example produced from the SELIPS model [33]. Fig. 3b depicts a zoomed area that is
+contained in the dotted black rectangle at Fig. 3a.
+optimization problem is formalized by
+max
+4
+�
+j=1
+log
+�����
+� �
+x∈X
+p (f(x))
+�
+∩ Sj
+����
+�
+�
+x∈X
+ℓ(p(f(x))
+s.t.
+x ∈ X
+f(x) ∈ S
+(5)
+In (5), the size of the union of different sets (denoted by the absolute function over a set)
+accounts for each state that is visited by some floater only once, and the loss function forces
+the solver to find initial states whose path must at least pass through one state from each of the
+subspaces {Si}4
+i=1 of S. The loss function aims to guide the solver to choose placements that
+doesn’t lead to infinite loops.
+1) Iterative WC segmentation: Our solution starts by identifying segments in the WC’s
+velocity field. Each segment contains a homogeneous set of WC vectors representing the WC’s
+magnitude and direction. The task is performed by oracle-based algorithms. The data is assumed
+to be “hidden” and only available to the oracle, and the user is allowed to ask the oracle questions
+
+Cm/s
+35
+35
+34.5
+30
+25
+Longitude
+34
+20
+33.5
+15
+33
+10
+32.5
+5
+32
+1
+0
+31.5
+32
+32.5
+33
+33.5
+LatitudeCm/s
+32.5
+35
+32.4
+30
+25
+Longitude
+32.3
+20
+15
+32.2
+10
+5
+32.1
+1
+32
+0
+32
+32.1
+32.2
+32.3
+32.4
+32.5
+Latitude(a) Map partitioning into grids
+(b) WC segmentation
+(c) Clustering set of sets
+Fig. 4: Illustration of running our model on the toy example Fig. 3. Fig. 4a presents a partitioning
+of the flow field such that from each grid cell, a representative is randomly selected. Fig. 4b
+presents our segmentation which entails grouping areas of similar direction and speed. Fig. 4c
+clusters the segments to ensure that the number of clusters is equal to the number of floaters.
+with “yes/no” responses. Such oracles are known by the term membership oracles. The motivation
+behind such decisions is scalable algorithms for segmentation. Using the oracle-based approach,
+the emphasis is to segment the map into a set of segments using minimal oracle queries. In
+addition, such approaches enable the handling of large-scale velocity maps in near-linear time.
+In our context, the oracle has the ability to distinguish between different current patches.
+With such an oracle, we find the minimal volume enclosing ellipsoid (or MVEE in short, see
+Definition 2) of each WC patch. Specifically, using the given membership oracle, a separation
+
+Cm/s
+35
+35
+34.5
+30
+25
+Longitude
+34
+20
+33.5
+15
+33
+10
+32.5
+5
+32
+1
+0
+31.5
+32
+32.5
+33
+33.5
+Latitude35.0
+34.5
+e
+34.0
+itude
+33.5
+g
+33.0
+32.5
+32.0
+31.5
+32.0
+32.5
+33.0
+33.5
+Latitude35.0
+34.5
+34.0
+itude
+33.5
+ngi
+33.0
+32.5
+32.0
+31.5
+32.0
+32.5
+33.0
+33.5
+Latitudeoracle can be constructed in polynomial time. The response of the separation oracle is “True” if
+a point lies inside the body of interest. If the point lies outside the body of interest, the oracle
+outputs a hyperplane, separating the point from the WC patch. Using the separation oracle,
+the ellipsoid method [34] can be leveraged to find a (1 + O (ε))-approximation for the optimal
+MVEE. The time complexity of such algorithms is O
+�
+nd4 logO(1) � d
+ε
+��
+; recall that in our setting,
+d ∈ O (1); hence, the time complexity of our algorithm is linear in the number of points n. We
+refer the reader to [35], [36], [37] for an extensive analysis of this method.
+Once a WC patch P has been enclosed by an ellipsoid, we proceed to obtain an εβ-approximation
+towards the volume of P, i.e., we aim to find C ⊆ P such that
+Vol (Conv (C))
+Vol (Conv (P)) ≥ 1 − εβ.
+(6)
+For this task, we first dissect P to Vol (P) εd
+β cells (see Definition 2). From each cell of this
+type, we uniformly choose a representative point at random. This ensures that the volume of the
+set of sampled points approximates the volume of P, which in turn approximates the structural
+properties of P [38].
+2) Clustering WC: A fundamental clustering approach is k-means, which can also be used
+here to cluster WC. However, k-means will disregard the connectivity between points (segment
+points). Instead, we use sets clustering [27], which is a generalization of k-means to cluster
+dependent sets of points.
+Each approximated WC patch is partitioned into a set of triplets based on distance. More
+specifically, each point is associated with the closest two points to it based on Euclidean distance.
+The result is a (3, n)-set P, where P ∈ P is a set of 3 WC velocity vectors, and n denotes
+the number of all such sets (see Definition 3). The time complexity for finding a “sub-optimal”
+solution for such clustering is O
+�
+n log n (nk)dk�
+[27], where n denotes the number of sets of
+points, k denotes the number of desired clusters, and d denotes the dimension of each point in
+the sets of points. Such a solution is, at most, worse than the optimal solution by a multiplicative
+factor of O (log n). Leveraging the use of coresets, we can reduce the running time to n log nk3+
+� log n
+ε dk3�O(dk), while maintaining a solution that is associated with an approximation factor of
+O ((1 + ε) log n) [27]. It can be shown that solving the clustering problem on the coreset admits
+an approximation towards the optimal clustering obtained on all of the data, as follows.
+
+Claim 6. Let P be an (n, 3), ε ∈ (0, 0.5), and (C, w) denote its ε-coreset as in Definition 5. Let
+XC denote the optimal clustering with respect to the coreset (C, w) and XP denote the optimal
+clustering with respect to P. Then
+�
+P∈P
+D (P, XC) ∈ (1 + O(ε))
+�
+P∈P
+D (P, XP) .
+Proof. Observe that
+�
+P∈P
+D (P, XP) ≤
+�
+P∈P
+D (P, XC)
+≤
+1
+1 − ε
+�
+P∈C
+w(P)D (P, XC)
+≤
+1
+1 − ε
+�
+P∈C
+w(P)D (P, XP)
+≤ 1 + ε
+1 − ε
+�
+P∈P
+D (P, XP) ,
+(7)
+where the first inequality holds by definition of XP, the second and last inequality follows from
+Definition 5, and the third inequality holds by definition of XC.
+The claim holds since 1+ε
+1−ε ≤ 1 + 4ε due to the fact that ε ∈ (0, 0.5).
+Since the input of our algorithm is a discrete map, usually represented via a grid, the input
+to the clustering must also incorporate the coordinates of each point in any WC patch in the
+given map. For such a task, to each point p in each triplet P, we concatenate the corresponding
+coordinates in the map, resulting in ˆp, while the set P is then referred to as ˆP. To ensure fairness
+across the two feature vectors that ˆp is composed of, we ensure that their norm is roughly equal
+through scaling. The set of all ˆP is then passed to the sets clustering scheme, and the result is
+treated at the clustering on the original set P.
+3) Towards optimal floater deployment while seeking maximal coverage: Once the WC map
+has been clustered, we determine the deployment position of the floaters to obtain the best WC
+velocity field estimation. We consider two types of solutions to the deployment strategy. The
+first is a heuristic approach, referred to as heuristic, where each cluster is assigned a unique
+floater. The floater’s location is then set at the farthest point along the negative of the dominating
+direction from the cluster, thereby obtaining the longest traversal inside the cluster. Here, the
+
+dominating direction of a cluster refers to the direction that most points in the cluster either
+point to, or are very close to in terms of cosine similarity. This approach ensures that each
+cluster is covered, while allowing for additional data collection from the deployment position
+to the cluster’s boundaries. However, the scheme is not optimal for the probable case where the
+number of clusters is lower than or equal to the number of floaters.
+A more rigorous approach would be to employ concepts from graph theory, and we refer to
+it as graph-based approach. Each cluster S is represented as a directed graph GS := (VS, ES),
+where each point in S is assigned a vertex in GS. As for the set of edges of GS, an edge
+e := v → u exists in GS if q is reachable from p following the velocity vector of p, where p
+and q are the corresponding points in S of that of v and u, respectively. In other words, an edge
+exists if, and only if, (i) one can move from p to q using the velocity vector that is associated
+with p, and (ii) q ∈ B (p, 1) where B(x, r) denotes a ball centered at x, with a radius of r. At this
+stage, we have K disconnected graphs for K floaters. For each graph, we compute the longest
+path from the set of shortest paths between each pair of graph nodes by applying a breadth-first
+search (BFS) algorithm [39] each time from a different node. The running time of this procedure
+is O
+�
+|VS|2 + |VS| |ES|
+�
+for any graph GS. Our above algorithm can be generalized by taking
+into account weights, in which case, the BFS algorithm is replaced by Johnson’s algorithm [40].
+A modification of this graph-based approach, referred to as the inter-graph scheme, connects
+these graphs by checking whether the roots and leaves of one graph can be connected to another
+graph. The connectivity is applied to each pair of graphs, and the resulting graph is denoted by
+Gall. The BFS algorithm is then used again to compute the K largest non-intersecting paths in
+Gall. Finally, the floaters’ deployment positions are set to be the starting vertices of the selected
+paths.
+V. EXPERIMENTAL ANALYSIS
+In this section, we evaluate the performance of our three strategies for determining the floaters’
+deployment positions, namely, heuristics, graph-based and inter graph-based. Without alternative
+benchmark for determining the initial position of the floats for WC prediction, we compare the
+performance of our schemes to the common approach of sampling the initial deployment locations
+uniformly at random. We follow [7] to evaluate the performance of the different deployment
+
+schemes in terms of the velocity field prediction. For any location p(x, y) in the velocity field,
+we denote the ground truth velocity vector at p(x, y) by
+�
+�vx
+vy
+�
+�, and the predicted velocity
+vector at p(x, y) by
+�
+�vpred
+x
+vpred
+y
+�
+�. The latter is obtained by applying the method in [7], each time for
+the different the floaters’ trajectories as obtained after deployment based on the four different
+deployment strategies. The prediction error is defined by
+ρspeed =
+��
+vx − vpred
+x
+�2
++
+�
+vy − vpred
+y
+�2
+.
+(8)
+Note that the method in [7] interpolates the floaters information towards the prediction of the
+flow field away from the floaters’ trajectories. As such, the prediction error in (8) is calculated
+for each location in the explored area.
+A. Experimental Settings
+The method in [7] offers a linear and a non-linear prediction models. Here, since we aim
+for complex flow fields with non-homogeneous sections, we choose the latter that is based on
+support vector regression (SVR) model with a radial bases function (RBF) kernel. To train the
+RBF-SVR model, we used a grid-search approach with cross validation [41], [42] to determine
+the best model parameters. The tuned parameters are (i) C – a regularization parameters, (ii) ϵ
+– an optimization-related parameter with respect to the SVR model, and (iii) γ – the exponent
+which controls the deviation of the spread of the radial basis function. For more details, we refer
+the reader to “Scikit-Learn” [43].
+As a WC model (and ground truth), we use 48 WC maps, as produced by the SELIPS
+model [33] for the Gulf of Haifa, Israel. The maps span over a time period of 12 months.
+Model SELIPS is an operational forecasting system based on POM, a 3D numerical model for
+the simulation of ocean dynamics, with a horizontal resolution of about 1 km. The output was
+given as 3 hour averages of the velocity components. We explore the results for two options:
+1) clean map: the WC model is the same as the velocity field used to simulate the drift of the
+floaters, and 2) noisy map: the velocity field used for the simulation is a noisy version of the
+
+WC model. We explore the results for different number of floaters, K, and for different model
+parameters.
+B. Experimental Analysis
+1) The Toy Example: We start by showing the performance of each of our proposed deploy-
+ment strategies on the toy example in Fig. 3a. In Fig. 5, we present the empirical cumulative
+distribution function (CDF) results of the velocity error vector (8), as generated by predicting
+the flow field for each point in the map. We observe that the performance of our proposed
+strategies exceeds that of the uniformly choosing deployment strategy. This indicates that, from
+a statistical point of view, our method forms better candidates for deployment position strategies
+than sampling uniformly. Comparing the performance of our three schemes, we conclude that
+the inter-graph-based approach is better, mostly because of the complexity in the structure of
+the velocity field, which induces diversity in the WC. The inter-graph approach, which is more
+rigorous, can better capture this diversity.
+Fig. 5: CDF of the norm of velocity error (8). The results show our advantage upon using
+uniform sampling for determining deployment positions.
+
+1.0
+w
+V
+20.8
+error
+velocity
+0.6
+0.4
+Uniform sampling
+0.2
+our heuristic
+our graph based
+our inter-graph based
+0.0
+0
+5
+10
+15
+20
+25
+[曾]2) Choosing the “best” parameters for our model: Our model relies on a predefined set of
+parameters. Specifically, the approximation error εβ with respect to the volume of the explored
+area, which is required for the iterative segmentation stage, and the coreset size µ with respect
+to the clustering phase. To explore the sensitivity of the graph-based and inter-graph-based
+approaches to the choice of these parameters, we use the clean simulation setup for all 48 WC
+maps to show that our model is robust against changes in these parameters. As seen in Fig 6, best
+results are associated with µ := 1500 and εβ := 0.05. Still, the difference is rather small, which
+reflects on the robustness of our approach. In the below we choose µ := 1500 and εβ := 0.05
+for both the clean maps, for the noisy maps.
+(a)
+(b)
+Fig. 6: CDF of the averaged prediction error across 48 “clean” maps when using different model
+parameters with respect to our graph-based approach (left-most), and with respect to our inter-
+graph based (right-most).
+3) Comparison Against Uniform Sampling on a Variety of Maps: We now explore the efficacy
+of the three schemes for different WC maps in the clean map setup against Uniform sampling.
+Fig. 7 presents the CDF of the averaged prediction errors across the 48 maps. We observe that
+the heuristic-based strategy for deployment outperforms most of the deployment strategies. This
+is due to the fact that there are no obstacles in each of the 48 maps. That is an area in the
+WC flow field with zeroed-out velocity vectors, e.g., an island. Observe that while the graph-
+
+1.0
+(m)
+V
+20.8
+error
+Ivelocity
+0.6
+8g = 5e -02. u= 500
+8g = 5e-02. μ= 1000
+0.4
+3 = 5e - 02. μ= 1500
+β = 8e - 02, μ= 500
+0.2
+8β = 8e - 02, μ= 1000
+8g = 8e- 02. μ= 1500
+0.0
+0
+10
+20
+30
+40
+50
+m1.0
+)
+V
+20.8
+error
+0.6
+8g = 5e-02. μ= 500
+8g= 5e-02.μ= 1000
+0.4
+3 = 5e - 02. μ= 1500
+8β = 8e - 02, μ= 500
+0.2
+8β = 8e - 02, μ= 1000
+83 = 8e- 02.μ= 1500
+0.0
+2
+3
+4
+5Fig. 7: CDF of the averaged prediction error across 48 maps.
+based approach is comparable to the heuristic-based approach, at some point it starts being
+less efficient. This is due to the fact that the graph-based method needs denser clusters, i.e.,
+the parameter εβ needs to be smaller leading to larger and denser clusters, e.g., the effect of
+εβ is best observed visually in Fig. 4c. In turn, the graphs generated from the clusters hold
+more information regarding the flow field. This will yield better results than our heuristic-based
+approach. In addition, the same behavior appears also when observing the inter-graph-based
+approach.
+4) Robustness Against Noise: We explore the performance of the three schemes when the
+noisy map setup is considered. For our toy map M, we produce its noisy map M ′ by adding a
+Gaussian noise with zero mean, and a standard deviation equal to σ% of the standard deviation
+of M. The added noise is added to a fraction of the WC map, denoted by η ∈ (0, 1), representing
+the corruption ratio of the WC map.
+To generate a difference between the WC map and the velocity field used, we determine the
+deployment positions of the floaters, p(0), based on the given WC model (i.e., without the added
+noise), but calculate the trajectories of the floaters based on the noisy WC map. As a result,
+the assumed map is mismatched with the noisy one. The effect of η and σ on our toy example
+
+1.0
+(m)
+V
+20.8
+error
+0.6
+0.4
+--→-. Uniform sampling
+ourheuristic
+0.2
+our graph based
+our inter-graph based
+0.0
+5
+10
+15
+20
+mη%
+σ%
+15
+133
+15
+100
+TABLE I: The effect of added noise on our toy example.
+are presented in Table I. With a small corruption percent (small η), we observe that the resulted
+map is not that different from its clean version; see Figure 3a. As η and σ increase, the map
+loses its underlying structure almost entirely, as depicted at the rightmost lower cell of Table I.
+The results are given in Fig. 8. We observe that the average error increases with σ%. We argue
+that, as σ% increases, the correlation between the generated deployment positions on the clean
+map and the resulted noisy map becomes less strong leading to an increase in the average error.
+The heuristic-based and graph-based approaches still outperform the uniform sampling, but for
+the inter-graph-based approach which is sensitive to the smoothness of the maps, becomes less
+efficient in the presence of noise.
+5) Assessing the effect of K: In this experiment, we explore the effect of increasing the
+number of floaters on the error of reconstructing flow fields. Fig. 9 presents the average error
+
+Cm/s
+35
+35
+34.5
+30
+25
+Longitude
+34
+20
+33.5
+15
+33
+10
+32.5
+5
+32
+1
+0
+31.5
+32
+32.5
+33
+33.5
+LatitudeCm/s
+35
+35
+34.5
+30
+25
+Longitude
+34
+20
+33.5
+15
+33
+10
+32.5
+5
+32
+1
+0
+31.5
+32
+32.5
+33
+33.5
+LatitudeCm/s
+35
+35
+34.5
+30
+25
+Longitude
+34
+20
+33.5
+15
+33
+10
+32.5
+5
+32
+1
+0
+31.5
+32
+32.5
+33
+33.5
+LatitudeCm/s
+35
+35
+34.5
+30
+25
+Longitude
+34
+20
+33.5
+15
+33
+10
+32.5
+5
+32
+1
+0
+31.5
+32
+32.5
+33
+33.5
+Latitude(a) η = 15%
+(b) η = 40%
+(c) η = 95%
+Fig. 8: The averaged norm of the velocity error on our toy map as a function σ, when using
+three different corruption percents η ∈ {0.15, 0.4, 0.95}.
+(a)
+(b)
+Fig. 9: On the left graph, the averaged norm of the velocity error across the 48 maps as a function
+of the number of floaters K. On the right graph, we zoom in Fig. 9a showing the averaged norm
+of the velocity error across the 48 maps as a function of the number of floaters K. In both
+figures, the shaded regions denote a 95% confidence bar.
+across the 48 WC maps. We observe that as the number of floaters increases, the average error
+for each of the 4 deployment strategies decreases. This is due to the fact that the amount of
+collective data also increases as more floaters are available, hinging upon a larger discovery of
+the underlying structure of the flow fields. The results show that graph-based and heuristic-based
+
+25
+23
+ Uniform sampling
+our heuristic
+22
+our graph based
+our inter-graph based
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+[.] 
+a24
+.... Uniform sampling
+our heuristic
+22
+our graph based
+our inter-graph based
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+[曾]
+a23
+..... Uniform sampling
+our heuristic
+our graph based
+our inter-graph based
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+["]
+a12
+T.... Uniform sampling
+.... our heuristic
+10
+... our graph based
+our inter-graph based
+4
+2
+6
+8
+10
+12
+14
+Number of floaters3.4
+our heuristic
+our graph based
+3.2
+3.0
+2.8
+2.6
+2.4
+6
+8
+10
+12
+14
+Number of floaters(a)
+(b)
+Fig. 10: On the right, we present the average error of each of our deployment methods with
+respect to WC reconstruction as a function of εβ. On the left, we present the running time
+needed to generate the positions for our graph-based approach εβ. Shaded regions denote the
+standard deviation with respect to the y-axis.
+outperform uniform sampling by at least 225%. This is due to the nature of our approaches,
+which rely on information about the structural properties of the WC maps. On the other hand,
+the performance of our inter-graph-based approach is sometimes weaker than uniform sampling.
+This is mainly due to the fact that the former method requires denser graphs, ultimately leading
+to lower εβ.
+6) When to use each of our deployment strategies: Finally, we explore the best setups that
+fit best each of our deployment strategies.
+a) When accuracy matters more than run-time: Figure 10a presents the effect of εβ on each
+of our proposed strategies. When εβ decreases, the best strategy is the graph-based approach. It
+lines well with the observation that this method uses the underline structure of the map. However,
+the cost of using such low εβ is reflected in Figure 10b. Using an AMD Ryzen Threadripper
+3990X 2.9 GHz 64-Core with 128GB RAM, the run time increases from minutes to hours. The
+run time for the heuristic-based approach ranges between 3000 seconds (at εβ = 0.01) and 4500
+seconds (at εβ = 0.001), while the run time of the inter-graph-based approach is similar to that
+
+lvelocity errorll2
+our heuristic
+our graph based
+our inter-graph based
+22
+21
+0.010
+0.008
+0.006
+0.004
+0.002our graph-based
+12000
+10000
+8000
+6000
+4000
+0.010
+0.008
+0.006
+0.004
+0.002
+EβFig. 11: The averaged norm of the velocity error across the 48 maps as a function σ, where the
+corruption percent η is 100%. The shaded regions denote a 95% confidence bar.
+of the graph-based approach.
+b) Corrupted maps: We explore the performance of the three schemes when the given
+model is different than the real channel, i.e., using the noisy map setup. For each map M from
+our set of 48 flow field maps, we produce its noisy map M ′ by adding a Gaussian noise with
+zero mean, and a standard deviation equals to σ% of the standard deviation of M. Here the
+corruption percentage, η is 100%. The results are given in Fig. 11 for different σ2 values. We
+observe that the average error decreases as the added noise increases. This rather non-intuitive
+result is due to the fact that, as noise increases, the obtained map becomes similar to a Gaussian
+distributed. Consequently, the distribution of the map’s entries can be better estimated from the
+learning phase, i.e., the path traversed by the floaters. That is, the impact of the floater’s initial
+location becomes less dominant as the noise increases and the mismatch between the model and
+the actual map increases. That said, since the structure of the noise field still dominates over
+the added noise, we observe that our inter-graph-based approach is on par with the uniform
+deployment approach. This is because the former is the least information-collective approach
+among our three deployment strategies.
+
+23
+Uniform sampling
+our heuristic
+22 
+our graph based
+our inter-graph based
+21
+20
+2~1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+["]VI. CONCLUSIONS AND FUTURE WORK
+In this paper, we explored how to determine the initial deployment positions of a group of
+floaters to best evaluate the WC flow field. Our approach relies on clustering a given model of
+the WC into segments, each of which is represented by a coreset, and determining the floaters’
+initial deployment positions with the aim of visiting all coresets under constraints: the number
+of floaters, and the time frame used for evaluation. We analyzed the results of our scheme over
+a database of 48 WC maps that span over a year of measurements in the Gulf of Haifa, Israel.
+Compared to the uniform sampling benchmark, the results show that our scheme is more accurate
+in terms of the WC’s prediction, and is more robust to mismatches between the given WC model
+and the actual one. Future work will identify gaps in the given model and complete them by
+guiding the floaters to visit these locations.
+VII. ACKNOWLEDGEMENTS
+This work was supported in part by the MOST action for Agriculture, Environment, and Water
+for the 490 year 2019 (Grant # 3-16728) and by the the University of Haifa’s Data Science
+Research Center.
+REFERENCES
+[1] “Open source code for all the algorithms presented in this paper,” 2022, Link for open-source code.
+[2] L. Kuznetsov, K. Ide, and C. K. Jones, “A method for assimilation of Lagrangian data,” Monthly Weather Review, vol.
+131, no. 10, pp. 2247–2260, 2003.
+[3] R. N. Miller, Numerical modeling of ocean circulation.
+Cambridge University Press, 2007.
+[4] M. Santoki, S. Ratheesh, R. Sharma, K. Joshipura, and S. Basu, “Assimilation of drifter data in a circulation model of the
+Indian Ocean,” IEEE Geoscience and Remote Sensing Letters, vol. 9, no. 1, pp. 100–103, 2011.
+[5] S. Castellari, A. Griffa, T. M. Özgökmen, and P.-M. Poulain, “Prediction of particle trajectories in the adriatic sea using
+Lagrangian data assimilation,” Journal of Marine Systems, vol. 29, no. 1-4, pp. 33–50, 2001.
+[6] M. J. Carrier, H. Ngodock, S. Smith, G. Jacobs, P. Muscarella, T. Ozgokmen, B. Haus, and B. Lipphardt, “Impact of
+assimilating ocean velocity observations inferred from Lagrangian drifter data using the NCOM-4DVAR,” Monthly Weather
+Review, vol. 142, no. 4, pp. 1509–1524, 2014.
+[7] R. Diamant, “Prediction of water current using a swarm of submerged drifters,” IEEE Sensors Journal, vol. 20, no. 19,
+pp. 11 598–11 607, 2020.
+[8] A. Molcard, L. I. Piterbarg, A. Griffa, T. M. Özgökmen, and A. J. Mariano, “Assimilation of drifter observations for the
+reconstruction of the Eulerian circulation field,” Journal of Geophysical Research: Oceans, vol. 108, no. C3, 2003.
+
+[9] J. L. Callaham, K. Maeda, and S. L. Brunton, “Robust flow reconstruction from limited measurements via sparse
+representation,” Physical Review Fluids, vol. 4, no. 10, p. 103907, 2019.
+[10] J. S. Jaffe, P. J. Franks, P. L. Roberts, D. Mirza, C. Schurgers, R. Kastner, and A. Boch, “A swarm of autonomous miniature
+underwater robot drifters for exploring submesoscale ocean dynamics,” Nature communications, vol. 8, no. 1, pp. 1–8,
+2017.
+[11] M. Tukan, C. Baykal, D. Feldman, and D. Rus, “On coresets for support vector machines,” in International Conference
+on Theory and Applications of Models of Computation.
+Springer, 2020, pp. 287–299.
+[12] G. Umgiesser, D. M. Canu, A. Cucco, and C. Solidoro, “A finite element model for the venice lagoon. development, set
+up, calibration and validation,” Journal of Marine Systems, vol. 51, no. 1-4, pp. 123–145, 2004.
+[13] H. Salman, K. Ide, and C. K. Jones, “Using flow geometry for drifter deployment in Lagrangian data assimilation,” Tellus
+A: Dynamic Meteorology and Oceanography, vol. 60, no. 2, pp. 321–335, 2008.
+[14] C. Chapman and J.-B. Sallée, “Can we reconstruct mean and eddy fluxes from Argo floats?” Ocean Modelling, vol. 120,
+pp. 83–100, 2017.
+[15] H. Bai, “Motion-dependent estimation of a spatial vector field with multiple vehicles,” in 2018 IEEE conference on decision
+and control (CDC).
+IEEE, 2018, pp. 1379–1384.
+[16] L. Shi, R. Zheng, S. Zhang, and M. Liu, “Cooperative estimation to reconstruct the parametric flow field using multiple
+AUVs,” IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1–10, 2021.
+[17] M. Rafiee, Q. Wu, and A. M. Bayen, “Kalman filter based estimation of flow states in open channels using Lagrangian
+sensing,” in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese
+Control Conference.
+IEEE, 2009, pp. 8266–8271.
+[18] G. A. Hollinger and G. S. Sukhatme, “Sampling-based motion planning for robotic information gathering.” in Robotics:
+Science and Systems, vol. 3, no. 5.
+Citeseer, 2013.
+[19] A. Molcard, A. C. Poje, and T. M. Özgökmen, “Directed drifter launch strategies for Lagrangian data assimilation using
+hyperbolic trajectories,” Ocean Modelling, vol. 12, no. 3-4, pp. 268–289, 2006.
+[20] J. Hansen and G. Dudek, “Coverage optimization with non-actuated, floating mobile sensors using iterative trajectory
+planning in marine flow fields,” in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).
+IEEE, 2018, pp. 1906–1912.
+[21] K. Kim, D. Lee, and I. Essa, “Gaussian process regression flow for analysis of motion trajectories,” in 2011 International
+Conference on Computer Vision.
+IEEE, 2011, pp. 1164–1171.
+[22] K.-F. Dagestad, J. Röhrs, Ø. Breivik, and B. Ådlandsvik, “Opendrift v1. 0: a generic framework for trajectory modelling,”
+Geoscientific Model Development, vol. 11, no. 4, pp. 1405–1420, 2018.
+[23] H.-P. Tan, R. Diamant, W. K. Seah, and M. Waldmeyer, “A survey of techniques and challenges in underwater localization,”
+Ocean Engineering, vol. 38, no. 14-15, pp. 1663–1676, 2011.
+[24] T. Alexandri, M. Walter, and R. Diamant, “A time-difference-of-arrival based target motion analysis for localization of
+underwater vehicles,” IEEE Transactions on Vehicular Technology, 2021.
+[25] F. John, “Extremum problems with inequalities as subsidiary conditions,” in Traces and emergence of nonlinear
+programming.
+Springer, 2014, pp. 197–215.
+
+[26] M. J. Todd and E. A. Yıldırım, “On Khachiyan’s algorithm for the computation of minimum-volume enclosing ellipsoids,”
+Discrete Applied Mathematics, vol. 155, no. 13, pp. 1731–1744, 2007.
+[27] I. Jubran, M. Tukan, A. Maalouf, and D. Feldman, “Sets clustering,” in International Conference on Machine Learning.
+PMLR, 2020, pp. 4994–5005.
+[28] Y.-Y. Lin, C.-C. Ni, N. Lei, X. David Gu, and J. Gao, “Robot coverage path planning for general surfaces using quadratic
+differentials,” in 2017 IEEE International Conference on Robotics and Automation (ICRA), 2017, pp. 5005–5011.
+[29] X. Zheng, S. Jain, S. Koenig, and D. Kempe, “Multi-robot forest coverage,” in 2005 IEEE/RSJ International Conference
+on Intelligent Robots and Systems.
+IEEE, 2005, pp. 3852–3857.
+[30] D. P. Woodruff, “Sketching as a tool for numerical linear algebra,” arXiv preprint arXiv:1411.4357, 2014.
+[31] A. Badanidiyuru, B. Mirzasoleiman, A. Karbasi, and A. Krause, “Streaming submodular maximization: Massive data
+summarization on the fly,” in Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery
+and data mining, 2014, pp. 671–680.
+[32] D. Feldman, “Core-sets: Updated survey,” in Sampling techniques for supervised or unsupervised tasks.
+Springer, 2020,
+pp. 23–44.
+[33] R. Goldman, E. Biton, E. Brokovich, S. Kark, and N. Levin, “Oil spill contamination probability in the southeastern
+Levantine basin,” Marine Pollution Bulletin, vol. 91, no. 1, pp. 347–356, 2015.
+[34] M. Grötschel, L. Lovász, and A. Schrijver, “The ellipsoid method,” in Geometric Algorithms and Combinatorial
+Optimization.
+Springer, 1993, pp. 64–101.
+[35] M. Grötschel, L. Lovász, and A. Schrijver, Geometric algorithms and combinatorial optimization.
+Springer Science &
+Business Media, 2012, vol. 2.
+[36] Y. T. Lee, A. Sidford, and S. S. Vempala, “Efficient convex optimization with membership oracles,” in Conference On
+Learning Theory.
+PMLR, 2018, pp. 1292–1294.
+[37] M. Tukan, A. Maalouf, D. Feldman, and R. Poranne, “Obstacle aware sampling for path planning,” in 2022 IEEE/RSJ
+International Conference on Intelligent Robots and Systems (IROS), 2022, pp. 13 676–13 683.
+[38] S. Har-Peled, Geometric approximation algorithms.
+American Mathematical Soc., 2011, no. 173.
+[39] A. Bundy and L. Wallen, “Breadth-first search,” in Catalogue of artificial intelligence tools.
+Springer, 1984, pp. 13–13.
+[40] D. B. Johnson, “Efficient algorithms for shortest paths in sparse networks,” Journal of the ACM (JACM), vol. 24, no. 1,
+pp. 1–13, 1977.
+[41] P. Refaeilzadeh, L. Tang, and H. Liu, “Cross-validation.” Encyclopedia of database systems, vol. 5, pp. 532–538, 2009.
+[42] M. Claesen and B. De Moor, “Hyperparameter search in machine learning,” arXiv preprint arXiv:1502.02127, 2015.
+[43] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss,
+V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine
+learning in Python,” Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011.
+

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@@ -0,0 +1,2503 @@
+Draft version January 16, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Toward accurate measurement of property-dependent galaxy clustering:
+II. Tests of the smoothed density-corrected Vmax method
+Lei Yang (杨蕾)
+1 and Zhigang Li (李志刚)2
+1South-Western Institute for Astronomy Research, Yunnan University
+Kunming, Yunnan 650500, China
+2College of Physics and Electronic Engineering, Nanyang Normal University
+Nanyang, Henan, 473061, China
+ABSTRACT
+We present a smoothed density-corrected Vmax technique for building a random catalog for property-
+dependent galaxy clustering estimation. This approach is essentially based on the density-corrected
+Vmax method of Cole (2011), with three improvements to the original method. To validate the improved
+method, we generate two sets of flux-limited samples from two independent mock catalogs with different
+k + e corrections. By comparing the two-point correlation functions, our results demonstrate that the
+random catalog created by the smoothed density-corrected Vmax approach provides a more accurate
+and precise measurement for both sets of mock samples than the commonly used Vmax method and
+redshift shuffled method. For flux-limited samples and color-dependent subsamples, the accuracy for
+the projected correlation function is well constrained within 1% on the scale 0.07h−1Mpc to 30h−1Mpc.
+The accuracy of the redshift-space correlation function is less than 2% as well. Currently, it is the
+only approach that holds promise for achieving the high-accuracy goal of clustering measures for next-
+generation surveys.
+Keywords: Galaxies(573) — Galaxy evolution(594) — Large-scale structure of the universe(902) —
+Two-point correlation function(1951)
+1. INTRODUCTION
+Over the last couple of decades, the successful observa-
+tion of galaxy redshift surveys (e.g., Two Degree Field
+Galaxy Redshift Survey, 2dFGRS, Colless et al. 2003;
+the Sloan Digital Sky Survey, SDSS, York et al. 2000; the
+Baryon Oscillation SpectroscopicSurvey, BOSS, Eisen-
+stein et al. 2011; the VIMOS Public Extragalactic Red-
+shift Survey, VIPERS, Garilli et al. 2012) enable sig-
+nificant progress has been achieved on our understand-
+ing of galaxy formation and evolution (Madgwick et al.
+2003; Berlind et al. 2006; Guo et al. 2011, 2018; Zu et al.
+2021), the galaxy-halo connection (Jing et al. 1998; Yang
+et al. 2003, 2008, 2012; Zheng et al. 2005, 2009; Vale &
+Ostriker 2004; Alam et al. 2021a; Wechsler & Tinker
+2018; Behroozi et al. 2019), and the nature of grav-
+ity and dark energy ( Peacock et al. 2001; Weinberg
+Corresponding author: Lei Yang
+[email protected]
+et al. 2013; Samushia et al. 2013; Alam et al. 2021b and
+reference therein).
+In the upcoming years, the next-
+generation surveys, such as the Dark Energy Spectro-
+scopic Instrument (DESI; Levi et al. 2013; DESI Col-
+laboration et al. 2016a,b), the Legacy Survey of Space
+and Time (LSST; LSST Dark Energy Science Collabo-
+ration 2012), the space mission Euclid (Amendola et al.
+2013) and CSST (Cao et al. 2018; Gong et al. 2019),
+will map the 3D galaxy distribution in an unprecedent-
+edly volume, leading to about an order of magnitude
+more extragalactic spectroscopic redshifts than that of
+SDSS, BOSS and eBOSS have achieved (Zarrouk et al.
+2021; Yuan et al. 2022b; Myers et al. 2022; Schlegel
+et al. 2022). Massive amounts of data from deeper in
+the sky will provide new insights into the physics of
+galaxy formation, as well as the nature of dark matter
+and dark energy (Hahn et al. 2022). Galaxy two-point
+statistics, being one of the most fundamental tools, will
+continue to play a crucial role in future data analysis
+(Valluri et al. 2022; Amin et al. 2022), as they have in
+the past (Zehavi et al. 2011; Nuza et al. 2013; Skibba
+arXiv:2301.05520v1  [astro-ph.CO]  13 Jan 2023
+
+ID2
+Yang et al.
+et al. 2014; Samushia et al. 2014; Guo et al. 2014; Planck
+Collaboration et al. 2016; Shi et al. 2018). Due to dif-
+ferent systematics, it is still difficult to reliably mea-
+sure small-scale property-dependent galaxy clustering at
+the present time.
+These systematics include redshift-
+dependent completeness, the missing galaxies in ob-
+servations (Reid et al. 2016; Bianchi & Percival 2017;
+Bianchi & Verde 2020), the incorrect estimation of the
+radial selection model (Ross et al. 2012; Yang et al.
+2020), among others (Breton & de la Torre 2021; Farrow
+et al. 2021; Merz et al. 2021). Fortunately, the coming
+big data will considerably reduce random errors in clus-
+tering determination, but to reach the high accuracy of
+clustering analysis required by the next generation sur-
+veys, we must eliminate systematic errors in measure-
+ment (Beutler et al. 2014; Reid et al. 2016; Glanville
+et al. 2021; D´avila-Kurb´an et al. 2021). In this study, the
+systematic bias produced by the radial selection model
+is investigated in greater detail.
+To measure the galaxy two-point correlation function
+(hereafter 2PCF), we must build a random catalog with
+the same angular and radial selection functions as the
+observed sample, but with a random distribution in the
+observed space (Davis & Peebles 1983; Hamilton 1993).
+The angular selection function is easy to obtain from
+observation, but the radial selection function is difficult
+to estimate accurately. As the sample has a fixed num-
+ber density and the redshift distribution of a random
+catalog is straightforward to construct (Tegmark et al.
+2004), previous works often use a volume-limited sample
+for clustering analysis (Norberg et al. 2002; Zehavi et al.
+2002, 2005, 2011; McBride et al. 2011; Shi et al. 2016;
+Mohammad et al. 2018). However, due to the need of
+excluding a substantial number of galaxies, the statisti-
+cal precision of the clustering measurement is reduced
+(Zehavi et al. 2005; Xu et al. 2016).
+Alternatively, a
+flux-limited sample may optimize the utilization of ob-
+served galaxies, but since its radial selection function
+φ(z) changes with redshift, it is not easy to build the
+redshifts of random galaxies for a flux-limited sample
+unless we know the galaxy luminosity function (here-
+after LF) Φ(Mr) (Loveday et al. 2015; Karademir et al.
+2021).
+The radial selection function for the flux-limited sam-
+ple has been recovered using a number of ways. For in-
+stance, the smooth spline fit approach utilizes a ‘spline’
+model to fit the redshift distribution of a galaxy sam-
+ple (Reid et al. 2010; Wang et al. 2017).
+The Vmax
+method populates random galaxies within the maximum
+viewable volume of a real galaxy, which is dependent
+on the galaxy’s observational limitations. The redshift
+‘shuffled’ technique is a commonly employed alternative
+(Guo et al. 2013; Zu & Mandelbaum 2015; Wang et al.
+2021). This approach chooses redshifts at random from
+the real galaxy sample and assigns them to the ran-
+dom galaxy catalog. Through clustering analysis of the
+VIPERS data, de la Torre et al. (2013) showed that the
+spline fit approach underestimates the predicted 2PCF
+in comparison to the Vmax method, particularly on scales
+larger than 3 h−1Mpc.
+In the BOSS systematics in-
+vestigation, Ross et al. (2012) revealed that the shuf-
+fled technique had a minor bias in BAO measurement
+compared to the spline fit method (Ross et al. 2015).
+However, de Mattia & Ruhlmann-Kleider (2019) demon-
+strated that the shuffled approach suffers from the ‘inte-
+gral constraint’ effect when measuring the power spec-
+trum. Using mocks from a high-resolution simulation,
+Yang et al. (2020) (hereafter Paper I) found that both
+the redshift shuffled technique and the Vmax method un-
+derestimate galaxy clustering by 30% and 20%, respec-
+tively, on scales ≳ 10h−1Mpc for flux-limited samples.
+Consequently, as long as we continue to use the afore-
+mentioned radial selection methods to construct the red-
+shifts for random catalogs for a flux-limited sample, our
+clustering measurement will contain an unavoidable sys-
+tematic deviation from the true galaxy clustering.
+Cole (2011) proposes a density-corrected Vmax tech-
+nique for concurrently estimating LF and generating a
+random catalog for a flux-limited sample.
+Unlike the
+conventional Vmax method, this technique can success-
+fully eliminate density fluctuations.
+In Cole (2011),
+they examine the radial distribution of random galax-
+ies, which is in excellent agreement with the input
+galaxy sample. This method has been employed to de-
+termine property-dependent galaxy clustering (Farrow
+et al. 2015) and clustering analysis (de la Torre et al.
+2017; Pezzotta et al. 2017; Loveday et al. 2018; John-
+ston et al. 2021). However, its clustering measurement
+performance has not been assessed. The purpose of this
+study is to test the Cole (2011) technique for clustering
+measurements using mock data. In addition, some mod-
+ifications are made to the original approach in order to
+improve its measurement accuracy.
+This paper is structured as follows.
+In Section 2,
+we review the Cole (2011) method and introduce the
+smoothed density-corrected Vmax method.
+The con-
+structions of mock galaxy catalogs are detailed in Sec-
+tion 3.
+We present the testing results of the correla-
+tion functions in Section 4. In Section 5, we assess the
+smoothed density-corrected Vmax method and discuss
+the potential sources of uncertainty in estimate.
+We
+conclude the paper in Section 6.
+
+The smoothed Vmax method
+3
+2. THE SMOOTHED DENSITY-CORRECTED Vmax
+METHOD
+To address the difficulty of recovering the radial selec-
+tion function of property-dependent galaxy sample, Cole
+(2011) developed a density-corrected Vmax approach for
+galaxy clustering estimate. This section starts with a
+briefly overview of the Cole (2011) technique. Follow-
+ing that, we detail the improvements to the original
+Cole (2011) methodology, which we call the smoothed
+density-corrected Vmax method.
+2.1. The Cole (2011) method
+On the basis of the standard Vmax approach, Cole
+(2011) presented a weighted Vmax method based on a
+joint stepwise maximum likelihood method, which effec-
+tively eliminates the influence of density fluctuation. In
+this method, a density-weighted maximum volume V DC
+max
+1 is defined, which is the normal Vmax weighted by the
+estimated galaxy overdensities ∆(z) and the LF density
+evolution P(z). They further define a weight
+wα ≡
+Vα,max
+V dc
+α,max + µVα,max
+,
+(1)
+where Vα,max and V dc
+α,max are the normal Vmax and
+density-corrected Vmax for the αth galaxy in the ob-
+served sample.
+µ is a Lagrange multiplier providing
+constraints with ⟨
+Vα,max
+V DC
+α,max+µVα,max ⟩ = 1 when estimating
+LF for the galaxy sample. Lastly, a random catalog can
+be created by replicating individual galaxies nα = nwα
+times and distributing them at random across the Vα,max
+volume. Note that, unlike the standard Vmax approach,
+nα is no longer the same for all galaxies and the selec-
+tion rate of random galaxies is adjusted by weight wα.
+The brightness of the galaxy may be over- or under-
+represented in the observed sample as a result of the
+density variation in the Vmax volume being appropri-
+ately compensated by the weight wα. By comparing the
+output redshift distribution to that of the input galaxy
+sample, Cole (2011) proved that the random catalog cre-
+ated by this density-weighted Vmax technique could re-
+cover the genuine galaxy selection function. While this
+approach has not yet been tested on galaxy clustering
+using mock galaxy catalog, it remains to be validated
+using mocks.
+2.2. The smoothed density-corrected Vmax method
+Before testing the Cole (2011) method, we perform
+three modifications to the original public code 2. The
+1 See their equation (11) and (16) in Cole (2011).
+2 http://astro.dur.ac.uk/∼cole/random cats/
+original algorithm is only applicable to galaxy sample
+with a single faint flux cut, but by adding zmin estimate,
+our first update makes the code applicable to a generic
+double flux-cut sample 3. The maximum(minimum) red-
+shifts zmax(min) in our updated code is same as Paper I
+which are determined as follows:
+zmax = min[zmag,max, zsample,max],
+(2)
+zmin = max[zmag,min, zsample,min],
+(3)
+where zsample,max(min) is the redshift limits of galaxy
+sample, and zmag,max(min) is derived by
+mfaint = M + DM(zmag,max) + k(z) − E(z),
+(4)
+mbright = M + DM(zmag,min) + k(z) − E(z),
+(5)
+where the flux limits are set by apparent magnitude
+mbright(faint), M is the absolute magnitude, the distance
+modulus is DM = 5log10(dL) + 25 − 5log10h, k(z) is the
+k-correction, and E(z) is the luminosity evolution cor-
+rection (e-correction). Our second code improvement is
+the k-correction. In the original code, the k-correction is
+performed for all galaxies depending on the input func-
+tion k(z), which hinders the method’s ability to apply
+to a real galaxy sample whose k-correction is depen-
+dent not just on redshift but also on galaxy properties
+(e.g., color). We modify the code to take a k(z, color)
+model as input, allowing k-correction to be conducted
+on individual galaxies based on their redshifts and col-
+ors. This makes the technique more applicable to ob-
+servable galaxies. Following the aforementioned mod-
+ifications, the output cloned random catalog from the
+updated algorithm is basically consistent with the gen-
+uine radial distribution of the galaxy number density
+ntrue(z). However, there are small fluctuations in the
+output radial distribution that have a considerable in-
+fluence on the final clustering estimate. Our final mod-
+ification to the algorithm is to smooth the radial dis-
+tribution of the output cloned random galaxies. In the
+smooth procedure, we begin by generating a histogram
+of comoving distance d for the random galaxies. We set
+a bin size of ∆d = 5h−1Mpc, and N(d)hist represents
+the number of random galaxies in each bin. Secondly,
+we employ a convolution operator to smooth the his-
+togram as N s
+hist = [Nhist ∗ ∆smooth], where ∆smooth = 5
+is the smooth box size in 1D and N s
+hist is the smooth
+radial distribution of random galaxies. Final redshifts
+for random galaxies are generated based on the profile
+of N s
+hist that has been smoothed. In Section 4.2, we will
+3 This modification primarily changes the step-function S from
+S(Lmin|L) to S(Lmin, Lmax|L) in equation(5) and the lower limit
+of Vmax integration in equation (11) and (39) in Cole (2011).
+
+4
+Yang et al.
+observe that our modifications enhance the clustering
+measurement accuracy significantly.
+Farrow et al. (2015) recently developed the Cole
+(2011) technique to quantify the property-dependent
+galaxy clustering of GAMA II data (Driver et al. 2011;
+Liske et al. 2015). They found that the Cole (2011) tech-
+nique yields a redshift distribution that is too broad for
+cloned random galaxies, which may be the result of lumi-
+nosity evolution. To mitigate this unanticipated impact,
+Farrow et al. (2015) developed a Gaussian window func-
+tion to restrict the redshift distribution of the cloned
+galaxies. In the first place, the mock galaxy catalogs
+that we construct in this study resemble the low red-
+shift SDSS data, as opposed to the GAMA data, which
+encompass a relatively broad redshift range of 0∼0.5. In
+our mock galaxies, luminosity evolution is expected to
+have negligible effects. Second, our first adjustment to
+the zmin calculation narrows the distribution of cloned
+random galaxies. Our test findings in Section 4.2 will
+demonstrate that the smoothed density-corrected Vmax
+approach is adequate for obtaining accurate galaxy clus-
+tering determination.
+3. THE MOCK GALAXY CATALOGS
+In this part, we describe the construction of mock
+galaxy catalogs for a robust test of the smooth density-
+corrected Vmax approach on clustering estimation. We
+build two sets of mock samples, one with simple k + e-
+corrections and the other with complex k+e-corrections
+for galaxies.
+The first group of mock galaxy catalogs is created in
+a manner similar to Paper I. For the halo catalog, we
+adopt the WMAP 3072 600 cosmological N-body sim-
+ulation from the CosmicGrowth simulation suite (Jing
+2019). This simulation starts at redshift 144 with 30723
+particles evolving in a 600 h−1Mpc cube box. The sim-
+ulation assumes a standard flat ΛCDM cosmology with
+{Ωm = 0.268, Ωb = 0.045, σ8 = 0.83, ns = 0.968} and
+h = H0/(100 km s−1Mpc−1) = 0.71, which are compati-
+ble with the Nine-Year Wilkinson Microwave Anisotropy
+Probe (WMAP 9) observations (Hinshaw et al. 2013;
+Bennett et al. 2013). This simulation has a mass reso-
+lution of 5.54 × 108 h−1M⊙. To identify halos for each
+output snapshot, the friends-of-friends technique is used
+with a linking length of 0.2 in units of the mean parti-
+cle separation (Davis et al. 1985). Hierarchical Bound-
+Tracing technique is used to find subhalos and their
+merger histories. In this study, the snapshot at z = 0
+is utilized to build the halo catalog, and each halo con-
+tains at least 50 particles. The “orphan” halos are also
+maintained in the catalog 4 (Yang et al. 2019).
+We use the subhalo abundance matching (SHAM)
+method to establish the connection between galaxies
+and subhalos. Based on the galaxy absolute magnitude
+M 0.1
+r
+and the peak mass Mpeak of subhalos, a mono-
+tonic relationship between the cumulative number den-
+sity n(< M 0.1
+r
+) = n(> Mpeak) has been constructed
+(Conroy et al. 2006; Hearin et al. 2014; Wechsler & Tin-
+ker 2018; Contreras et al. 2021).
+We employ the lu-
+minosity function of the SDSS DR7 full 1 sample of
+the New York University Value-Added catalog (NYU-
+VAGC)5 (Blanton et al. 2001, 2003, 2005), for which
+the r−band absolute magnitude M 0.1
+r
+of galaxies has
+been k− and e−corrected to z = 0.1.
+The Mpeak is
+the maximum mass ever attained by a subhalo over its
+entire evolutionary history.
+Once a subhalo has been
+matched to a galaxy, its position and velocity are given
+to the galaxy. By periodically rotating and stacking the
+mock box, we generate 60 mock galaxy catalogs from
+the parent catalog. Random sites are assigned to the ob-
+server. The observed redshift zobs is determined by the
+galaxy’s position and velocity relative to the observer.
+To obtain the apparent magnitude mr, the k− correc-
+tion and e− correction, as described in equation (4) and
+equation (5), must be provided. Real data processing
+determines these values by fitting the observed galaxy
+flux to a library of synthetic spectrum models, which is
+generally inapplicable to mock galaxies and also beyond
+the scope of this work. For the sake of simplicity, we
+consider two simple k− and e−correction cases here. In
+the first case, no k+e corrections are applied to the mock
+galaxies. In the second case, we suppose that all galax-
+ies follow a simple k− and e−correction model. For the
+k−correction, we take the model of Smith et al. (2017):
+k0.1(z) =
+4
+�
+i=0
+Ai(z − 0.1)4−i.
+(6)
+Smith et al. (2017) fit the above fourth-order polynomial
+to individual GAMA galaxies, where Ai is the polyno-
+mial’s fitting coefficient (McNaught-Roberts et al. 2014).
+There are seven color-dependent k(z) models (see sec-
+tion below) and we adopt the (g − r)0.1
+med = 0.603 model
+with the following fitting coefficients:
+A0 = −3.428,
+A1 = 9.478, A2 = −2.703, and A3 = 0.7646. For the
+4 In the evolution process, some subhalos go below the resolution
+limit due to the tidal stripping.
+We keep subhalos whose in-
+fall time is shorter than the merger time, and those subhalos do
+not merge into the core of the host halo and host the “orphan”
+galaxies.
+5 lfvmax − q2.00a − 1.00.dr72full1.fits.
+
+The smoothed Vmax method
+5
+e−correction, we use the SDSS model (Blanton 2006) :
+E(z) = q0[1 + q1(z − z0))](z − z0),
+(7)
+where z0 = 0.1 is the zero point redshift for evolu-
+tion correction, q0 = 2 denotes the evolution of mag-
+nitude per redshift, q1 = −1 is the nonlinear param-
+eter in redshift evolution. After applying the k− and
+e−corrections to the mock galaxies, our final samples
+are constructed as follows.
+For each mock catalog in
+each k + e correction case, we first generate a flux-
+limited sample with flux cuts at mr = [15, 17] and a
+sky coverage of ∼ 5950 deg2.
+The flux-limited cata-
+log is then divided into two luminosity-dependent sam-
+ples, named LC1 with M 0.1
+r
+= [−19, −22] and LC2 with
+M 0.1
+r
+= [−20, −23]. Using these selection criteria, the
+galaxy sample’s number density changes as a function
+of redshift. Figure 13 in Appendix A displays the av-
+erage number density n(z) of 60 samples for two lumi-
+nosity cuts in each k + e correction case. This redshift-
+dependent number density typically prevents us from
+obtaining an accurate measurementof galaxy clustering
+particularly at scales ≤ 30h−1Mpc for flux-limited sam-
+ples (Yuan et al. 2022a). In the following text, the above
+mock samples generated from the simulation of (Jing
+2019) are referred to as LC samples.
+The second group of mock galaxy catalogs is built
+from the lightcone catalog of Smith et al. (2017) 6. It
+is essential to access the radial selection model using a
+catalog of galaxies that closely resembles the observed
+galaxies. The Smith et al. (2017) catalog is constructed
+using the MXXL simulation (Angulo et al. 2012), which
+assumes a ΛCDM cosmology with WMAP1 parameters
+{Ωm = 0.25, σ8 = 0.9, ns = 0.968, h = 0.73} and
+operates in a 3h−1Gpc box. The mass of the particle
+is 6.17 × 109h−1M⊙.
+Smith et al. (2017) created the
+lightcone catalog by applying the halo occupation dis-
+tribution method to link galaxies to subhalos. To assign
+colors to the galaxies, they utilize an enhanced redshift-
+dependent Skibba & Sheth (2009) model. The galaxy
+k+e corrections in their lightcone catalog are more com-
+plicated than the ones we use for LC samples. They em-
+ploy color-dependent k−corrections obtained from the
+GAMA survey for the k−corrections. In brief, they esti-
+mate the k−corrections for individual galaxies in GAMA
+data by fitting with equation (6), and they determine the
+median k−correction in seven evenly spaced color bins to
+construct seven k−correction models. These models are
+(g −r)0.1
+med = 0.131, 0.298, 0.443, 0.603, 0.785, 0.933, 1.067
+with different polynomial coefficients. The k(z, color) is
+6 http://icc.dur.ac.uk/data/
+then interpolated for the lightcone catalog using seven
+median color (g − r)0.1
+med models based on the galaxy’s
+color and redshift 7.
+For the LF evolution, they em-
+ployed the evolving Schechter function derived from
+GAMA data. In the low redshift region z ≲ 0.13, the LF
+of their catalog coincides with the LF of Blanton et al.
+(2003), which we employ for LC samples, , and in the
+median redshift region, the LF evolves to the GAMA
+LF. The luminosity(color)-dependent galaxy clusterings
+in Smith et al. (2017) catalog are generally consistent
+with the SDSS DR7 results measured by Zehavi et al.
+(2011) at low redshift, as well as the GAMA results mea-
+sured by Farrow et al. (2015) at the median redshift.
+Therefore, this catalog is suitable for testing different
+radial selection models for property-dependent cluster-
+ing measurement. We construct ten flux-limited sam-
+ples from the full-sky lightcone catalog by rotating the
+sky, using the galaxy selection criteria (mr = [15, 17])
+and sky coverage (∼ 5950 deg2).
+Two luminosity-
+dependent galaxies, LS1 (M 0.1
+r
+= [−19, −22]) and LS2
+(M 0.1
+r
+= [−20, −23]), are generated from each flux-
+limited sample, much as we did for the LC samples.
+As our sample selection resembles the SDSS DR7 data,
+we further divide the luminosity-dependent sample into
+blue subsample and red subsample using the color-cut
+equation (g − r)0.1
+cut = 0.21 − 0.03M 0.1
+r
+of Zehavi et al.
+(2011). In the rest of this study, we refer to the mock
+galaxy samples built from Smith et al. (2017) catalog as
+LS samples.
+In summary, we generate two sets of mock samples
+from two simulations using the same selection criterion
+for galaxies.
+For the LC samples, flux-limited sam-
+ples are constructed from sixty mocks with two abso-
+lute magnitude cuts. Two cases are considered for k + e
+corrections: (1) there are no k + e corrections; (2) all
+galaxies are assumed to follow a simple k + e correction
+model. Ten LS samples are created in the same man-
+ner as LC samples, but using a public lightcone cata-
+log. The LS samples, however, feature a color-dependent
+k−correction and a complex e−correction that are un-
+known to us. In order to examine the color-dependent
+clustering, the luminosity-dependent LS data are split
+into blue and red subsamples. We emphasize that nei-
+ther the LC samples nor the LS samples are subjected
+to any deliberate impact (e.g., fiber collision) in or-
+der to decrease unknown systematic uncertainty in our
+later tests.
+In addition, when calculating the comov-
+ing distance from redshift, we employ the cosmological
+7 For details see Setion4.3 in Smith et al. (2017)
+
+6
+Yang et al.
+model of the simulation from which the samples are con-
+structed, separately.
+4. TEST THE SMOOTHED
+DENSITY-CORRECTED Vmax METHOD WITH
+THE 2PCFS
+In this section, we describe the construction of ran-
+dom galaxy catalog, focusing on the radial distribution
+of random galaxies derived from various radial selec-
+tion models. Following that, we compare the correla-
+tion functions generated by the random catalogs used in
+these models.
+4.1. Construction of the random catalogs
+The random catalogs are constructed as follows. For
+the angular distribution, we first generate a large num-
+ber of random points that are uniformly dispersed across
+the surface of a unit sphere.
+For each mock sample
+and subsample, we extract a collection of points with
+the same sky coverage as the corresponding sample and
+subsample.
+We consider the positions of these points
+to be the angular distribution (ra, dec) of the random
+galaxies, with no angular selection effect or survey masks
+imposed. For the redshifts of random galaxies, the fol-
+lowing radial selection models are used in our tests:
+1. ntrue method, which generates the redshift distri-
+bution for random galaxies using the true galaxy
+number density n(z)true taken from the LF of the
+parent mock catalog.
+2. VSDC
+max method, in which redshifts for random cat-
+alog are generated using the smoothed density-
+corrected Vmax method.
+3. VDC
+max method, in which the density-corrected
+Vmax method of Cole (2011) is utilized, but with-
+out the smoothing procedure.
+4. Vmax method, where the normal Vmax method is
+adopted.
+5. Shuffled method, which applies the redshift shuf-
+fled method. In this method, galaxy redshifts of
+the sample are randomly assigned to the random
+galaxies.
+For LC samples, it is simple to incorporate the k + e
+corrections into the redshift generation process.
+En-
+abling the validation of the capacity of different radial
+selection models to restore the true radial selection func-
+tion n(z)true. Figure 1 shows comparison between the
+radial distributions of a single LC sample and random
+catalogs generated by aforementioned radial selection
+methods in the case of no k + e corrections. In the left
+and right panels, the comparisons for LC1 and LC2 sam-
+ples are presented, respectively. The second row of pan-
+els displays the deviation of random galaxy number rela-
+tive to the galaxy number in each comoving distance bin,
+which is defined as ∆g = (nr −ng)/ng. The third row of
+panels displays the deviation of the random galaxy num-
+ber of the other four techniques from the number of the
+ntrue approach, defined as ∆ntrue = (nr −nr,true)/nr,true.
+The black histograms in the top row of panels denote
+the distribution of galaxies in flux-limited samples. The
+radial distribution of random catalogs created by the
+ntrue method is represented by green lines, which in-
+dicate the distribution arising from the genuine selec-
+tion function. The purple dashed line is the distribution
+producing from the V DC
+max approach. We see small fluc-
+tuations in the radial distribution, which are notably
+clear in the bottom row of panels. These noisy fluctua-
+tions have been reduced by the smoothing process in the
+V SDC
+max approach; as indicated by the blue solid lines, the
+smoothed radial distribution is in excellent agreement
+with the distribution predicted by the ntrue method.
+The radial distributions from the Vmax method and
+the shuffled method are represented by red and yel-
+low lines, respectively.
+As shown in the bottom pan-
+els, ∆ntrue of the Vmax approach exhibits a systematic
+bias in both luminosity-dependent LC samples as a re-
+sult of the influence of large-scale structures in galaxy
+radial distribution. The Vmax approach creates an ex-
+cess of random galaxies near these structures; hence, the
+amount of random galaxies in the high-redshift tail have
+been decreased. Figure 2 shows the same comparison as
+Figure 1 for LC samples with the simple k + e correc-
+tions. The deviations of different approaches from the
+ntrue method shown in the bottom panels are similar to
+those in Figure 1.
+Figure 3 shows a comparison for the LS samples, em-
+ploying the same color-coded lines as Figure 1. The left
+panels compare an LS1 sample, whereas the middle and
+right panels compare its blue and red subsamples, re-
+spectively.
+For the ntrue method, the radial selection
+function derived from the LF of the lightcone catalog is
+applied. The k + e corrections are appropriately incor-
+porated into the redshift generation process for the ntrue
+and Vmax methods. For the V SDC
+max
+and V DC
+max methods,
+the same k−correction models that Smith et al. (2017)
+performed for their lightcone database are employed,
+which interpolate the k−correction from seven models
+based on the color and redshift of individual galaxies.
+The e−correction is also properly applied to LS samples
+and their color-dependent subsamples by using the evo-
+lutionary property of the lightcone catalog. The results
+
+The smoothed Vmax method
+7
+of the comparison are generally consistent with those of
+the LC samples. The redshifts generated by the Vmax
+technique are substantially influenced by the sample’s
+structures; the bias in ∆ntrue is greater than that of LC
+samples, which reaches 20% on the high redshift tail (red
+solid lines). The redshifts from the V SDC
+max approach suc-
+cessfully mitigate this impact, resulting in a relatively
+small deviation in ∆ntrue (blue solid lines).
+For both
+LC and LS samples, the redshifts of random catalogs
+obtained by the shuffled approach replicate the radial
+distribution of galaxies (yellow solid lines), hence, the
+structures are also cloned. In the following section, we
+will examine how galaxy clustering measurements are
+affected by the deviations in these radial distributions
+that differ from the expected distribution produced by
+the ntrue model.
+4.2. Comparison of the correlation functions
+This section introduces the 2PCF estimator that we
+employ to measure galaxy clustering. Then, we provide
+comparison of the projected 2PCFs and the redshift-
+space 2PCFs determined from random catalogs gener-
+ated by the aforementioned radial selection methods.
+4.2.1. Estimator
+We measure the 2PCF in the same way as Paper I.
+First, we define the redshift separation vector s and the
+line-of-sight vector l as s ≡ υ1 −υ2 and l ≡ (υ1 +υ2)/2,
+where υ1 and υ2 are redshift-space position vectors of
+a pair of galaxies (Hamilton 1992; Fisher et al. 1994).
+Separations that are parallel (π) and perpendicular (rp)
+to the line-of-sight direction are derived as
+π ≡ s · l
+|l| ,
+r2
+p ≡ s · s − π2.
+(8)
+We construct a grid of π and rp by taking 1 h−1Mpc
+as the bin size for π from 0 up to πmax = 40 h−1Mpc
+linearly, and a bin size of 0.2 for rp is adopted logarith-
+mically in the range of [0.01, 40] h−1Mpc. Estimator
+of Landy & Szalay (1993) is used to calculate the 2D
+correlation function as
+ξ(rp, π) = DD − 2DR + RR
+RR
+,
+(9)
+where DD, DR, and RR are the numbers of data-data,
+data-random, and random-random pairs. Given s2 =
+|s|2 = r2
+p + π2, we derive the redshift-space correlation
+function ξ(s). By integrating ξ(rp, π) along the line-of-
+sight direction, we estimate the projected 2PCF (Davis
+& Peebles 1983) by
+wp(rp) ≡ 2
+� ∞
+0
+ξ(rp, π) dπ ≈ 2
+� πmax=40
+0
+ξ(rp, π) dπ.
+(10)
+We employ the public code CORRFUNC (Sinha & Garri-
+son 2019) for pair counting in this work. To reduce the
+shot noise on small-scale clustering, we use 50 times the
+number of galaxies in the random catalogs for random
+galaxies.
+4.2.2. Comparison of projected 2PCFs
+The projected 2PCFs for LC samples without and
+with simple k + e corrections are compared in Figure 4
+and Figure 5, respectively. We compare the average pro-
+jected 2PCF estimated using random catalogs produced
+by the radial selection models outlined in Section 4.1. In
+the left and right panels for the LC1 and LC2 samples,
+respectively, the estimated mean wp of 60 mock samples
+are displayed. In the top panels, wp,true computed using
+random catalog from the ntrue model is represented by
+solid black points with errors representing the 1σ dis-
+persion across individual wp,trues of samples. The blue
+dashed lines, green dotted lines, red long-dashed lines,
+and orange lines represent wps estimated from random
+catalogs of the V SDC
+max
+technique, V DC
+max method, Vmax
+method, and shuffled method, respectively. The aver-
+age offsets [wp − wp,true] from wp,true for the models are
+shown in the middle row of panels, which are defined as
+[wp − wp,true] =
+1
+60
+�60
+i=1 wi
+p − wi
+p,true, where wi
+p is the
+projected 2PCF measured for the ith LC sample. The
+offsets increase when the scale drops below 1h−1Mpc for
+both the V DC
+max method (green dotted lines) and shuffled
+method (orange diamonds).
+When using the random
+catalogs of the V SDC
+max technique to measure wp, the little
+positive offsets in the blue open rolls with error bars in-
+dicate a slight overestimation on scale rp ≲ 0.4h−1Mpc.
+On a small scale, there are apparent offsets for the Vmax
+approach for LC1 samples in both k+e correction cases,
+as seen by the open red squares with error bars. For
+LC2 samples, there are extremely modest systematic
+offsets for the Vmax technique across all of the scales
+tested, and these offsets are smaller than those for the
+V SDC
+max method. Compared to the 1σtrue (gray solid lines)
+among 60 wp,trues, the V SDC
+max and Vmax methods’ offsets
+are essentially insignificant.
+In the bottom panels of Figure 4 and Figure 5, we dis-
+play the average deviation from wp,true for each model,
+using the same color-coded symbols and lines as the mid-
+dle panels. The mean deviation [(wp − wp,true)/wp,true]
+is calculated from 60 mock samples in the same manner
+as [(wp − wp,true)]. Clearly, wps derived using random
+catalogs from the V SDC
+max approach provide a mostly un-
+biased estimate of the genuine projected 2PCFs for both
+LC1 and LC2 samples in both no k + e correction case
+(Figure 4) and simple k + e correction case (Figure 5).
+The 1σ deviations among 60 samples for the V SDC
+max ap-
+
+8
+Yang et al.
+Figure 1. In the case of no k + e correction, a comparison of the radial distributions of one LC sample and its corresponding
+random catalogs. The bin size is ∆d = 5 h−1Mpc. The LC samples have a flux cut at mr = [15, 17] and two luminosity cuts
+at M 0.1
+r
+= [−19, −22] (left panels) and M 0.1
+r
+= [−20, −23] (right panels). The black histogram denotes galaxy distribution.
+Random catalogs generated by the n(z)true method, the V SDC
+max
+method, the V DC
+max method, the Vmax method, and the shuffled
+method are represented by the green line, the blue line, the purple dashed line, the red line, and the yellow line, respectively.
+The second row of panels displays the number bias ∆g in each bin of the random catalogs compared to the galaxies, calculated
+as ∆g = (nr − ng)/ng. The third row of panels displays the number bias of random catalogs compared to n(z)true, which is
+defined as ∆ntrue = (nr − nr,true)/nr,true.
+Figure 2. The same as Figure 1 but for the simple k + e correction case of LC samples.
+
+4000-
+Galaxies
+ntrue
+Vmax
+4000
+VSDC
+Shuffled
+3000
+max
+3000
+Number
+mr=[15,17]
+2000
+mr=[15,17]
+2000
+M9.1=[-19,-22]
+M9.1=[-20, -23]
+1000
+1000
+No k + e corrections
+0
+0-
+100
+200
+400
+500
+100
+400
+500
+300
+200
+300
+600
+0.5
+0.5
+0.0
+-0.5
+-0.5
+100
+200
+400
+500
+100
+200
+400
+600
+300
+300
+500
+0.1
+0.1
+0.0
+0.0
+-0.1
+-0.1
+200
+300
+400
+500
+100
+200
+400
+500
+600
+100
+300
+d h-1Mpc
+d h-1MpcGalaxies
+VDC
+max
+4000
+ntrue
+Vmax
+3000
+VSDC
+Shuffle
+max
+3000
+Number
+2000
+mr=[15,17]
+mr=[15,17]
+2000
+M9.1=[-20,-23]
+M9.1 =[-19, -22]
+1000
+1000
+Simple k + e corrections
+0
+0
+100
+200
+400
+500
+400
+500
+300
+100
+200
+300
+600
+0.5
+0.5
+0.0
+0.0
+0.5
+0.5-
+200
+400
+500
+100
+500
+600
+100
+300
+200
+300
+400
+0.1
+0.1
+Itrue
+0.0
+0.0
+-0.1
+-0.1
+200
+300
+100
+400
+500
+100
+200
+400
+500
+600
+300
+d h-1Mpc
+d h-1MpcThe smoothed Vmax method
+9
+Figure 3. The same as Figure 1 but for the LS1 samples.
+
+All
+Blue
+Red
+6000
+Vmax
+Galaxies
+2000-
+3000
+ntrue
+Shuffle
+VDC
+1500-
+max
+2000
+1000-
+2000
+1000
+Imr=[15,17]
+500
+M9.1 =[-19, - 22]
+0 -
+0-
+0
+200
+400
+600
+200
+800
+400
+600
+800
+200
+400
+600
+800
+0.5
+0.5-
+0.5
+0.0
+0.0
+0.5
+-0.5-
+-0.5
+600
+600
+600
+200
+400
+800
+200
+400
+800
+200
+400
+800
+0.2
+0.2
+0.2
+NM
+0.1
+0.1
+0.1
+0.0
+0.0
+0.0
+N
+K
+-0.1
+-0.1-
+-0.1-
+-0.2
+-0.2
+-0.2
+200
+400
+600
+800
+200
+400
+600
+800
+200
+400
+600
+800
+d h-1Mpc
+d h-1Mpc
+d h-1Mpc10
+Yang et al.
+Figure 4. Top panels: The average projected correlation functions wp for LC1 (left panel) and LC2 (right panel) samples in
+the case of no k + e corrections. LC1 samples have a flux-cut at mr = [15, 17] and a luminosity cut at M 0.1
+r
+= [−19, −22]. LC2
+samples have the same flux-cut as LC1 samples but a brighter luminosity cut at M 0.1
+r
+= [−20, −23]. The solid black points
+with error bars represent the wp,true and 1σ dispersion across 60 LS samples utilizing random catalogs generated by the ntrue
+approach. wp of the V SDC
+max
+method, V DC
+max method, Vmax method, and shuffled technique are shown by the blue dashed lines,
+the green dotted line, the red long-dashed lines, and the orange lines, respectively. Middle panels: The average deviations
+from wp,true for various techniques of assigning redshifts to random catalogs, as determined by wp of 60 LC samples. The blue
+open rolls with error bars represent the mean offset and 1σ deviations of wp for the V SDC
+max technique. The results of the Vmax
+technique are displayed as open red squares with error bars. The mean offsets computed from wp for the V DC
+max and shuffled
+methods are shown by green dashed lines and a yellow open diamond, accordingly. The gray lines represent the 1σ dispersion
+of wp,true among 60 LC samples. The horizontal dashed black lines indicate the zero offset. Bottom panels: The average bias
+of wp relative to wp,true for four radial selection models, defined as [(wp − wp,true)/wp,true]. The color-coded lines and symbols
+are identical to those in the middle panels.
+proach (blue error bars) are significantly smaller than
+those for the Vmax method (red error bars). For LC1
+samples in both k + e correction cases, the Vmax ap-
+proach underestimates wp by less than 1%, and this bias
+worsens as scale grows. At rp ∼ 30h−1Mpc, the bias
+reaches 13% with a substantial variance 8. For LC2 sam-
+8 This bias is marginally less than the 20% bias found for the Vmax
+approach by Paper I. This may be owing to the increase in the
+number of galaxies in the samples, as the LC samples cover twice
+as much sky as the flux-limited samples in Paper I.
+ples, the measurement accuracies for both the V SDC
+max and
+Vmax methods are equivalent at scale rp ≲ 4h−1Mpc for
+both methods. On a larger scale, deviation of the Vmax
+method grows to 4%, but remains within the margin of
+error.
+These discrepancies in wp from wp,true for the
+Vmax model are mostly attributable to density fluctua-
+tions in galaxy samples. wps measured using random
+catalogs from the V DC
+max approach are overestimated at
+scale rp ≲ 2h−1Mpc and underestimated at larger scales
+for both LC1 and LC2 samples as shown in the bottom
+panels (green dashed lines) of Figure 4 and Figure 5. As
+
+103
+No k + e corrections
+mr = [15,17]
+mr = [15, 17]
+M0.1 = [-19, -22]
+Mo.1 = [-20, -23]
+102
+10-
+ntrue
+Vmax
+dm
+VSDC
+ShufHed
+max
+100
+VDC
+max
+8
+dm
+true
+-8
+0.05
+0.00
+'dm
+-0.05
+_dm.
+-0.10
+豆
+Vmax
+VSDC
+0
+max
+Shuffed
+VDC
+max
+-0.15
+10
+100
+101
+10°
+100
+101
+Tp (h-1Mpc)
+p (h-1Mpc)The smoothed Vmax method
+11
+Figure 5. The same as Figure 4 but for the LC samples with a simple k + e corrections.
+seen in Figure 1 and Figure 2, this tendency of deviation
+is the result of small fluctuations in the radial distribu-
+tion of the random catalog generated by the V DC
+max model.
+In essence, the fluctuations increase the number of RR
+pairs at the fluctuation-scale, resulting in an underes-
+timating of wp. Due to the integral constraint effect,
+a small-scale overestimation of wp is unavoidable. Af-
+ter smoothing out the fluctuations, the V SDC
+max approach
+yields estimates that are almost unbiased of wp,true. The
+results of the shuffled technique are consistent with Pa-
+per I, which shows that an underestimating of wp grows
+as the scale increases.
+Due to the severe deviations of wps for the V DC
+max model
+in the tests using LC samples, the following compari-
+son for LS samples will focus on testing for the V SDC
+max
+method, Vmax method, and shuffled method. Figure 6
+and Figure 7 display comparison results for LS sam-
+ples with the two luminosity-cuts, respectively.
+The
+left, middle, and right panels, respectively, present wp
+comparisons for luminosity-dependent samples and their
+blue and red subsamples. From 10 mock galaxy samples,
+the mean wp, [wp − wp,true], [(wp − wp,true)/wp,true] are
+calculated (from top to bottom panels).
+The ntrue
+method, the V SDC
+max method, the Vmax method, and the
+shuffled method all utilize the same color-coded lines
+and symbols as those used for figures of LC samples.
+For the LS1 samples in Figure 6, the V SDC
+max model pro-
+duces tiny wp offsets from wp,true, which are consistent
+with the findings for LC samples.
+Significant offsets
+are seen for the Vmax and shuffled methods, notably
+for the LS1 samples and their blue subsamples, where
+the offsets are more than 1σ dispersion of wp,true at
+rp ≲ 3h−1Mpc scale. The average deviations displayed
+in the bottom panels clearly demonstrate the superior-
+ity of the V SDC
+max
+approach over the Vmax method and
+the shuffled method when measuring projected 2PCFs.
+∼ 0.5% deviations are detected for both LS1 samples
+and their color-dependent subsamples, which is essen-
+tially within the 1σ error margin.
+For the Vmax ap-
+proach, [(wp − wp,true)/wp,true]s deviate by 6%, 5%, and
+9% for LS1 samples, blue subsamples, and red subsam-
+ples, respectively, which are considerably larger than 1σ
+errors.
+At rp ≲ 10h−1Mpc, the mean deviations for
+the shuffled approach are marginally better than those
+
+103
+Simple k + e corrections mr = [15, 17]
+mr = [15,17]
+M0.1 = [-19, -22]
+M0.1 = [-20, -23]
+102
+10-
+Vmax
+dm
+ntrue
+VSDC
+ShufHed
+max
+100
+VDC
+max
+8
+true
+-8 
+0.05
+0.00
+'dm
+-0.05
+dm.
+-0.10
+Vmax
+VSDC
+豆
+0
+max
+Shuffed
+VDC
+max
+-0.15
+10
+100
+101
+100
+10°
+101
+rp (h-1Mpc)
+p (h-1Mpc)12
+Yang et al.
+Figure 6. Similar to Figure 4: comparison of wp for LS1 samples (left panels) and their blue (middle panels) and red (right
+panels) subsamples. The color-coded lines and symbols are identical to those in Figure 4, excluding the result of the V DC
+max
+technique.
+for the Vmax method, but worsen as the scale increases,
+which is consistent with the test results for LC samples.
+Figure 7 presents a comparison of wps for the LS2 sam-
+ples. The offsets from wp,true for the V SDC
+max
+technique
+are roughly comparable with the LS1 sample results.
+wps measured using random catalogs from the Vmax ap-
+proach exhibits large offsets from wp,true that are worse
+than the offsets for the shuffled method on small scales,
+particularly for LS2 samples (left middle panel) and red
+subsamples (right middle panel). In the bottom pan-
+els of Figure 7, the accuracy of measurement for three
+models is shown clearly. At scale rp < 1h−1Mpc, there
+is a ∼ 0.5% underestimate for the LS2 samples (bottom
+left panel). At a larger scale, this deviation becomes an
+overestimation, reaching 2% at rp ∼ 30h−1Mpc while
+being within the margin of error. The mean deviations
+for the blue and red subsamples are well constrained
+within 1%. The results of the Vmax approach exhibit
+larger mean deviations than the LS1 samples, which are
+even worse than the results of the shuffled method. The
+deviations for LS2 samples, blue subsamples, and red
+subsamples are roughly 9%, 8%, and 10%, respectively.
+wps determined for red subsamples exhibit more severe
+departures from wp,true for the Vmax technique for both
+LS1 and LS2 samples, demonstrating density fluctua-
+tions have a greater impact on clustering determination
+for red galaxies.
+To better quantify the measurement accuracy of pro-
+jected 2PCF for various radial selection models, we cal-
+culate the χ2 between wp and wp,true for the V SDC
+max tech-
+nique, the Vmax method, and the shuffled method, re-
+spectively, as shown in Table 1. χ2 is computed as fol-
+lows:
+χ2 =
+N
+�
+i=0
+(wi
+p − wp,true)2
+σ2
+true
+.
+(11)
+The number of mock samples N is 60 for LC samples
+and 10 for LS samples. For the LC samples, with the
+exception of the LC2 samples with simple k + e correc-
+tions for which χ2s of the V SDC
+max method and the Vmax
+
+103
+mr = [15, 17]
+Blue subsamples
+Red subsamples
+(h-1Mpc)
+M0.1
+=「19,—22]
+102
+FLLL
+ntrue
+dm
+VSDC
+max
+101
+Shuffled
+60 
+ToI
+VSDC
+ShufHed
+max
+40
+Vmax
+Otrue
+20
+0
+口
+口
+合
+合
+dm
+-20
+T
+40 
+?
+0.89
+0.00 -
+-0.02
+ant'dm.
+-0.04
+anut'dm
+-0.06
+_dm.
+-0.08
+-0.10
+-0.12
+100
+101
+10
+100
+101
+10-
+100
+10-
+-1
+101
+rp (h-1Mpc)
+Tp (h-1Mpc)
+rp (h-1Mpc)The smoothed Vmax method
+13
+Figure 7.
+The same as Figure 6 but for LS2 samples (left panels) and their blue (middle panels) and red (right panel)
+subsamples.
+method are essentially equal, wps of the V SDC
+max method
+exhibit the least χ2 from wp,true when compared to other
+two models. For all LS samples and their blue and red
+subsamples, the V SDC
+max approach also yields the least χ2
+among three methods. The χ2 values for the LS sam-
+ples are greater than those for the LC samples for all
+three models. This may probably due to the fact that
+the LS samples built from a lightcone catalog contain
+more complicated k + e corrections than LC samples.
+On the basis of the preceding figures and χ2 tests, we
+demonstrate that wps measured using the random cata-
+logs generated by the V SDC
+max approach result in the least
+deviation from wp,true for both flux-limited samples and
+their color-dependent subsamples. In Section 5, we pro-
+vide more discussion on the performance of the radial
+selection models for LC and LS samples
+4.2.3. Comparison of the redshift-space 2PCFs
+The redshift-space correlation functions are compared
+in the same manner as wp for both the LC and LS sam-
+ples, and the results for different radial selection models
+Table 1. χ2 of the projected 2PCFs for the mock samples
+Samples
+χ2
+V SDC
+max
+Vmax
+Shuffled
+LC1(no k + e)
+1.364
+6.264
+107.225
+LC2(no k + e)
+1.460
+4.254
+62.329
+LC1(simple k + e)
+3.531
+6.351
+108.770
+LC2(simple k + e)
+2.757
+2.667
+106.466
+LS1
+1.893
+1618.495
+977.362
+LS1 (blue)
+33.013
+161.187
+124.543
+LS1 (red)
+19.525
+2769.991
+1988.678
+LS2
+45.168
+3416.047
+857.843
+LS2 (blue)
+63.572
+925.400
+240.416
+LS2 (red)
+71.431
+5054.464
+1562.508
+are generally consistent with the comparisons for wps in
+the previous section. The mean ξ0, [ξ0 − ξ0,true], and
+[(ξ0 − ξ0,true)/ξ0,true] for LC samples with simple k + e
+corrections are shown in Figure 8, from top to bottom,
+
+103
+mr = [15, 17]
+Blue subsamples
+Red subsamples
+(h-1Mpc)
+M0.1
+=「—20,—23]
+102
+ntrue
+dm
+VSDC
+max
+101
+ShufHed
+60 
+ToI
+VSDC
+ShufHed
+max
+40
+Vmax
+true
+20
+0
+LOHO
+OHOO
+Ol
+口
+dm
+口
+口
+-40
+TOI
+-60
+0.02
+0.00 .
+-
+0.02
+nf'dm
+-0.04
+ut'dm
+-0.06
+dm.
+-0.08
+-0.10
+-0.12
+100
+101
+10
+100
+101
+10-
+100
+10-
+101
+rp (h-1Mpc)
+Tp (h-1Mpc)
+rp (h-1Mpc)14
+Yang et al.
+Figure 8. Similar to Figure 4, a comparison of ξ0s for the redshift-space 2PCFs of LC1 samples (left panels) and LC2 samples
+(right panels) with simple k + e corrections.
+respectively. Estimates of ξ0 derived from random cata-
+logs created by the V SDC
+max approach display the smallest
+offsets and deviations from ξ0,true for both LC1 (left
+panels) and LC2 (right panels) samples. For the V DC
+max
+technique, ξ0s at scale rp < 1h−1Mpc exhibit large off-
+sets and deviations compared to the findings of wp. For
+the Vmax method, ξ0 deviations are marginally attenu-
+ated compared to the results of wp, indicating that the
+impact of density fluctuations on clustering is less signif-
+icant in redshift space. The ξ0s for the shuffled approach
+exhibit the same offsets and deviations from ξ0,true as
+wp. As the results of LC samples without k + e correc-
+tions are similar to Figure 8, they are omitted here.
+Figure 9 illustrates a comparison of ξ0 for LS1 samples
+(left panels), their blue (middle panels), and red (right
+panels) subsamples, respectively. Compared to the Vmax
+and shuffled methods, the V SDC
+max approach produces the
+least offsets and deviations from ξ0,true for LS1 samples
+and red subsamples. For the blue subsamples, the V SDC
+max
+method’s mean offset at s ∼ 0.07h−1Mpc is slightly
+larger than the Vmax method’s mean offset, and both
+approaches have comparable deviations at that scale.
+This is not a worry because the amount of uncertainty
+at this scale is also high due to the shot noise. In general
+on ξ0 measurements, the V SDC
+max
+technique continues to
+outperform the other two radial selection models. Since
+the findings of LS2 samples are basically consistent to
+Figure 9, they are also excluded here.
+In Figure 10, the average 2D correlation functions
+ξ(rp, π) for LS samples are presented. ξ(rp, π)s for LS1
+samples (left panel), blue subsamples (middle panel),
+and red samples (right panel) are displayed in the up-
+per panels. ξ(rp, π)s for the ntrue method, the V SDC
+max
+method, the Vmax method, and the shuffled method are
+represented by black solid lines, blue dashed lines, red
+dashed lines, and yellow dashed lines, respectively. The
+1σtrue dispersion of ξtrue(rp, π) among 10 mock samples
+is denoted by dotted gray lines in places with shading.
+ξ(rp, π)s of the V SDC
+max model provide the best agreement
+with ξtrue(rp, π) for LS1 samples and color-dependent
+subsamples. For ξ(rp, π) of the Vmax method and the
+shuffled method, there are offsets of varying degrees;
+yet, the offsets stay within the 1σtrue error margins;
+however, the contour shapes are altered. In the lower
+panels displaying ξ(rp, π)s for LS2 samples, the major-
+ity of contours for the V SDC
+max model are consistent with
+ξtrue(rp, π). 1% ∼ 2% deviations seen in wp (bottom left
+panel in Figure 7) for both LS2 samples and blue sub-
+samples are also observed in contours at large scale. For
+the Vmax technique and the shuffled method, the offsets
+in the ξ(rp, π) contours are close to the error margins
+of 1σtrue; thus, the contour shapes are altered as well.
+
+102
+mr = [15, 17]
+mr = [15, 17]
+M0.1 = [-19, -22]
+M0.1
+=「—20,23]
+101
+Simple k + e corrections
+100
+ntrue
+VSDC
+ShufHed
+max
+VDC
+max
+10-2
+Tol
+VSDC
+5
+ShufHed
+max
+VDC
+true
+max
+-5
+0.05
+0.00
+50.
+-0.05
+So,
+-0.10
+-0.15
+10
+100
+101
+100
+101
+10
+s (h-1Mpc)
+s (h-1Mpc)The smoothed Vmax method
+15
+Figure 9. Similar to Figure 6, a comparison of ξ0s for the redshift-space 2PCFs of LS1 samples (left panels) and their blue
+(middle panels) and red (right panels) subsamples.
+Since the comparisons for LC samples are substantially
+identical to those in Figure 10, they are excluded here.
+5. DISCUSSION
+Our tests demonstrate that, for flux-limited sample
+with a redshift-dependent number density n(z), utilizing
+the random catalog generated by the V SDC
+max technique to
+measure galaxy clustering produces the least deviation
+from the true clustering when compared to the other
+radial selection methods. Some aspects of the perfor-
+mance of the V SDC
+max technique remain to be clarified and
+discussed, as detailed below.
+5.1. The impact of smoothness parameters on
+clustering estimation
+For the V SDC
+max
+approach, we add a smoothing step
+to eliminate the unanticipated small fluctuations in the
+redshift distribution of the cloned random galaxies gen-
+erated by the V DC
+max method.
+Previous comparison of
+2PCFs for the V SDC
+max
+and V DC
+max methods demonstrate
+the necessity of a smooth procedure for random catalog
+in order to produce a nearly unbiased clustering mea-
+surement for flux-limited sample.
+Smoothing requires
+a selection of histogram bin size ∆d and smooth box
+size ∆smooth.
+To determine the effect of varying ∆d
+and ∆smooth values on the final galaxy clustering de-
+termination, we vary these two smoothness parameters
+and regenerate random catalogs to perform the estimate.
+First, we set ∆d = 5h−1Mpc and ∆smooth = 5 as the
+fiducial case, which we have used for the V SDC
+max
+tech-
+nique in previous tests in Section 4.2. Second, we chose
+∆d = 2.5h−1Mpc and 10h−1Mpc for histogram bin size,
+with ∆smooth = 5 set to smooth.
+Thirdly, we select
+∆smooth = 3 and 7 for smooth with ∆d = 5h−1Mpc
+set.
+Figure 11 displays the average deviations of wp
+from wp,true for random catalogs created by the V SDC
+max
+technique with various ∆d and ∆smooth values. To sim-
+plify the assessment, we just test the projected 2PCFs
+of the LC samples here. In the absence of k + e correc-
+tions, the upper panels of Figure 11 depict the mean
+deviations of wp for the LC1 (left panel) and LC2
+(right panel) samples, respectively. We see that a finer
+
+Blue subsamples
+Red subsamples
+mr = [15, 17]
+EELL
+M0.1
+=「-19,-22]
+101
+LLL
+ntrue
+100
+VSDC
+max
+10-1
+Shuffled
+ToI
+VSDC
+ShufHed
+10 -
+max
+Vmax
+Otrue
+5
+0
+Y
+口
+口
+合
+中
+合
+立
+查
+-5
+-10
+0.03
+0.00
+-0.03
+Eo,tr
+-0.06
+-0.09
+-0.12
+100
+101
+10-
+10
+100
+101
+10-
+100
+101
+s (h-1Mpc)
+s (h-1Mpc)
+s (h-1Mpc)16
+Yang et al.
+Figure 10. Comparison of the average 2D correlation function ξ(rp, π) for the luminosity-dependent flux-limited samples. The
+L1-C1 sample, the L2-C1 sample, and the L3-C1 sample are shown from top to bottom accordingly, their blue/red subsamples
+are shown in the middle and right panels in each row. Here, ξ(rp, π) is the averaged ξ(rp, π) among 60 mock samples. The
+true ξ(rp, π) measured using the random catalog from the n(z)true method is in the black contour. The gray shaded region
+with dotted lines mark the 1σ scatter of the true ξ(rp, π) among 60 mock samples. The yellow, red, and blue dashed contours
+denote the ξ(rp, π) of the shuffled method, Vmax method, and the V SDC
+max method, respectively. The contour levels from outside-in
+correspond to ξ(rp, π) = [0.1, 0.2, 0.3, 0.5, 1.0, 2.0, 5.0]. The middle column and right column panels show the comparison of the
+blue/red subsamples
+value of ∆d = 2.5h−1Mpc (green dashed lines) and
+∆smooth = 3 (light blue lines) lead to a constant drop
+of [(wp − wp,true)/wp,true] on all test scales, resulting
+in reduced deviations at rp ≲ 2h−1Mpc and an un-
+derestimate on a larger scale, especially for LC1 sam-
+ples. In contrast, a coarser size of ∆smooth = 7 (orange
+long-dashed lines) results in an overall increase relative
+to the mean deviation in fiducial case (open blue rolls
+with error bars), resulting in an overestimation at scale
+rp ≲ 20h−1Mpc. A coarser size of ∆d = 10h−1Mpc (yel-
+low short-dashed lines) leads in a ∼ 1% increase in the
+mean deviation of wp relative to the deviation in fiducial
+case; this is the only mean deviation that exceeds the
+1σ errors but is still around ∼ 1%. In the lower panels,
+the test results for LC samples with simple k +e correc-
+tions are displayed, which are essentially identical to the
+findings in the above panels, suggesting that the smooth
+process is insensitive to galaxy samples when different
+k + e corrections are applied. Our tests indicate that
+the variation of ∆d and ∆smooth in the smooth process
+of the V SDC
+max technique affects the accuracy of clustering
+measurement, however the effect on deviations is much
+less than 1%. The advantage of the V SDC
+max technique over
+other radial selection models still stands.
+5.2. Difference in clustering uncertainty
+In prior tests, the uncertainties in clustering devia-
+tions among 60 LC samples are significantly larger than
+the uncertainties in 10 LS samples, which is not expected
+intuitively. In addition, the deviation uncertainties for
+the V SDC
+max approach are approximately a fourth of those
+for the Vmax method in LC samples. As seen in Fig-
+ure 12, we further investigate the radial distribution of
+the LC and LS samples in order to determine the prob-
+able distinct drivers of these discrepancies.
+Here, we
+take into account the LC samples without k + e correc-
+tions and the LS1 samples, which are sufficient to ex-
+plain the difference in uncertainty. Firstly, we compute
+the normalized radial distribution for galaxy samples
+and random catalogs created using the n(z)true method,
+the V SDC
+max
+method, and the Vmax method, respectively.
+
+501
+50 于
+50 
+mr = [15, 17]
+ntrue
+45
+45
+45.
+Mo.1 =[-19,-22]
+Otrue
+Blue subsamples
+Red subsamples
+40
+40
+40
+VSDC
+ max
+35
+35
+35
+Vmax
+1Mpc)
+30
+30
+30
+ShufHed
+C
+25
+25
+25
+s
+20
+20
+20
+15
+15
+15
+10
+10
+5
+5
+5
+0
+0:
+¥40 45 50
+5
+ 10 15 20 25 30 35
+0
+ 30 35 40 45 50
+ 20 25 30 35 40 45 50
+0
+5
+10　15
+20
+¥25
+0
+ 1015
+5
+50于
+50
+mr = [15, 17]
+45
+45 
+45
+M0.1 = [-20,-23]
+40 
+40
+40 
+35
+35
+35
+30
+30
+30 
+25
+25
+25
+20
+20
+20
+15
+15
+15
+10 
+10
+10
+5
+5
+1
+01
+0
+0
+5
+10　15
+20
+25
+3035
+ 404550
+5
+10
+15
+20
+25
+30
+4045
+50
+0
+1015
+2025
+ 3035
+404550
+0
+35
+Tp (h-1Mpc)
+Tp (h-1Mpc)
+rp (h-1Mpc)The smoothed Vmax method
+17
+Figure 11. The average deviations of wp from wp,true for the V SDC
+max
+method, in which alternative histogram bin sizes and
+smooth box sizes are adopted in the smooth process in order to assess the impact of multiple choices on clustering estimation.
+The fiducial bin size and smooth box size used in Section 4.2 are ∆d = 5h−1Mpc and ∆smooth = 5, respectively, as indicated
+by the open blue circles with error bars. The alternate histogram bin sizes are ∆d = 2.5h−1Mpc and ∆d = 10h−1Mpc , with
+the same smooth box size as the fiducial one, as indicated by the green dashed lines and the light blue lines, respectively. The
+alternate smooth box sizes are ∆smooth = 3 and ∆smooth = 7, with the same fixed histogram bin size as the fiducial one, as
+shown by the yellow short-dashed and orange long-dashed lines, respectively. The zero deviation is shown by the horizontal black
+dashed lines. Upper panels: Tests for the LC1 samples (left panel) and LC2 samples (right panel) for the no k + e correction
+case. Lower panels: Similar tests for LC1 and LC2 samples to those in the upper panels, but for the simple k + e correction
+case.
+To quantify the density fluctuations relative to the true
+smooth distribution created by n(z)true method, we esti-
+mate the average deviations ∆ and 1σ variances of these
+distributions from the genuine normalized distribution
+for sixty LC samples and ten LS1 samples separately, as
+shown in Figure 12 from top to bottom.
+The ∆ and 1σ variance for the galaxy samples are
+shown by the thick gray line and thin light gray line.
+For both LC1 (upper panel) and LC2 (middle panel)
+samples, the variations across sixty individual samples
+vary greatly, as indicated by 1σ variance, whereas ∆
+exhibits a relatively small deviation from the true nor-
+malized distribution. The light yellow and light orange
+regions denote the locations in which 90 percent and 60
+percent of the expected random galaxies are likely to
+be distributed, and we anticipate that the bulk of pairs
+used to estimate clustering are from 90% region. ∆ (red
+thick lines) and σ (light red thin lines) of the Vmax tech-
+nique reveal that this approach corrects the fluctuations
+in galaxy samples; nonetheless, the imprints of large-
+scale structures are still discernible. For instance, ∆ for
+LC1 samples shows a small but observable deviation at
+100 ∼ 450h−1Mpc where 90% of galaxies are located.
+This explains the consistent bias noticed in wp and ξ
+in previous testing.
+For LC2 samples, the systematic
+bias is almost imperceptible, with just a tiny overesti-
+mation at d ≳ 500h−1Mpc, indicating a clustering bias
+that has been detected in prior tests.
+For the V SDC
+max
+approach, there are noisy fluctuations in ∆ (blue thick
+lines) for both LC1 and LC2 samples, indicating that
+the smooth does not eliminate all noisy fluctuations in
+radial distribution and there is still room to improve the
+smooth. Fortunately, these fluctuations are complimen-
+tary in certain degree, yielding a substantially unbiased
+measurement for galaxy clustering. We observe that the
+1σ errors (light blue thin lines) for the V SDC
+max approach
+are less than those for the Vmax method, especially for
+the LC1 samples at 60% region. This is essentially the
+
+0.02
+No k + e corrections
+0.01
+0.00
+_dm
+-0.01
+mr = [15, 17]
+mr = [15, 17]
+-0.02
+M0.1 = [-19, -22]
+Mo.1 = [-20,-23]
+-0.03
+0.02
+Simple k + e corrections
+0.01
+0.00
+dm
+0.01
+VSDC (△d = 5, △smooth = 5)
+max
+△d = 10, △smooth = 5
+-0.02
+△d = 5, △smooth = 7
+△d = 2.5, △smooth = 5
+-0.03
+10-1
+100
+101
+100
+101
+rp (h-1Mpc)
+rp (h-1Mpc)18
+Yang et al.
+reason for the substantial difference in uncertainty found
+between the two techniques for wp and ξ, demonstrating
+once again that the V SDC
+max method can more successfully
+rectify the effect of density fluctuations on individual
+samples, and thus the clustering estimations converge
+to the genuine galaxy clustering.
+As demonstrated in the bottom panel, ∆ for the LS1
+samples deviates significantly from the genuine distri-
+bution when compared to the LC samples. By rotating
+the sky, just 10 LS1 samples are created from a single
+lightcone catalog. These samples have a significantly re-
+duced 1σ variance than LC samples, particularly at 60%
+region. In LS1 samples, the advantage of density cor-
+rection in the V SDC
+max approach is exhibited more clearly
+compared to the Vmax method. Both approaches have
+equal errors, but the ∆ of the V SDC
+max method deviates less
+from the true distribution, resulting in a more accurate
+clustering measurement. In contrast, the Vmax technique
+predicts too many random galaxies at d ≲ 400 and fewer
+galaxies at high d due to strong fluctuations in galaxy
+samples, hence exhibiting a greater deviation in ∆ in
+comparison to ∆ of LC samples. This also explains the
+extremely systematic bias in wp observed for the Vmax
+approach on all testing scales in earlier tests.
+Last but not least, the LC samples and LS samples
+are derived from distinct parent mock catalogs utilizing
+two simulations with different resolutions and galaxy-
+halo connection models. Both LC and LS samples are
+complete at M 0.1
+r
+≤ −18, however the simulation of
+(Jing 2019) used to generate LC samples has a mass
+resolution that is an order of magnitude higher than
+that of MXXL simulation (Angulo et al. 2012), imply-
+ing that more halo and galaxy structures are resolved
+in LC samples. Moreover, despite the fact that the LC
+samples are constructed using a simple galaxy-halo con-
+nection model with simple k + e corrections, the benefit
+is that all model parameters are clear and straightfor-
+ward; hence, the potential deviation and error sources
+are comprehendible. For LS samples, with a more so-
+phisticated galaxy evolution and k−correction, the light-
+cone catalog of Smith et al. (2017) is theoretically closer
+to actual observation data; the main drawback is a re-
+stricted number of samples. The test results of these two
+sample groups demonstrate that either the k + e correc-
+tions are based on simple or more complex and realistic
+mock catalogs, the Vmax technique may produce an in-
+accurate measurement of galaxy clustering, whereas the
+V SDC
+max method can always produce an accurate and pre-
+cise estimate of clustering.
+5.3. The effect of k + e corrections on galaxy clustering
+6. CONCLUSIONS
+Figure 12. Top panel: The average deviations ∆ and 1σ er-
+rors from the radial distribution of random catalog obtained
+by the n(z)true method. The mean deviation is computed
+using the equation ∆ = (ni − ni
+true)/ni
+true, where ni is the
+normalized radial distribution of the ith LC1 sample and
+random catalog produced using the V SDC
+max
+and Vmax meth-
+ods. The ∆ of the LC1 samples is shown by the thick gray
+lines, while the 1σ errors over 60 samples are represented
+by the thin gray lines. The thick blue lines and thin light
+blue lines represent ∆ and errors for the random catalogs
+generated by the V SDC
+max
+technique. The thick red lines and
+thin light red lines represent the Vmax algorithm. The light
+yellow and light orange regions indicate the locations of 90%
+and 60% of galaxies, respectively. Middle panel: Similar to
+the top panel, it presents the average deviations and errors
+for LC2 samples and their corresponding random catalogs.
+Bottom panel: Similar to top panel, it displays the average
+deviations and errors for LS1 samples and their correspond-
+ing random catalogs.
+In this paper, we provide a radial selection model, the
+V SDC
+max approach, for generating the redshifts of random
+catalogs in galaxy two-point statistics that allows for a
+high level of accuracy and precision in the estimation.
+
+0.10.
+LCl (noke)
+0.05
+ Galaxy
+0.00
+nax
+-0.05
+90%
+-0.10
+60%
+0.10
+LC2 (noke)
+0.05
+0.00
+-0.05
+-0.10.
+0.2
+LS1
+0.1
+△
+0.0
+-0.1-
+-0.2
+100
+200
+300
+400
+500
+600
+700
+d h-1MpcThe smoothed Vmax method
+19
+This method is an improvement on the density-corrected
+Vmax method proposed by Cole (2011), and it consists
+mostly of three modifications: (1) Adding estimate of
+zmin and expanding the code’s application to a general
+flux-limited sample; (2) Support for a redshift and color
+dependent k−correction model applicable to individual
+galaxies; (3) Adding a smooth step to the output cloned
+radial distribution of random galaxies. These modifica-
+tions are crucial for obtaining a smooth radial distribu-
+tion for a random catalog that is unaffected by galaxy
+density fluctuations, which is the key to a clustering
+measure with high precision and accuracy.
+We measure 2PCFs using two groups of flux-limited
+samples, designated LC and LS, to validate the V SDC
+max
+approach. The flux-limited LC samples are constructed
+from sixty mock catalogs with two luminosity cuts and
+two simple k+e correction cases. Using the same sample
+selection criteria and luminosity thresholds as for the LC
+samples, ten LS samples are generated using the light-
+cone catalog of Smith et al. (2017). To test property-
+dependent clustering, LS samples are subdivided into
+blue and red subsamples.
+We compare the projected
+and redshift-space 2PCFs using random catalogs cre-
+ated from the ntrue method, the V SDC
+max
+method, the
+V DC
+max method, the Vmax method, and the redshift shuffled
+method. Our test results demonstrate that the V SDC
+max ap-
+proach is the only reliable radial selection model capable
+of achieving sub-percent accuracy for wp measurement
+on scales ranging from 0.07h−1Mpc to ∼ 40h−1Mpc. A
+2% deviation arises on a large scale for the LS2 sample,
+however it is still less than the deviations of other radial
+selection models. In general, the V SDC
+max
+technique can
+constrain the measure accuracy of wp to within 1% for
+color-dependent galaxy clustering, validating its supe-
+riority over the Vmax method and the redshift shuffled
+method.
+The next generation of spectroscopic surveys, specif-
+ically the DESI experiment, will obtain the spectra of
+around 40 million galaxies and quasars over 14,000 deg2,
+which is almost an order of magnitude more than the
+previous observed galaxies (Myers et al. 2022). These
+extra-galactic objects include 13 million bright galaxy
+sample (2 magnitude deeper than SDSS main sam-
+ple) (Lan et al. 2022), 8 million luminous red galaxies
+(LRGs), 16 million emission line galaxies (ELG), and
+3 million quasars (Levi et al. 2013; DESI Collabora-
+tion et al. 2016a,b; Raichoor et al. 2022). On the one
+hand, the two-point statistics of these up-coming galax-
+ies will surely afford us an unprecedented opportunity to
+comprehend the physics of galaxy formation and evolu-
+tion, improve the galaxy-halo connection, and shed light
+on the role of the halo environment in determining the
+galaxy’s physical properties (Ferreira et al. 2022). On
+the other hand, how to fully exploit these galaxies, par-
+ticularly with the assistance of galaxy 2PCFs, remains
+a challenge. Using volume-limited catalogs to conduct
+the 2PCF analysis will not only result in the rejection
+of a considerable number of galaxies, but it may also
+lead to the loss of crucial information imprinted in clus-
+tering. The density-corrected Vmax approach proposed
+by (Cole 2011) solves this problem, and our improve-
+ments and tests confirm that the V SDC
+max method is a vi-
+able technique for accurately measuring clustering for
+flux-limited and color-dependent samples, hence maxi-
+mizing the use of galaxies. Our present tests are pre-
+liminary, concentrating mostly on low redshift galaxies.
+In the future, we will continue to improve this approach
+and conduct more tests on various properties of galax-
+ies (e.g., stellar mass, star-formation rate, and so forth)
+as well as tests employing relative high redshift galaxies
+(e.g., CMASS, BOSS and eBOSS) and mocks.
+We appreciate the referee’s insightful comments and
+suggestions, which substantially improve this article.
+We would like to thank Yipeng Jing for carefully read-
+ing the manuscript and providing valuable comments.
+We are also grateful to Yipeng Jing for generously pro-
+viding the simulation data.
+Lei Yang expresses grat-
+itude to Chun Xia for assisting with the use of the
+Yunnan University Astronomy Supercomputer.
+This
+work is sponsored by grants from Yunnan Univer-
+sity’s Launching Research Fund for Postdoctoral Fel-
+low (C176220200) and the China Postdoctoral Science
+Foundation (2020M683387).
+The majority of calcula-
+tions were performed on the Yunnan University Astron-
+omy Supercomputer.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+
+20
+Yang et al.
+Figure 13. The mean number density n(z) among 60 LC mock samples. These samples have a flux cut at mr = [15, 17] and
+two luminosity cuts at M 0.1
+r
+= [−19, −22] for LC1 samples and M 0.1
+r
+= [−20, −23] for LC2 samples. In the case of no k + e
+corrections, the light blue and dark blue points with error bars represent the n(z) and 1σ variance for the LC1 and LC2 samples,
+respectively. The light orange and orange lines show the input nDR7 derived by the input LF and sample selection criteria. In
+the case of simple k + e corrections, the orange and red triangles with errors indicate n(z) and 1σ for LC1 and LC2 samples,
+respectively. The inputs nDR7 are shown in light gray and gray lines.
+APPENDIX
+A. THE MOCK SAMPLES
+As an example, Figure 13 displays the estimated average galaxy number density n(z) for 60 LC samples. The n(z)
+of these flux-limited samples changes as a function of comoving distance. The n(z)s of the LC samples are in excellent
+agreement with the predicted input nDR7 derived from the input luminosity function and the corresponding sample
+selection criteria. As predicted, n(z) for LS2 samples contains more brighter and high redshift galaxies than n(z) for
+LS1 samples. In addition, the n(z) for the samples with simple k + e corrections exhibits a slight evolution toward
+higher redshift when compared to samples without k + e corrections.
+Figure 14 displays the LS samples on the redshift-magnitude diagram (left panel) and color-magnitude diagram
+(right panel), respectively. The flux-limited LS samples are constructed from a lightcone catalog with two luminosity
+cuts. At low-redshift regions, the lightcone catalog mimic the SDSS DR7 data and, hence, has a LF of Blanton et al.
+(2003). We use the method described in Zehavi et al. (2011) to divide the galaxies into blue and red galaxies, as
+indicated by the red line in the right panel. Additionally, the LS samples have a redshift-dependent number density
+identical to that observed in Figure 13 and spanning a broader redshift range.
+REFERENCES
+Alam, S., de Mattia, A., Tamone, A., et al. 2021a, MNRAS,
+504, 4667, doi: 10.1093/mnras/stab1150
+Alam, S., Aubert, M., Avila, S., et al. 2021b, PhRvD, 103,
+083533, doi: 10.1103/PhysRevD.103.083533
+Amendola, L., Appleby, S., Bacon, D., et al. 2013, Living
+Reviews in Relativity, 16, 6, doi: 10.12942/lrr-2013-6
+Amin, M. A., Cyr-Racine, F.-Y., Eifler, T., et al. 2022,
+arXiv e-prints, arXiv:2203.07946.
+https://arxiv.org/abs/2203.07946
+Angulo, R. E., Springel, V., White, S. D. M., et al. 2012,
+MNRAS, 426, 2046,
+doi: 10.1111/j.1365-2966.2012.21830.x
+Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C.
+2019, MNRAS, 488, 3143, doi: 10.1093/mnras/stz1182
+Bennett, C. L., Larson, D., Weiland, J. L., et al. 2013,
+ApJS, 208, 20, doi: 10.1088/0067-0049/208/2/20
+Berlind, A. A., et al. 2006, ApJS, 167, 1,
+doi: 10.1086/508170
+
+((h-1Mpc)
+10
+No k + e corrections
+Input nDR7(z):[-19,-22]
+Mock n(z) :
+「-19,-22]
+Input nDR7(z) :[-20,-23]
+n(
+Mock n(z) :「-20,-23l
+Simple k + e corrections
+Input nDR7(z) :[-19,-22]
+10-6,
+Mock n(z) :
+[19,—22]
+Input nDR7(z) :[-20,-23]
+Mock n(z) :
+[—20,—23]
+10-7
+60
+100
+200
+500
+d (h-1Mpc)The smoothed Vmax method
+21
+Figure 14. Left panel: LS samples on the magnitude-redshift diagram. The light blue points denote galaxies in one of LS1
+samples with a flux cut at mr = [15, 17] and luminosity cut at M 0.1
+r
+= [−19, −22]. The dark blue points stand for one of LS2
+samples with cuts of mr = [15, 17] and M 0.1
+r
+= [−20, −23]. The samples have a sky coverage of 5950 deg2. Right panel: LS
+samples on the color-magnitude diagram. LS1 and LS2 samples are color-coded similarly to the panel on the left. The red line
+referred to by Zehavi et al. (2011) splits galaxies into blue and red subsamples.
+Beutler, F., Saito, S., Seo, H.-J., et al. 2014, MNRAS, 443,
+1065, doi: 10.1093/mnras/stu1051
+Bianchi, D., & Percival, W. J. 2017, MNRAS, 472, 1106,
+doi: 10.1093/mnras/stx2053
+Bianchi, D., & Verde, L. 2020, MNRAS, 495, 1511,
+doi: 10.1093/mnras/staa1267
+Blanton, M. R. 2006, ApJ, 648, 268, doi: 10.1086/505628
+Blanton, M. R., et al. 2001, AJ, 121, 2358,
+doi: 10.1086/320405
+Blanton, M. R., Hogg, D. W., Bahcall, N. A., et al. 2003,
+ApJ, 592, 819, doi: 10.1086/375776
+Blanton, M. R., et al. 2005, AJ, 129, 2562,
+doi: 10.1086/429803
+Breton, M.-A., & de la Torre, S. 2021, A&A, 646, A40,
+doi: 10.1051/0004-6361/202039603
+Cao, Y., Gong, Y., Meng, X.-M., et al. 2018, MNRAS, 480,
+2178, doi: 10.1093/mnras/sty1980
+Cole, S. 2011, MNRAS, 416, 739,
+doi: 10.1111/j.1365-2966.2011.19093.x
+Colless, M., et al. 2003, ArXiv Astrophysics e-prints
+Conroy, C., Wechsler, R. H., & Kravtsov, A. V. 2006, ApJ,
+647, 201, doi: 10.1086/503602
+Contreras, S., Angulo, R. E., & Zennaro, M. 2021,
+MNRAS, 508, 175, doi: 10.1093/mnras/stab2560
+D´avila-Kurb´an, F., S´anchez, A. G., Lares, M., & Ruiz,
+A. N. 2021, MNRAS, 506, 4667,
+doi: 10.1093/mnras/stab1622
+Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M.
+1985, ApJ, 292, 371, doi: 10.1086/163168
+Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465,
+doi: 10.1086/160884
+de la Torre, S., et al. 2013, A&A, 557, A54,
+doi: 10.1051/0004-6361/201321463
+de la Torre, S., Jullo, E., Giocoli, C., et al. 2017, A&A, 608,
+A44, doi: 10.1051/0004-6361/201630276
+de Mattia, A., & Ruhlmann-Kleider, V. 2019, JCAP, 2019,
+036, doi: 10.1088/1475-7516/2019/08/036
+DESI Collaboration, Aghamousa, A., Aguilar, J., et al.
+2016a, ArXiv e-prints. https://arxiv.org/abs/1611.00036
+—. 2016b, ArXiv e-prints.
+https://arxiv.org/abs/1611.00037
+Driver, S. P., Hill, D. T., Kelvin, L. S., et al. 2011,
+MNRAS, 413, 971, doi: 10.1111/j.1365-2966.2010.18188.x
+Eisenstein, D. J., Weinberg, D. H., Agol, E., et al. 2011,
+AJ, 142, 72, doi: 10.1088/0004-6256/142/3/72
+Farrow, D. J., Cole, S., Norberg, P., et al. 2015, MNRAS,
+454, 2120, doi: 10.1093/mnras/stv2075
+Farrow, D. J., S´anchez, A. G., Ciardullo, R., et al. 2021,
+MNRAS, 507, 3187, doi: 10.1093/mnras/stab1986
+Ferreira, L., Adams, N., Conselice, C. J., et al. 2022, ApJL,
+938, L2, doi: 10.3847/2041-8213/ac947c
+Fisher, K. B., Davis, M., Strauss, M. A., Yahil, A., &
+Huchra, J. 1994, MNRAS, 266, 50
+Garilli, B., Paioro, L., Scodeggio, M., et al. 2012, PASP,
+124, 1232, doi: 10.1086/668681
+Glanville, A., Howlett, C., & Davis, T. M. 2021, MNRAS,
+503, 3510, doi: 10.1093/mnras/stab657
+Gong, Y., Liu, X., Cao, Y., et al. 2019, ApJ, 883, 203,
+doi: 10.3847/1538-4357/ab391e
+Guo, H., Yang, X., & Lu, Y. 2018, ApJ, 858, 30,
+doi: 10.3847/1538-4357/aabc56
+Guo, H., Zehavi, I., Zheng, Z., et al. 2013, ApJ, 767, 122,
+doi: 10.1088/0004-637X/767/2/122
+Guo, H., et al. 2014, MNRAS, 441, 2398,
+doi: 10.1093/mnras/stu763
+
+-23.0 -
+-22.5
+-22.0
+-21.5-
+-21.0
+M
+-20.5
+Sky coverage: 5950 deg?
+-20.0
+mr=[15,17]
+-19.5
+M9.1 =[-20,-23]
+M9.1=[-19,-22]
+-19.0
+0.05
+0.10
+0.15
+0.20
+0.25
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Z
+g-r22
+Yang et al.
+Guo, Q., White, S., Boylan-Kolchin, M., et al. 2011,
+MNRAS, 413, 101, doi: 10.1111/j.1365-2966.2010.18114.x
+Hahn, C., Wilson, M. J., Ruiz-Macias, O., et al. 2022,
+arXiv e-prints, arXiv:2208.08512.
+https://arxiv.org/abs/2208.08512
+Hamilton, A. J. S. 1992, ApJL, 385, L5,
+doi: 10.1086/186264
+—. 1993, ApJ, 417, 19, doi: 10.1086/173288
+Hearin, A. P., Watson, D. F., Becker, M. R., et al. 2014,
+MNRAS, 444, 729, doi: 10.1093/mnras/stu1443
+Hinshaw, G., Larson, D., Komatsu, E., et al. 2013, ApJS,
+208, 19, doi: 10.1088/0067-0049/208/2/19
+Jing, Y. 2019, Science China Physics, Mechanics, and
+Astronomy, 62, 19511, doi: 10.1007/s11433-018-9286-x
+Jing, Y. P., Mo, H. J., & B¨orner, G. 1998, ApJ, 494, 1,
+doi: 10.1086/305209
+Johnston, H., Joachimi, B., Norberg, P., et al. 2021, A&A,
+646, A147, doi: 10.1051/0004-6361/202039682
+Karademir, G. S., Taylor, E. N., Blake, C., et al. 2021,
+arXiv e-prints, arXiv:2109.06136.
+https://arxiv.org/abs/2109.06136
+Lan, T.-W., Tojeiro, R., Armengaud, E., et al. 2022, arXiv
+e-prints, arXiv:2208.08516.
+https://arxiv.org/abs/2208.08516
+Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64,
+doi: 10.1086/172900
+Levi, M., Bebek, C., Beers, T., et al. 2013, ArXiv e-prints.
+https://arxiv.org/abs/1308.0847
+Liske, J., Baldry, I. K., Driver, S. P., et al. 2015, MNRAS,
+452, 2087, doi: 10.1093/mnras/stv1436
+Loveday, J., Norberg, P., Baldry, I. K., et al. 2015,
+MNRAS, 451, 1540, doi: 10.1093/mnras/stv1013
+Loveday, J., Christodoulou, L., Norberg, P., et al. 2018,
+MNRAS, 474, 3435, doi: 10.1093/mnras/stx2971
+LSST Dark Energy Science Collaboration. 2012, arXiv
+e-prints, arXiv:1211.0310.
+https://arxiv.org/abs/1211.0310
+Madgwick, D. S., et al. 2003, MNRAS, 344, 847,
+doi: 10.1046/j.1365-8711.2003.06861.x
+McBride, C. K., Connolly, A. J., Gardner, J. P., et al. 2011,
+ApJ, 726, 13, doi: 10.1088/0004-637X/726/1/13
+McNaught-Roberts, T., Norberg, P., Baugh, C., et al. 2014,
+MNRAS, 445, 2125, doi: 10.1093/mnras/stu1886
+Merz, G., Rezaie, M., Seo, H.-J., et al. 2021, MNRAS, 506,
+2503, doi: 10.1093/mnras/stab1887
+Mohammad, F. G., Granett, B. R., Guzzo, L., et al. 2018,
+A&A, 610, A59, doi: 10.1051/0004-6361/201731685
+Myers, A. D., Moustakas, J., Bailey, S., et al. 2022, arXiv
+e-prints, arXiv:2208.08518.
+https://arxiv.org/abs/2208.08518
+Norberg, P., et al. 2002, MNRAS, 332, 827,
+doi: 10.1046/j.1365-8711.2002.05348.x
+Nuza, S. E., et al. 2013, MNRAS, 432, 743,
+doi: 10.1093/mnras/stt513
+Peacock, J. A., Cole, S., Norberg, P., et al. 2001, Nature,
+410, 169
+Pezzotta, A., de la Torre, S., Bel, J., et al. 2017, A&A, 604,
+A33, doi: 10.1051/0004-6361/201630295
+Planck Collaboration, Ade, P. A. R., Aghanim, N., et al.
+2016, A&A, 594, A13, doi: 10.1051/0004-6361/201525830
+Raichoor, A., Moustakas, J., Newman, J. A., et al. 2022,
+arXiv e-prints, arXiv:2208.08513.
+https://arxiv.org/abs/2208.08513
+Reid, B., Ho, S., Padmanabhan, N., et al. 2016, MNRAS,
+455, 1553, doi: 10.1093/mnras/stv2382
+Reid, B. A., Percival, W. J., Eisenstein, D. J., et al. 2010,
+MNRAS, 404, 60, doi: 10.1111/j.1365-2966.2010.16276.x
+Ross, A. J., Samushia, L., Howlett, C., et al. 2015,
+MNRAS, 449, 835, doi: 10.1093/mnras/stv154
+Ross, A. J., Percival, W. J., S´anchez, A. G., et al. 2012,
+MNRAS, 424, 564, doi: 10.1111/j.1365-2966.2012.21235.x
+Samushia, L., et al. 2013, MNRAS, 429, 1514,
+doi: 10.1093/mnras/sts443
+—. 2014, MNRAS, 439, 3504, doi: 10.1093/mnras/stu197
+Schlegel, D. J., Ferraro, S., Aldering, G., et al. 2022, arXiv
+e-prints, arXiv:2209.03585.
+https://arxiv.org/abs/2209.03585
+Shi, F., Yang, X., Wang, H., et al. 2016, ApJ, 833, 241,
+doi: 10.3847/1538-4357/833/2/241
+—. 2018, ApJ, 861, 137, doi: 10.3847/1538-4357/aacb20
+Sinha, M., & Garrison, L. 2019, in Software Challenges to
+Exascale Computing, ed. A. Majumdar & R. Arora
+(Singapore: Springer Singapore), 3–20.
+https://doi.org/10.1007/978-981-13-7729-7 1
+Skibba, R. A., & Sheth, R. K. 2009, MNRAS, 392, 1080,
+doi: 10.1111/j.1365-2966.2008.14007.x
+Skibba, R. A., et al. 2014, ApJ, 784, 128,
+doi: 10.1088/0004-637X/784/2/128
+Smith, A., Cole, S., Baugh, C., et al. 2017, MNRAS, 470,
+4646, doi: 10.1093/mnras/stx1432
+Tegmark, M., Strauss, M. A., Blanton, M. R., et al. 2004,
+PhRvD, 69, 103501, doi: 10.1103/PhysRevD.69.103501
+Vale, A., & Ostriker, J. P. 2004, MNRAS, 353, 189,
+doi: 10.1111/j.1365-2966.2004.08059.x
+Valluri, M., Chabanier, S., Irsic, V., et al. 2022, arXiv
+e-prints, arXiv:2203.07491.
+https://arxiv.org/abs/2203.07491
+Wang, S.-J., Guo, Q., & Cai, R.-G. 2017, MNRAS, 472,
+2869, doi: 10.1093/mnras/stx2183
+
+The smoothed Vmax method
+23
+Wang, Z., Xu, H., Yang, X., et al. 2021, Science China
+Physics, Mechanics, and Astronomy, 64, 289811,
+doi: 10.1007/s11433-021-1707-6
+Wechsler, R. H., & Tinker, J. L. 2018, ARA&A, 56, 435,
+doi: 10.1146/annurev-astro-081817-051756
+Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al.
+2013, PhR, 530, 87, doi: 10.1016/j.physrep.2013.05.001
+Xu, H., Zheng, Z., Guo, H., Zhu, J., & Zehavi, I. 2016,
+MNRAS, 460, 3647, doi: 10.1093/mnras/stw1259
+Yang, L., Jing, Y., Yang, X., & Han, J. 2019, ApJ, 872, 26,
+doi: 10.3847/1538-4357/aafc22
+Yang, L., Jing, Y.-P., Li, Z.-G., & Yang, X.-H. 2020,
+Research in Astronomy and Astrophysics, 20, 054,
+doi: 10.1088/1674-4527/20/4/54
+Yang, X., Mo, H. J., & van den Bosch, F. C. 2003, MNRAS,
+339, 1057, doi: 10.1046/j.1365-8711.2003.06254.x
+—. 2008, ApJ, 676, 248, doi: 10.1086/528954
+Yang, X., Mo, H. J., van den Bosch, F. C., Zhang, Y., &
+Han, J. 2012, ApJ, 752, 41,
+doi: 10.1088/0004-637X/752/1/41
+York, D. G., et al. 2000, AJ, 120, 1579, doi: 10.1086/301513
+Yuan, S., Hadzhiyska, B., & Abel, T. 2022a, arXiv e-prints,
+arXiv:2211.02068. https://arxiv.org/abs/2211.02068
+Yuan, S., Hadzhiyska, B., Bose, S., & Eisenstein, D. J.
+2022b, arXiv e-prints, arXiv:2202.12911.
+https://arxiv.org/abs/2202.12911
+Zarrouk, P., Ruiz-Macias, O., Cole, S., et al. 2021, arXiv
+e-prints, arXiv:2106.13120.
+https://arxiv.org/abs/2106.13120
+Zehavi, I., et al. 2002, ApJ, 571, 172, doi: 10.1086/339893
+—. 2005, ApJ, 630, 1, doi: 10.1086/431891
+—. 2011, ApJ, 736, 59, doi: 10.1088/0004-637X/736/1/59
+Zheng, Z., Zehavi, I., Eisenstein, D. J., Weinberg, D. H., &
+Jing, Y. P. 2009, ApJ, 707, 554,
+doi: 10.1088/0004-637X/707/1/554
+Zheng, Z., et al. 2005, ApJ, 633, 791, doi: 10.1086/466510
+Zu, Y., & Mandelbaum, R. 2015, MNRAS, 454, 1161,
+doi: 10.1093/mnras/stv2062
+Zu, Y., Shan, H., Zhang, J., et al. 2021, MNRAS, 505,
+5117, doi: 10.1093/mnras/stab1712
+

99FRT4oBgHgl3EQfrDen/content/tmp_files/2301.13619v1.pdf.txt

@@ -0,0 +1,1011 @@
+Brillouin and Kerr nonlinearities of a low-index silicon oxynitride platform
+Kaixuan Ye,1 Yvan Klaver,1 Oscar A Jimenez Gordillo,2 Roel Botter,1
+Okky Daulay,1 Francesco Morichetti,2 Andrea Melloni,2 and David Marpaung1, ∗
+1Nonlinear Nanophotonics, MESA+ Institute of Nanotechnology,
+University of Twente, Enschede, the Netherlands
+2Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, 20133, Italy
+(Dated: February 1, 2023)
+Nonlinear optical effects including stimulated Brillouin scattering (SBS) and four-wave mixing
+(FWM) play an important role in microwave photonics, optical frequency combs, and quantum
+photonics. Harnessing SBS and FWM in a low-loss and versatile integrated platform would open
+the path to building large-scale Brillouin/Kerr-based photonic integrated circuits. In this letter, we
+investigate the Brillouin and Kerr properties of a low-index (n=1.513 @ 1550 nm) silicon oxynitride
+(SiON) platform. We observed, for the first time, backward SBS in SiON waveguides with a Brillouin
+gain coefficient of 0.3 m−1W−1, which can potentially be increased to 0.95 m−1W−1 by just tailoring
+the waveguide cross-section. We also performed FWM experiments in SiON rings and obtained the
+nonlinear parameter γ, of 0.02 m−1W−1. Our results point to a low-loss and low-index photonic
+integrated platform that is both Brillouin and Kerr active.
+INTRODUCTION
+Stimulated Brillouin scattering (SBS), which is an in-
+teraction between optical and acoustic waves, is currently
+revolutionizing photonic integrated circuit designs [1–8].
+Featuring a narrow-band (tens of MHz) gain resonance
+shifted around tens of GHz away from the pump light, the
+on-chip SBS plays a significant role in microwave photon-
+ics [9–11], narrow-linewidth integrated lasers [7, 12, 13],
+and on-chip nonreciprocal light propagation [3, 14].
+Efficient on-chip SBS process requires simultaneously
+guiding both the optical and gigahertz acoustic waves
+in a waveguide, making it challenging to be realized in
+most integrated platforms. Several encouraging results
+have been demonstrated recently in various platforms,
+including chalcogenide [2], silicon [5], doped silica [15],
+aluminum gallium arsenide [16], and aluminum nitride
+[17]. In addition, SBS has also been observed in silicon
+nitride-based waveguides [7, 8, 18], opening the pathway
+to intersect Brillouin scattering with Kerr nonlinearities
+in low-loss and mature platforms.
+Silicon oxynitride (SiON) is another highly-developed
+integrated platform that has appealing properties includ-
+ing low propagation loss, wide transparency window, ab-
+sence of multi-photon absorption effects, and stress-free
+fabrication [19, 20].
+The optical and mechanical properties of SiON could
+be tuned continuously between those of SiO2
+and
+Si3N4 at different nitrogen/oxygen (N/O) ratios [21,
+22].
+For example, a variety of SiON, known as Hy-
+dex (n=1.7 @ 1550 nm), has been widely used for
+Kerr-based nonlinear optic applications including opti-
+cal frequency comb [23], optical neural network [24],
+and quantum photonics [25].
+A slightly higher in-
+dex SiON (n=1.83 @ 1550 nm) was also proposed in
+∗ Corresponding author: [email protected]
+Fig. 1. (a) Artistic representation of the SiON waveguides,
+showing the four-wave mixing process in an all-pass microring
+resonator and the backward stimulated Brillouin scattering
+(SBS) in a spiral waveguide.
+(b) The cross-section of the
+SiON platform in our work. (c) The chip photograph of the
+SiON microring resonators with a FSR of 50 GHz. (d) The
+chip photograph of the 5-cm SiON straight waveguide.
+[20, 26] for Kerr-based applications. In both cases, the
+SiON platforms have a refractive index close to sili-
+con nitride (n=1.98 @ 1550 nm) instead of silicon oxide
+(n=1.45 @ 1550nm). The relatively high refractive in-
+dex induces a high nonlinear index, making it useful for
+Kerr-based nonlinear optic applications.
+But from the Brillouin perspectives, high refractive in-
+dex SiON is less attractive due to the high content of the
+nitrogen that leads to a meager photoelastic coefficient
+p12 because of the weak p12 of the Si3N4 [18]. Moreover,
+high-index SiON also has similar mechanical properties
+arXiv:2301.13619v1  [physics.optics]  31 Jan 2023
+
+fp
+a
+fs
+fp
+probe
+fs
+Amplified probe
+SiON chip
+pump
+SiO2 (1.45)
+7um
+SiON
+(1.513)
+2.2um
+2.2um
+15um
+b
+Si (3.45)
+dFig. 2. (a) Simulated optical mode of the SiON waveguide.
+(b) Simulated acoustic response of the SiON waveguide. (c)-
+(h) Measured SBS gain spectra of the 2.0 µm, 2.2 µm, 2.3 µm,
+2.4 µm, 2.6 µm, and 3.5 µm-wide SiON waveguides, respec-
+tively. (i) Brillouin gain coefficients and linewidth of the SiON
+waveguides with different widths.
+to Si3N4, such as high acoustic velocity that prevents
+acoustic confinement when cladded with SiO2 [7, 8, 18].
+In this paper, we investigate the Brillouin and Kerr
+properties of a SiON integrated platform with a rela-
+tively lower refractive index (n=1.513 @ 1550 nm). Con-
+trasting to SiON platforms mentioned above, the SiON
+platform investigated here has a larger photoelastic co-
+efficient p12, lower acoustic velocity, and a larger cross-
+section, all of which lead to an enhanced SBS effect. We
+experimentally observed, for the first time to our knowl-
+edge, backward SBS in SiON waveguides. We also char-
+acterized the Brillouin gain coefficient gb of the SiON
+waveguides with different widths. We found out the gb
+of this SiON waveguide can potentially be increased to
+around 0.95 m−1W−1 by simply tailoring the waveguide
+cross-section. This sufficiently large Brillouin gain coef-
+ficient, together with the low propagation loss, makes it
+possible to generate decent SBS gain for a plethora of
+Brillouin-based applications in this SiON platform.
+Furthermore, we also measured the nonlinear param-
+eter γ and nonlinear index n2 of this SiON platform
+through four-wave mixing (FWM) experiments in a ring
+resonator. While the measured γ is an order of magni-
+tude lower when compared to that of high-index SiON,
+we expect that with lower losses and higher pump power,
+the unique interplay between the SBS and Kerr effect
+such as Brillouin-assisted Kerr frequency comb [27, 28]
+could be observed in this integrated platform.
+RESULTS
+We performed the backward SBS and four-wave mix-
+ing experiments in single-pass (spiral or straight) wave-
+guides and microring resonators respectively, as shown in
+Fig. 1(a). The cross-section of this platform is shown in
+Fig. 1(b) [29, 30]. The 2.2 µm-thick SiON layer has a
+refractive index n of 1.513 at 1550 nm. It is on top of a
+15-µm SiO2 layer and is covered by a 7 µm-thick SiO2 up-
+per cladding. The refractive index contrast ∆n between
+the core and the cladding is 4.4%, enabling a bending ra-
+dius of 600 µm with negligible radiation losses. Fig. 1(c)
+shows the photograph of the microring resonators in this
+platform with a free spectral range (FSR) of 50 GHz and
+coupling coefficients varying from 0.05 to 0.8. Fig. 1(d)
+shows the photograph of several groups of 5-cm straight
+waveguides with different widths. The measured prop-
+agation loss of those straight waveguides is 0.25 dB/cm
+with coupling loss to lensed-tip fibers of approximately
+3 dB/facet.
+Stimulated Brillouin Scattering in SiON Waveguides
+We developed a finite element model [8] in COMSOL to
+estimate the SBS response of the SiON waveguides. The
+simulated optical field and the corresponding acoustic re-
+sponse of the 2.2 µm-wide SiON waveguide are shown
+in Fig. 2(a) and (b), respectively.
+The optical field is
+well confined around the SiON core area because of the
+total internal reflection (TIR). However, the TIR condi-
+tion does not hold for the acoustic response because the
+acoustic velocity of the SiON (∼ 6.2 km/s) is higher than
+that of the SiO2 (∼ 5.9 km/s). As a result, part of the
+acoustic field would leak into the cladding as shown in
+Fig. 2(b). Nevertheless, most of the acoustic field still
+
+Normalized electric field
+Normalized displacement field
+a
+2.2 um
+2.2 um
+2.2 um
+2.2 um
+0
+0.8
+c
+2.0 um
+d
+2.2 um
+8
+0.
+j0.10 m-1 w-1
+FWHM
+0.15 m-1 W-1
+FWHM
+g
+@14.22 GHz
+358 MHz
+@14.31 GHz
+282 MHz
+0.4
+0,2
+M
+0
+0
+[a.u.]
+13.5
+14.0
+14.5
+15.0
+13.5
+14.0
+14.5
+15.0
+Normalized amplitude 
+e
+2.3 um
+2.4 um
+0.8
+0.8
+0.17 m-1 w-1
+FWHM
+0.19 m-1 w-1
+FWHM
+0.6
+@14.40 GHz
+280 MHz
+@14.44 GHz
+269 MHz
+0.4
+0.4
+0,2
+umy
+0
+13.5
+14.0
+14.5
+15.0
+13.5
+14.0
+14.5
+15.0
+0.8
+g
+2.6 um
+0.8
+h
+3.5 um
+0.24 m-1 w-1
+FWHM
+0.32 m-1 w-1
+0.6
+FWHM
+9
+@14.43 GHz
+224 MHz
+@14.48 GHz
+105 MHz
+0.4
+0.4
+0,2
+0
+0
+13.5
+14.0
+14.5
+15.0
+13.5
+14.0
+14.5
+15.0
+Brillouin frequency shift [GHz]
+350
+0.30
+300
+Linewidth [MHz]
+0.25
+[m-1 W-1]
+250 200 150
+0.15 0.20
+0.10
+100
+2.0
+2.5
+3.0
+3.5
+Waveguide width [μm]Fig. 3. (a) Simulated optical mode and (b) simulated acous-
+tic response and (c) simulated Brillouin gain spectrum of the
+optimized SiON waveguide. (d) Estimated SBS gain from the
+optimized and current SiON waveguides.
+remains inside the SiON core because of the relatively
+large cross-section area [31]. This results in a large over-
+lap between the optical and acoustic fields that leads to
+improved Brillouin gain coefficient. Extensive simulation
+results of the SBS gain coefficients are included in the
+Supplementary.
+To verify the simulation results, we characterized the
+SBS responses of the SiON waveguides with a pump-
+probe experimental apparatus [8, 18].
+The pump and
+probe light are intensity-modulated and coupled into the
+opposite facets of the waveguide. We keep the pump fre-
+quency fixed at 1561 nm while sweeping the probe at fre-
+quencies down shifted from the pump by about 15 GHz.
+When the frequency difference between the pump and
+the probe is close to the Brillouin frequency shift of the
+SiON waveguide, the probe will experience the SBS gain
+and a peak will be detected at the lock-in amplifier (See
+the Supplementary for more details about the SBS ex-
+periment).
+Several 5 cm-long SiON waveguides are characterized
+to investigate the influence of waveguide width on the
+Brillouin gain spectra.
+The measured SBS responses
+of the 2.0 µm, 2.2 µm, 2.3 µm, 2.4 µm, 2.6 µm, and
+3.5 µm-wide waveguides are shown in Fig. 2(c) to (h),
+respectively. All waveguides show a clear SBS peak well
+above the noise floor with the Brillouin frequency shift in-
+creases from 14.22 GHz for the 2.0 µm-wide waveguide to
+14.48 GHz for the 3.5 µm-wide waveguide. Fig. 2(i) plots
+the measured Brillouin gain coefficient gb and the SBS
+linewidth of the SiON waveguides with different widths
+(See the Supplementary for more details about the Bril-
+louin gain coefficient calculation). The Brillouin gain co-
+efficient gb increases from 0.1 m−1W−1 to 0.32 m−1W−1
+when the waveguide width increases from 2.0 µm to
+3.5 µm.
+In the meantime, the linewidth of the SBS
+peak reduces from 358 MHz to 105 MHz. The increasing
+Brillouin gain coefficient and the narrowing of the SBS
+linewidth indicate an improvement in acoustic confine-
+ment when the SiON waveguides become wider.
+The Brillouin gain coefficient can be further increased
+by optimizing the cross-section of the waveguide through
+the genetic algorithm [8]. Fig. 3 (a) and (b) show the
+simulated optical mode and the acoustic response of a
+SiON waveguide with the same core refractive index but
+with an optimized cross-section for SBS gain. The di-
+mension of such a waveguide is 4.0 µm × 3.2 µm with a
+top cladding of 3 µm and a bottom cladding of 10 µm.
+Compared to the optical and acoustic fields of the wave-
+guide structure in this work, less acoustic field is scat-
+tered into the cladding while the optical field is still well
+confined in the optimized waveguide structure. The Bril-
+louin gain spectrum of the optimized waveguide structure
+is shown in Fig. 3 (c). The simulated peak Brillouin gain
+coefficient of this waveguide is 0.95 m−1W−1, which is
+3× higher than the waveguide structure measured in this
+work. Furthermore, the propagation loss in this SiON
+platform can also be significantly lowered by reducing
+sidewall roughness and improving the thermal anneal-
+ing process [30], allowing for longer effective waveguide
+length for the SBS process. Fig. 3 (d) estimates the SBS
+gain of both the measured and the optimized SiON wave-
+guides with different propagation losses. The optimized
+Brillouin gain coefficient (around 0.95 m−1W−1), along
+with the improved propagation loss (around 0.1 dB/cm),
+can enhance the SBS gain from less than 0.5 dB to near
+1.5 dB for a 60-cm waveguide, which is sufficient for ap-
+plications like SBS-based narrow-bandwidth microwave
+photonic notch filters [8, 10].
+Four-wave mixing in SiON Waveguides
+We further investigate the Kerr nonlinearities of this
+SiON platform. High-index SiON platforms are widely
+used for Kerr-based nonlinear optics applications because
+of the relatively large nonlinear parameter γ [19]. How-
+ever, the nonlinear parameter γ is highly dependent on
+the refractive index and the geometry of the waveguide.
+The SiON waveguide in this work has a relatively lower
+refractive index and a larger cross-section compared with
+other SiON platforms [19, 20], and the nonlinear index
+n2 and nonlinear parameter γ of the SiON waveguide in
+this platform has never been characterized before.
+We devised a four-wave mixing (FWM) experiment
+for the nonlinear parameter characterization.
+Because
+of the limited effective length of the available samples,
+the FWM conversion efficiency of the straight waveguide
+is comparable with that of the fiber pigtails, making it
+difficult to accurately measure the n2 and the γ.
+We
+use the all-pass ring resonators to enhance the FWM in
+the SiON waveguide so that the contribution from fibers
+in the setup can be neglected [32]. The ring resonator
+
+Normalized electric field
+Normalized displacement field
+a
+b
+4.0 um
+4.0 um
+3.2um
+3.2um
+c
+d
+5
+0.10 dB/cm
+SBS gain [dB]
+0.15 dB/cm
+0.95 m-1 W-1
+[m-1 W-1]
+0.25 dB/cm
+@ 15.76 GHz
+5.
+optimized
+0
+This work
+0.0
+0.
+0
+15.5
+15.7
+15.9
+16.1
+0
+20
+40
+60
+Brillouin frequency shift [GHz]
+Waveguide length [cm]Fig. 4. (a) Measured resonance response of the SiON ring
+resonator.
+(b) Measured four-wave mixing response of the
+SiON ring resonator. (c) Conversion efficiency of the four-
+wave mixing at different pump power.
+(d) The estimated
+nonlinear parameter γ of the SiON waveguides with different
+widths.
+applied in our experiment is made of the 2.2 µm-wide
+SiON waveguide and it has a free spectral range (FSR)
+of 50 GHz and a power coupling coefficient of 0.05. The
+pump laser is locked close to the resonance of the ring
+resonator to mitigate the thermal influence on the ring
+resonator. The signal laser is set close to 2 × FSR away
+from the pump signal and is combined with the pump
+light with a 99:1 coupler. The combined pump and sig-
+nal are coupled into the all-pass ring resonator with a
+lensed fiber with a spot size of 2 µm.
+The generated
+idler is then coupled out from the chip and sent to the
+optical spectrum analyzer to measure the conversion ef-
+ficiency from the signal to the generated idler (See the
+Supplementary for details of the FWM experiment).
+To determine the field enhancement factor of the FWM
+process in the ring resonator, we first characterized the
+resonance response of the ring resonator with a vector
+network analyzer, as shown in Fig. 4 (a) (See the Supple-
+mentary for details of the characterization). The mea-
+sured full-width at half-maximum (FWHM) is 612 MHz
+with an extinction ratio of 8.9 dB, corresponding to a
+loaded Q-factor of 330,000 and a propagation loss of
+0.27 dB/cm. Fig. 4 (b) shows the measured FWM re-
+sponse of the 50 GHz SiON ring resonator. A clear peak
+is shown at 2 × FSR down shifted from the pump fre-
+quency, which is the idler generated from the FWM pro-
+cess between the pump and signal in the ring resonator.
+The nonlinear index n2 and nonlinear parameter γ of
+the SiON waveguide in this platform can be estimated
+from the conversion efficiency between the signal and
+the idler (See the supplementary for details of the cal-
+culation). Fig. 4 (c) shows the measured conversion ef-
+ficiency of the FWM process at different pump power.
+Based on this measurement, the calculated γ and n2
+of the 2.2 µm-wide SiON waveguide are 0.024 m−1W−1
+and 4.16 ×10−20 m2/W, respectively. We also estimated
+the nonlinear parameter γ of the SiON waveguides with
+different widths based on the measured value of n2, as
+shown in Fig. 4 (d).
+The γ decreases from around
+0.025 m−1W−1 to 0.020 m−1W−1 when the waveguide
+width reduces from 2.0 µm to 3.5 µm.
+DISCUSSION
+For Brillouin-Kerr interactions, the balance between
+the nonlinearities needs to be considered. In microcavi-
+ties, it is generally preferred to have larger Brillouin gain,
+as it is easier to inhibit cascading or other unwanted in-
+teractions via mode manipulation. Comparing the values
+of the measured gb in Fig. 2 (i) and γ in Fig. 4 (a), the
+SiON waveguides reported here have an order of magni-
+tude larger Brillouin gain compared to Kerr nonlinearity.
+This gb/γ ratio is similar to previous demonstrations of
+Brillouin-assisted Kerr frequency combs in [27, 28], show-
+ing the potential to realize it in an integrated platform.
+In conclusion, we have investigated the Brillouin and
+Kerr properties of a SiON integrated platform with a
+relatively low refractive index. We observed, for the first
+time, the backward SBS response of those SiON wave-
+guides.
+We also measured its nonlinear index n2 and
+nonlinear parameter γ. These SiON waveguides can be
+fabricated in a versatile and low-loss integrated platform,
+and can potentially lead to a plethora of Brillouin and
+Kerr-based applications,
+including narrow-bandwidth
+microwave photonic filters, and narrow-linewidth lasers,
+and optical frequency combs.
+AUTHOR CONTRIBUTIONS
+D.M. and K.Y. developed the concept and proposed
+the physical system.
+K.Y. and Y.K. developed and
+performed numerical simulations.
+K.Y. performed the
+SBS characterisation with input from R.B., K.Y., and
+O.D. Y.K. and K.Y. performed the FWM experiments.
+O.A.J.G., F.M., and A.M. developed and fabricated the
+samples. K.Y., D.M., and Y.K. wrote the manuscript.
+D.M. led and supervised the entire project.
+FUNDING INFORMATION
+This project is funded by the European Research
+Council Consolidator Grant (101043229 TRIFFIC) and
+Nederlandse Organisatie voor Wetenschappelijk Onder-
+zoek (NWO) projects (740.018.021 and 15702).
+
+Pump
+-80 -70 -60 -50 -40 -30
+a
+Signal
+2
+Power [dBm]
+FWHM:
+2 x FSR
+612 MHz
+S
+Idler
+8
+2
+4
+6
+-100
+-50
+50
+100
+Frequency [GHz]
+Frequency [GHz]
+d
+c
+0.021 0.023 0.025
+-34
+n [dB]
+-38
+-42
+-46
+21
+23
+25
+27
+2.0
+2.5
+3.0
+3.5
+Pump power [dBm]
+Waveguide width [μm][1] B. J. Eggleton, C. G. Poulton, P. T. Rakich, M. J. Steel,
+and G. Bahl, “Brillouin integrated photonics,” Nature
+Photonics, 2019.
+[2] R. Pant, C. G. Poulton, D.-Y. Choi et al., “On-chip stim-
+ulated brillouin scattering,” Optics Express, vol. 19, pp.
+8285–8290, 4 2011.
+[3] E. A. Kittlaus, N. T. Otterstrom, P. Kharel, S. Gertler,
+and P. T. Rakich, “Non-reciprocal interband brillouin
+modulation,” Nature Photonics, vol. 12, pp. 613–619,
+2018.
+[4] E. A. Kittlaus, N. T. Otterstrom, and P. T. Rakich, “On-
+chip inter-modal brillouin scattering,” Nature Communi-
+cations, vol. 8, pp. 1–9, 2017.
+[5] E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large bril-
+louin amplification in silicon,” Nature Photonics, vol. 10,
+pp. 463–467, 2016.
+[6] P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and
+Z. Wang, “Giant enhancement of stimulated brillouin
+scattering in the subwavelength limit,” Physical Review
+X, vol. 2, p. 11008, 2012.
+[7] S. Gundavarapu, G. M. Brodnik, M. Puckett et al., “Sub-
+hertz fundamental linewidth photonic integrated bril-
+louin laser,” Nature Photonics, vol. 13, pp. 60–67, 12
+2018.
+[8] R. Botter, K. Ye, Y. Klaver et al., “Guided-acoustic stim-
+ulated brillouin scattering in silicon nitride photonic cir-
+cuits,” Science Advances, vol. 8, p. 2196, 10 2022.
+[9] D. Marpaung, J. Yao, and J. Capmany, “Integrated mi-
+crowave photonics,” Nature Photonics, vol. 13, pp. 80–90,
+2019.
+[10] D. Marpaung, B. Morrison, M. Pagani et al., “Low-
+power, chip-based stimulated brillouin scattering mi-
+crowave photonic filter with ultrahigh selectivity,” Op-
+tica, vol. 2, p. 76, 2015.
+[11] L. McKay, M. Merklein, A. C. Bedoya et al., “Brillouin-
+based phase shifter in a silicon waveguide,” Optica, Vol.
+6, Issue 7, pp. 907-913, vol. 6, pp. 907–913, 7 2019.
+[12] N. T. Otterstrom, R. O. Behunin, E. A. Kittlaus,
+Z. Wang, and P. T. Rakich, “A silicon brillouin laser,”
+Science, vol. 360, pp. 1113–1116, 6 2018.
+[13] N. Chauhan, A. Isichenko, K. Liu et al., “Visible light
+photonic integrated brillouin laser,” Nature Communica-
+tions 2021 12:1, vol. 12, pp. 1–8, 8 2021.
+[14] J.
+Kim,
+M.
+C.
+Kuzyk,
+K.
+Han,
+H.
+Wang,
+and
+G. Bahl, “Non-reciprocal brillouin scattering induced
+transparency,” Nature Physics, vol. 11, pp. 275–280, 1
+2015.
+[15] S. Li, X. Li, W. Zhang, J. Chen, and W. Zou, “Inves-
+tigation of brillouin properties in high-loss doped silica
+waveguides by comparison experiment,” IEEE Photon-
+ics Technology Letters, vol. 32, pp. 948–951, 2020.
+[16] W. Jin, L. Chang, W. Xie et al., “Stimulated brillouin
+scattering in algaas on insulator waveguides,” Conference
+on Lasers and Electro-Optics, paper SM4L.7, 2020.
+[17] Q. Liu, H. Li, and M. Li, “Electromechanical brillouin
+scattering in integrated optomechanical waveguides,”
+Optica, vol. 6, pp. 778–785, 2019.
+[18] F. Gyger, J. Liu, F. Yang et al., “Observation of stim-
+ulated brillouin scattering in silicon nitride integrated
+waveguides,” Physical Review Letters, vol. 124, 2020.
+[19] D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson,
+“New cmos-compatible platforms based on silicon nitride
+and hydex for nonlinear optics,” Nature Photonics, vol. 7,
+pp. 597–607, 7 2013.
+[20] A. Trenti, M. Borghi, S. Biasi et al., “Thermo-optic coef-
+ficient and nonlinear refractive index of silicon oxynitride
+waveguides,” AIP Advances, vol. 8, p. 025311, 2 2018.
+[21] T. Baak, “Silicon oxynitride; a material for grin optics,”
+Applied Optics, Vol. 21, Issue 6, pp. 1069-1072, vol. 21,
+pp. 1069–1072, 3 1982.
+[22] H. T. Grahn, H. J. Maris, J. Tauc, and K. S. Hat-
+ton, “Elastic properties of silicon oxynitride films deter-
+mined by picosecond acoustics,” Applied Physics Letters,
+vol. 53, p. 2281, 12 1998.
+[23] M. Rowley, P. H. Hanzard, A. Cutrona et al., “Self-
+emergence of robust solitons in a microcavity,” Nature
+2022 608:7922, vol. 608, pp. 303–309, 8 2022.
+[24] X. Xu, M. Tan, B. Corcoran et al., “11 tops photonic con-
+volutional accelerator for optical neural networks,” Na-
+ture 2020 589:7840, vol. 589, pp. 44–51, 1 2021.
+[25] C. Reimer, M. Kues, P. Roztocki et al., “Generation of
+multiphoton entangled quantum states by means of in-
+tegrated frequency combs,” Science, vol. 351, pp. 1176–
+1180, 3 2016.
+[26] G. Piccoli,
+M. Sanna,
+M. Borghi,
+L. Pavesi,
+and
+M. Ghulinyan, “Silicon oxynitride platform for linear and
+nonlinear photonics at nir wavelengths,” Optical Materi-
+als Express, vol. 12, 2022.
+[27] Y. Bai, M. Zhang, Q. Shi et al., “Brillouin-kerr soliton
+frequency combs in an optical microresonator,” Phys.
+Rev. Lett., vol. 126, p. 063901, Feb 2021.
+[28] M. Nie, K. Jia, Y. Xie et al., “Synthesized spatiotem-
+poral mode-locking and photonic flywheel in multimode
+mesoresonators,” Nature Communications 2022 13:1,
+vol. 13, pp. 1–9, 10 2022.
+[29] F. Morichetti, S. Grillanda, S. Manandhar et al., “Alpha
+radiation effects on silicon oxynitride waveguides,” ACS
+Photonics, vol. 3, pp. 1569–1574, 9 2016.
+[30] F. Morichetti, A. Melloni, A. Breda et al., “A reconfig-
+urable architecture for continuously variable optical slow-
+wave delay lines,” Optics Express, vol. 15, pp. 17 273–
+17 282, 12 2007.
+[31] C. G. Poulton, R. Pant, and B. J. Eggleton, “Acoustic
+confinement and stimulated brillouin scattering in inte-
+grated optical waveguides,” JOSA B, vol. 30, pp. 2657–
+2664, 10 2013.
+[32] P. P. Absil, J. V. Hryniewicz, B. E. Little et al., “Wave-
+length conversion in gaas micro-ring resonators,” Optics
+Letters, vol. 25, pp. 554–556, 4 2000.
+
+SUPPLEMENTARY NOTE A: DETAILS OF THE SBS EXPERIMENTS
+Experiment setup
+We applied the double-intensity-modulation pump-probe technique to characterize the Brillouin gain coefficient of
+the SiON 5-cm straight waveguides with different widths. Fig. S1 shows the schematic of the experimental setup. The
+pump laser (Agere D2525P22) operates at 1562 nm and is modulated by an intensity modulator (Thorlabs LN81S-FC)
+with a 10.075 MHz sine signal generated by an RF signal generator (Keysight EDU33212A). The pump signal is then
+amplified by an Erbium-doped fiber amplifier (EDFA, Amonics AEDFA-33-B-FA) to 28.7 dBm. After that, the pump
+signal passes an optical circulator (Thorlabs 6015-3-APC) and a polarization controller (Thorlabs FPC032) before it
+is coupled into the chip with an AR-coated polarization maintaining lensed fiber with a spot size of 2 µm (OZ optics).
+The transmitted pump power is monitored with a power meter (Thorlabs S144C). The coupling loss of the sample is
+3 dB per facet.
+Fig. S1. Schematic of the setup used for the SBS characterization. EDFA: erbium-doped fiber amplifier, PM: optical power
+meter, PC: fiber polarization controller, PD: photodetector, RF: radiofrequency signal generator.
+The probe laser (Newport TLB-6728-P) sweeps at frequencies down shifted from the pump laser by about 15 GHz.
+The probe is modulated by an intensity modulator of the same model (Thorlabs LN81S-FC) with a 10 MHz signal
+generated by the other channel of the RF signal generator (Keysight EDU33212A). It is then amplified by an EDFA
+(Amonics AEDFA-PA-35-B-FA) to 21.6 dBm. After that, the probe signal passes an optical circulator (Thorlabs
+6015-3-APC) and a polarization controller (Thorlabs FPC032) before it is coupled into the other side of the chip with
+an identical lensed fiber. The transmitted probe signal passes through an optical bandpass filter (EXFO XTM-50) to
+filter out the reflected pump. After that, 1% of the probe signal is tapped into the power meter (Thorlabs S144C).
+The remaining probe signal is sent into a photodiode (Optilab PD-23-C-DC). The detected electrical signal is then
+analyzed with a lock-in amplifier (Z¨urich Instruments, MFLI 500 kHz) that is synchronized with the RF source.
+TABLE S1 lists the experimental parameters of the setup.
+TABLE S1. The experimental parameters of the SBS characterization setup.
+Parameter Value
+Unit
+Description
+Pprobe
+21.6
+dBm
+Probe optical power after amplification
+Ppump
+28.7
+dBm
+Pump optical power after amplification
+Vπ,probe
+7.2
+V
+Vπ @ DC of the probe modulator
+Vπ,pump
+6.4
+V
+Vπ @ DC of the pump modulator
+Pmod,probe
+0
+dBm
+RF power sent to probe modulator
+Pmod,pump
+0
+dBm
+RF power sent to pump modulator
+rpd
+1.05
+A/W
+Photodiode sensitivity
+Ppd
+6.0
+dBm
+Optical power detected at the photodiode
+αc
+3
+dB/facet Coupling loss per facet, including fiber components
+
+Optical path
+RF path
+sync. clk
+RF
+Lock-in
+amplifier
+source
+PD
+99%
+1%
+Z
+PM 2
+10 MHz
+10 MHz + 75 kHz
+PM 1
+Optical
+bandpass filter
+PC1
+PC2
+Intensity
+Intensity
+Probe w1
+EDFA 1
+SiON waveguides
+EDFA 2
+zm dwnd
+modulator 1
+modulator 2TABLE S2. Simulated and measured Brillouin properties of SiON waveguides.
+Waveguide
+Frequency shift
+Linewidth
+Gain coefficient
+width
+Simulated Measured Simulated Measured Simulated Measured
+(µm)
+(GHz)
+(GHz)
+(MHz)
+(MHz)
+(m−1W−1) (m−1W−1)
+2.0
+15.70
+14.22
+154
+358
+0.35
+0.10
+2.2
+15.70
+14.31
+139
+282
+0.40
+0.15
+2.3
+15.71
+14.40
+131
+280
+0.42
+0.17
+2.4
+15.71
+14.44
+125
+269
+0.44
+0.19
+2.6
+15.71
+14.43
+115
+213
+0.47
+0.24
+3.5
+15.73
+14.48
+92
+105
+0.52
+0.32
+Brillouin gain coefficient calculation
+The SBS gain in a waveguide is determined by:
+G = egBLeffPpump,
+(S1)
+where gB is the Brillouin gain coefficient in m−1W−1, Leff is the effective length of the waveguide, and Ppump is the
+on-chip pump power.
+The effective length of a waveguide is calculated using:
+Leff = 1 − e−αL
+α
+,
+(S2)
+where α is the propagation loss, and L is the actual waveguide length.
+By using (S1), and taking the small signal approximation we can calculate the gain coefficient using:
+gB,SDS = VSDS
+Vfiber
+gB,fiberLeff,fiberPpump,fiber
+Leff,SDSPpump,SDS
+(S3)
+Here V denotes the signal voltage measured by the lock-in amplifier, the subscripts fiber and SDS refer to the
+properties of the fiber and chip used in this experiment.
+Comparison between the measurement and simulation results
+The simulated SBS responses of the 2.0 µm, 2.2 µm, 2.3 µm, 2.4 µm, 2.6 µm, and 3.5 µm-wide SiON waveguides
+are shown in Fig. S2. The peak of the simulated SBS responses increases and shifts towards higher frequencies as the
+waveguide becomes wider, in the meantime, the linewidth also becomes narrower, all of which are coherent with the
+trend of our measurement results.
+We compared the Brillouin frequency shift, linewidth, and the Brillouin gain coefficient of different waveguides
+between the simulation and the measurement results in TABLE S2.
+The simulated Brillouin frequency shift is
+1.3 GHz higher than the measured results, which is less than 10% of the total frequency shift. The measured SBS
+linewidth of different waveguides are constantly larger than the simulations, however, the discrepancy keeps reducing
+as the waveguide becomes wider. The broader linewidth we measured could be contributed to the non-uniformity of
+the waveguides. There is also a discrepancy between the measured and the simulated Brillouin gain coefficients of
+different waveguides. The lower measurement values could come from the increasing coupling loss when we pump
+higher power into the samples. Nevertheless, the measured value also matches better with the simulation results for
+the wider waveguides.
+
+Fig. S2. Simulated SBS responses of the 2.0 µm, 2.2 µm, 2.3 µm, 2.4 µm, 2.6 µm, and 3.5 µm-wide SiON waveguides.
+SUPPLEMENTARY NOTE B: DETAILS OF THE FWM EXPERIMENTS
+Ring resonator characterization
+Before we performed the four-wave mixing (FWM) experiment, we first characterized the ring resonator with the
+experimental setup shown in Fig. S3.
+The laser (Agere D2525P22) was modulated with an intensity modulator
+(Thorlabs LN05S-FC) and was sent to the optical bandpass filter (EXFO XTM-50) to filter out the lower sideband.
+After that, the light is coupled into the sample with the lensed fiber and coupled out from the other side of the
+chip. The signal was then amplified by the EDFA (Amonics AEDFA-33-B-FA) and converted to the RF domain
+with a photodiode (Optilab PD-23-C-DC). We swept the sideband of the light across the resonance response of the
+ring resonator using the vector network analyzer (VNA, Keysight P5007A) with an RF power of -5 dBm. From the
+measured S21 parameter, we then can get the linewidth of the ring resonator with MHz-level resolution, in addition,
+we can use phase information to confirm the ring is over-coupled.
+
+0.5
+2.0 um
+1
+1
+0.0
+-
+0.5
+-
+2.2 um
+0.0
+-
+0.5
+2.3 um
+[m-1 W-1]
+0.0
+b
+60
+0.5
+2.4 um
+0.0
+0.5
+2.6 um
+0.0
+0.5
+3.5 um
+F0'0
+15.3
+15.5
+15.7
+15.9
+16.1
+Brillouin frequency shift [GHz]Fig. S3. Schematic of the setup used for ring resonator characterization. PC: polarization controller, PD: photodetector,
+EDFA: erbium-doped fiber amplifier, VNA: vector network analyzer.
+TABLE S3. The experimental parameters of the FWM characterization setup.
+Parameter Value Unit
+Description
+Ppump
+20.3
+dBm
+Input pump optical power
+Psignal
+6.3
+dBm
+Input signal optical power
+Pider
+-38.7
+dBm
+Output idler optical power
+δvpump
+110
+MHz
+Detuning of the pump light
+δvsignal
+197.5
+MHz
+Detuning of the signal light
+δvidler
+417.5
+MHz
+Detuning of the idler light
+FEpump
+31
+-
+Enhancement factor of the pump light
+FEsignal
+25.2
+-
+Enhancement factor of the signal light
+FEider
+12.4
+-
+Enhancement factor of the idler light
+Leff
+386
+mm
+Effective length of the ring resonator
+FWM experiment setup
+We measured the nonlinear index n2 and nonlinear parameter γ of the SiON waveguide with the FWM experiments
+in the all-pass ring resonator. The experimental setup was shown in Fig.S4. The pump laser (Santec TSL-210) operates
+at 1562 nm and is amplified with an EDFA (Amonics AEDFA-33-B-FA). After that, the pump is sent to an optical
+bandpass filter (EXFO XTM-50) to filter out the amplified spontaneous emission. The signal laser (Agere D2525P22)
+is set close to 2xFSR (100 GHz) away from the pump and is amplified with an EDFA (Amonics AEDFA-37-R-FA).
+The pump is thermally locked close to the resonance of the ring resonator, while the frequency of the probe is tuned
+manually. The pump and the signal are combined with a 99:1 optic coupler (Thorlabs TN1550R1A2) and coupled
+into the all-pass ring resonator with an AR-coated lensed fiber with a spot size of 2 µm (OZ optics). The generated
+idler together with the pump and signal is then coupled out from the chip and sent to the optical spectrum analyzer
+(Finisar Waveanalyzer 1500S) to measure the conversion efficiency from the signal to the idler. The experimental
+parameters are listed in TABLE S3.
+The conversion efficiency η of the four-wave mixing in an all-pass ring resonator is [32]:
+Fig. S4. Schematic of the setup used for four-wave mixing experiment. PC: polarization controller, EDFA: erbium-doped fiber
+amplifier, OSA: optical spectrum analyzer.
+
+VNA
+Optical path
+RF path
+88
+PC
+PD
+Laser
+Intensity
+Optical
+SiON all-pass
+EDFA
+modulator
+bandpass filter
+ring resonatorPC1
+EDFA 1
+Optical
+Pump w1
+bandpass filter
+99%
+88
+OSA
+1%
+PC2
+SiON all-pass ring resonator
+Signal w2
+EDFA 2η = (γLeffPp)2 |FE (ωp)|4 |FE (ωs)|2 |FE (ωi)|2 ,
+(S4)
+where the L is the circumference of the ring resonator, Pp is the on-chip pump power, and the FE(ωp,s,i) is the field
+enhancement factor of the pump, signal, and idler, respectively.
+The effective length considers both the attenuation and the phase mismatch of the four-wave mixing process:
+L2
+eff = L2e−αL
+����
+1 − exp(−α + i∆kL)
+αL − i∆kL
+����
+2
+,
+(S5)
+where α is the propagation loss, and the ∆k is the phase mismatch between the pump, signal, and idler.
+The field enhancement factor can be calculated by:
+|FE| =
+�����
+√κ
+1 − √1 − κ exp(−αL/2) cos
+�
+− 2πδv
+F SR
+�
+����� ,
+(S6)
+where the κ is the power coupling coefficient and the δv is the detuning. We calculated the detuning of the pump and
+the signal by comparing the extinction ratio of the light and the resonance response of the ring resonator. Assuming
+negligible dispersion, the detuning of the idler can be obtained based on:
+��vidler = 2δvpump + δvsignal
+(S7)
+The nonlinear parameter γ can be calculated by combining (S4 - S7). Moreover, the nonlinear index n2 can be
+calculated from the nonlinear parameter γ by:
+n2 = cAeffγ
+ω
+,
+(S8)
+where the ω is the angular frequency of the pump, c is the speed of light in the vacuum and Aeff is the effective mode
+area of the waveguide.
+

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+FedDebug: Systematic Debugging for Federated
+Learning Applications
+Waris Gill
+Computer Science Department
+Virginia Tech
+Blacksburg, USA
+[email protected]
+Ali Anwar
+Computer Science and Engineering Department
+University of Minnesota
+Minneapolis, USA
+[email protected]
+Muhammad Ali Gulzar
+Computer Science Department
+Virginia Tech
+Blacksburg, USA
+[email protected]
+Abstract—In Federated Learning (FL), clients train a model
+locally and share it with a central aggregator to build a global
+model. Impermissibility to access client’s data and collaborative
+training makes FL appealing for applications with data-privacy
+concerns such as medical imaging. However, these FL charac-
+teristics pose unprecedented challenges for debugging. When a
+global model’s performance deteriorates, finding the round and
+the clients responsible is a major pain point. Developers resort
+to trial-and-error debugging with subsets of clients, hoping to
+increase the accuracy or let future FL rounds retune the model,
+which are time-consuming and costly.
+We design a systematic fault localization framework, FEDDE-
+BUG, that advances the FL debugging on two novel fronts. First,
+FEDDEBUG enables interactive debugging of realtime collabora-
+tive training in FL by leveraging record and replay techniques
+to construct a simulation that mirrors live FL. FEDDEBUG’s
+breakpoint can help inspect an FL state (round, client, and
+global model) and seamlessly move between rounds and clients’
+models, enabling a fine-grained step-by-step inspection. Second,
+FEDDEBUG automatically identifies the client responsible for
+lowering global model’s performance without any testing data
+and labels–both are essential for existing debugging techniques.
+FEDDEBUG’s strengths come from adapting differential testing
+in conjunction with neurons activations to determine the precise
+client deviating from normal behavior. FEDDEBUG achieves
+100% to find a single client and 90.3% accuracy to find multiple
+faulty clients. FEDDEBUG’s interactive debugging incurs 1.2%
+overhead during training, while it localizes a faulty client in only
+2.1% of a round’s training time. With FEDDEBUG, we bring
+effective debugging practices to federated learning, improving
+the quality and productivity of FL application developers.
+Index Terms—software debugging, federated learning, testing,
+fault localization
+I. INTRODUCTION
+Many machine learning models today require private user
+information for training accurate models. However, users are
+naturally reluctant to share such data to minimize the risk
+of privacy violation. To address the above needs, Federated
+Learning (FL) [42] enables individual participating clients
+(e.g., smart-home edge devices) to train a machine learning
+(ML) model on their local data in a privacy-preserving envi-
+ronment, and then send the trained model (e.g., the weights
+of the neural network) to a central aggregator to build a
+global model. FL trains highly accurate models without ever
+accessing user data, keeping clients’ data privacy intact [33].
+With the advent of frameworks like Fedml [15], and IBMFL
+[38], FL is actively used in solving real-world problems [20],
+[37], [47], [73].
+Problems.
+The
+support
+for
+collaborative
+yet
+privacy-
+preserving training on FL frameworks comes at the cost
+of transparency and comprehension, making debugging pro-
+hibitively complicated. For instance, a faulty client can send
+an inaccurate model to the aggregator either due to noisy labels
+[18], [22], [28], [30], [31], [44], [72] in the training data or
+malicious intent to deteriorate the global model’s performance
+[1]–[3], [7]. Finding such a faulty client is challenging due
+to a large number of unpredictable clients, that may not have
+participated in every round because of a poor network connec-
+tion or low battery power [54], [63]. The FL training process
+also spans numerous rounds which significantly increases the
+search space for identifying the true, culprit round. None of
+the existing FL frameworks provide debugging and testing
+support to assist the developers building FL applications using
+FL frameworks [24]. These developers rely on guesswork and
+expensive trail-error debugging to find a fault-inducing client.
+Challenges. FL poses two fundamental challenges when de-
+signing a debugging technique. First, in FL deployments, train
+and test data are kept private and strictly reside with clients.
+Access to such data could allow developers to evaluate indi-
+vidual clients’ models sent to the aggregator and identify the
+lowest performing model as the culprit, similar to traditional
+ML model testing. Neither test data nor labels are available
+to an FL application developer and, therefore, existing ML
+debugging approaches [4], [13], [45], [46], [52], [55], [57],
+[59], [61], [62], [71] are also inapplicable.
+Second, due to the unpredictability of clients’ participation
+in a round and the ephemeral nature of their contributions
+in the global model, reproducing a fault (i.e., faulty client)
+and then debugging it is not feasible. Traditional breakpoint
+debugging will pause the entire training process in FL across
+all clients, causing severe side-effects such as data loss as
+clients may not have persistent storage to store data. Live
+postmortem or trial-error debugging may lead to a new set of
+clients for each round, based on client availability and quorum,
+thus making debugging even more ineffective. Considering the
+above limitations and challenges, we must design a debugging
+approach that does not rely on clients’ data, can debug a live
+arXiv:2301.03553v1  [cs.SE]  9 Jan 2023
+
+FL application without any interference, and can localize a
+faulty client precisely.
+Contributions. We take inspiration from traditional debug-
+gers, such as gdb, and redesign traditional debugging con-
+structs that are tailored to the needs of an FL application
+developer. Our approach, FEDDEBUG, selectively records an
+FL application’s telemetry data to enable realtime interactive
+debugging on a simulation that mirrors a live FL application.
+With FEDDEBUG’s breakpoint, a developer can spawn a
+simulation of a live FL application and inspect the current
+state containing information such as clients’ models and their
+reported metrics (e.g., their training loss or hyperparameters).
+It also allows a seamless transition between the rounds and
+clients at a given breakpoint, enabling a fine-grained step-by-
+step inspection of the application’s state. When a developer
+finds a suspicious state (e.g., multiple clients report high
+training loss), FEDDEBUG’s novel automated fault localization
+approach precisely identifies the faulty client without ever
+needing any test data or labels. Once a faulty client is iden-
+tified, FEDDEBUG’s fix and replay repairs the global training
+by retroactively removing the client and resumes the live FL
+training.
+Key Insights. FEDDEBUG leverages several insights to enable
+systematic FL debugging while preserving clients’ privacy. We
+observe that instead of debugging a live FL application, we
+can record a set of runtime metrics essential to regenerate a
+given state in an FL application. Thus, FEDDEBUG performs
+debugging on a regenerated simulated state equivalent to a
+live state. To have a measurable impact on the global model,
+a faulty client’s model must behave differently than the regular
+clients. Every client in an FL application has the same model
+architecture, so their internal behaviors are comparable. Based
+on this insight, FEDDEBUG proposes an inference-guided
+test selection method to select high-quality and diverse test
+data from a pool of randomly generated input images using
+Kaiming Initialization [17]. However, an auto-generated data
+does not include the class label i.e., an oracle. To address the
+oracle problem with such data, FEDDEBUG adapts differential
+testing to FL domain. It captures differences in the models’
+execution via neuron activations instead of output labels to
+identify diverging behavior of a faulty client.
+Evaluations. We perform large-scale, extensive evaluation of
+FEDDEBUG on popular models, two large-scale datasets, two
+well-established FL data distributions, and a real-world fault-
+injection technique in a total of 68 different FL configurations.
+We evaluate FEDDEBUG on fault localizability, debugging
+time, performance overhead over a vanilla FL framework
+(IBMFL), and scalability. FEDDEBUG shows remarkable suc-
+cess in identifying faulty clients. It can localize the real-
+world faulty client with 100% accuracy within 2.1% of a
+round’s training time. When faced with multiple faulty clients,
+FEDDEBUG retains the high fault localization accuracy i.e.,
+90.3%. FEDDEBUG’s debugging constructs incur an overhead
+of 48% of the aggregation time to record telemetry data for
+state regeneration. Surprisingly, this time is only 1.2% of
+a single round’s training time in our experiments. Through
+Hospital1
+2
+HospitalN
+2
+Hospital2
+2
+Hospital3
+2
+Central 
+Aggregator 
+Server
+4
+…
+1
+1
+1
+1
+3
+3
+3
+3
+Aggregation
+Developer
+Fig. 1: In a centralized FL architecture, an aggregator sends a
+global model to clients (step 1). Each client trains the model
+on local data (step 2) and sends the locally trained model back
+to the server (step 3). The server aggregates all models to form
+a new global model (step 4).
+our evaluation, we demonstrate that FEDDEBUG effectively
+conducts interactive debugging and efficiently automates fault
+localization without incurring high runtime costs. FEDDEBUG
+augments the IBMFL framework, but its underlying insights
+can be adapted for other FL frameworks.
+We summarize FEDDEBUG’s contributions below:
+• Originality: To the best of our knowledge, FEDDEBUG
+is the first general-purpose debugging framework for fed-
+erated learning applications that is not limited by access
+to clients’ data. It addresses open debugging challenges
+in FL [24].
+• Approach: Traditional ML trains a single model, whereas
+FL involves distributed training across hundreds of clients
+over multiple rounds. Thus, existing ML debugging ap-
+proaches are inapplicable on FL. FEDDEBUG’s novelty
+lies in observations about FL and the exploitation of in-
+sights on reproducibility, inference guided test generation,
+and differential testing that do not impede performance
+or violate FL privacy principles.
+• Benchmark: We evaluate FEDDEBUG in 68 FL config-
+urations derived from well-established datasets, models,
+varying clients, data distribution, and fault-injections.
+We package our experiment environment into a public
+benchmark for future research use.
+• Usefulness: Our extensive experiments demonstrate that
+FEDDEBUG successfully locates faulty client(s) with-
+out impeding the FL workflow. On a wide range of
+experiments, FEDDEBUG exhibits robust results against
+multiple faulty clients, challenging data distributions, and
+a large number of clients.
+II. BACKGROUND AND MOTIVATION
+A. Federated Learning
+In Federated Learning, multiple clients locally train a model
+on their data and share it with a central server (also called
+an aggregator) to construct a global model. During this col-
+laborative training, clients’ training data never leaves their
+premises [24]. Figure 1 shows an FL setting where multiple
+hospitals collaboratively train a global model on their local
+labeled medical imaging data.
+1) In the first step, the aggregator sends copies of the
+current global model, i.e., the global model weights,
+
+and hyperparameters (e.g., learning rate and epochs) to
+participating clients (Step 1 of Figure 1).
+2) Using the global model as initial parameters, each client
+trains a model on its local data similar to traditional ML
+training (Step 2 of Figure 1).
+3) Once trained, each client sends its local model, in the
+form of updated weights, back to the aggregator as
+shown in Step 3 of Figure 1. Additionally, clients share
+performance metrics such as training loss and qual-
+ity/quantity of training data with the central aggregator.
+4) After receiving model updates, the server aggregates the
+updated weights from all clients using established model
+aggregations such as FedAvg [42] to form a new global
+model (Step 4 in Figure 1).
+The four steps are repeated for a fixed number of rounds
+or until the global model meets some convergence criteria, for
+example, when training loss is close to zero. Note that not
+every client participates in every round. There are additional
+variants of federated learning (FL) such as vertical FL [36],
+FL with differential privacy [60], decentralized FL [43], and
+personalized FL [53]. Our work mainly focuses on the standard
+FL, where the goal is to train a single global model.
+B. Motivating Scenario
+Suppose that an FL application developer trains a global
+neural network model, ResNet [16], on chest X-ray images
+from hospitals across the country to diagnose respiratory
+diseases (e.g., Covid-19). We use the term developer to refer to
+a person who writes, deploys, and monitors the FL application
+at the central server as shown in Figure 1. Every participating
+hospital collects X-rays of patients labelled by radiologists and
+trains a local ResNet model on that data. Hospitals periodically
+share their locally trained models with a central server. The
+central server then aggregates these shared models into one
+global model using FedAvg [42]. After aggregation, the central
+server sends the updated global model to each hospital to
+incorporate in local training in the next round, as shown in
+Figure 1.
+The developer observes that multiple hospitals are reporting
+a high training loss from their preceding training rounds.
+One plausible reason is that one of the hospitals performed
+training on noisy, mislabelling by inexperienced staff [8], [22],
+[30], [31] and continuously impacted the global model during
+aggregation. Thus, when the global model is shared back with
+the other hospitals, it influences their training.
+Limitations for FL Debugging. After noticing an increase
+in training loss, the developer must investigate the root cause,
+as misdiagnosis can lead to ill-treatment. To debug the FL ap-
+plication at this scale, the developer starts manually inspection
+of the collected logs such as global model weights, shared
+local models from hospitals, response/training time of each
+hospital, at the central server. Due to patient’s privacy, the hos-
+pitals refrain from sharing their labelled training data, which
+is critical for correctly evaluating the quality of a model and
+thus essential for localizing the faulty round and model. Even
+if she can find the problematic round, she cannot isolate the
+hospital(s) responsible for affecting the global model without
+test data. One option is cross validating each client’s model
+by requesting that other clients test the model on their local
+data. This is prohibited in practice, as it adds computational
+burden on clients (e.g., edge devices) and can potentially cause
+data privacy violation. Lastly, statically inspecting hospitals’
+models does not provide any meaningful information. Without
+any debugging techniques at her disposal, she resorts to using
+guesswork to identify the hospital with noisy labels (i.e., faulty
+client).
+FedDebug. The developer decides to use FEDDEBUG to
+investigate the root cause behind high training loss. When
+enabled, FEDDEBUG allows a developer to set a breakpoint
+at any round with training loss. This breakpoint separately
+invokes a debugging session, a simulation of the original FL
+service, without stopping the live training. In the debugging
+session, the developer uses FEDDEBUG’s step-back and step-
+next constructs to move between rounds, inspecting the global
+and local models of hospitals recorded by FEDDEBUG. Upon
+inspecting the training rounds, she finds the specific round,
+e.g., Round-8, where the performance starts to deteriorate.
+This round can be different from the breakpoint enabled
+round, as performance issues can manifest in earlier rounds but
+surface later. During this inspection, FEDDEBUG also reports
+the list of hospitals that participated in that round. Next, she
+invokes FEDDEBUG’s fault localization algorithm to precisely
+identify the hospital responsible for deteriorating the global
+model, leading to lower performance. After finding the hospital
+with noisy labels, developer removes it from the problematic
+round (i.e., Round-8) and onwards. FEDDEBUG’s fix and
+replay starts retraining from round Round-8 to the current
+round, and then replaces the impacted global model with
+the retrained global model and continues the original FL
+application.
+III. FEDDEBUG’S DEBUGGING CONSTRUCTS
+The goal of FEDDEBUG is to facilitate an FL application
+developer in isolating a faulty client responsible for dete-
+riorating the global training. Recent studies emphasize the
+need for debugging techniques in FL applications and the
+challenges associated with providing debugging support in FL
+frameworks [24]. To this end, we must overcome the following
+major challenges in designing FEDDEBUG. First, the privacy
+concerns of FL put restrictions on any client-side interference.
+Second, the unpredictable and ephemeral nature of clients in
+FL threatens reproducibility, which is critical for debugging
+a live system. Third, the distributed nature with hundreds of
+participating clients makes traditional breakpoint debugging
+ineffective in FL. Pausing the entire FL application at this
+scale will be prohibitively expensive. Therefore, traditional
+debugging approaches, such as gdb, are not suited for the
+FL systems’ scale and architecture.
+In FEDDEBUG, we address the above challenges and ad-
+vance the systematic FL application debugging. We enable
+realtime, interactive debugging on a simulation of the live
+FL application. To do so, FEDDEBUG’s continuously collects
+
+G
+Algo 2: Faulty Client Localization
+Step back
+C1
+C3
+C8
+C20
+FedAvg
+C1
+C3
+C5
+C8
+C20
+G
+R20
+FedAvg
+C1
+C3
+C8
+C20
+FedAvg
+C1
+C3
+C5
+C8
+C20
+G
+R21
+FedAvg
+C1
+C3
+C8
+C20
+G
+R23
+FedAvg
+C1
+C3
+C5
+C8
+C20
+R22
+FedAvg
+C1
+C3
+C8
+C20
+G
+FedAvg
+Step next
+Step next
+C5
+C5
+Step in
+G
+G
+Resume
+Resume
+G
+3
+2
+4
+5
+Breakpoint
+1
+C5
+Debugging Interface
+IBMFL Interface
+Fig. 2: Using FEDDEBUG, a developer can set a breakpoint
+at round R20. When the FL application finishes round R20,
+FEDDEBUG launches a Debugging Interface, reflected on the
+right. Step next (—) takes the developer to the next step (round
+or client). Step-in increases the granularity of computation,
+e.g., round to client level. Resume (˜) will re-join the current
+execution status of the FL application if no intrusive actions
+are taken. At a given round, FEDDEBUG can automatically
+localize the faulty client (™) and then resume (š) upon which
+the global model will be recomputed without the faulty client.
+This model will replace the corresponding round's model, and
+FEDDEBUG will start retraining from that round, R22, in the
+FL interface.
+and stores concise telemetry data from a live FL application.
+Whenever a debugging need arises, the developer can interact
+with the FEDDEBUG’s debugging interface, which uses the
+telemetry data to regenerate an FL application state.
+A. Selective Telemetry
+FEDDEBUG collects critical FL execution metrics to repro-
+duce an FL application's state for the developer to interact with
+it while investigating the root cause of a problem. Existing FL
+frameworks are carefully architected to refrain from revealing
+private data. As a result, most debugging data is private and
+cannot be investigated.
+FEDDEBUG’s debugging approach takes inspiration from
+replay debugging. As with any other replay debugging, it
+is essential that FEDDEBUG stores the necessary runtime
+metrics to reproduce an FL application's state if requested
+by the developer. We design a highly selective FL event
+telemetry technique that records the concise execution data
+available at the central aggregator that is vital for generating
+any prior FL application state. FEDDEBUG is different from
+traditional replay debugging as it only tracks the information
+needed to recreate an observable event and does not log the
+information unavailable to the developer in a live application.
+This design reduces the size of continuously growing telemetry
+data and minimizes the likelihood of information leakage.
+FEDDEBUG mainly stores the information available after
+step 3 of Figure 1 which is clients’ models, their reported
+metrics such as response time, training loss, validation loss,
+performance metric (e.g., F1 score), hyperparameters (e.g.,
+learning rates, epochs, weight decay), and round ID. Note that
+the FL application, including client-side training, will continue
+uninterrupted in the background with FEDDEBUG’s telemetry
+module continuously collecting execution traces.
+B. Interactive Replay Debugging
+To start the interactive debugging process, a developer can
+invoke FEDDEBUG’s debugging constructs that let the devel-
+oper leverage the telemetry data to investigate the root cause.
+Breakpoint debugging is the de-facto method of debugging a
+program. It pauses the program when the execution reaches
+it. At that point, a developer can inspect the values assigned
+to different variables, both local and global, and examine
+the method stack. Such debugging features are not applicable
+in FL. The traditional breakpoint will pause the distributed
+training (i.e., none of the clients will be able to train its model),
+resulting in unnecessary idling. Additionally, since the state
+of a round is not saved, it is currently impossible for the
+developer to inspect previous rounds. For instance, a developer
+may want to debug a latent issue that was introduced by a
+client five rounds ago but surfaced in the current round when
+the same client participated in training again.
+We make the following observation about FL frameworks.
+An FL application only reveals aggregator's events to a de-
+veloper. In contrast, events on the client's side are entirely
+hidden from the developer except the ones relayed to the
+aggregator by the client. Building on this observation and the
+telemetry data captured by FEDDEBUG, our insight is that
+instead of debugging a system in real-time, we can recreate
+its observable behavior in a simulated environment, giving
+an illusion of debugging an FL application in real-time. By
+doing so, inspections with FEDDEBUG are side-effect free,
+i.e., they will not interfere or interrupt the live FL application,
+thus eliminating the need to pause client-side training or halt
+aggregator execution.
+Breakpoint. To this end, FEDDEBUG offers breakpoint that
+can help a developer inspect intermediate states of an FL
+application in real-time without stopping the training process.
+FEDDEBUG’s breakpoint operates on computation units of
+rounds and clients. A developer can set a breakpoint on either
+a round (e.g., round R20) or on both a round and a client
+(e.g., round R20, client C5) to inspect the state of FL training
+using metrics such as training loss, clients’ participation, and
+response time. When the live FL application arrives on a
+breakpoint, FEDDEBUG spawns a new debugging interface
+on the aggregator side, as shown in – in Figure 2, while
+continuing the live FL training in the background.
+Step in/Step out. While at a breakpoint in a debugging
+session, a developer can use its step-in and step-out actions to
+switch between different granularities of computational units.
+Traditionally, these two actions are used to go one-level deeper
+in the stack (e.g., inside a function call) and move one level up
+in the stack (e.g., outside the function call), respectively. Based
+on this convention, we define a round as a coarse-grained unit
+of computation that can be decomposed into a subset of clients
+participating in that round. Suppose the current breakpoint is
+at round R20. In that case, step-in will take the developer
+to the client-level granularity (— in Figure 2) where trained
+models from clients are being aggregated incrementally. Step-
+out will take the developer back to round, where they can
+
+inspect the global trained model at the granularity of round.
+Inspecting a state at client-level granularity entails evaluating
+the performance of a partially-aggregated global model. For
+example, in Figure 2, step-in at — will take the execution
+between C1 and C3, where the global model has yet to
+incorporate the local models of client C3 and onwards.
+Step Next/Step Back. Similar to step-in/out, step next and
+step back helps a developer transition from one state to
+another. For instance, if the current breakpoint is at round
+R20, step next will take the execution to round R21 in the
+debugging interface, showing all the information correspond-
+ing to that round only. Similarly, if the current breakpoint is at
+client C5, step back will take the execution state to a partial
+global model after aggregating models from clients C1 and
+C3 only (Step back in Figure 2).
+Resume. Unlike resume in gdb, FEDDEBUG’s resume does
+not resume any paused execution. Instead, resume gives the
+illusion to the developer that execution is being continued
+from where it left off. FEDDEBUG creates this environment by
+replaying the telemetry data that was captured while the FL
+application was being inspected using breakpoints, in case the
+developer does not find any faults in the round under inspec-
+tion. Once the sequence of events in telemetry catches up with
+the live execution of the FL application, FEDDEBUG switches
+to the FL interface and shuts down the debugging interface.
+This three-step process is nearly indistinguishable from an FL
+application with FEDDEBUG disabled, giving the impression
+of debugging a real-time FL application interactively. Resume
+is also illustrated in Figure 2 - ˜.
+C. Fix and Replay
+When the developer successfully identifies a faulty client
+in any round, FEDDEBUG offers Fix and Replay to allow a
+developer to roll back the training and provide a retrained
+global model (the one without a faulty client). We describe
+the technique to identify a faulty client in Section IV. A faulty
+client may have a compound effect on the global model, as
+it may have begun to share its noisy model updates latently
+several rounds ago, which only later becomes noticeable. In
+such cases, it is important to rectify the impact of a faulty
+client's inclusion in prior training rounds by removing its
+contributions. This requires retraining over multiple rounds,
+which is not possible as clients may not store the data used in
+training in the prior rounds. Figure 2-™ shows the removal of
+a faulty client (C5) in round R21. FEDDEBUG recomputes the
+global model in the debugging interface and then replaces the
+actual global model in round R22 with the newly recomputed
+global model, after the fix and replay action in Figure 2-š. By
+default, FEDDEBUG forbids the faulty client from participating
+in the FL training. However, it is up to the developer to weigh
+the benefits of including the faulty client in future rounds.
+IV. FAULTY CLIENT LOCALIZATION
+Faults in a client’s model can arise due to measurement
+errors, human labeling errors, data poisoning, communication
+problems, or subjective biases of labellers [10], [11], [29],
+Algorithm 1: Inference-Guided Test Input Selection
+Input: shape: dimension of the random input to be generated.
+Input: κ: number of inputs to be generated.
+Input: η: minimum number of clients for same prediction.
+Output: X: a list containing auto-generated test inputs.
+1 rand inputs = lazilyGenerateRandInputs(shape)
+2 X = list()// a list for inference guided test inputs
+3 seen clients sequences = list()
+4 while length(X) < κ do
+5
+r input = pop(rand inputs)
+6
+clients preds = getP redictions(clients, r input)
+7
+for label ∈ class labels do
+8
+clients seq = sameP redClients(clients preds, label)
+9
+if clients seq ̸∈ seen clients sequences and
+length(clients seq) ≥ η then
+10
+seen sequences.append(clients seq)
+11
+X.append(r input) // valid test input
+12
+break
+13
+if length(rand inputs) < 1 then
+14
+rand inputs = lazilyGenerateRandInputs(shape)
+15 return X
+[51]. To achieve optimal performance of the global model, it
+is critical to correctly identify a faulty client and potentially
+restrict its participation. Manually identifying faulty clients
+is neither scalable nor effective due to a large number of
+participating clients in FL and their uninterpretable models i.e.,
+model parameters do not provide any meaningful debugging
+information. To automate faulty client localization, we must
+define a feedback mechanism to guide our search for faulty
+clients efficiently. Automated debugging tools [27], [66] for
+regular software address this problem by relying on multiple
+test inputs and a test oracle. For example, unit tests can guide
+the search toward concise input leading to incorrect program
+output [66]. In FL, the two (i.e., inputs and oracle) translate
+into diverse test data and the corresponding accurate labels;
+both of which are unavailable in FL applications.
+FEDDEBUG addresses the challenges of automated fault
+localization with a two-pronged approach. First, it generates
+a pool of random test inputs and applies a novel inference-
+guided test input selection to construct a suite of test inputs, as
+shown in Figure 3-A. Since the test inputs are autonomously
+generated, and they are not accompanied with ground truth
+labels, and hence metrics such as F1 score or accuracy cannot
+be used as oracle feedback to find a faulty client. Instead,
+FEDDEBUG performs differential testing of clients’ models to
+measure similarities and differences among models’ behaviors
+on selected inputs (Figure 3-B). FEDDEBUG fingerprints a
+neural network behavior on an input by profiling the internal
+neurons’ contributions towards a prediction of the model.
+Subsequently, it accurately recognizes a client as faulty if
+its behavior deviates from the norm i.e., the majority of the
+clients’ behavior. Our insight is that a faulty client’s model
+will show a noticeable difference in its internal neuron values
+compared to benign clients’ models, based on the principle
+that faulty executions are intrinsically different from correct
+ones. The same principle is behind popular fault localization
+techniques, such as Spectra-based Fault Localization [23] and
+Delta Debugging [66].
+
+Select
+Clients’ Models
+Test Input
+Differential Execution of Clients’ Models
+Client 1
+Client 2
+Client 3
+Client 4
+Faulty Client
+Clients 1-4 are benign because 
+they have the highest 
+common activated neurons. 
+Activated Neuron
+Inactivated Neuron
+(A): Inference Guided Input Selection
+(B): Fault Localization
+. . .
+Infer
+Criteria 1: Minimum  
+(η=3) clients predict 
+same label 
+Criteria 2: Unique 
+(η=3) clients’ 
+combination
+Random Input Pool
+If met
+If not met
+✓
+x
+Client 5
+Fig. 3: An overview of FEDDEBUG’s fault localization ap-
+proach. Firstly, it selects a random input that invokes diverse
+model behavior (A). Secondly, it applies differential execution
+on clients’ models to localize a faulty client (B).
+Inference-Guided Test Input Selection. As shown in Fig-
+ure 3-A, FEDDEBUG first lazily generates a pool of random
+test inputs (e.g., 32x32 images constructed from random values
+within the RGB scale) using Kaiming Initialization [17]. It
+then automatically selects only those inputs that lead to a con-
+sensus on predictions among a unique subset of clients. FED-
+DEBUG selects up to κ test inputs (default is κ = 10) among
+the pool of 1000 random inputs. The goal is to minimize any
+overlapping behavior between clients while inferring unique
+class labels on selected test inputs. This is similar to achieving
+maximum code coverage in regular software with minimum
+tests. Algorithm 1 selects a test input (line 5) if at least (η ≥ 5)
+clients predict the same label and that subset of clients has not
+been seen in the previously selected input (line 6-11). On the
+next random input, if the previously observed subset of clients
+(i.e., clients seq ∈ seen clients sequences) predict the
+same class label, we discard this input. If a unique combination
+of clients predicts an unseen label, we include the input in the
+test suite. This process is repeated until we collect a user-
+defined, κ, number of test inputs.
+Differential Execution of Clients Models. In the absence
+of correct labels of generated test inputs, FEDDEBUG adapts
+differential testing to find behavioral differences and sim-
+ilarities among clients’ models, as shown in Figure 3-B.
+FEDDEBUG profiles the contributions of individual neurons
+during model inference on an input and uses it to identify
+models with common behavior. Note that clients’ models
+in FL are comparable due to having a similar architecture.
+Algorithm 2 describes the faulty client localization process.
+For a selected test input, FEDDEBUG exhaustively iterates all
+possible combinations of potentially non-faulty clients (i.e.,
+�n
+1
+�
+combinations). For each combination, it performs model
+inference on the test input and captures its neuron profiles. It
+aims to find one combination of clients that has the highest
+overlap in behavior, representing the true n − 1 benign clients
+and consequently isolating the precise faulty client. This is a
+Algorithm 2: Faulty Client Localization using Differ-
+ential Testing
+Input: clients: a list of clients participated in the given FL round.
+Input: x: a random input belongs to X.
+Input: na t: a threshold to profile neuron activations.
+Output: faulty client: the faulty client who has abnormal behaviour.
+1 all clients combinations = nChooseK(clients, 1)
+2 benign clients = set()
+3 max common activations = −1
+4 for t clients ∈ all clients combinations do
+5
+neuron ids = ActivatedNeurons(t clients, x, na t)
+6
+t clients common neurons = intersection(neuron ids)
+7
+temp n = length(t clients common neurons)
+8
+if temp n > max common activations then
+9
+max common activations = temp n
+10
+benign clients = t clients
+11 faulty client = clients − benign clients
+12 return faulty client
+lightweight process due to the negligible model inference time
+and the iterations’ linear time (O(n)) complexity.
+Our insight is that among all possible combinations of
+clients, only one represents true benign clients’ subset. The
+remaining combinations contain the faulty client with abnor-
+mal neuron activations, reducing the model behavior overlap
+within that set. In summary, at a given ill-performing round in
+FL, FEDDEBUG takes in all participating clients’ models as the
+only input. It automatically generates test inputs and employs
+differential testing on clients’ models to monitor abnormal
+behavior to precisely identify a faulty client.
+V. EVALUATION
+We evaluate FEDDEBUG on (1) runtime performance over-
+head, (2) debugging time, (3) fault localizability and (4)
+scalability. Our evaluation targets to answer the following
+research questions:
+• RQ1. What impact does FEDDEBUG have on the baseline
+FL framework’s performance?
+• RQ2. How accurate is FEDDEBUG in identifying a faulty
+client?
+• RQ3. Can FEDDEBUG identify multiple faulty clients?
+• RQ4. Can FEDDEBUG scale to large number of clients
+and find a faulty client efficiently?
+Datasets, Model, & FL Framework. We evaluate FEDDE-
+BUG on CIFAR-10 and FEMNIST. Both are considered as gold
+standard to evaluate both FL frameworks [5], [9], [35], [49],
+[58] and deep learning testing techniques [13], [46], [57], [61],
+[62]. FEMNIST is a modified version of MNIST presented in
+the FL LEAF Benchmark [6] and the Non-IID Bench [34]. The
+FEMNIST dataset contains over 340K training and over 40K
+testing grayscale, 28x28 images spanning ten different classes.
+CIFAR-10 contains 50K training 32x32 RGB images that
+span ten different classes and 10K instances for testing. We
+adopt popular CNN models i.e., ResNet, VGG, and DenseNet
+architectures [16], [19], [50]. We set the learning rate between
+0.0001 and 0.001, the number of epochs between 10 and 25,
+the batch size from 512 to 2048, and the weight to 0.0001.
+We realize FEDDEBUG’s design in the IBMFL library [38] due
+to its ease-of-use, open documentation, and publicly available
+
+codebase. These techniques should be equally applicable to
+other FL frameworks.
+Evaluation Environment Specifications. We run our exper-
+iments on an AMD 16-core processor, with 128 GB RAM and
+an NVIDIA Tesla T4 GPU. To measure the performance of
+FEDDEBUG in terms of runtime and debugging overhead, we
+simulate IBMFL framework deployment on a MacBook Pro
+with Quad-core Intel Core i5 processor and 16 GB RAM.
+Federated Learning Experimental Settings. Prior FL lit-
+erature [6], [34] establishes two data distribution strategies
+among FL clients: IID (independent and identically distributed
+data), and Non-IID (non-independent and identically dis-
+tributed data)
+[34]. For Non-IID, we use the quantity base
+imbalance [34] where clients have an unequal quantity of data,
+and the class distribution is random. In IID, the clients receive
+the same quantity of data. None of the clients share the same
+data points in both settings. We simulate FL with varying
+quantities of clients, ranging from 10 to 400 clients.
+Fault Injection. Since there is no existing FL benchmark
+with faulty clients, FEDDEBUG adopts a standard noisy labels
+approach from prior machine learning literature to inject a
+faulty client into experiments [10], [18], [21], [32], [64].
+Similar to prior work [11], [30], [41], we arbitrarily add noise
+by changing training data labels (e.g., changing label “bird”
+to “cat”). When such a client’s model is merged with the
+global model, the global model’s performance (e.g., accuracy)
+deteriorates. We define different strengths of noise with a noise
+rate that controls the amount of labels modified in a faulty
+client. Noise rate is defined as a ratio between changed labels
+and original labels (change labels/original labels).
+Figure 4 shows the impact of different noise rates on the
+global model’s accuracy, with one faulty client and nine benign
+clients. Low noise rates, ranging from 0.2 to 0.7, barely affect
+the global model performance. With a 0.7 noise rate, the
+accuracy is lowered by 4.8% and 5.5% in CIFAR-10 and
+FEMNIST, respectively. A noise rate of 0.9 incurs a 16.2% and
+9.9% reduction in the global model accuracy in both settings.
+Thus, to have a measurable impact on the global model’s
+performance, we select a noise rate of one for a faulty client.
+0.0 0.2 0.4 0.6 0.8
+0
+20
+40
+60
+80
+100
+Noise Rate
+Global Model
+Accuracy (%)
+(a) CIFAR-10
+0.0 0.2 0.4 0.6 0.8
+Noise Rate
+(b) FEMNIST
+Fig. 4: Global model (ResNet-34) prediction accuracy in the
+presence of a faulty client with different noise rates. Lower
+noise rates hardly degrade global model performance.
+Neuron Activation Threshold. We adopt the method from
+Harel-Canada et al. [14] to profile neuron activations. We
+empirically find 0.003 to be the optimal value for the default
+activation threshold. A neuron is considered active when its
+value crosses this threshold.
+5
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+10
+20
+30
+40
+0.3
+0.6
+1.9
+2.5
+3.4
+4.7
+4.8
+13.4
+18
+21
+23.8
+0.6
+1.2
+2.2
+3.9
+5.8
+6.6
+8.7
+15.5
+19.7
+27.5
+28.3
+Number of Clients in an FL Setting
+Aggregation Time (s)
+Vanilla-IBMFL
+FEDDEBUG-IBMFL
+Fig. 5: FEDDEBUG’s runtime overhead as a comparison be-
+tween vanilla FL framework’s aggregation time with FEDDE-
+BUG enabled FL aggregation.
+Faulty Client Localization Accuracy. We calculate faulty
+client localization accuracy as the ratio between (a) the number
+of test inputs on which faulty clients are correctly identified
+and (b) the total number of test inputs. For instance, if
+FEDDEBUG identifies the correct set of faulty clients on four
+out of ten test inputs generated by Alogrithm 1, we report 40%
+fault localization accuracy.
+A. FEDDEBUG’s Performance
+Capturing telemetry data in realtime may slow down the
+performance of the FL application’s aggregator. In this sub-
+section, we present our evaluation results of FEDDEBUG’s
+runtime overhead as well as the fault localization time. These
+experiment settings employ ResNet-18 with CIFAR-10.
+Runtime Overhead (RQ1). To evaluate the impact on the
+FL application’s performance, we measure the slowdown in
+the running time that FEDDEBUG incurs. We compare the
+cumulative processing time of the vanilla IBMFL’s aggregator
+(baseline) against that of the FEDDEBUG-enabled aggregator
+on a variety of client combinations i.e., from 5 clients to
+100 clients, simulating a real-world FL deployment. The
+aggregation time varies with the model’s architecture and the
+number of clients participating in a round, but it is completely
+independent of the models’ quality. Therefore, we create up
+to 100 pre-trained ResNet-18 models and perform the FL
+aggregation.
+Figure 5 compares the baseline’s aggregation time with the
+FEDDEBUG enabled aggregation time. The X-axis represents
+the number of clients ranging from 5 to 100 clients, and
+the Y-axis represents the average time across two FL rounds.
+For instance, with 30 clients, FEDDEBUG takes 3.9 seconds
+compared to the 2.5 seconds for the baseline to aggregate
+30 trained models into a global model. Overall, FEDDEBUG
+takes approximately 48% additional aggregation time across
+all experiments. However, in an end-to-end round, the training
+phase on the clients’ end occupies the majority (up to 97.8% in
+our experiments) of the round’s time. Compared to the training
+time of a round, the aggregation time is almost negligible, as
+low as 1.2% in our experiments.
+
+10
+30
+50
+100
+102
+2.4
+0.2
+0.5
+0.1
+0.5
+0.8
+48
+132.2
+279.5
+Number of Clients in an FL Setting
+Time (s), Log Scale
+Input Time
+Localization Time
+Training Time
+Fig. 6: FEDDEBUG’s debugging time contains input generation
+time and faulty client detection time and is compared against
+a round’s training time.
+Summary: Considering the training and aggregation time
+of each FL round, FEDDEBUG’s runtime overhead is a very
+small fraction, 1.2%, of the training time. Hence, capturing
+telemetry data for replay debugging does not impede the FL
+application’s runtime performance.
+Debugging Time (RQ1). To assess the localizability of FED-
+DEBUG, we design experiments to measure FEDDEBUG’s
+debugging time, the time it takes to localize a faulty client.
+We then compare this time with the training time of that
+round. Since there is no comparable approach to localize a
+faulty client, we use training time as a baseline to provide a
+meaningful scale for the cost of debugging.
+Figure 6 shows the results of these experiments. The X-
+axis represents the number of clients, and the Y-axis shows
+the debugging time in seconds on a logarithmic scale. For 30
+clients, FEDDEBUG’s input generation and selection takes 0.2
+seconds to find high-quality test input, and its fault localization
+takes approximately 0.5 seconds to localize a faulty client. In
+a ten clients setting, input selection takes more time due to
+the stricter constraint of (i.e., η = 4) for criteria 1 in Figure 3,
+i.e., at least four previously unseen clients should predict the
+same label on newly selected test input.
+Overall, our results show an increasing debugging time
+when the number of clients increases, which is expected as
+increasing the number of clients increases the search space.
+Note that the debugging time is still in the order of seconds,
+even for 50 clients. This is because 1) for n clients, the search
+space has at most n possible combinations of potentially
+benign n-1 clients, representing linear complexity, and 2) on a
+given input, FEDDEBUG only profiles neuron activations once
+while iterating over the n combinations.
+Summary: On average, FEDDEBUG can efficiently identify
+a faulty client in 2.1% of the total training time of a round.
+B. Localization of Faulty Client
+To answer RQ2, we measure how accurate FEDDEBUG is
+in localizing a faulty client. We automatically inject a faulty
+client that is representative of a real-world scenario and can
+cause a measurable change in the global model’s performance.
+By varying the number of clients, datasets, models, and data
+Clients
+Dataset
+Architecture
+Accuracy
+% (IID)
+Accuracy
+% (Non-
+IID)
+Avg.
+Input
+Time (s)
+Avg. Lo-
+calization
+Time (s)
+10
+CIFAR10
+DenseNet-121
+100
+100
+2.41
+0.44
+10
+CIFAR10
+ResNet-50
+100
+100
+2.40
+0.22
+10
+CIFAR10
+VGG-16
+100
+100
+2.40
+0.21
+30
+CIFAR10
+DenseNet-121
+100
+100
+2.42
+1.29
+30
+CIFAR10
+ResNet-50
+100
+100
+1.18
+0.70
+30
+CIFAR10
+VGG-16
+100
+100
+2.41
+0.47
+50
+CIFAR10
+DenseNet-121
+100
+100
+2.42
+3.26
+50
+CIFAR10
+ResNet-50
+100
+100
+1.37
+1.24
+50
+CIFAR10
+VGG-16
+100
+100
+2.43
+0.91
+10
+FEMNIST
+DenseNet-121
+100
+100
+2.40
+0.47
+10
+FEMNIST
+ResNet-50
+100
+100
+2.40
+0.25
+10
+FEMNIST
+VGG-16
+100
+100
+2.40
+0.18
+30
+FEMNIST
+DenseNet-121
+100
+100
+2.41
+1.37
+30
+FEMNIST
+ResNet-50
+100
+100
+0.91
+0.68
+30
+FEMNIST
+VGG-16
+100
+100
+2.41
+0.55
+50
+FEMNIST
+DenseNet-121
+100
+100
+2.24
+2.44
+50
+FEMNIST
+ResNet-50
+100
+100
+1.42
+1.24
+50
+FEMNIST
+VGG-16
+100
+100
+2.40
+1.25
+TABLE I: FEDDEBUG’s debugging time and accuracy when
+localizing a faulty client in 36 different FL settings with 100
+test inputs.
+0.2
+0.4
+0.6
+0.8
+0
+20
+40
+60
+80
+100
+Noise Rate
+Fault Localization
+Accuracy (%)
+(a) ResNet-34
+0.2
+0.4
+0.6
+0.8
+0
+20
+40
+60
+80
+100
+Noise Rate
+(b) DenseNet-121
+Fig. 7: FEDDEBUG localization performance when a faulty
+client has varying fault strength (i.e., low noise rate).
+distributions (IID and Non-IID), we create 36 different FL
+configurations for FEDDEBUG’s evaluation.
+Column 4 and 5 of Table I show the accuracy of FEDDEBUG
+in the IID and Non-IID settings, respectively. We repeat each
+experiment on 100 generated test inputs and take the average
+of each metric to generalize the results. FEDDEBUG correctly
+identifies a faulty client with 100% accuracy in both IID and
+Non-IID settings.
+Varying Noise Rate. Figure 4 shows the impact of different
+noise rates on the global model prediction accuracy. We
+observe that a faulty client has measurable impact on the
+global model with a noise rate of > 0.8. The global model’s
+accuracy merely drops from 73.8% to 71.1% when the faulty
+client has 0.6 noise rate and drops to 57% when the noise
+is close to one. FEDDEBUG accurately localizes a faulty
+client with low noise rates, showing its robustness. Figure 7
+shows the evaluations on varying noise rates in 10 clients FL
+settings with ResNet and DenseNet architectures. The X-axis
+shows the faulty client’s noise rate, and the Y-axis represents
+the average fault localization accuracy on the CIFAR-10 and
+FEMNIST datasets. The results, as seen in Figure 7, indicate
+that FEDDEBUG has the capability to identify low noise faults–
+it successfully localizes a faulty client with 0.4 noise rate
+with approximately 58% and 87.5% accuracy in DenseNet and
+ResNet settings, respectively.
+
+Summary: FEDDEBUG achieves 100% fault localization
+accuracy on average on a total of 3600 test inputs, when
+the faulty client significantly deters the global model per-
+formance in both IID and Non-IID settings.
+Detecting Multiple Faulty Clients (RQ3). We evaluate FED-
+DEBUG’s ability to identify multiple faulty clients in an FL
+application. To this end, we inject up to seven faulty clients
+in the following experiment settings. We train ResNet-50 and
+DenseNet-121 on the CIFAR-10 and FEMNIST datasets in
+30 and 50 clients FL settings. Each setting is evaluated on
+10 test inputs. By default, FEDDEBUG’s fault localization
+technique finds a single faulty client. We apply FEDDEBUG in
+an iterative manner to find multiple faulty clients by removing
+one faulty client on each iteration, similar to traditional bug
+repair process, where one bug is fixed first before the next one
+is investigated.
+Table II presents the results of finding multiple faulty clients
+in 32 FL configurations. For instance, when 7 out of 30
+clients are faulty and the model is ResNet-50, FEDDEBUG
+finds all seven faulty clients with 100% accuracy on CIFAR-
+10 and 97.1% accuracy on FEMNIST. Compared to ResNet,
+FEDDEBUG performs relatively better with DenseNet. This
+behavior is expected because, compared to ResNet, DenseNet
+learns better features due to dense concatenation among its
+layers, resulting in better performance [69]. Thus, FEDDEBUG
+performs well in localizing multiple faults with DenseNet with
+an average accuracy of 99.7% on both datasets compared to
+ResNet’s 80.8%.
+Table II also reveals that, generally, FEDDEBUG’s local-
+ization performance is positively correlated to the number of
+training data points per client. Large, high-quality training
+data promotes better feature learning among neurons and thus,
+yields better performance. Since the number of data points
+in FEMNIST (340K) is large compared to CIFAR-10 (40K),
+clients in FEMNIST have significantly larger training data
+than clients in CIFAR-10. As a result, FEDDEBUG average
+localization accuracy is 78.5% in ResNet-CIFAR experiment,
+while it has 83.1% localization accuracy in the ResNet-
+FEMNIST experiment. FEDDEBUG finds multiple faults with
+linear time complexity, as shown in Figure 8 with 50 clients.
+The input generation time is almost constant, as the number
+of clients is fixed. However, the localization time increases as
+we increase the number of faults from 2 to 7. For instance,
+it localizes two faulty clients in 3.6 seconds and five faulty
+clients in 4 seconds.
+Scalability (RQ4): Our findings also show that FEDDEBUG is
+scalable to larger datasets and an increasing number of clients
+in FL. Figure 9 summarizes the impact on FEDDEBUG’s
+ability to identify a faulty client when the number of clients
+changes from 25 to 400 and the training data size per client
+changes. We perform this experiment with two faulty clients
+in the FEMNIST-DenseNet configuration. Figure 9-(a) verifies
+that FEDDEBUG’s fault localization accuracy only reduces
+to 75% even when the number of clients increases to 400.
+FEDDEBUG’s debugging time increases linearly as the number
+TABLE II: FEDDEBUG’s fault localization in 32 FL configu-
+rations with multiple faulty clients, ranging from two to seven.
+Faulty
+Clients
+Total
+Clients
+Architecture
+Accuracy %
+(CIFAR-10)
+Accuracy %
+(FEMNIST)
+2
+30
+ResNet-50
+100
+100
+3
+30
+ResNet-50
+100
+100
+5
+30
+ResNet-50
+100
+98
+7
+30
+ResNet-50
+100
+97.1
+2
+30
+DenseNet-121
+100
+100
+3
+30
+DenseNet-121
+100
+100
+5
+30
+DenseNet-121
+100
+100
+7
+30
+DenseNet-121
+100
+100
+2
+50
+ResNet-50
+50
+80
+3
+50
+ResNet-50
+66.7
+66.7
+5
+50
+ResNet-50
+54
+60
+7
+50
+ResNet-50
+57.1
+62.9
+2
+50
+DenseNet-121
+100
+100
+3
+50
+DenseNet-121
+100
+100
+5
+50
+DenseNet-121
+100
+100
+7
+50
+DenseNet-121
+100
+95.7
+2
+3
+4
+5
+6
+7
+0
+2
+4
+6
+# of Faulty Clients
+Time (s)
+(a) DenseNet-121 and CIFAR-10
+Input Generation Time
+Fault Localization Time
+2
+3
+4
+5
+6
+7
+0
+2
+4
+6
+# of Faulty Clients
+Time (s)
+(b) ResNet-50 and CIFAR-10
+Input Generation Time
+Fault Localization Time
+Fig. 8: FEDDEBUG finds multiple faulty clients in a linear
+time. Total clients are 50 in each graph.
+of clients increases, consistent with the scale-up properties
+of general distributed systems, as shown in Figure 9-(b).
+When the number of clients increases, less data is used to
+train a client’s model, which may reduce the accuracy of
+clients’ models. Figure 9-(c) also shows that FEDDEBUG’s
+fault localizability also increases when the number of data
+points per client increases, and it is also robust against low
+performing client models. For instance, when the number of
+data points increases from 850 to 1700, FEDDEBUG’s local-
+ization accuracy also changes from 75% to 85%, respectively.
+100 200 300 400
+0
+20
+40
+60
+80
+100
+Clients
+Localization Accuracy
+(a)
+100 200 300 400
+101
+102
+103
+104
+Clients
+Time (s), Log Scale
+(b)
+Input Generation Time
+Fault Localization Time
+0
+0.5
+1
+·104
+0
+20
+40
+60
+80
+100
+Data/Client
+Localization Accuracy
+(c)
+Fig. 9: FEDDEBUG retains scalability on a large number of
+clients.
+Summary: Our experiment results provide concrete evi-
+dence that FEDDEBUG preserves scalability properties both
+in terms of time overhead and in the presence of multiple
+faults. It successfully identifies multiple faulty clients in
+32 different FL configurations with an average accuracy of
+90.3%.
+
+0
+0.2 0.4 0.6 0.8
+60
+70
+80
+90
+100
+Localization Accuracy (%)
+(a) ResNet-50 and CIFAR-10
+0
+0.2 0.4 0.6 0.8
+20
+40
+60
+80
+100
+(b) ResNet-50 and FEMNIST
+0
+0.2 0.4 0.6 0.8
+20
+40
+60
+80
+100
+Neuron Activation Threshold
+Localization Accuracy (%)
+(c) DenseNet-121 and CIFAR-10
+0
+0.2 0.4 0.6 0.8
+20
+40
+60
+80
+100
+Neuron Activation Threshold
+(d) DenseNet-121 and FEMNIST
+Fig. 10: FEDDEBUG performance at neuron activation thresh-
+old on 30 clients, including five faulty clients.
+C. Neuron Activation Threshold
+There is no standard threshold of neuron activations [46]
+and prior work uses experiential value for different use
+cases [14]. We evaluate the impact of different activation
+thresholds on FEDDEBUG’s faulty client localizability. We
+take 30 clients including five faulty clients, and train ResNet-
+50 and DenseNet-121 on both the CIFAR-10 and FEMNIST
+datasets. We repeat each experiment on 10 different inputs
+generated by Algorithm 1.
+Figure 10 shows the result of these experiments. The X-
+axis represents the neuron activation thresholds, ranging from
+0 to 0.9. The Y-axis shows the FEDDEBUG’s localization
+accuracy in a given experiment setting. For instance, at the
+0.003 threshold, the average localization accuracy across four
+settings is 100%. On the other hand, at 0.5 threshold, the
+average accuracy decreases significantly to 73.5% across these
+configurations. Specifically, for DenseNet-121 and FEMNIST
+experiment in Figure 10-(d), the localization drops to 64%
+at the 0.5 neuron activation threshold. We observe that FED-
+DEBUG performs better at lower thresholds (< 0.01) across
+different models and datasets. This behavior is expected be-
+cause lower thresholds increase the sensitivity of FEDDEBUG’s
+localization approach. It starts monitoring the majority of the
+neurons compared to a higher threshold, where FEDDEBUG
+profiles only a few neurons crossing the threshold.
+D. Threats to Validity
+To alleviate threats to external validity, we use established
+state-of-the-art FL experimental models (ResNet-18, ResNet-
+34, ResNet-50, DenseNet-121, and VGG-16), two standard-
+ized datasets from FL benchmarks, two real-world data dis-
+tributions, and an industrial scale FL framework. Similarly,
+we remove bias in fault injection using standard noisy labels
+technique from the ML literature, to make a fault reflective
+of real-world scenarios. We also experiment with varying
+noise rates for better evaluations, transparency, and fairness.
+Another source of external threats to validity is randomness in
+the FEDDEBUG’s input selection method. We minimize such
+randomness by evaluating each configuration on at least 10
+and 100 test inputs and reporting the average results.
+VI. RELATED WORK
+Debugging ML models has been extensively explored in
+recent works [4], [13], [45], [46], [57], [59], [62]. The primary
+objectives of these approaches are interpretability, generating
+new test cases by carefully perturbing the real-world training
+inputs to improve performance and find bugs and corner cases
+in the given model. These approaches require access to the
+training and testing data, and some are limited to testing a sin-
+gle neural network; hence, such approaches cannot be directly
+imported in FL. Lack of access to client data and resources in
+FL settings makes testing and debugging FL more challenging.
+If applied to FL, these testing approaches will find every
+client’s model defective. Clients’ models are architecturally
+similar, but trained on local clients’ data, and thus their models
+are semantically different from each other. Identifying defects
+in an isolated model is not practical either. Every client’s
+model has weaknesses that will surface on carefully selected
+test data. FEDDEBUG overcomes these problems by focusing
+on the commonality of models instead of differences.
+Most relevant work to FEDDEBUG primarily focuses on
+finding clients’ contributions to a global model without ex-
+posing the private data to a central server [67]. In practice,
+individual clients report information about training such as
+dataset size and performance metrics to the central aggregator
+[12], [25], [26], [48], [65], [68], [70]. Existing approaches
+use prior information e.g., previous task performance and data
+quality obtained via third-party services to evaluate clients’
+models [56]. Some approaches recommend cross-validating
+clients’ models on another client’s local dataset [40]. Another
+alternate is to maintain a validation dataset at the central server
+to evaluate clients’ models [8], [39]. A major limitation of the
+above FL-related approaches is that the aggregator server is
+entirely dependent on the client's reported information or test
+data to evaluate clients’ models. The aggregator also assumes
+that all clients are trustworthy about their performance in these
+approaches, which invites adversarial clients to exploit FL
+in order to retrieve clients’ private data. Cross validation is
+also prohibited due to limited computing resources for edge
+devices such as smart home sensors. FEDDEBUG overcomes
+the limitations of debugging faulty clients with interactive and
+automated approaches that are privacy preserving.
+VII. CONCLUSION
+Federated learning promotes accurate and collaborative
+model training across millions of clients–a type of learning
+that was previously impossible due to privacy concerns related
+to user data. However, FL poses unprecedented challenges
+in debugging a faulty client responsible for deterring global
+training. With minimal information about the training process
+and non-existent debugging techniques, such issues are often
+
+left untreated. FEDDEBUG enables interactive and automated
+fault localization in FL. It adapts conventional debugging
+practices in FL with its breakpoint and fix and replay feature.
+It offers a novel differential testing technique to automatically
+identify the precise faulty clients. We demonstrate that FED-
+DEBUG identifies a faulty client with 100% accuracy within
+2.1% of a round’s training time, advocating for FEDDEBUG’s
+efficacy and efficiency. With FEDDEBUG, we pave the way
+for advanced software debugging techniques to be adapted
+in the emerging area of federated learning and the broader
+community of machine learning practitioners.
+REFERENCES
+[1] Eugene Bagdasaryan, Andreas Veit, Yiqing Hua, Deborah Estrin, and
+Vitaly Shmatikov. How to backdoor federated learning. In International
+Conference on Artificial Intelligence and Statistics, pages 2938–2948.
+PMLR, 2020.
+[2] Arjun Nitin Bhagoji, Supriyo Chakraborty, Prateek Mittal, and Seraphin
+Calo.
+Analyzing federated learning through an adversarial lens.
+In
+International Conference on Machine Learning, pages 634–643. PMLR,
+2019.
+[3] Battista Biggio, Blaine Nelson, and Pavel Laskov. Poisoning attacks
+against support vector machines. arXiv preprint arXiv:1206.6389, 2012.
+[4] Houssem Ben Braiek and Foutse Khomh.
+Deepevolution: A search-
+based testing approach for deep neural networks. In 2019 IEEE Inter-
+national Conference on Software Maintenance and Evolution (ICSME),
+pages 454–458. IEEE, 2019.
+[5] Lukas Burkhalter, Hidde Lycklama, Alexander Viand, Nicolas K¨uchler,
+and Anwar Hithnawi. Rofl: Attestable robustness for secure federated
+learning. arXiv preprint arXiv:2107.03311, 2021.
+[6] Sebastian Caldas, Sai Meher Karthik Duddu, Peter Wu, Tian Li,
+Jakub Koneˇcn`y, H Brendan McMahan, Virginia Smith, and Ameet
+Talwalkar. Leaf: A benchmark for federated settings. arXiv preprint
+arXiv:1812.01097, 2018.
+[7] Xinyun Chen, Chang Liu, Bo Li, Kimberly Lu, and Dawn Song. Tar-
+geted backdoor attacks on deep learning systems using data poisoning.
+arXiv preprint arXiv:1712.05526, 2017.
+[8] Yiqiang Chen, Xiaodong Yang, Xin Qin, Han Yu, Biao Chen, and Zhiqi
+Shen. Focus: Dealing with label quality disparity in federated learning.
+arXiv preprint arXiv:2001.11359, 2020.
+[9] Liam Collins, Hamed Hassani, Aryan Mokhtari, and Sanjay Shakkottai.
+Exploiting shared representations for personalized federated learning.
+In International Conference on Machine Learning, pages 2089–2099.
+PMLR, 2021.
+[10] Benoˆıt Fr´enay and Michel Verleysen. Classification in the presence of
+label noise: a survey. IEEE transactions on neural networks and learning
+systems, 25(5):845–869, 2013.
+[11] Aritra Ghosh, Himanshu Kumar, and P. S. Sastry. Robust loss functions
+under label noise for deep neural networks.
+In Proceedings of the
+Thirty-First AAAI Conference on Artificial Intelligence, AAAI’17, page
+1919–1925. AAAI Press, 2017.
+[12] Jack Goetz, Kshitiz Malik, Duc Bui, Seungwhan Moon, Honglei
+Liu, and Anuj Kumar.
+Active federated learning.
+arXiv preprint
+arXiv:1909.12641, 2019.
+[13] Jianmin Guo, Yu Jiang, Yue Zhao, Quan Chen, and Jiaguang Sun.
+Dlfuzz: Differential fuzzing testing of deep learning systems.
+In
+Proceedings of the 2018 26th ACM Joint Meeting on European Software
+Engineering Conference and Symposium on the Foundations of Software
+Engineering, pages 739–743, 2018.
+[14] Fabrice Harel-Canada, Lingxiao Wang, Muhammad Ali Gulzar, Quan-
+quan Gu, and Miryung Kim. Is neuron coverage a meaningful measure
+for testing deep neural networks? In Proceedings of the 28th ACM Joint
+Meeting on European Software Engineering Conference and Symposium
+on the Foundations of Software Engineering, pages 851–862, 2020.
+[15] Chaoyang He, Songze Li, Jinhyun So, Xiao Zeng, Mi Zhang, Hongyi
+Wang, Xiaoyang Wang, Praneeth Vepakomma, Abhishek Singh, Hang
+Qiu, et al.
+Fedml: A research library and benchmark for federated
+machine learning. arXiv preprint arXiv:2007.13518, 2020.
+[16] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep
+residual learning for image recognition, 2015.
+[17] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving
+deep into rectifiers: Surpassing human-level performance on imagenet
+classification. In Proceedings of the IEEE international conference on
+computer vision, pages 1026–1034, 2015.
+[18] Dan Hendrycks, Mantas Mazeika, Duncan Wilson, and Kevin Gimpel.
+Using trusted data to train deep networks on labels corrupted by severe
+noise. Advances in neural information processing systems, 31, 2018.
+[19] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q
+Weinberger. Densely connected convolutional networks. In Proceedings
+of the IEEE conference on computer vision and pattern recognition,
+pages 4700–4708, 2017.
+[20] Ji Chu Jiang, Burak Kantarci, Sema Oktug, and Tolga Soyata. Federated
+learning in smart city sensing: Challenges and opportunities. Sensors,
+20(21):6230, 2020.
+[21] Lu Jiang, Di Huang, Mason Liu, and Weilong Yang. Beyond synthetic
+noise: Deep learning on controlled noisy labels.
+In International
+Conference on Machine Learning, pages 4804–4815. PMLR, 2020.
+[22] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-
+Fei. Mentornet: Learning data-driven curriculum for very deep neural
+networks on corrupted labels. In International conference on machine
+learning, pages 2304–2313. PMLR, 2018.
+[23] James A. Jones, Mary Jean Harrold, and John Stasko.
+Visualization
+of test information to assist fault localization. In Proceedings of the
+24th International Conference on Software Engineering, ICSE ’02, pages
+467–477, New York, NY, USA, 2002. ACM.
+[24] Peter Kairouz, H. Brendan McMahan, Brendan Avent, Aur´elien Bellet,
+Mehdi Bennis, Arjun Nitin Bhagoji, Kallista Bonawitz, Zachary Charles,
+Graham Cormode, Rachel Cummings, Rafael G. L. D’Oliveira, Hubert
+Eichner, Salim El Rouayheb, David Evans, Josh Gardner, Zachary
+Garrett, Adri`a Gasc´on, Badih Ghazi, Phillip B. Gibbons, Marco Gruteser,
+Zaid Harchaoui, Chaoyang He, Lie He, Zhouyuan Huo, Ben Hutchinson,
+Justin Hsu, Martin Jaggi, Tara Javidi, Gauri Joshi, Mikhail Kho-
+dak, Jakub Koneˇcn´y, Aleksandra Korolova, Farinaz Koushanfar, Sanmi
+Koyejo, Tancr`ede Lepoint, Yang Liu, Prateek Mittal, Mehryar Mohri,
+Richard Nock, Ayfer ¨Ozg¨ur, Rasmus Pagh, Mariana Raykova, Hang Qi,
+Daniel Ramage, Ramesh Raskar, Dawn Song, Weikang Song, Sebas-
+tian U. Stich, Ziteng Sun, Ananda Theertha Suresh, Florian Tram`er,
+Praneeth Vepakomma, Jianyu Wang, Li Xiong, Zheng Xu, Qiang Yang,
+Felix X. Yu, Han Yu, and Sen Zhao. Advances and open problems in
+federated learning, 2021.
+[25] Jiawen Kang, Zehui Xiong, Dusit Niyato, Han Yu, Ying-Chang Liang,
+and Dong In Kim.
+Incentive design for efficient federated learning
+in mobile networks: A contract theory approach. In 2019 IEEE VTS
+Asia Pacific Wireless Communications Symposium (APWCS), pages 1–
+5. IEEE, 2019.
+[26] Tra Huong Thi Le, Nguyen H Tran, Yan Kyaw Tun, Minh NH Nguyen,
+Shashi Raj Pandey, Zhu Han, and Choong Seon Hong. An incentive
+mechanism for federated learning in wireless cellular networks: An
+auction approach.
+IEEE Transactions on Wireless Communications,
+20(8):4874–4887, 2021.
+[27] Claire Le Goues, ThanhVu Nguyen, Stephanie Forrest, and Westley
+Weimer.
+Genprog: A generic method for automatic software repair.
+Ieee transactions on software engineering, 38(1):54–72, 2011.
+[28] Kuang-Huei Lee, Xiaodong He, Lei Zhang, and Linjun Yang. Cleannet:
+Transfer learning for scalable image classifier training with label noise.
+In Proceedings of the IEEE conference on computer vision and pattern
+recognition, pages 5447–5456, 2018.
+[29] Bo Li, Yining Wang, Aarti Singh, and Yevgeniy Vorobeychik. Data poi-
+soning attacks on factorization-based collaborative filtering. Advances
+in neural information processing systems, 29, 2016.
+[30] Junnan Li, Richard Socher, and Steven CH Hoi.
+Dividemix: Learn-
+ing with noisy labels as semi-supervised learning.
+arXiv preprint
+arXiv:2002.07394, 2020.
+[31] Junnan Li, Yongkang Wong, Qi Zhao, and Mohan S Kankanhalli.
+Learning to learn from noisy labeled data.
+In Proceedings of the
+IEEE/CVF Conference on Computer Vision and Pattern Recognition,
+pages 5051–5059, 2019.
+[32] Junnan Li, Caiming Xiong, and Steven CH Hoi.
+Learning from
+noisy data with robust representation learning. In Proceedings of the
+IEEE/CVF International Conference on Computer Vision, pages 9485–
+9494, 2021.
+[33] Mu Li, David G Andersen, Jun Woo Park, Alexander J Smola, Amr
+Ahmed, Vanja Josifovski, James Long, Eugene J Shekita, and Bor-
+Yiing Su.
+Scaling distributed machine learning with the parameter
+
+server. In 11th {USENIX} Symposium on Operating Systems Design
+and Implementation ({OSDI} 14), pages 583–598, 2014.
+[34] Qinbin Li, Yiqun Diao, Quan Chen, and Bingsheng He.
+Federated
+learning on non-iid data silos: An experimental study.
+In IEEE
+International Conference on Data Engineering, 2022.
+[35] Tian Li, Shengyuan Hu, Ahmad Beirami, and Virginia Smith. Ditto: Fair
+and robust federated learning through personalization. In International
+Conference on Machine Learning, pages 6357–6368. PMLR, 2021.
+[36] Yang Liu, Yan Kang, Liping Li, Xinwei Zhang, Yong Cheng, Tianjian
+Chen, Mingyi Hong, and Qiang Yang.
+A communication efficient
+vertical federated learning framework. Scanning Electron Microsc Meet
+at, 2019.
+[37] Guodong Long, Yue Tan, Jing Jiang, and Chengqi Zhang. Federated
+learning for open banking.
+In Federated learning, pages 240–254.
+Springer, 2020.
+[38] Heiko Ludwig, Nathalie Baracaldo, Gegi Thomas, Yi Zhou, Ali Anwar,
+Shashank Rajamoni, Yuya Ong, Jayaram Radhakrishnan, Ashish Verma,
+Mathieu Sinn, et al. Ibm federated learning: an enterprise framework
+white paper v0. 1. arXiv preprint arXiv:2007.10987, 2020.
+[39] Lingjuan Lyu, Xinyi Xu, Qian Wang, and Han Yu. Collaborative fairness
+in federated learning. In Federated Learning, pages 189–204. Springer,
+2020.
+[40] Lingjuan Lyu, Jiangshan Yu, Karthik Nandakumar, Yitong Li, Xingjun
+Ma, Jiong Jin, Han Yu, and Kee Siong Ng. Towards fair and privacy-
+preserving federated deep models. IEEE Transactions on Parallel and
+Distributed Systems, 31(11):2524–2541, 2020.
+[41] Xingjun Ma, Yisen Wang, Michael E Houle, Shuo Zhou, Sarah Erfani,
+Shutao Xia, Sudanthi Wijewickrema, and James Bailey. Dimensionality-
+driven learning with noisy labels.
+In International Conference on
+Machine Learning, pages 3355–3364. PMLR, 2018.
+[42] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and
+Blaise Aguera y Arcas.
+Communication-efficient learning of deep
+networks from decentralized data. In Artificial intelligence and statistics,
+pages 1273–1282. PMLR, 2017.
+[43] Vaikkunth Mugunthan, Ravi Rahman, and Lalana Kagal.
+Blockflow:
+An accountable and privacy-preserving solution for federated learning.
+arXiv preprint arXiv:2007.03856, 2020.
+[44] Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and
+Ambuj Tewari.
+Learning with noisy labels.
+Advances in neural
+information processing systems, 26:1196–1204, 2013.
+[45] Augustus Odena, Catherine Olsson, David Andersen, and Ian Good-
+fellow. Tensorfuzz: Debugging neural networks with coverage-guided
+fuzzing. In International Conference on Machine Learning, pages 4901–
+4911. PMLR, 2019.
+[46] Kexin Pei, Yinzhi Cao, Junfeng Yang, and Suman Jana. Deepxplore:
+Automated whitebox testing of deep learning systems. In proceedings
+of the 26th Symposium on Operating Systems Principles, pages 1–18,
+2017.
+[47] Nicola Rieke, Jonny Hancox, Wenqi Li, Fausto Milletari, Holger R
+Roth, Shadi Albarqouni, Spyridon Bakas, Mathieu N Galtier, Bennett A
+Landman, Klaus Maier-Hein, et al. The future of digital health with
+federated learning. NPJ digital medicine, 3(1):1–7, 2020.
+[48] Yunus Sarikaya and Ozgur Ercetin.
+Motivating workers in federated
+learning: A stackelberg game perspective.
+IEEE Networking Letters,
+2(1):23–27, 2019.
+[49] Aviv Shamsian, Aviv Navon, Ethan Fetaya, and Gal Chechik.
+Per-
+sonalized federated learning using hypernetworks.
+arXiv preprint
+arXiv:2103.04628, 2021.
+[50] Karen Simonyan and Andrew Zisserman.
+Very deep convolu-
+tional networks for large-scale image recognition.
+arXiv preprint
+arXiv:1409.1556, 2014.
+[51] Jacob Steinhardt, Pang Wei W Koh, and Percy S Liang.
+Certified
+defenses for data poisoning attacks.
+Advances in neural information
+processing systems, 30, 2017.
+[52] Youcheng Sun, Xiaowei Huang, Daniel Kroening, James Sharp, Matthew
+Hill, and Rob Ashmore.
+Deepconcolic: testing and debugging deep
+neural networks.
+In 2019 IEEE/ACM 41st International Conference
+on Software Engineering: Companion Proceedings (ICSE-Companion),
+pages 111–114. IEEE, 2019.
+[53] Canh T Dinh, Nguyen Tran, and Josh Nguyen. Personalized federated
+learning with moreau envelopes.
+Advances in Neural Information
+Processing Systems, 33:21394–21405, 2020.
+[54] Shunpu Tang, Wenqi Zhou, Lunyuan Chen, Lijia Lai, Junjuan Xia, and
+Liseng Fan. Battery-constrained federated edge learning in uav-enabled
+iot for b5g/6g networks. Physical Communication, 47:101381, 2021.
+[55] Yuchi Tian, Kexin Pei, Suman Jana, and Baishakhi Ray.
+Deeptest:
+Automated testing of deep-neural-network-driven autonomous cars. In
+Proceedings of the 40th international conference on software engineer-
+ing, pages 303–314, 2018.
+[56] Muhammad Habib ur Rehman, Ahmed Mukhtar Dirir, Khaled Salah,
+Ernesto Damiani, and Davor Svetinovic.
+Trustfed: a framework for
+fair and trustworthy cross-device federated learning in iiot.
+IEEE
+Transactions on Industrial Informatics, 17(12):8485–8494, 2021.
+[57] Muhammad Usman, Yannic Noller, Corina S P˘as˘areanu, Youcheng Sun,
+and Divya Gopinath.
+Neurospf: A tool for the symbolic analysis of
+neural networks.
+In 2021 IEEE/ACM 43rd International Conference
+on Software Engineering: Companion Proceedings (ICSE-Companion),
+pages 25–28. IEEE, 2021.
+[58] Saeed Vahidian, Mahdi Morafah, and Bill Lin. Personalized federated
+learning by structured and unstructured pruning under data hetero-
+geneity.
+In 2021 IEEE 41st International Conference on Distributed
+Computing Systems Workshops (ICDCSW), pages 27–34. IEEE, 2021.
+[59] Mohammad Wardat, Wei Le, and Hridesh Rajan. Deeplocalize: fault
+localization for deep neural networks.
+In 2021 IEEE/ACM 43rd
+International Conference on Software Engineering (ICSE), pages 251–
+262. IEEE, 2021.
+[60] Kang Wei, Jun Li, Ming Ding, Chuan Ma, Howard H Yang, Farhad
+Farokhi, Shi Jin, Tony QS Quek, and H Vincent Poor.
+Federated
+learning with differential privacy: Algorithms and performance analysis.
+IEEE Transactions on Information Forensics and Security, 15:3454–
+3469, 2020.
+[61] Xiaofei Xie, Lei Ma, Felix Juefei-Xu, Minhui Xue, Hongxu Chen, Yang
+Liu, Jianjun Zhao, Bo Li, Jianxiong Yin, and Simon See. Deephunter:
+a coverage-guided fuzz testing framework for deep neural networks. In
+Proceedings of the 28th ACM SIGSOFT International Symposium on
+Software Testing and Analysis, pages 146–157, 2019.
+[62] Xiaofei Xie, Lei Ma, Haijun Wang, Yuekang Li, Yang Liu, and Xiaohong
+Li. Diffchaser: Detecting disagreements for deep neural networks. In
+IJCAI, pages 5772–5778, 2019.
+[63] Zichen Xu, Li Li, and Wenting Zou. Exploring federated learning on
+battery-powered devices. In Proceedings of the ACM Turing Celebration
+Conference-China, pages 1–6, 2019.
+[64] Jiangchao Yao, Jiajie Wang, Ivor W Tsang, Ya Zhang, Jun Sun,
+Chengqi Zhang, and Rui Zhang.
+Deep learning from noisy image
+labels with quality embedding. IEEE Transactions on Image Processing,
+28(4):1909–1922, 2018.
+[65] Dongdong Ye, Rong Yu, Miao Pan, and Zhu Han. Federated learning
+in vehicular edge computing: A selective model aggregation approach.
+IEEE Access, 8:23920–23935, 2020.
+[66] Andreas Zeller and Ralf Hildebrandt. Simplifying and isolating failure-
+inducing input. Software Engineering, IEEE Transactions on, 28(2):183–
+200, 2002.
+[67] Rongfei Zeng, Chao Zeng, Xingwei Wang, Bo Li, and Xiaowen Chu.
+A comprehensive survey of incentive mechanism for federated learning.
+arXiv preprint arXiv:2106.15406, 2021.
+[68] Rongfei Zeng, Shixun Zhang, Jiaqi Wang, and Xiaowen Chu. Fmore:
+An incentive scheme of multi-dimensional auction for federated learning
+in mec. In 2020 IEEE 40th International Conference on Distributed
+Computing Systems (ICDCS), pages 278–288. IEEE, 2020.
+[69] Chaoning Zhang, Philipp Benz, Dawit Mureja Argaw, Seokju Lee,
+Junsik Kim, Francois Rameau, Jean-Charles Bazin, and In So Kweon.
+Resnet or densenet? introducing dense shortcuts to resnet. In Proceed-
+ings of the IEEE/CVF winter conference on applications of computer
+vision, pages 3550–3559, 2021.
+[70] Jingfeng Zhang, Cheng Li, Antonio Robles-Kelly, and Mohan Kankan-
+halli.
+Hierarchically
+fair
+federated
+learning.
+arXiv
+preprint
+arXiv:2004.10386, 2020.
+[71] Mengshi Zhang, Yuqun Zhang, Lingming Zhang, Cong Liu, and Sar-
+fraz Khurshid.
+Deeproad: Gan-based metamorphic testing and input
+validation framework for autonomous driving systems. In 2018 33rd
+IEEE/ACM International Conference on Automated Software Engineer-
+ing (ASE), pages 132–142. IEEE, 2018.
+[72] Zhilu Zhang and Mert Sabuncu.
+Generalized cross entropy loss for
+training deep neural networks with noisy labels. Advances in neural
+information processing systems, 31, 2018.
+
+[73] Zhaohua Zheng, Yize Zhou, Yilong Sun, Zhang Wang, Boyi Liu, and
+Keqiu Li.
+Applications of federated learning in smart cities: recent
+advances, taxonomy, and open challenges. Connection Science, pages
+1–28, 2021.
+

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+arXiv:2301.00073v1  [cs.IT]  30 Dec 2022
+1
+Fluid Antenna System: New Insights on
+Outage Probability and Diversity Gain
+Wee Kiat New, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Xu Hao, Member,
+IEEE, Kin-Fai Tong, Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE
+Abstract
+To enable innovative applications and services, both industry and academia are exploring new
+technologies for sixth generation (6G) communications. One of the promising candidates is fluid antenna
+system (FAS). Unlike existing systems, FAS is a novel communication technology where its antenna
+can freely change its position and shape within a given space. Compared to the traditional systems, this
+unique capability has the potential of providing higher diversity and interference-free communications.
+Nevertheless, the performance limits of FAS remain unclear as its channels and system properties are
+highly peculiar to be analyzed. To address this, we approximate the outage probability and diversity
+gain of FAS in closed-form expressions. We then propose a suboptimal FAS with N ∗ ports, where a
+significant gain can be obtained over FAS with N ∗ − 1 ports whilst FAS with N ∗ + 1 ports only yields
+marginal improvement over the proposed suboptimal FAS. In this paper, we also provide analytical and
+simulation results to unfold the key factors that affect the performance of FAS. Limited to systems
+with one active radio frequency (RF)-chain, we show that the proposed suboptimal FAS outperforms
+single-antenna (SISO) system and selection combining (SC) system in terms of outage probability.
+Interestingly, when the given space is λ
+2 , the outage probability of the proposed suboptimal FAS with
+one active RF-chain achieves near to that of the maximal ratio combining (MRC) system with multiple
+active RF-chains.
+Index Terms
+6G, fluid antenna system, outage probability, diversity gain, performance analysis.
+The work is supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/W026813/1. For
+the purpose of open access, the authors will apply a Creative Commons Attribution (CC BY) licence to any Author Accepted
+Manuscript version arising. (Corresponding author: Kai-Kit Wong)
+Wee Kiat New (email: [email protected]), Kai-Kit Wong (email: [email protected]), Xu Hao (email:[email protected]), and
+Kin-Fai Tong (email: [email protected]) are with the Department of Electronic and Electrical Engineering, University College
+London, London WC1E 6BT, United Kingdom.
+Kai-Kit Wong and Chan-Byoung Chae (email: [email protected]) are with School of Integrated Technology, Yonsei
+University, Seoul, Korea.
+
+2
+I. INTRODUCTION
+Fifth generation (5G) wireless networks have recently been deployed worldwide and thus the
+industry and academia are now looking for new technologies to maximize the potentials of sixth
+generation (6G) wireless networks. One of the promising candidates is fluid antenna system
+(FAS). Unlike traditional antenna systems, FAS is a software-controllable fluidic, conductive, or
+dielectric structure that can freely adjust its position and shape within a given space [1].
+For example, the most basic single fluid antenna consists of one radio frequency (RF)-chain
+and N ports that are distributed in a given space. The radiating element of the fluid antenna
+can freely switch its position among the ports (e.g., the strongest port) to obtain a stronger
+channel gain, lower interference, and other desirable performance [2]. This is achievable due
+to the recent advancement of using liquid metals (e.g., Galistan and Eutectic Gallium Indium)
+and ionized solutions (e.g., sodium chloride and potassium chloride) for antennas. Note that
+software-controlled pixel antennas, moveable antennas and other flexible antenna structures are
+also considered as fluid antenna [3]. Besides, FAS can co-exist with other 6G candidates such
+as re-configurable intelligent surfaces [4], surface-wave communications [5], intelligent massive
+multiple-input multiple-output [6], and terahertz communications [7].
+Despite its advantages, the fundamental limits of FAS and key factors that affect its perfor-
+mance remain unclear. One of the reasons is because the channels of FAS are strongly correlated
+since the ports can be closely placed to each other. Consequently, the probability density function
+(PDF) and cumulative distribution function (CDF) of FAS channels are intractable [8]. As a result,
+the outage probability and diversity gain of FAS are not known in closed-form expressions. In
+addition, increasing the number of ports of FAS has an inherit diminishing gain due to one active
+RF-chain [9].1 Thus, a suboptimal number of ports that are required to achieve a satisfactory
+performance is not known. Yet, this number is practically and theoretically important as it reduces
+the implementation challenges and analysis complexity.
+Researchers might argue that FAS resembles a traditional selection combining (SC) system as
+the strongest antenna is selected in a point-to-point setting. From this viewpoint, some similarities
+are observed as there is a set of antennas/ports to select from and both systems only use one
+active RF-chain for communications. Nevertheless, FAS can have infinitely many ports (e.g.,
+1Throughout this paper, we refer to an active RF-chain as the RF-chain used for communications. In contrast, the term
+RF-chains refers to a collection of RF-chains that are connected to each antenna for it to work as intended.
+
+3
+when using liquid metals) which makes the implementation and analysis much more challenging.
+In addition, the unique capability of freely switching the radiating element among the ports can
+be exploited to mitigate multi-user interference. These features are impractical or too costly in
+traditional SC systems.
+State-of-the-arts show that FAS outperforms maximal ratio combining (MRC) system if the
+number of ports is sufficiently large [3]. In fact, [3] proves that FAS achieves arbitrarily small
+outage for a fixed rate/signal-to-noise ratio (SNR) as N → ∞. In [10], the authors reveal that the
+ergodic capacity of FAS increases with N and thus FAS can outperform MRC in terms of ergodic
+capacity. Interestingly, FAS can also be used for multiple access. Specifically, [11] proposes a
+fluid antenna multiple access (FAMA) system which leverages the moment of deep fades in space
+to reduce multi-user interference. Motivated by these works, [12] employs stochastic geometry
+to analyze the outage probability of FAS in large-scale downlink cellular networks and [13]
+analyzes the performance of FAS in a more general correlated fading channel.
+Nevertheless, [14] alludes that the channel modeling in the previous works might be inaccurate.
+To address this, [8] proposes a highly complicated channel model to follow closely the spatial
+correlation of the Jake’s model. Using this channel model, they highlight that FAS has limited
+performance gain as N increases. Yet, the key reasons that limit the performance of FAS remain
+ambiguous. This is because the eigenvalue and eigenvector entries that are used in the analytical
+PDF/CDF expressions provide limited insights.
+It is important to highlight that deriving the PDF/CDF of FAS channels is extremely challeng-
+ing [8]. This is because the channels of FAS are strongly correlated and thus they have to be
+formulated in terms of multivariate distributions. Over the past few decades, extensive efforts have
+been dedicated to this problem [15]. However, most of the works only obtain the bivariate [16],
+[17], trivariate [16], [18], [19], or quadvariate [19], [20] distributions while other works restrict
+the correlation matrix to certain forms (e.g., equally correlated [21] and exponentially correlated
+[22]). Fortunately, the multivariate PDF/CDF of arbitrarily correlated Rayleigh distributions are
+recently derived in [23]–[25]. Nevertheless, the assumption of non-singular correlation matrix
+is retained. In this paper, we omit this assumption (i.e., our correlation matrix could be near-
+singular) and address the computation problem via a suboptimal approximation.2
+2The computational problem of a near-singular correlation matrix is much harder to address than that of a singular matrix.
+This is because we can obtain an independent matrix from a singular matrix by removing the dependent entries [26]. But in the
+near-singular case, this approach cannot be applied. Instead, we need to rely on approximations.
+
+4
+In addition to the above works, [27] develops a port selection algorithm that can approach the
+performance of optimal FAS when only the received SNR of a few ports are observed. Further-
+more, [28] considers a field-response channel model while omitting the spatial correlation effect
+and [29] extends the model to a multiple-input multiple-output (MIMO) scenario. Moreover,
+FAMA can be categorized into i) slow-FAMA and ii) fast-FAMA. The earlier switches its port
+when the channel changes [30] while the latter switches its port on a symbol-by-symbol basis
+[31]. The analytical outage probability of two-user FAMA is also derived in [32].
+Motivated by the aforementioned works, this paper aims to understand the fundamental limits
+of FAS as well as the key factors that affect its performance. To this end, we approximate
+the outage probability and diversity gain of FAS in closed-form expressions via a simple and
+accurate channel model that follows closely the spatial correlation of Jake’s model. In addition,
+we propose a suboptimal FAS with N∗ ports as well as an algorithm to approximate N∗. The
+main contributions of our paper are summarized as follows:
+• We employ a simple and accurate channel model that follows the spatial correlation of Jake’s
+model. Based on this channel model, we approximate the outage probability in closed-form
+expressions. By applying Taylor series approximation, we simplify the outage probability at
+high SNR into a simpler and more meaningful expression. Using this result, we also obtain
+the diversity gain of FAS.
+• We propose a suboptimal FAS with N∗ ports. The proposed suboptimal FAS plays an
+important role as it enables FAS to achieve near-optimal performance with minimal number
+of ports. In particular, one may define εtol to adjust the sub-optimality of the proposed FAS.
+For example, if εtol is small, the proposed FAS is quantifiably near-optimal at a cost of more
+ports. In addition, we develop a polynomial-time algorithm to approximate N∗. Besides,
+N∗ can be used to address the near-singular correlation matrix problem.
+• We provide analytical and simulation results to demonstrate the key parameters that affect the
+performance of FAS. Our discussions include intuitive insights on the system characteristics
+as well as practical guidelines for efficient FAS design.
+The rest of the paper is organized as follows: Section II details the system model and performance
+metrics. Section III presents the outage probabiility and diversity gain of FAS. The details of
+suboptimal FAS and the algorithm to approximate N∗ are discussed in Section IV. Section V
+provides our numerical results and we conclude the paper in Section VI.
+Notations: Scalar variables are denoted by italic letters (e.g., c), vectors are denoted by boldface
+
+5
+italic small letters (e.g., c) and matrices are denoted by boldface italic capital letters (e.g., C).
+Besides, (·)T denotes transpose, (·)H denotes conjugate transpose while |·| and ∥·∥F denotes
+absolute and Frobenius norm, respectively. Throughout this paper, log(·) denotes logarithm with
+base 2, E [·] denotes the expectation and P {·} denotes the probability of an event. In addition,
+fc (·) denotes the PDF of c, and Fc (·) denotes the CDF of c. The notation 1c {·} is an indication
+function for condition c and [·]+/−
+c
+outputs the argument that is lower/upper bounded by c.
+II. SYSTEM MODEL
+In this paper, we consider a point-to-point FAS where the transmitter is equipped with a
+conventional antenna and the receiver is equipped with a fluid antenna. The fluid antenna consists
+of N ports, which are evenly distributed along a linear dimension of length Wλ where λ is the
+wavelength of the operating system. Since the ports are closely packed together, there is a strong
+spatial correlation among them. Based on Jake’s model [33], the spatial correlation between the
+mth and nth ports is given by
+Jm,n = σ2J0
+�
+2π(m − n)
+N − 1 W
+�
+,
+(1)
+where σ2 accounts for the large-scale fading effect and J0 (·) is the zero-order Bessel function
+of the first kind.
+For ease of analysis, we introduce the correlation matrix J where
+J =
+
+
+J1,1
+· · ·
+J1,N
+...
+...
+...
+JN,1
+· · ·
+JN,N
+
+ .
+(2)
+In (2), we have Jm,n = Jn,m. Therefore, using eigenvalue decomposition, we can obtain J =
+UΛU H where U is an N × N matrix whose n-th column (denoted by un) is the eigenvector
+of J and Λ = diag (λ1, . . . , λN) is an N × N diagonal matrix whose n-th diagonal entries are
+the corresponding eigenvalues of un. Without loss of generality, we assume that the values of
+the eigenvalues in Λ are arranged in descending order. i.e., λ1 ≥ · · · ≥ λN.
+Throughout this paper, we assume there is only one RF chain in FAS and thus only one port
+can be activated for communications. The received signal of the nth port is expressed as
+yn = hnx + wn, n = 1, . . . , N,
+(3)
+
+6
+where hn is the complex channel coefficient of the nth port, x is the information signal with
+E
+�
+|x|2�
+= P and wn ∼ CN (0, N0) , ∀n is the additive white Gaussian noise of the nth port.
+Due to the spatial correlation of the ports, hn can be modeled as
+hn =
+N
+�
+m=1
+un,m
+�
+λmzm,
+(4)
+where un,m is the (n, m)-th entry of U, zn = an + jbn, where an, bn, ∀n, are i.i.d. Gaussian ran-
+dom variables with zero mean and variance of 1
+2. According to [8], (4) can also be approximated
+as
+ˆhn = Ψvn +
+ǫ-rank
+�
+m=1
+un,m
+�
+λmzm,
+(5)
+where ǫ-rank is a modeling parameter, Ψ =
+�
+σ2 − �ǫ-rank
+m=1 u2
+n,mλm, vn = cn + jdn and
+cn, dn, ∀n, are i.i.d. Gaussian random variables with zero mean and variance of 1
+2.
+To obtain the global optimum performance, FAS activates a port with the maximum signal
+envelope [3],3 i.e.,
+|hFAS| = max {|h1| , . . . , |hN|} .
+(6)
+The average received SNR of the receiver is found as
+Θ = |hFAS|2 P
+N0
+= |hFAS|2 SNR,
+(7)
+where SNR =
+P
+N0 is the transmit SNR and its outage probability is defined as
+P {log (1 + Θ) < q} = P {|hFAS| < Ω} ,
+(8)
+where Ω =
+�
+2q−1
+SNR and q is the minimum required rate. In addition, the diversity gain of FAS
+can be defined as [34]
+lim
+SNR→∞ − log Pe (SNR)
+log (SNR)
+(a)
+=
+lim
+SNR→∞ − log P
+�
+log
+�
+1 + |hFAS|2 SNR
+�
+< q
+�
+log (SNR)
+= d,
+(9)
+where (a) follows from the fact that error probability and outage probability differ by a constant
+shift at high SNR [35].
+3Due to the port spatial correlation, it is shown in [30] that only a small number of observed ports/training is required to
+obtain the full channel state information.
+
+7
+III. OUTAGE PROBABILITY AND DIVERSITY GAIN OF FAS
+As it is seen in (4), the complex channel coefficients h = [h1, . . . , hN]T are correlated.
+Therefore, |h| is a correlated Rayleigh random vector. We present the following lemmas to
+obtain the closed-form outage probability and diversity gain of FAS.
+Lemma 1. The PDF of |h| can be approximated as
+f|h| (r1, . . . , rN)
+≈ η
+s0
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+�1
+2
+��T
+t=1 s∗
+t
+T�
+t=1
+β (t, s∗
+t)
+�
+v∈V
+
+
+T�
+t=1
+
+ s∗
+t
+vt
+
+
+
+
+�
+(2π)N
+N
+�
+i=1
+1{∆i=0}
+�
+.
+(10)
+Proof: See Appendix A.
+In (10), η =
+N
+�
+n=1
+|hn|
+πNdet(J) exp
+�
+−
+�N
+n=1|hn|2Kn,n
+det(J)
+�
+, T = N(N−1)
+2
+, β (t, st) ≜ ζst
+t
+st! , ζt = −2Km,n|hn||hm|
+det(J)
+,
+and s∗
+t = st − st+1 with sT+1 = 0. The subscript t and m, n are related as follows: t =
+n + (m − 1) N − m(m+1)
+2
+, m < n, while m, n can be obtained from t with m = min m′ ∈ Z
+subject to �m′
+i=1 (N − i) > t and n = t − (m − 1) N + m(m+1)
+2
+.
+Note that s0 is a finite constant which has to be large for the approximation to be accurate.
+In addition, v = [v1, . . . , vT]T, V denotes the set of all the possible permutations, and ∆i =
+�N
+n=1 Gi,n + �N
+n=1 Gn,i − Gi,i. Furthermore, Km,n is the (m, n)-th entry of K where K is the
+co-factor of J, and Gm,n is the (m, n)-th entry of G where G is defined as
+G =
+
+
+0
+γ1
+γ2
+· · ·
+γN−1
+γN
+· · ·
+γ2N−3
+...
+...
+...
+γT
+0
+· · ·
+0
+
+
+,
+(11)
+and γt = 2vt − j∗
+t ∈ Z.
+Lemma 2. The CDF of |h| can be approximated as
+F|h| (R1, . . . , RN) ≈
+j0
+�
+j1=0
+j1
+�
+j2=0
+. . .
+jp−1
+�
+jp=0
+g (s∗)
+πNdet(J)
+T
+�
+t=1
+(−Kt)s∗
+t
+s∗
+t!det(J)s∗
+t ×
+N
+�
+n=1
+� Kn,n
+det(J)
+�− ¯sn
+2 −1 �
+Γ
+�
+1 + ¯sn
+2
+�
+− Γ
+�
+1 + ¯sn
+2 , Kn,nR2
+n
+det(J)
+��
+.
+(12)
+
+8
+Proof: See Appendix B.
+In (12), ¯sn is the sum of s∗
+t→m,n affecting |hn| and
+g (s∗) =
+�1
+2
+��T
+t=1 s∗
+t �
+v∈V
+
+
+T�
+t=1
+
+ s∗
+t
+vt
+
+
+
+ (2π)N
+N
+�
+i=1
+1{∆i=0}.
+(13)
+The expressions in (10) and (12) are extremely complicated. Nevertheless, they enable us to
+obtain more insightful derivations as shown later in this paper. Using the above lemmas, we
+present the following theorems.
+Theorem 3. The outage probability of FAS can be approximated in a closed-form expression as
+P {|hFAS| < Ω} = F|h| (Ω, . . . , Ω)
+(14)
+≈
+j0
+�
+j1=0
+j1
+�
+j2=0
+. . .
+jp−1
+�
+jp=0
+g (s∗)
+πNdet(J)
+T�
+t=1
+(−Kt)s∗
+t
+s∗
+t!det(J)s∗
+t ×
+N
+�
+n=1
+� Kn,n
+det(J)
+�− ¯sn
+2 −1 �
+Γ
+�
+1 + ¯sn
+2
+�
+− Γ
+�
+1 + ¯sn
+2 , Kn,nΩ2
+det(J)
+��
+.
+Proof: The result can be obtained using Lemma 2 and substituting R1 = · · · = RN = Ω.
+Remark 4. According to [8], h can be modeled using ˆh =
+�
+ˆh1, . . . , ˆhN
+�T
+and using the latter
+model, they show that the outage probability of FAS can be approximated by
+F|hFAS| (Ω) ≈
+
+
+N
+�
+n=1
+∞
+ˆ
+0
+1
+�ǫ−rank
+m=1
+u2n,mλm
+exp
+�
+−
+r
+�ǫ−rank
+m=1
+u2n,mλm
+�
+×
+�
+1 − Q1
+�√
+2r
+Ψ ,
+√
+2Ω
+Ψ
+��L
+dr
+
+
+1
+L
+,
+(15)
+where Q1 (·, ·) is the Marcum-Q function and L = min
+�
+1.52(N−1)
+2πW
+, N
+�
+. Note that (15) is a
+remarkable expression as each n term only has a single integral. Nevertheless, we found that it
+is challenging to obtain deeper insights from this expression.
+Theorem 5. The outage probability of FAS at high SNR is given by
+P {|hFAS| < Ω} =
+1
+det(J)ΩN + o
+�
+1
+SNRN
+�
+.
+(16)
+Proof: See Appendix C.
+
+9
+Theorem 6. The diversity gain of FAS is approximately expressed as
+DFAS ≈ min {N, N′} ,
+(17)
+where N′ is the rank of J ′ such that J ′ is the covariance matrix as defined in (2) with N → ∞
+for a fixed W.
+Proof: See Appendix D.
+In Theorem 5, we can interpret det
+�
+J −1�
+as the penalty term and Ω as gain of FAS that scales
+exponentially w.r.t. N. Meanwhile, the term with little-o can be ignored as it approaches zero if
+the SNR is high. Nevertheless, in Theorem 6, we can see that the diversity gain is limited by
+min {N, N′}. Thus, increasing N over N′ might not be useful. Notice that these interpretations
+cannot be directly obtained from (15).
+IV. SUBOPTIMAL SOLUTION: FAS WITH N∗ PORTS
+At a fundamental level, [9] showed that increasing the number of channels (or ports) would
+yield a diminishing gain (i.e., the average received SNR gain is �N
+n
+1
+n.). In fact, [8] showed that
+for a fixed W, the outage probability of FAS might remain similar after some N. For ease of
+expositions, we denote this N as N∗ where N∗ ≤ N.
+To the best of our knowledge, little is known about N∗. In fact, it is very challenging to
+obtain N∗ as it varies with the parameter W or more precisely the correlation matrix J.4 Yet,
+finding N∗ is essential in both theory and practice since it helps FAS to achieve an efficient
+performance with a minimal number of ports. In this section, we present a simple method to
+approximate N∗ for a given W.
+To begin with, we present the following theorem.
+Theorem 7. Suppose the channels of FAS with N ports are denoted by h. Then h can be
+well-approximated by ˜h =
+�
+˜h1, . . . , ˜hN
+�T
+where
+˜hn =
+˜
+N
+�
+m=1
+un,m
+�
+λmzm,
+(18)
+where ˜N is the numerical rank of J. That is, the PDF and CDF of h and ˜h are similar.
+4Referring to (1) and (2), we can see that N ∗ depends on the parameter W .
+
+10
+Proof: Let ˜N be the numerical rank of J where ˜N ≤ N. Using the definition of numerical
+rank, we have λn < ǫ for n ∈
+�
+˜N + 1, . . . , N
+�
+where ǫ ≈ 0. According to Eckart-Young-Mirsky
+theorem [36], the optimal ˜J that minimizes the Frobenius norm between matrix J and ˜J subject
+to the constraint that rank
+�
+˜J
+�
+≤ ˜N is ˜J = U ˜ΛU H where ˜Λ = diag (λ1, . . . , λ ˜
+N, 0, . . . 0).
+Using this insight, we introduce ˜h as defined in Theorem 7 where the covariance of ˜h is ˜J
+(i.e., the best approximation of J for rank
+�
+˜J
+�
+≤ ˜N). As a result, we can well-approximate h
+using ˜h since the Frechet distance between the two distributions is [37]
+W2
+�
+CN (0N×1, J) , CN
+�
+0N×1, ˜J
+��
+=
+����(Λ)
+1
+2 −
+�
+˜Λ
+� 1
+2
+����
+2
+F
+≈ 0.
+(19)
+Corollary 8. If we have the exact eigenvalues and rank of J, then h = ˜h.
+Proof: Let Λ and ˜N be the exact rank and eigenvalues of J. Using the definition of rank,
+we have λn = 0 for n ∈
+�
+˜N + 1, . . . , N
+�
+. It then follows that the Frechet distance between the
+distributions of h and ˜h is zero.
+As seen in (19), it is the eigenvalues of correlation matrix that plays a critical role in the
+channel approximation. Motivated by this insight, we introduce a new formula as follows:
+εN∗ = SN − SN∗ = σ2 − SN∗,
+(20)
+where SN∗ = 1
+N
+�N∗
+n=1 λn. Note that (20) is analogous to (19) in the sense that the left hand side
+of (20) measures the gap between the distributions of h and h∗, where h∗ is similarly defined
+as in (18) but we instead replace ˜N with N∗ and impose that N∗ ≤ ˜N. Meanwhile, on the right
+hand side of (20), we consider the average eigenvalues of J ∗, where J ∗ is the covariance of h∗.
+To reduce the number of required ports, we define εtol > 0 and find the smallest integer N∗
+such that εtol ≥ εN∗. Since J ∗ only has N∗ dominant eigenvalues, we propose to employ a
+suboptimal FAS with N∗ ports. Interestingly, εtol has a nice heuristic interpretation in practice.
+Specifically, it defines the sub-optimality of the proposed FAS, i.e., the proposed FAS is near
+optimal if εtol is small and less optimal if εtol is large.
+By fixing εtol appropriately,5 we observe that FAS with N∗ ports yields considerable improve-
+ment over all FAS with N < N∗ ports while most of the FAS with N > N∗ ports yields
+5We recommend to set εtol = 0.01σ2 (i.e., the average eigenvalues of J ∗ is 99% of that of J)
+
+11
+Algorithm 1 Method of approximating N∗ given W
+1: Input: W, εtol; Output: N∗
+2: Compute J = UΛU H
+3: Define n = 1 and compute εn
+4:
+While εtol < εn and n < ˜N
+5:
+n = n + 1
+6:
+εn = σ2 − Sn
+7: end
+8: Return n as N∗
+marginal improvement over FAS with N − 1 ports. Note that we usually have N∗ < ˜N if J is
+ill-conditioned and N∗ = ˜N if J is well-conditioned.
+The method of approximating N∗ is given in Algorithm 1. To measure the computational
+complexity of our algorithm, we consider the floating-point operations (flops). A flop is defined
+as one addition, subtraction, multiplication or division of two floating point numbers [38].
+In Algorithm 1, computing J and UΛU H requires 6N2 and 21N3 flops, respectively [39].
+Computing εn requires n + 1 flops for each n. Therefore, the total flops of Algorithm 1 is
+21N3 +6N2 + 1
+2N∗2 + 3
+2N∗, which has a polynomial time-complexity of O (N3) since N∗ ≤ N.
+In other words, Algorithm 1 is only dominated by the computation of UΛU H.
+Note that N∗ is also useful in theory. For example, Lemma 1 and 2 and Theorem 3, 5, and
+6 are incalculable if J is near-singular. To address this, we present the following theorem.
+Theorem 9. If J is near-singular, then we can approximate the channels of FAS with N ports
+using N∗ ports from a computational perspective. Nevertheless, a small gap between the channel
+distributions of FAS with N ports and that of N∗ ports might exist.
+Proof: If J is near-singular, then one or more entries are almost linear combinations of
+the other entries. Thus, we can remove these nearly-dependent entries and only consider N∗
+independent entries. Since FAS with N∗ ports has N∗ dominant eigenvalues, Lemma 1 and 2
+and Theorem 3, 5, and 6 are calculable. Nevertheless, there might be a small gap between the
+channel distributions of FAS with N ports and that of N∗ ports since the entries are nearly-
+dependent only.
+
+12
+(a)
+(b)
+Figure 1: FAS with 2 ports: (a) joint PDF; (b) joint CDF.
+V. RESULTS AND DISCUSSION
+In this section, we present simulation results to better understand the performance of FAS.
+We focus on the design of an efficient FAS as well as the factors that limit its performance.
+Unless stated otherwise, we assume that σ2 = 1, N = 50, W = 0.5, q = 10 and SNR = 30dB.
+Firstly, we demonstrate the accuracy of (10) and (12). In order to visualize the joint PDF and
+CDF of |h|, we consider a FAS with 2 ports (i.e., N = 2). In Fig. 1, the red grid represents
+the numerical PDF/CDF while the solid surface is the analytical PDF/CDF. As observed, the
+approximation of the PDF/CDF of |h| matches closely with the numerical ones over all the
+distributed region. Still, it is worth noting that (10) and (12) are very complicated. Thus,
+approximations with simpler expressions remain desirable.
+In Fig. 2, we compute the outage probability of FAS versus SNR for different N and W.
+Comparing Fig. 2(a) and Fig. 2(b), we can clearly see that the outage probability is mainly
+limited by W. In particular, if W is small and N is large, the outage probability remains similar
+which is in alignment with the findings of [8]. Nevertheless, if W is sufficiently large, the outage
+probability decreases significantly as N increases.
+To better understand this, we further compare the outage probability of FAS to (15) and (16).
+In Fig. 3, we can see that (15) is less accurate while (16) is accurate as SNR increases. From
+(16), we learn that det
+�
+J −1�
+plays a critical role in the performance of FAS. In particular, J
+has to be well-conditioned in order for ΩN to be the dominant term. If J is near-singular, then
+
+30.8
+0.6
+CDF
+0.4
+0.2
+0
+4
+3
+2
+1
+1
+0
+0
+[h2]
+[hi]0.03Q.0
+0.6
+PDF
+0.4
+0.2
+0
+4
+3
+2
+2
+1
+1
+[h2]
+0
+0
+[hi]13
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+(a)
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+(b)
+Figure 2: Outage probability of FAS versus SNR for different N and W: (a) W = 0.5; (b)
+W = 10.
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+SNR
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+Figure 3: Outage probability of FAS at high SNR.
+the parameter N is no longer important. This is because det
+�
+J −1�
+cannot be compensated by
+ΩN. To make J a well-conditioned matrix, we can either increase W for a fixed N or decrease
+N for a fixed W. Nevertheless, we believe that larger N does not cause any harm to the system
+in practice. It only makes the theoretical analysis harder.
+As shown in Fig. 4(a), we compare the outage probability of FAS with N ports and that of
+N′ ports for different W where N < N′. As it is seen, the outage probability of the earlier
+is lower bounded by the latter regardless of W. In Fig. 4(b), we investigate the opposite case
+
+14
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+(a)
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+(b)
+Figure 4: Outage probability of FAS with N ports versus ˜N ports: (a) N = 3 <
+˜N; (b)
+N = 50 > ˜N .
+where N > N′. As observed, the outage probability of FAS with N ports and that of N′ ports
+are the same for different W. Thus, the diversity gain of FAS is limited by min {N, N′}, which
+verifies Theorem 6. Theorem 6 also suggests that increasing the ports beyond N′ provides no
+improvement in a point-to-point setting.
+Fig. 5(a) presents the CDF of h and ˜h where we fix R1 = · · · = RN = R. In the result, no
+significant variation is observed between h and ˜h regardless of R, N and W. This is because
+the Frechet distance between the two distributions is always near zero. This confirms Theorem
+7 and suggests that one can always use ˜h instead of h. In addition, Fig. 5(b) shows the CDF of
+h and h∗. Unlike the previous result, there is a small gap between the two distributions as W
+increases. Despite having some gaps, the approximation is still fairly good. This result verifies
+Theorem 9.
+Next, we investigate the accuracy of Algorithm 1 and the efficiency of the proposed suboptimal
+FAS. The parameter N∗ for different W using Algorithm 1 is summarized in Table I. As it is
+seen, the outage probability of FAS with N∗ ports is promising. Specifically, FAS with N∗ ports
+yields a significant improvement over FAS with N∗ −1 ports. Meanwhile, FAS with N +1 ports
+provides negligible improvement over FAS with N∗ ports. Thus, we may use the suboptimal
+FAS for an efficient performance.
+Finally in Fig. 7, we compare the outage probability of the proposed suboptimal FAS, the
+
+15
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+CDF
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+CDF
+(b)
+Figure 5: CDF between: (a) h and ˜h; (b) h and h∗.
+Table I: Parameter N∗ generated by algorithm 1 for different W.
+W
+0.5
+1
+2
+3
+4
+N ∗
+3
+4
+6
+8
+10
+optimal FAS, the single antenna (SISO) system, the N-branch SC system, and the N-branch
+MRC system. In SC and MRC systems, we assume there are N RF-chains where each antenna
+has to be at least λ
+2 apart and their spatial correlations are considered. Results show that the
+proposed suboptimal FAS outperforms SISO and SC systems. This improvement is due to the
+ability of FAS switching to the best port within a finite W.
+In addition, MRC has the lowest outage probability. It outperforms optimal FAS. This su-
+periority is due to the power gain where a larger number of active RF-chains (i.e.,
+� W
+0.5
+�
++ 1)
+is utilized. Although MRC is more superior than the suboptimal FAS, the latter can achieve a
+similar performance as compared to the earlier when W = 0.5. Yet, it is important to recall
+that MRC has one additional RF-chain as compared to the suboptimal FAS in this case. Thus, it
+will be very interesting to compare the performance of MIMO-FAS and MIMO with the same
+number of RF-chains.
+
+16
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+W=4
+W=2
+W=1
+W=0.5
+W=3
+Figure 6: Outage probability of suboptimal FAS.
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Outage probability
+Figure 7: Outage probability of suboptimal FAS vs. SISO, SC, and MRC.
+VI. CONCLUSIONS
+In this paper, we consider FAS and approximate its outage probability and diversity gain in
+closed-form expressions. New meaningful insights are obtained from the analytical results, and
+simulation results are given to better understand the factors that limit the performance of FAS.
+Our results show that the performance of FAS strongly depends on the spatial correlation matrix
+J. Specifically, increasing the ports beyond N′ yields no diversity gain in a point-to-point setting.
+Instead, increasing N causes the correlation matrix J to be ill-conditioned. To address this, one
+can either increase W for a fixed N or decrease N for a fixed W. In addition, we propose a
+suboptimal FAS with N∗ ports. By fixing an appropriate εtol, the proposed scheme enables us to
+
+17
+obtain a significant gain over FAS with N∗ − 1 while it nearly achieves the same performance
+as FAS with N∗ + 1 ports. Thus, the approximation of N∗ is pragmatically useful since a larger
+number of ports yields diminishing gains and additional costs. Furthermore, N∗ can be used to
+approximate the channels of FAS with N ports if the correlation matrix J is near-singular. Last
+but not least, the proposed suboptimal FAS outperforms SISO and SC systems but falls behind
+MRC due to having a single active RF-chain. Nevertheless, it is discovered that suboptimal FAS
+and MRC achieve similar performance when W = 0.5. Thus, it will be interesting to study the
+performance of MIMO-FAS and MIMO in the future.
+APPENDIX A: APPROXIMATED PDF OF |h|
+The exact PDF of |h| is first derived in [23]–[25]. In this paper, we employ similar steps
+and further approximate the PDF of |h| by introducing G: an N × N matrix, using an accurate
+binomial theorem, and truncating the infinite series to a finite one for ease of computation.
+According to [34], the PDF of a circularly symmetric complex Gaussian random variables is
+known as
+f (h) =
+1
+πNdet(J) exp
+�
+−hHJ −1h
+�
+,
+(21)
+where J −1 =
+KT
+det(J) via Crammer rule. Using [40, (7-8) & (7-9)], the PDF of (21) in terms of
+its amplitude and phase can be obtained as
+f|h|,θ (|h1| , θ1, . . . , |hN| , θN) = η
+T�
+t=1
+exp
+�
+ζt cos
+�¯θt
+��
+,
+(22)
+where η =
+N
+�
+n=1
+|hn|
+πNdet(J) exp
+�
+−
+�N
+n=1|hn|2Kn,n
+det(J)
+�
+, T = N(N−1)
+2
+, ζt = −2Km,n|hn||hm|
+det(J)
+and ¯θt = θn − θm.
+In (22), we use the mapping function where t = n + (m − 1) N − m(m+1)
+2
+, m < n, while
+(m, n) can be obtained from t by setting m = min m′ ∈ Z subject to �m′
+i=1 (N − i) > t and
+n = t − (m − 1) N + m(m+1)
+2
+.
+
+18
+Integrating (22) w.r.t. θn, ∀n over [0, 2π], we have
+f|h| (|h1| , . . . , |hN|)
+=
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+f (|h1| , θ1, . . . , |hN| , θN) dθ1 . . . dθN
+(23)
+(a)
+=η
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+T�
+t=1
+∞
+�
+st=0
+ζst
+t
+st! cos
+�¯θt
+�st dθ1 . . . dθN
+(24)
+(b)
+=η
+∞
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+T�
+t=1
+β (t, s∗
+t)
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+cos
+�¯θt
+�s∗
+t dθ1 . . . dθN
+(25)
+(c)
+=η
+∞
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+�1
+2
+��T
+t=1 s∗
+t
+T�
+t=1
+β (t, s∗
+t) ×
+(26)
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+T�
+t=1
+�
+exp
+�
+j¯θt
+�
++ exp
+�
+−j¯θt
+��s∗
+t dθ1 . . . dθN
+(d)
+=η
+∞
+�
+s1=0
+. . .
+sT −1
+�
+sT =0
+�1
+2
+��T
+t=1 s∗
+t
+T�
+t=1
+β (t, s∗
+t)
+�
+v∈V
+T
+�
+t=1
+
+ s∗
+t
+vt
+
+ ×
+(27)
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+exp
+�
+j
+T
+�
+t=1
+γt¯θt
+�
+dθ1 . . . dθN,
+where (24) is obtained by using exp {x} = �∞
+s=0
+xs
+s! and (25) is obtained using Cauchy product
+of power series where β (t, st) ≜
+ζst
+t
+st! and s∗
+t = st − st+1 with sT+1 = 0. Furthermore, (26)
+is obtained using cos (x) = exp(jx)+exp(−jx)
+2
+and (27) is obtained using binomial theorem where
+v = [v1, . . . , vT]T, V denotes the set of all the possible permutations and γt = 2vt − j∗
+t ∈ Z.
+Note that
+´ 2π
+0
+· · ·
+´ 2π
+0
+exp
+�
+j �T
+t=1 γt¯xt
+�
+dθ1 . . . dθN = (2π)N if and only if �T
+t=1 γt¯xt = 0,
+and otherwise zero. Therefore, we introduce a new matrix G as defined in (11) and the matrix
+¯Θ given by
+¯Θ =
+
+
+0
+¯θ1
+¯θ2
+. . .
+¯θN−1
+¯θN
+. . .
+¯θ2N−3
+...
+...
+...
+¯θT
+0
+. . .
+0
+
+
+=
+
+
+0
+θ2 − θ1
+θ3 − θ1
+. . .
+θN − θ1
+θ3 − θ2
+. . .
+θN − θ2
+...
+...
+...
+θN − θN−1
+0
+. . .
+0
+
+
+.
+(28)
+Using ¯Θ and G, we can easily integrate (27) w.r.t. to θi by taking the sum of the same entries
+
+19
+of G as that of ¯Θ with θi, i.e., ∆i = �N
+n=1 Gi,n + �N
+n=1 Gn,i − Gi,i. Therefore, (27) leads to
+(27) =η
+∞
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+�1
+2
+��T
+t=1 s∗
+t
+T
+�
+t=1
+β (t, s∗
+t)
+�
+v∈V
+
+
+T�
+t=1
+
+ s∗
+t
+vt
+
+
+
+
+�
+(2π)N
+N
+�
+i=1
+1{∆=0}
+�
+(29)
+(a)
+≈η
+s0
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+�1
+2
+��T
+t=1 s∗
+t
+T
+�
+t=1
+β (t, s∗
+t)
+�
+v∈V
+
+
+T�
+t=1
+
+ s∗
+t
+vt
+
+
+
+
+�
+(2π)N
+N
+�
+i=1
+1{∆i=0}
+�
+,
+(30)
+where (a) can be obtained since β (t, s∗
+t) ≈ 0 if s∗
+t is sufficiently large.
+APPENDIX B: APPROXIMATED CDF OF |h|
+Using (10), the CDF of |h| can be obtained as
+F (R1, . . . , RN)
+≈
+ˆ R1
+0
+. . .
+ˆ RN
+0
+f|h| (|h1| , . . . , |hN|) d |h1| · · · d |hN|
+(31)
+=
+s0
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+g (s∗)
+πNdet(J)
+T�
+t=1
+(−2Km,n→t)s∗
+t
+s∗
+t!det(J)s∗
+t
+ˆ R1
+0
+. . .
+ˆ RN
+0
+×
+(32)
+N
+�
+n=1
+|hn|
+N
+�
+n=1
+N
+�
+m<n
+(|hn| |hm|)s∗
+t→m,n exp
+�
+−
+�N
+n=1 |hn|2 Kn,n
+det(J)
+�
+d |h1| · · · d |hN|
+=
+s0
+�
+s1=0
+s1
+�
+s2=0
+. . .
+sT −1
+�
+sT =0
+g (s∗)
+πNdet(J)
+T�
+t=1
+(−2Km,n→t)s∗
+t
+s∗
+t!det(J)s∗
+t
+×
+(33)
+N
+�
+n=1
+ˆ Rn
+0
+|hn|¯sn+1 exp
+�
+−|hn|2 Kn,n
+det(J)
+�
+d |h1| · · · d |hN|
+=
+j0
+�
+j1=0
+j1
+�
+j2=0
+. . .
+jp−1
+�
+jp=0
+g (s∗)
+πNdet(J)
+T�
+t=1
+(−Kt)s∗
+t
+s∗
+t!det(J)s∗
+t ×
+(34)
+N
+�
+n=1
+� Kn,n
+det(J)
+�− ¯sn
+2 −1 �
+Γ
+�
+1 + ¯sn
+2
+�
+− Γ
+�
+1 + ¯sn
+2 , Kn,nR2
+n
+det(J)
+��
+,
+where ¯sn is the sum of s∗
+t→m,n affecting (|hn| |hm|)s∗
+t→m,n and
+g (s∗) =
+�1
+2
+��T
+t=1 s∗
+t �
+v∈V
+
+
+T�
+t=1
+
+ s∗
+t
+vt
+
+
+
+ (2π)N
+N
+�
+i=1
+1{∆i=0}.
+
+20
+APPENDIX C: OUTAGE PROBABILITY OF FAS AT HIGH SNR
+According to [35], the outage probability of a wireless communication system at high SNR can
+be obtained via the PDF of its fading channels. In particular, suppose the PDF of the channels
+at high SNR can be approximated as
+f|hFAS| (Ω) = 2ξΩ2M+1 + o
+�
+Ω2M+1�
+.
+(35)
+Then the outage probability at high SNR is found as
+P {|hFAS| < Ω} =
+ξ
+M + 1ΩM+1 + o
+�
+1
+SNRM+1
+�
+.
+(36)
+Before approximating the PDF of FAS at high SNR, we highlight that the PDF of (21) in
+terms of its amplitude and phase can be rewritten as
+f|h|,θ (|h1| , θ1, . . . , |hN| , θN) =
+N
+�
+n=1
+|hn| Hn
+πNdet(J)
+(37)
+where
+Hn = exp
+�
+−Kn,n |hn|2 + 2 �N
+m=n+1 Km,n |hn| |hm| cos (θn − θm)
+det(J)
+�
+.
+(38)
+Using (37), the approximated PDF of FAS at high SNR can be derived as
+f|hFAS| (Ω) =∂F|hFAS| (Ω)
+∂Ω
+(39)
+(a)
+=N
+ˆ Ω
+0
+. . .
+ˆ Ω
+0
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+f|h|,θ (|h1| , θ1, . . . , |hN−1| , θN−1, Ω, θN)
+(40)
+d |h1| · · · d |hN−1| dθ1 . . . dθN
+(b)
+=
+NΩ
+πNdet(J)
+ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+HN
+ˆ Ω
+0
+|hN−1|
+�
+HN−1 × . . .
+(41)
+�ˆ Ω
+0
+|h2| H2
+�ˆ Ω
+0
+|h1| H1d |h1|
+�
+d |h2|
+�
+· · · d |hN−1|
+�
+dθ1 . . . dθN,
+where (a) is obtained using Leibniz integral and (b) is obtained using (37).
+According to [41], the term
+´ Ω
+0 |hn| Hnd |hn| can be solved by applying Taylor series approx-
+imation at around zero. Specifically, we have
+ˆ Ω
+0
+|hn| Hnd |hn| = Ω2
+2 + o
+�
+Ω2�
+, n = {1, . . . , N − 1}
+(42)
+and the Taylor series approximation of HN at zero is
+HN = 1 + o (1) .
+(43)
+
+21
+Substituting (42) and (43) into (41), we have
+f|hFAS| (Ω) =
+NΩ
+πNdet(J)
+�Ω2
+2 + o
+�
+Ω2��N−1 ˆ 2π
+0
+· · ·
+ˆ 2π
+0
+dθ1 . . . dθN
+(44)
+= 2N
+det(J)Ω2N−1 + o
+�
+Ω2N−1�
+.
+(45)
+Comparing (45) to (35), we have M = N − 1 and ξ =
+N
+det(J). Applying (36), we have
+P {|hFAS| < Ω} ≈
+ΩN
+det(J) + o
+�
+1
+SNRN
+�
+.
+(46)
+APPENDIX D: DIVERSITY GAIN OF FAS
+Let us consider the case where W → ∞. According to [35], the diversity gain of a wireless
+communication system can be obtained via the PDF of its fading channels at high SNR. Specif-
+ically, suppose the PDF of the channels at high SNR can be approximated as in (35). Then
+diversity gain of such system is given by
+D = M + 1.
+(47)
+In Appendix C, we have M = N − 1. Thus, it is straightforward that the diversity gain of
+FAS as W → ∞ is N. Nevertheless, if W is finite, J might be near to being singular. To
+see this, let us consider FAS with N → ∞ ports within a finite W where each port is equally
+separated, and they are indexed as 1, 2, . . .. Without loss of generality, let us focus on two
+ports: n-th and (n + 1)-th port. The correlation between the n-th port and (n + 1)-th port is
+J n,n+1 = lim
+N→∞σ2J0
+�
+2π
+1
+N−1W
+�
+= σ2J0 (0), and we have hn+1 = hn. Thus, the joint CDF of hn
+and hn+1 is Fhn,hn+1 (g1, g2) = Fhn (min {g1, g2}), which implies that they reduce to singularity.
+Since there are many such ports, we can use a finite N′ ports to approximate the channels of
+FAS with N ports, where N′ is the rank of J′ such that J ′ is covariance matrix as defined in
+(2) with N → ∞ for a fixed W. As a result, the diversity gain of FAS is approximately limited
+by min {N, N′}. If N is large, the same observation can be obtained. To remove the nearly-
+dependent entries of J, one may employ rank-revealing QR factorization [42] or Gauss-Jordan
+elimination with a given tolerance.
+REFERENCES
+[1] A. Shojaeifard, K.-K. Wong, K.-F. Tong, Z. Chu, A. Mourad, A. Haghighat, I. Hemadeh, N. T. Nguyen, V. Tapio, and
+M. Juntti, “MIMO evolution beyond 5G through reconfigurable intelligent surfaces and fluid antenna systems,” Proceedings
+of the IEEE, vol. 110, no. 9, pp. 1244–1265, 2022. I
+
+22
+[2] K. K. Wong, K.-F. Tong, Y. Shen, Y. Chen, and Y. Zhang, “Bruce Lee-inspired fluid antenna system: Six research topics
+and the potentials for 6G,” Frontiers in Communications and Networks, p. 5, 2022. I
+[3] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” IEEE Transactions on Wireless
+Communications, vol. 20, no. 3, pp. 1950–1962, 2021. I, II
+[4] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through
+reconfigurable intelligent surfaces,” IEEE Access, vol. 7, pp. 116 753–116 773, 2019. I
+[5] K.-K. Wong, K.-F. Tong, Z. Chu, and Y. Zhang, “A vision to smart radio environment: Surface wave communication
+superhighways,” IEEE Wireless Communications, vol. 28, no. 1, pp. 112–119, 2021. I
+[6] O. Elijah, S. K. Abdul Rahim, W. K. New, C. Y. Leow, K. Cumanan, and T. Kim Geok, “Intelligent massive MIMO
+systems for beyond 5G networks: An overview and future trends,” IEEE Access, vol. 10, pp. 102 532–102 563, 2022. I
+[7] C.-X. Wang, J. Wang, S. Hu, Z. H. Jiang, J. Tao, and F. Yan, “Key technologies in 6G terahertz wireless communication
+systems: A survey,” IEEE Vehicular Technology Magazine, vol. 16, no. 4, pp. 27–37, 2021. I
+[8] M. Khammassi, A. Kammoun, and M.-S. Alouini, “A new analytical approximation of the fluid antenna system channel,”
+arXiv preprint arXiv:2203.09318, 2022. I, II, 4, IV, V
+[9] D. G. Brennan, “Linear diversity combining techniques,” Proceedings of the IRE, vol. 47, no. 6, pp. 1075–1102, 1959. I,
+IV
+[10] K. K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Performance limits of fluid antenna systems,” IEEE Communi-
+cations Letters, vol. 24, no. 11, pp. 2469–2472, 2020. I
+[11] K.-K. Wong and K.-F. Tong, “Fluid antenna multiple access,” IEEE Transactions on Wireless Communications, vol. 21,
+no. 7, pp. 4801–4815, 2022. I
+[12] C. Skouroumounis and I. Krikidis, “Large-scale fluid antenna systems with linear MMSE channel estimation,” in ICC 2022
+- IEEE International Conference on Communications, 2022, pp. 1330–1335. I
+[13] L. Tlebaldiyeva, G. Nauryzbayev, S. Arzykulov, A. Eltawil, and T. Tsiftsis, “Enhancing QoS through fluid antenna
+systems over correlated Nakagami-m fading channels,” in 2022 IEEE Wireless Communications and Networking Conference
+(WCNC), 2022, pp. 78–83. I
+[14] K. Wong, K. Tong, Y. Chen, and Y. Zhang, “Closed-form expressions for spatial correlation parameters for performance
+analysis of fluid antenna systems,” Electronics Letters, vol. 58, no. 11, pp. 454–457, 2022. I
+[15] K. N. Le, “A review of selection combining receivers over correlated rician fading,” Digital Signal Processing, vol. 88,
+pp. 1–22, 2019. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1051200418307176 I
+[16] K. S. Miller, “Complex Gaussian processes,” Siam Review, vol. 11, no. 4, pp. 544–567, 1969. I
+[17] C. Tan and N. Beaulieu, “Infinite series representations of the bivariate Rayleigh and Nakagami-m distributions,” IEEE
+Transactions on Communications, vol. 45, no. 10, pp. 1159–1161, 1997. I
+[18] P. Dharmawansa, N. Rajatheva, and C. Tellambura, “On the trivariate Rician distribution,” IEEE Transactions on
+Communications, vol. 56, no. 12, pp. 1993–1997, 2008. I
+[19] Y. Chen and C. Tellambura, “Infinite series representations of the trivariate and quadrivariate Rayleigh distribution and
+their applications,” IEEE Transactions on Communications, vol. 53, no. 12, pp. 2092–2101, 2005. I
+[20] M. Tekinay and C. Beard, “Moments of the quadrivariate Rayleigh distribution with applications for diversity receivers,”
+Annals of Telecommunications, vol. 75, no. 7, pp. 447–459, 2020. I
+[21] Y. Chen and C. Tellambura, “Distribution functions of selection combiner output in equally correlated Rayleigh, Rician,
+and Nakagami-m fading channels,” IEEE Transactions on Communications, vol. 52, no. 11, pp. 1948–1956, 2004. I
+[22] G. Karagiannidis, D. Zogas, and S. Kotsopoulos, “On the multivariate Nakagami-m distribution with exponential
+correlation,” IEEE Transactions on Communications, vol. 51, no. 8, pp. 1240–1244, 2003. I
+
+23
+[23] M. Wiegand and S. Nadarajah, “A series representation for multidimensional Rayleigh distributions,” International
+Journal
+of
+Communication
+Systems,
+vol.
+31,
+no.
+6,
+p.
+e3510,
+2018,
+e3510
+dac.3510.
+[Online].
+Available:
+https://onlinelibrary.wiley.com/doi/abs/10.1002/dac.3510 I, VI
+[24] ——, “Series approximations for Rayleigh distributions of arbitrary dimensions and covariance matrices,” Signal
+Processing, vol. 165, pp. 20–29, 2019. I, VI
+[25] ——,
+“New
+generalised
+approximation
+methods
+for
+the
+cumulative
+distribution
+function
+of
+arbitrary
+multivariate
+Rayleigh
+random
+variables,”
+Signal
+Processing,
+vol.
+176,
+p.
+107664,
+2020.
+[Online].
+Available:
+https://www.sciencedirect.com/science/article/pii/S0165168420302073 I, VI
+[26] R. G. Gallager, Principles of digital communication.
+Cambridge University Press Cambridge, UK, 2008, vol. 1. 2
+[27] Z. Chai, K.-K. Wong, K.-F. Tong, Y. Chen, and Y. Zhang, “Port selection for fluid antenna systems,” IEEE Communications
+Letters, vol. 26, no. 5, pp. 1180–1184, 2022. I
+[28] L. Zhu, W. Ma, and R. Zhang, “Modeling and performance analysis for movable antenna enabled wireless communications,”
+arXiv preprint arXiv:2210.05325, 2022. I
+[29] W. Ma, L. Zhu, and R. Zhang, “MIMO capacity characterization for movable antenna systems,” arXiv preprint
+arXiv:2210.05396, 2022. I
+[30] N. Waqar, K.-K. Wong, K.-F. Tong, and S. Adrian, “Deep learning enabled slow fluid antenna multiple access,” Submitted
+to IEEE Communications Letters, 2022. I, 3
+[31] K.-K. Wong, K.-F. Tong, Y. Chen, and Y. Zhang, “Fast fluid antenna multiple access enabling massive connectivity,”
+Submmited to IEEE Communications Letters, 2022. I
+[32] H. Xu, K.-K. Wong, W. K. New, and K.-F. Tong, “On the outage probability for two-user fluid antenna multiple access,”
+Submitted to 2023 IEEE International Conference on Communications (ICC), 2023. I
+[33] G. L. Stüber and G. L. Steuber, Principles of mobile communication.
+Springer, 1996, vol. 2. II
+[34] D. Tse and P. Viswanath, Fundamentals of wireless communication.
+Cambridge university press, 2005. II, VI
+[35] Z. Wang and G. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE
+Transactions on Communications, vol. 51, no. 8, pp. 1389–1398, 2003. II, VI, VI
+[36] G. Golub, A. Hoffman, and G. Stewart, “A generalization of the eckart-young-mirsky matrix approximation
+theorem,”
+Linear
+Algebra
+and
+its
+Applications,
+vol.
+88-89,
+pp.
+317–327,
+1987.
+[Online].
+Available:
+https://www.sciencedirect.com/science/article/pii/0024379587901145 IV
+[37] D.
+Dowson
+and
+B.
+Landau,
+“The
+Frechet
+distance
+between
+multivariate
+normal
+distributions,”
+Journal
+of
+Multivariate
+Analysis,
+vol.
+12,
+no.
+3,
+pp.
+450–455,
+1982.
+[Online].
+Available:
+https://www.sciencedirect.com/science/article/pii/0047259X8290077X IV
+[38] S. Boyd, S. P. Boyd, and L. Vandenberghe, Convex optimization.
+Cambridge university press, 2004. IV
+[39] W. Ford, Numerical linear algebra with applications: Using MATLAB.
+Academic Press, 2014. IV
+[40] A. Papoulis and S. U. Pillai, “Probability, random variables, and stochastic processes,” 2002. VI
+[41] S. Liu, J. Cheng, and N. C. Beaulieu, “Asymptotic error analysis of diversity schemes on arbitrarily correlated Rayleigh
+channels,” IEEE Transactions on Communications, vol. 58, no. 5, pp. 1351–1355, 2010. VI
+[42] G. Golub, “Numerical methods for solving linear least squares problems,” Numerische Mathematik, vol. 7, no. 3, pp.
+206–216, 1965. VI
+

CtAzT4oBgHgl3EQfGfuT/content/tmp_files/2301.01029v1.pdf.txt

@@ -0,0 +1,1261 @@
+arXiv:2301.01029v1  [cond-mat.str-el]  3 Jan 2023
+Ab-initio study of phononic thermal conduction in ScAgC half-Heusler
+Vinod Kumar Solet1,∗ and Sudhir K. Pandey1,†
+1School of Mechanical and Materials Engineering,
+Indian Institute of Technology Mandi, Kamand - 175075, India
+(Dated: January 4, 2023)
+We present a first-principles lattice calculations to comprehend the thermal expansion
+α(T ) and lattice thermal conductivity κph of ScAgC. The obtained positive frequencies of phonon
+dispersion shows the dynamical stability of ScAgC in FCC structure. The estimated α(T ) from
+quasi-harmonic approximation (QHA) at 300(1200) K is ∼4(4.6)×10−6 K−1. The predicted value
+of total κph from phonon-phonon interaction (PPI) at 300(1200) K is ∼7.4(1.8) Wm−1K−1. The
+highest group velocity for acoustic & optical branches (AB & OB) is ∼6.7 and ∼3.5 km/s, respec-
+tively. The predicted average phonon lifetime (τλ) for AB(OB) is ∼2.5(1.65) ps at 300 K, whereas it
+is ∼0.6(0.4) ps at 1200 K. The estimated highest heat capacity (Cλ) at 200 K for AB(OB) is ∼23.5
+(19.5) meV/K. We fitted the equation AκT−xκ(AτT−xτ ) in the κph(τλ) curve to gain a thorough
+understanding of temperature-dependent κph trend. The xκ for total branches is calculated to be
+∼1.02. The xτ value due to total AB(OB) is estimated to be ∼1.04(1.02), while it is ∼1.03 for total
+branches. The calculated xκ for total AB(OB) is ∼1.04(0.95), implying that AB contributes more
+to the total κph. This research could be important for enhancing the properties of ScAgC regarding
+thermoelectric and photovoltaic applications.
+I.
+INTRODUCTION
+Energy demand is continuously growing in to-
+day’s technological world. Understanding the transport
+properties of materials, and in particular, the capacity
+to conduct heat throughout the crystal, is critical due
+to its technological influence on energy-related devices.
+The subject of heat transportation is vast in solid state
+physics [1], but we have focused attention on thermal
+transport by phonons here. Basically, the knowledge of
+phonons is very critical in accounting for many physical
+properties and behaviours of crystals, such as thermal
+transport properties, thermal expansion, phase transi-
+tion, mechanical properties, and certain electrical proper-
+ties (superconductivity), etc., [1]. Studying the thermal
+behaviour of heat energy transported by the motion of
+atoms has always been a challenging task for researchers.
+The starting point for almost all theories related to lat-
+tice dynamics is harmonic approximation (HA), which
+ignores the concept of PPI [1, 2].
+However, HA is no
+longer sufficient for accessing transport properties such
+as κph due to the infinite lifetime of phonons. We can
+cope with this problem by taking into account the scat-
+tering of the harmonic phonons by other phonons (PPI),
+defects, and crystal boundaries, etc., [1, 3] because it is
+well known that the κph is a crucial property in semicon-
+ducting thermoelectrics [4], solar photovoltaic (PV) cells
+[5], nuclear reactors [6], and heat management systems
+[7, 8], etc. At near or above room temperature, the PPI is
+an important factor to consider in the quantitative study
+of the κph of non-metallic solids, which is an anharmonic
+phenomenon [1].
+This anharmonicity causes complica-
+tions in the prediction of these transport coefficients at a
+∗ [email protected]
+† [email protected]
+finite temperature. However, high-performance comput-
+ers have contributed to some improvements in the calcu-
+lation of the transport properties of solids.
+In the last few decades, interesting developments
+toward describing the lattice dynamics and related prop-
+erties of materials have been made from first-principles
+molecular dynamics simulations [9, 10]. But such calcu-
+lations carry a large computational effort, and their im-
+plementation is not straightforward. In this regard, one
+can cope with these problems by using first-principles
+density functional theory (DFT) based methods, which
+are the appropriate and generally less expensive ways to
+analyse the phonon-based properties [11]. However, the
+use of methods beyond DFT is required for exploring
+the temperature-dependent transport coefficients.
+Re-
+cently, many-body theory based computational methods
+have become useful in exploring the PPI effects in crys-
+tals [12]. Another physical property, the τλ of phonons, is
+most important to take into account in seeking the mech-
+anism of phononic thermal conduction. The imaginary
+part of phonon self-energy enables us to see the strength
+of coupling between phonons as well as calculate the τλ of
+phonons. Nowadays, the study of anharmonic effects and
+the calculation of τλ is a highly active fields of research.
+The knowledge of α(T ) and κph is helpful for ma-
+terials used in thermoelectric (TE) applications. TE ma-
+terials are not only helpful to generate electricity from
+waste heat but also provide cooling power by allowing
+an electric current to flow through them [4, 13]. These
+materials are particularly attractive for refrigerators, air
+conditioners, heat pumps, automobile applications, and
+etc., since they are reliable, quiet, and devoid of any mov-
+ing parts [13]. The most difficult challenge for researchers
+working with TE compounds is increasing the nondimen-
+sional parameter, figure-of-merit ZT [14], which is de-
+fined as ZT = S2σT/κ. Where S, σ, κ and T are known
+as the material’s Seebeck coefficient, electrical conduc-
+
+2
+tivity, total thermal conductivity, and absolute temper-
+ature, respectively. Total κ has two parts: an electronic
+part (κe) and a lattice part (κph). The ZT value of an
+efficient TE material should be greater than one [15]. As
+a result, ZT can be improved by increasing S2σ or de-
+creasing κ. Obtaining a high ZT is actually quite difficult
+due to the strong correlation between S, σ, and κe via
+charge carriers [1, 16]. Hence, an understanding of κph
+is necessary to know the efficiency of TE materials. The
+α(T ) provides the idea of a change in the length of TE
+materials during heating or cooling processes. In many
+instances, some TE materials are subjected to enough
+thermal or mechanical stress, which is generated during
+a large number of heating and cooling cycles [17]. This
+mechanical stress is usually referred to by the term “ther-
+mal fatigue ”. The product of elastic modulus and linear
+thermal expansion coefficient is an important quantity to
+be taken into account just before analysing the TE mate-
+rials for thermal fatigue [17, 18]. The lower value of this
+product gives the lower value of thermal fatigue in the
+materials. The microcracking and porosity in TE ma-
+terials, which have an impact on their performance, are
+also caused by α(T ) [17]. Zhang et al. [19] have reported
+the effect of microcracking on S2σ of skutterudite TE
+material in the 20-800 K temperature range. The α(T )
+and κph are directly related to the phonon calculations.
+Apart from electronic properties, phonon properties cal-
+culated from DFT provide reliable accuracy, which was
+understood in many previous works [20–22].
+With all
+this in mind, the ScAgC half-Heusler (HH) compound
+has been chosen to explore the phonon-based properties.
+Heusler compounds have recently received much
+interest from researchers due to their great capabilities
+in different energy fields, including TE [22–25], PV so-
+lar cell [26, 27], topological insulators [23, 28],etc. This
+type of material has a general formula of XY Z (HH)
+or X2Y Z (full-Heuslers), where X and Y are mainly the
+transition elements and Z belongs to the p-block element.
+Our earlier study predicts that ScAgC is a promising HH
+TE compound and the highest predicted ZT is ∼0.53 at
+1200 K temperature [29]. One can achieve a high ZT
+by lowering the κ. Further, ScAgC can also be used to
+make solar cell devices because it has a strong absorp-
+tion of ∼1.7×106 cm−1 at photon energy ∼8.5 eV and a
+low reflectivity of ∼0.24 at ∼4.7 eV [29]. At 300 K, the
+highest PV efficiency of ∼33% is also observed at ∼1 µm
+thickness. The remaining absorbed solar energy will be
+transformed into thermal energy inside the cell and could
+raise the temperature at the junction until the heat is not
+dissipated effectively to the environment [30]. This rise
+in temperature reduces the mobility of charge carriers,
+which may be one reason for the decreased efficiency of
+solar cells [5]. Therefore, κph is critical in determining
+the efficiency of solar cell devices. Our earlier work [29]
+gives full information about the electronic related prop-
+erties, which is not sufficient for materials to design the
+TE and solar cell devices. Although the DFPT method
+was used to investigate phonon-based thermodynamical
+properties but κph was not calculated accurately in this
+work [29]. In this direction, it is mandatory to know a
+more accurate κph in order to use ScAgC for making PV
+and TE devices.
+Hence, the present research includes phonon-
+based properties estimated by first-principles calcula-
+tions.
+First of all, phonon dispersion
+�
+with and with-
+out including non-analytic term correction (NAC)
+�
+and
+phonon DOS have been calculated by considering super-
+cell and finite displacement approaches under HA. Both
+dispersion plots have no negative branches, which means
+that ScAgC is stable in the FCC structure. Next, α(T )
+has also been estimated via QHA. The expected room
+temperature value of α(T ) is ∼4×10−6 K−1, whereas
+it reaches a value of ∼4.6×10−6 K−1 at 1200 K. Sim-
+ilarly, first-principle lattice calculations with an anhar-
+monic force constant are used to capture the κph by as-
+suming PPI only. The observed value of κph at 300 K is
+∼7.4 Wm−1K−1, while it decreases to ∼1.8 Wm−1K−1
+at 1200 K. Furthermore, τλ and Cλ of each branch at
+different temperatures, as well as group velocity vλ of all
+phonon branches, are also calculated. The equation of
+temperature-dependent κph(AκT−xκ) and τλ(AτT−xτ )
+is also fitted in the respective curve.
+The value of xκ
+is found to be almost 1.02 for total branches, while it
+is ∼1.04(0.95) for AB(OB). The estimated value of xτ
+for total branches is ∼1.03, whereas it is ∼1.04(1.02) for
+AB(OB), which aids in understanding the temperature-
+dependent κph trend.
+II.
+COMPUTATIONAL DETAILS
+Phonon dispersion relations are carried out by
+means of the supercell approach and finite displace-
+ment method (FDM) [31] in the Phonopy code [32]. A
+2×2×2 supercell is constructed to obtain displacements
+from the equilibrium positions of atoms in the conven-
+tional unit cell. Then forces on these supercells (with 96
+atoms) are calculated from ABINIT software [33] within
+the projector-augmented wave (PAW) method [34] under
+DFT. These atoms are displaced with a fixed harmonic
+distance of 0.01 ˚A from their equilibrium positions [32].
+The PBE-GGA type of XC functional has been consid-
+ered in our calculations [35]. Converged results have been
+obtained by using cutoff and PAW cutoff kinetic energy
+for plane wave basis sets of 25 and 50 Ha, respectively.
+In the supercell, a 4×4×4 k-point grid is integrated over
+the Brillouin zone. The lattice parameter value of 5.6 ˚A
+is used in the entire calculations [29]. A force conver-
+gance criteria is set to be 5 × 10−8 Ha/Bohr. The DFPT
+method, as implemented in the ABINIT code [36, 37], has
+been used to obtain born effective charges (BEC) and
+static dielectric constants of atoms for including long-
+range interactions within primitive cells. The QHA [32]
+based method in the Phonopy code is used to evaluate
+the coefficient of α(T ). Next, the phono3py [12] package
+is used to calculate κph, and a 2×2×2 supercell is built to
+
+3
+Γ
+X
+Γ
+L
+W
+Γ
+0
+10
+20
+30
+40
+50
+60
+Energy (meV)
+Γ
+X
+Γ
+L
+W
+Γ
+0
+10
+20
+30
+40
+50
+60
+Energy (meV)
+without NAC
+with NAC
+(b)
+(a)
+FIG. 1: The harmonic phonon dispersion (a) before
+including NAC and (b) after including NAC of ScAgC.
+obtain the second and third order force constants. But
+here only those supercells are considered that have inter-
+actions between only three neighbouring atoms (uto 7.5
+Bohr), and the forces on these supercells have been ob-
+tained from the ABINIT code within the PAW method.
+Finally, these force constants are used to calculate the τλ
+from imaginary part of phonon self-energy and then κph
+by using a heavy q-mesh size of 21×21×21 under single
+mode relaxation time approximation [12].
+III.
+RESULTS AND DISCUSSION
+A.
+Phonon properties
+This section presents the phonon dispersion curve
+and phonon DOS of ScAgC estimated under HA. To con-
+firm the stability of this compound, the phonon disper-
+sion is calculated in the first Brillouin zone along the
+high symmetry direction of Γ-X-Γ-L-W-Γ. The disper-
+sion plot of Fig. 1(a) does not embody NAC. This plot
+contains a total of nine positive phonon branches, cor-
+responding to three atoms in the primitive unit cell.
+Among them, three are AB, and the remaining six are
+OB. The AB have energies varying from 0 to ∼15.85
+meV, with one being longitudinal AB (LAB) and two be-
+ing transverse AB (TAB). Similarly, two longitudinal OB
+(LOB) and four transverse OB (TOB) are contributed in
+OB modes. The two AB are degenrate along the X-Γ
+and Γ-L directions. The maximum phonon energy is es-
+timated as ∼58 meV. One can also observe the minimum
+energy gap of ∼8.6 meV between AB and OB. Three
+of the OB modes, which are located in the middle en-
+ergy range (∼24.5 meV to ∼35 meV), are well separated
+with an energy gap of ∼10 meV from the remaining three
+modes. These remaining branches are found in a higher
+energy range of ∼45 meV to ∼58 meV. Due to this gap,
+the coupling between the AB (OB) and OB (OB) may
+be weaker, which is expected to be the largest contribu-
+tor to κph. The OB modes are doubly degenerate along
+the X-Γ-L direction, and they are triply degenerate at
+the Γ-point. The AB are almost linear near the Γ-point,
+implying that group velocity and phase velocity will be
+the same in this region [1]. The slop of AB is used to
+0
+10
+20
+30
+40
+50
+60
+Energy (meV)
+0
+0.3
+0.6
+0.9
+1.2
+1.5
+1.8
+2.1
+2.4
+Phonon DOS (states/meV)
+Total
+Sc
+Ag
+C
+FIG. 2: The total phonon DOS per unit cell and partial
+DOS per atom of ScAgC.
+estimate sound velocity, which is an important factor in
+calculating the κph of solids [1].
+We proceed now to see the effect of long-range
+Coulomb interactions or dipole-dipole interactions on
+the ions present in the ionic crystals. It is well known
+that polar crystals become polarized by taking small
+atomic displacements from their equilibrium positions,
+and the resulting macroscopic field modifies the force con-
+stants close to the Γ-point [38]. Basically, LOB create a
+macroscopic electric field near the Γ-point in non-metallic
+solids, and thus NAC is calculated to take this contribu-
+tion into harmonic phonon dispersion. As a result, the
+LOB is lifted up, and TOB and LOB split close to the
+Γ-point. One can clearly observe this LO-TO splitting
+at the Γ-point in Fig. 1(b). Now the maximum energy
+of phonons at Γ-point is ∼58 meV and an energy gap of
+∼10 meV is also created. The splitting of OB also creates
+a minimum energy gap of ∼3 meV. In practice, the BEC
+of ions and the dielectric constant are required quantities
+for NAC at OB frequencies [38]. The calculated dielec-
+tric constant is ∼13.9, which is the same in all directions
+due to the cubic symmetry of ScAgC, while the BEC of
+Sc, Ag, and C ions is ∼2.5, ∼0.4, ∼ −2.9, respectively.
+According to this, the inclusion of NAC can play an im-
+portant role in deciding the phonon properties. However,
+in the presence of NAC, the other part of the dispersion
+is barely affected when compared to Fig.
+1(a). These
+theoretical aspects can be probed and verified if someone
+measures them by experiment.
+Further, the phonon DOS and a partial DOS have
+been studied in order to investigate the effect of AB and
+OB in ScAgC during heat transfer processes.
+Fig.
+2
+indicates the obtained graph of total phonon DOS per
+unit cell and partial phonon DOS per atom. The total
+DOS plot has three main peaks around the energies of
+∼14.5, ∼30 and ∼48 meV, respectively. One can see the
+gap in the DOS around 25 (40) meV, which separates
+the states corresponding to AB (OB) and OB (OB). In
+the figure, the AB in the lower energy region (below 25
+meV) have the major contributions due to the vibrations
+of the heavier mass Ag atoms. The middle energy range
+of OB is mainly influenced by the atomic vibrations of Sc
+atoms. While the main contribution of lighter C atoms
+
+4
+to vibrations is seen in higher (above 45 meV) energetic
+OB modes.
+B.
+Thermal expansion
+We shall now investigate the features of how the
+lattice can expand as a result of the thermal motion
+of ions at finite temperatures.
+The definition of α(T )
+and κph of compounds breaks down at the harmonic re-
+gion for crystal potential.
+The normal-mode frequen-
+cies of purely harmonic crystals are unaffected by the
+change of equilibrium volume and therefore do not lead
+to α(T ) [1].
+Further, anharmonic interaction in solid
+gives the α(T ) because it leads to asymmetry in crys-
+tals.
+In this regard, it has been discovered that the
+QHA [2] is a respectably good approximation for cap-
+turing the α(T ). It is known that normal mode frequen-
+cies do not always have volume dependence, but QHA
+considers volume-dependent phonon properties here [1].
+Accordingly, the thermal coefficient of linear expansion
+α(T ) of ScAgC is estimated under QHA. In this way,
+the total free energy as a function of primitive cell vol-
+ume is calculated at various temperatures ranging from
+0 to 1200 K with a step size of 100 K, as shown in
+Fig. 3(a). For this, ten different supercells are created
+around the equilibrium lattice constant with different ex-
+pansions and compressions. The total F at a given tem-
+perature and volume can be estimated as, F(T ; V ) =
+[Uel(V )−Uel(V0)]+Fph(T ; V ). Where Uel(V )−Uel(V0) is
+the relative DFT energy of the electronic system, V0 is
+the equilibrium volume at 0 K. Fph(T ; V ) is related to the
+phonon contribution to Helmholtz free energy. At each
+temperature, free energy has one minima corresponding
+to the equilibrium volume of a primitive cell, which is
+estimated after fitting the Birch-Murnaghan equation of
+states [39] to the F versus volume plot. In Fig.
+3(a),
+the solid red line connects every such energy point at
+equilibrium volume for a given temperature. Then, Fig.
+3(b) presents these equilibrium volumes as a function of
+temperature up to 1200 K. The calculated equilibrium
+volume at 0 K is ∼175.7 ˚A3, while it increases to ∼176.5
+˚A3 at 1200 K. When compared to its ground state vol-
+ume, the volume increases by up to ∼0.4% at 1200 K.
+From knowing the minimum free energy for equi-
+librium primitive cell volume, one can easily calculate
+linear coefficient of thermal expansion α(T ) from the ex-
+pression of α(T ) = 1
+3β(T ) [1]. Here, the term β(T ) is
+known as the volumetric thermal expansion coefficient,
+which is estimated as follows: β(T ) =
+1
+V (T )
+∂V (T )
+∂T
+. Where
+V (T ) represents the volume of a primitive cell as a func-
+tion of temperature, as illustrated in Fig. 3(b). Because
+our compound has cubic symmetry, the expansion is uni-
+form in all three directions, and thus α(T ) is one-third of
+β(T ) [1]. Fig. 3(c) shows a plot of the calculated α(T )
+versus temperature from 0-1200 K. The corresponding
+figure depicts a rapid increase in α(T ) up to ∼200 K, fol-
+lowed by a slow increment in temperature up to ∼500
+160
+170
+180
+190
+Volume (Å3)
+-6
+-4
+-2
+0
+2
+4
+6
+8
+Free energy F (eV)
+175.6
+175.8
+176
+176.2
+176.4
+176.6
+Volume (Å
+3)
+0
+200
+400
+600
+800
+1000
+1200
+Temperature (K)
+0
+1
+2
+3
+4
+5
+α(Τ) (×10
+−6 Κ
+−1)
+0 K
+1200 K
+(a)
+(b)
+(c)
+FIG. 3: (a) Variation of total free energy F with
+primitive cell volume. (b) Change in primitive cell
+volume with temperature. (c) The coefficient of linear
+thermal expansion α(T ) with respect to temperature for
+ScAgC.
+K. The rate of volume change in the crystal is high-
+est in the 0-200 K temperature range.
+From ∼500 K
+to highest studied temperature, the α(T ) shows almost
+constant behaviour with respect to temperature. At low
+temperatures, α(T ) varies as ∼T 3 and becomes nearly
+constant at higher temperatures, exhibiting nearly the
+same temperature dependence behaviour to specific heat
+Cv in both cases [1, 29]. The predicted value of α(T ) at
+room temperature is ∼4×10−6 K−1, while at 1200 K, it
+is ∼4.6×10−6 K−1. ScAgC has lower α(T ) values than
+other HH compounds like FeVSb [22], and ZrNiSn [24].
+From an application standpoint, the information about
+the α(T ) of materials is very helpful if one wants to utilize
+them for making real TE devices.
+C.
+Lattice thermal conductivity
+The obtained total κph for ScAgC in the temper-
+ature range of 300-1200 K is presented in Fig.
+4(a).
+One can notice the decreasing behaviour of κph with in-
+creasing temperature. The expected value of total κph
+at 300 K is ∼7.4 Wm−1K−1, whereas it decreases to
+∼1.8 Wm−1K−1 at 1200 K. One can generally expect
+this decreasing behaviour since the phonon-phonon scat-
+tering rate increases with increasing temperature. The
+branch–dependent κph is also calculated to know the per-
+centage weight of κph of AB and OB in the total κph,
+which is shown in Fig. 4(b). In this figure, the first three
+AB1-AB3 are the AB, while OB1-OB6 indicate the six
+OB. The AB2 shows largest κph among all the branches
+and the value is ∼2.5(0.6) Wm−1K−1 at 300(1200) K.
+One can also observe that the AB (OB) contribute nearly
+77–80 (20–23)% of the total κph in the studied temper-
+ature window.
+This percentage decreases for AB and
+increases for OB as temperature rises.
+This is due to
+the fact that the number of AB (OB) modes decreases
+(increases) as the temperature rises. Basically, this cal-
+culation of κph considers only PPI. But in reality, the κph
+also affected by the phonon-electron interactions (PEI),
+
+5
+300
+450
+600
+750
+900
+1050
+1200
+Temperature (K)
+0
+0.5
+1
+1.5
+2
+2.5
+κph (W/mK)
+AB1
+AB2
+AB3
+OB1
+OB2
+OB3
+OB4
+OB5
+OB6
+300
+450
+600
+750
+900
+1050
+1200
+Temperature (K)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+κph (W/mK)
+AB
+OB
+TB
+(a)
+(b)
+FIG. 4: The calculated κph for (a) acoustic branches
+(AB), optical branches (OB), and total branches (TB)
+(b) nine phonon branches as a function of temperature.
+phonon-defect interactions, etc.
+Apart from this, the
+DFT-based phonon band structure has been used in the
+estimation of temperature-dependent κph here. However,
+in the real world, phonon band structure is temperature
+dependent. Therefore, one can get a more realistic result
+of κph with the inclusion of all the above aspects. But
+large computational efforts are required to address these
+challenges.
+Now we shall focus on understanding the different
+physical parameters that contribute to the prediction of
+κph. Recalling Eq. (1), the variation of vλ with phonon
+frequency, and Cλ, τλ with respect to temperature have
+been studied. The calculated vλ, which is directly pro-
+portional to κph, is presented in Fig. 5(a). In this figure,
+each data point represents a phonon mode for a par-
+ticular q-point in the irreducible part of the Brillouin
+zone (IBZ). The AB3 has the highest vλ among all the
+branches, and the value is ∼6.7 km/s, which is almost a
+double value of the highest vλ (∼3.5 km/s) of the OB9.
+The relatively flat dispersion of OB in Fig.
+1(b) may
+be one of the reasons for the low vλ of OB compared
+to AB. Here, we have not considered the temperature-
+dependent vλ. Nextly, the calculated Cλ for all branches
+as a function of temperature is plotted in Fig. 5(b). In
+the studied temperature window, the AB have relatively
+large Cλ compared to the OB. At 200 K, the Cλ of AB1-
+AB3(OB1-OB3) branches is calculated to be ∼23.5(19.5)
+meV/K. At same temperature, the calculated Cλ of OB4-
+OB5(OB6) branches is ∼13.5(11.5) meV/K. This can be
+viewed from the Bose-Einstein distribution function, in
+which the phonon mode population is decreased by in-
+creasing the mode’s frequency at a fixed temperature.
+Consequently, OB have lower heat capacities, contribut-
+ing less to κph. Indeed, Cλ stays almost constant with a
+small ∼2–11% deviation (∼2% for 630 K and ∼11% for
+300 K) from a constant value (∼24.5 meV/K) of a clas-
+sical limit of Cλ at higher temperatures
+�
+T≫ΘD(∼630K
+[29])
+�
+, where ΘD is the Debye temperature. This obser-
+vation reveals that AB modes are the dominant phonon
+modes and therefore make a large contribution to κph.
+The consideration of temperature-independent vλ
+and nearly non-varying Cλ behaviour in the tempera-
+0
+3
+6
+9
+12
+15
+Frequency (THz)
+0
+1
+2
+3
+4
+5
+6
+7
+Group velocity (km/s)
+AB1
+AB2
+AB3
+OB1
+OB2
+OB3
+OB4
+OB5
+OB6
+0
+200
+400
+600
+800
+1000
+1200
+Temperature (K)
+0
+5
+10
+15
+20
+25
+Heat capacity (meV/K)
+AB1
+AB2
+AB3
+OB1
+OB2
+OB3
+OB4
+OB5
+OB6
+(a)
+(b)
+FIG. 5: The calculated mode dependent phonon (a)
+group velocity vλ and (b) heat capacity Cλ for nine
+phonon branches.
+300
+450
+600
+750
+900
+1050
+1200
+Temperature (K)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+Phonon lifetime (ps) 
+AB1
+AB2
+AB3
+OB1
+OB2
+OB3
+OB4
+OB5
+OB6
+300
+450
+600
+750
+900
+1050
+1200
+Temperature (K)
+0.3
+0.6
+0.9
+1.2
+1.5
+1.8
+2.1
+2.4
+Phonon lifetime (ps)
+AB
+OB
+TB
+(a)
+(b)
+FIG. 6: The phonon lifetime τλ for (a) nine phonon
+branches (b) acoustic branches (AB), optical branches
+(OB), and total branches (TB) as a function of
+temperature.
+ture range of 300-1200 K motivates us to look closely
+the temperature variation of τλ for all phonon branches.
+Here, from Eq. (4), τλ due to only PPI is estimated to
+understand the behaviour of κph in the 300-1200 K tem-
+perature range, which is shown in Fig. 6. In Fig. 6(a),
+τλ of all nine branches is obtained by taking the average
+weight of all q-points in the IBZ. Here, the same method
+is used for the calculation of τλ as employed in earlier
+work by Shastri et. al [40]. In the figure, one can clearly
+notice that OB1 and OB2 have the highest τλ, signify-
+ing the lowest phonon-phonon scattering among all the
+branches.
+The value of these branches is found to be
+∼3.9(0.95) ps at 300(1200) K. The last OB9 branch has
+a shorter τλ with a value of ∼0.07(0.02) ps at 300(1200)
+K, indicating a higher scattering rate than other phonon
+branches. All of the branches show decreasing τλ with in-
+creasing temperature, indicating that phonons feel more
+scattering at high temperatures than phonons at low
+temperatures. It is also clear from the figure that the
+decrement rate of τλ for the last three OB4-OB6 is much
+slower than for other phonons with increasing temper-
+ature that produce low κph.
+This can be understood
+through the PDOS plot in Fig. 2, in which the phonon
+number densities of the last three OB are comparably
+larger than other branches.
+The corresponding region
+contains strong phonon-phonon scattering interactions.
+The τλ of TB, AB, and OB is also estimated by aver-
+
+6
+TABLE I: The variation of τλ (κph) with a relation of
+AτT−xτ (AκT−xκ) [1] for acoustic branches (AB), optical
+branches (OB), and total branches (TB).
+Phonon branches Aτ(ps×K) xτ
+Aκ (W/m)
+xκ
+AB1
+1050
+1.06
+786
+1.05
+AB2
+1113
+1.03
+893
+1.03
+AB3
+628
+1.01
+463
+1.02
+OB1
+1263
+1.02
+95
+0.95
+OB2
+1288
+1.02
+119
+0.94
+OB3
+609
+1.03
+127
+0.95
+OB4
+59
+1.02
+0.81
+0.85
+OB5
+54
+1.02
+1.12
+0.84
+OB6
+17
+0.98
+0.76
+0.75
+AB
+924
+1.04
+2133
+1.04
+OB
+548
+1.02
+342
+0.95
+TB
+672
+1.03
+2413
+1.02
+aging the corresponding number of phonon branches, as
+shown in Fig. 6(b). The τλ of TB due to PPI is obtained
+by taking the average of all phonon branches. The figure
+presents that OB have lower τλ than AB and the values
+are ∼1.65(2.5) ps and ∼0.4(0.6) ps for OB(AB) branches
+at 300 and 1200 K, respectively. This means that the
+AB transport more heat energy than OB in the form of
+κph. The calculated room temperature value of τλ of TB
+is ∼1.93 ps, while it decreases as temperature increases,
+with a value of ∼0.46 ps at 1200 K.
+The total number of phonons in a solid is pro-
+portional to temperature T at higher temperatures (T≫
+ΘD), which can be understood from Eq. (3). Since a
+phonon that contributes to thermal conduction is more
+likely to be scattered with other phonons that are present
+in crystal, one should expect τλ to exhibit a decreas-
+ing behaviour as temperature rises.
+At high temper-
+atures, the Cλ follows the Dulong-Petit law and be-
+comes temperature-independent, which can also be ob-
+served in Fig. 5(b). vλ is assumed to be a temperature-
+independent constant in the Debye model, and even in
+more accurate models, it will not have a significant con-
+tribution to κph [1]. Therefore, in the high-temperature
+regime, the κph should decrease as the temperature
+rises. This temperature-dependence behaviour is con-
+firmed by the experiment, and the rate of decline is
+AκT−xκ [1]. The variables Aκ and xκ are temperature-
+independent and the value of xκ lies between 1 and 2 [1].
+Aκ and xκ are calculated by fitting the above relation in
+the respective κph curve, which is shown in Table 1. The
+calculated value of xκ for TB is ∼1.02, which is within
+the experimental range. From the above discussion, it
+is important to note that this kind of behaviour is valid
+for T≫ ΘD. However, in our case, ΘD is estimated to
+be ∼630 K [29], but the calculated κph follows this trend
+at temperatures above 300 K. The xκ value for each AB
+is greater than one and less than one for each OB. Also,
+the xκ value for AB(OB) is ∼1.04(0.95), indicating that
+AB controls the behaviour of xκ in TB. To deeply under-
+stand the κph behaviour with temperature, the equation
+τλ = AτT−xτ is also fitted in the respective plots. Here,
+Aτ and xτ are also temperature-independent variables.
+The value of xτ for TB is ∼1.03, while the values for AB
+and OB are ∼1.04 and ∼1.02, respectively. Each phonon
+branch (except OB6) has a xτ value greater than one,
+while OB6 has a value of ∼0.98. Also, xτ and xκ due to
+AB1-AB3 are nearly identical, whereas these values differ
+slightly for OB1-OB6. This means that xτ values for all
+AB contribute significantly more to xκ than the xτ val-
+ues of all OB. The OB2 has highest Aτ of ∼1288 ps×K,
+whereas AB2 has highest Aκ of ∼893 W/m.
+Finally,
+the conclusion can be drawn that acoustic modes are
+the dominant heat carriers in the temperature-dependent
+κph.
+As a result, AB are dominant carriers in all three
+previously studied parameters (vλ, Cλ and τλ), becom-
+ing the larger contributor to the heat transfer process
+in ScAgC in terms of κph. Among threes, the reduced
+trend of τλ with increasing temperature leads directly to
+the gradual decrement of κph in Fig. 4 with temperature.
+One can get a high ZT by reducing the κph as much as
+possible. Since τλ is higher for AB than OB, one can re-
+duce the τλ of AB by accounting for the extra scattering
+centres in the form of alloying, nanostructuring, and etc.,
+[15, 41] in the energy range of AB. These extra scatter-
+ings can provide a low κph and hence may give rise to a
+high ZT , which is a positive sign for compounds used in
+TE applications.
+IV.
+CONCLUSIONS
+Here, we have performed the DFT calculations to
+understand the lattice transport mechanisms of ScAgC
+combined with the HA and QHA. The phonon band dis-
+persion and phonon DOS are calculated under HA. The
+obtained positive frequencies (with and without NAC)
+of dispersion suggest the mechanical stability of ScAgC
+in the FCC structure. The value of α(T ) is found to be
+∼4×10−6 K−1 and ∼4.6×10−6 K−1 at 300 K and 1200
+K, respectively. Similarly, first-principles based anhar-
+monic phonon calculations have been used to analyse the
+κph and the value is obtained as ∼7.4(1.8) Wm−1K−1 at
+300(1200) K. The value of total τλ at 300 K is calculated
+to be ∼1.93 ps, whereas it is observed as ∼0.46 ps at
+1200 K. The highest calculated vλ for AB (OB) is ∼6.7
+(3.5) km/s. At 200 K, the AB (OB) have the highest
+(lowest) Cλ with a value of ∼23.5 (11.5) meV/K. By fit-
+ting the equation of AκT−xκ(AτT−xτ ) in κph(τλ) curve,
+the temperature-dependent behaviour is also understood.
+The xκ value for AB(OB) is predicted to be ∼1.04(0.95),
+while it is ∼1.02 for TB. Similarly, the obtained value of
+xτ is ∼1.04(AB), ∼1.02(OB), and ∼1.03(TB). Our study
+can be helpful in order to use ScAgC for renewable energy
+sources such as TE and PV applications.
+
+7
+[1] N. W. Ashcroft and N. D. Mermin, Solid State Physics,
+Vol. 239 (Saunders College Publishing, New York, 1976).
+[2] G. P. Srivastava, The physics of phonons (Taylor and
+Francis Group, New York, 1990).
+[3] L. Chaput, Phys. Rev. Lett. 110, 265506 (2013).
+[4] F. J. DiSalvo, Science 285, 703 (1999).
+[5] V.
+L.
+Dalal
+and
+A.
+R.
+Moore,
+J. Appl. Phys. 48, 1244 (1977).
+[6] M.
+Lambert
+and
+L.
+Fletcher,
+J Thermophys Heat Trans. 11, 129 (1997).
+[7] S.
+E.
+Kim,
+F.
+Mujid,
+A.
+Rai,
+and
+et
+al.,
+Nature 597, 660 (2021).
+[8] H. Song, J. Liu, B. Liu, and et al., Joule 2, 442 (2018).
+[9] O.
+Hellman,
+I.
+Abrikosov,
+and
+S.
+Simak,
+Phys. Rev. B 84, 180301 (2011).
+[10] T.
+Tadano
+and
+S.
+Tsuneyuki,
+Phys. Rev. B 92, 054301 (2015).
+[11] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
+[12] A.
+Togo,
+L.
+Chaput,
+and
+I.
+Tanaka,
+Phys. Rev. B 91, 094306 (2015).
+[13] L. E. Bell, Science 321, 1457 (2008).
+[14] Y.
+Pei,
+X.
+Shi,
+A.
+LaLonde,
+and
+et
+al.,
+Nature 473, 66 (2011).
+[15] G. Snyder and E. Toberer, Nat. Mater. 7, 105 (2008).
+[16] S.
+Sk,
+P.
+Devi,
+S.
+Singh,
+and
+et
+al.,
+Mater. Res. Express 6, 026302 (2018).
+[17] E. D. Case, J. Electron. Mater. 41, 1811 (2012).
+[18] D.
+Music,
+R.
+W.
+Geyer,
+and
+P.
+Keuter,
+Appl. Phys. Lett. 109, 223903 (2016).
+[19] L.
+Zhang,
+A.
+Grytsiv,
+B.
+Bonarski,
+and
+et
+al.,
+J. Alloys Compd. 494, 78 (2010).
+[20] S.
+Sk
+and
+S.
+K.
+Pandey,
+EPL (Europhys. Lett.) 137, 66002 (2022).
+[21] S. Sk and S. K. Pandey, arXiv preprint arXiv:2204.11198
+(2022).
+[22] S.
+S.
+Shastri
+and
+S.
+K.
+Pandey,
+J. Phys. Condens. Matter 33, 085704 (2020).
+[23] T.
+Graf,
+C.
+Felser,
+and
+S.
+S.
+Parkin,
+Prog. Solid State Chem. 39, 1 (2011).
+[24] S.
+S.
+Shastri
+and
+S.
+K.
+Pandey,
+J. Phys. Condens. Matter 32, 355705 (2020).
+[25] J.
+Shiomi,
+K.
+Esfarjani,
+and
+G.
+Chen,
+Phys. Rev. B 84, 104302 (2011).
+[26] L.
+Yu
+and
+A.
+Zunger,
+Phys. Rev. Lett. 108, 068701 (2012).
+[27] T. Gruhn, Phys. Rev. B 82, 125210 (2010).
+[28] V.
+Pandey,
+A.
+Sihi,
+and
+S.
+K.
+Pandey,
+J. Phys. Condens. Matter 33, 475503 (2021).
+[29] V.
+K.
+Solet,
+S.
+Sk,
+and
+S.
+K.
+Pandey,
+Phys. Scr. 97, 105711 (2022).
+[30] A.
+Royne,
+C.
+J.
+Dey,
+and
+D.
+R.
+Mills,
+Sol. Energy Mater. Sol. Cells 86, 451 (2005).
+[31] G.
+Kresse,
+J.
+Furthm¨uller,
+and
+J.
+Hafner,
+Europhys. Lett. 32, 729 (1995).
+[32] A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).
+[33] X. Gonze,
+J.-M. Beuken,
+R. Caracas, and et al.,
+Comput. Mater. Sci. 25, 478 (2002).
+[34] P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994).
+[35] J.
+P.
+Perdew,
+K.
+Burke,
+and
+M.
+Ernzerhof,
+Phys. Rev. Lett. 77, 3865 (1996).
+[36] X. Gonze, B. Amadon, P.-M. Anglade, and et al.,
+Comput. Phys. Comm. 180, 2582 (2009).
+[37] J. Zwanziger, J. Galbraith, Y. Kipouros, and et al.,
+Comput. Mater. Sci. 58, 113 (2012).
+[38] F.
+Detraux,
+P.
+Ghosez,
+and
+X.
+Gonze,
+Phys. Rev. Lett. 81, 3297 (1998).
+[39] F. Birch, Phys. Rev. 71, 809 (1947).
+[40] S.
+S.
+Shastri
+and
+S.
+K.
+Pandey,
+J. Phys. Condens. Matter 33, 265702 (2021).
+[41] D. J. Singh and I. Terasaki, Nat. Mater. 7, 616 (2008).
+[42] A. Maradudin and A. Fein, Phys. Rev. 128, 2589 (1962).
+
+8
+Supplementary material for “Ab-initio study of
+phononic thermal conduction in ScAgC
+half-Heusler”
+V.
+METHOD OF CALCULATION OF LATTICE
+THERMAL CONDUCTIVITY
+After getting the full solution of LBTE using the
+single mode relaxation time (SMRT) approximation [2],
+the closed tensor form of lattice thermal conductivity κph
+is written as follows [12]:
+κph =
+1
+NV0
+�
+λ
+Cλvλ ⊗ vλτ SMRT
+λ
+.
+(1)
+Here, N and V0 are the number of unit cells and vol-
+ume of unit cell, respectively. Cλ and vλ is the model
+specific heat and phonon group velocity of phonon mode
+λ, respectively. Here λ is the phonon mode denoted by
+a set of (q, j) with wave vector q in branch j. τ SMRT
+λ
+is the relaxation time of corresponding phonon mode λ.
+τ SMRT
+λ
+is approximately considered to be phonon lifetime
+τλ in order to further calculate κph. Then, τλ is obtained
+from the imaginary part of phonon self-energy Γλ(ωλ) by
+considering only phonon-phonon interaction (PPI). The
+anharmonic third-order force constant is used to calcu-
+late the Γλ(ωλ), which is obtained from the many-body
+perturbation theory as [12],
+Γλ(ω) = 18π
+ℏ2
+�
+λ′ λ′′
+��Φ−λλ′ λ′′ ��2
+��
+nλ′ + nλ′′ + 1
+�
+×δ
+�
+ω − ωλ′ − ωλ′′ �
++
+�
+nλ′ − nλ′′ �
+×
+�
+δ
+�
+ω + ωλ′ − ωλ′′ �
+−δ
+�
+ω − ωλ′ + ωλ′′ ���
+.
+(2)
+Where nλ represents the Bose–Einstein thermal distribu-
+tion function at the equilibrium of a particular phonon
+mode λ, which is given as,
+nλ =
+1
+exp(ℏωλ/kBT ) − 1
+(3)
+Φ−λλ′ λ′′ indicates the all possible three-phonon
+interaction strengths between modes of λ, λ
+′ and λ
+′′ in-
+volving in the scattering, which can be obtained from
+anharmonic third-order force constants.
+From Eq. (2), one can calculate the τλ of phonon
+branch λ as [12, 42],
+τ SMRT
+λ
+≡ τλ =
+1
+2Γλ(ωλ)
+(4)
+Where 2Γλ(ωλ) is the phonon linewidth and ωλ denotes
+the harmonic phonon frequency of a mode λ.
+The mode-dependent Cλ and vλ can be estimated
+directly from the solution of eigan-value problem [12],
+Cλ = kB
+� ℏωλ
+kBT
+�2
+exp(ℏωλ/kBT )
+[exp(ℏωλ/kBT ) − 1]2 ,
+(5)
+and,
+vα(λ) ≡ ∂ωλ
+∂qα
+=
+1
+2ωλ
+�
+kk′ βγ
+Wβ(k, λ)∂Dβγ(kk
+′, q)
+∂qα
+Wγ(k
+′, λ).
+(6)
+where α, β, and γ are the Cartesian indices. Wβ(k, λ) is
+the polarization vector of kth atom in a unit cell, which
+is obtained after solving the eigan-value equation of a
+dynamical matrix D(q) [12].
+

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+Moduli Stabilisation, de Sitter Vacua and Hybrid Inflation
+in Large Volume Compactifications
+Waqas Ahmeda 1, Athanasios Karozasb 2, George K. Leontarisb 3, Ilias Tavellarisb 4
+a School of Mathematics and Physics, Hubei Polytechnic University
+Huangshi 435003, China
+b Physics Department, University of Ioannina
+45110, Ioannina, Greece
+Abstract
+We study the cosmological implications of an effective field theory model derived within a con-
+figuration of D7 brane stacks in the framework of type-IIB string theory. We consider a suitable
+geometric setup where the Kähler moduli fields are stabilised and the parametric space is constrained
+so that a de Sitter vacuum is ensured. In addition to the moduli fields we also take into account the
+usual Higgs and matter fields included in the effective field theory. In this background we implement
+the standard hybrid inflation scenario with a singlet scalar field acting as the inflaton and the Higgs
+states serving as waterfall fields. Radiative corrections and soft supersymmetry breaking terms play
+an essential role in the realisation of a successful inflationary scenario consistent with the present
+cosmological data. Small tensor-to-scalar ratio values are predicted, which can be probed in future
+planned experiments. Further constraints on the model’s parameters are derived from bounds on
+dark radiation which is measured as a contribution to the effective number of neutrino species Nef f .
+In particular, we find an excess of ∆Nef f ≤ 0.95 at 2σ confidence level with natural values of the
+involved couplings.
+1E-mail: [email protected]
+2E-mail: mailto:[email protected]
+3E-mail: [email protected]
+4E-mail: [email protected]
+arXiv:2301.00329v1  [hep-ph]  1 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Description of the model and its constituents
+3
+3
+The effective potential
+5
+3.1
+Inflationary phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2
+Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4
+Reheating and dark radiation
+12
+5
+Conclusions
+15
+1
+Introduction
+Cosmological inflation is one of the most successful candidate scenarios in explaining the evolution
+of the Universe and its large scale structure observed today. Meanwhile, numerous effective quantum
+field theory models have been built aimed to conciliating cosmic inflation with particle physics models
+describing the low energy observables. A key criterion on the quest for appropriate models would be to
+have an ultra-violet (UV) completion in a quantum theory of gravity, valid up to the Planck scale MP. In
+this context, at present, string theory appears to be the only promising candidate for a consistent quantum
+theory at such a high scale which incorporates the Standard Model (SM) and its minimal supersymmetric
+extension (MSSM). String theory, however, is formulated in a ten dimensional spacetime framework
+and, therefore, compactification of the six extra dimensions is required to achieve a four-dimensional
+effective field theory compatible with the observed world. The reduction of the corresponding higher
+dimensional string action to four spacetime dimensions, however, entails an immense set of string vacua
+which is commonly nicknamed the string landscape. Yet, starting from a successful effective field theory
+which describes adequately the known physics phenomena, we cannot always embed it in a string theory
+framework.
+On the other hand, consistent effective field theory models emerging after compactification should
+possess a number of important features. Amongst them, they should predict a positive tiny cosmological
+constant Λ of the order Λ ≈ 10−120 M4
+P which could account for the dark energy, as suggested by cos-
+mological observations. A simple way to realise such a scenario is within an effective model involving
+a scalar field φ with a potential V(φ) which displays a (possibly metastable) positive minimum equal to
+the cosmological constant Λ. In fact, effective field theory models from strings compactified on Calabi-
+Yau (CY) manifolds contain a vast number of moduli fields and some of them could play the role of the
+inflaton φ. In this context, it is inferred that the cosmological issues are intertwined with the well known
+problem of moduli stabilisation. In fact, moduli stabilisation and (metastable) de Sitter vacua, play a key
+role for the successful implementation of the cosmological inflationary scenario in effective field theory
+
+(EFT) models of string origin. Therefore, reconciling these two issues is essential in the quest for a suit-
+able non-vanishing effective potential of some scalar field enacting as the inflaton φ which rolls down
+to the minimum of its (relatively shallow) potential. This enables the required exponential growth of the
+Universe, provided that the trajectory length ∼ ∆φ of the field φ to reach the minimum is sufficiently
+long to trigger inflation.
+A particular class of models involves constructions with large volume compactification scenarios [1]
+and inflatons associated with the Kähler moduli fields Tk = τk +iak. In previous scenarios [2, 3] consid-
+ered in the context of type-IIB theory it was shown that the internal volume modulus V expressed in terms
+of the real components of Kähler moduli, ReTk = τk, is a suitable candidate for this role, (φ ∝ logV ).
+Radiative corrections associated with intersecting space-filling D7 branes on the other hand, provide a
+stabilisation mechanism for Kähler moduli, and an uplift to their scalar potential through their universal
+abelian factors, this way leading to a positive cosmological constant. In particular, Kähler moduli stabil-
+isation is achieved through a non-zero potential generated by α′ and radiative (logarithmic) corrections
+induced when closed string loops traverse their codimension-two bulk towards localised gravity sources.
+Furthermore, the dS vacuum is obtained due to the positive D-term contributions, (originally proposed
+in [4]) from the intersecting D7-branes.
+In the large volume limit, the induced effective potential for the Kähler moduli receives a simple
+structure possessing two local extrema (a minimum and a maximum) and approaches to zero for φ → ∞.
+The separation distance of its two local extrema is of the order, ∆φ = φmax − φmin ∝ log(Vmax/Vmin).
+In the minimal effective model involving only moduli fields, it can be parametrised in terms of a single
+non-negative parameter while the larger possible separation ∆φ occurs at a critical value of this parameter
+where beyond this point only AdS solutions appear. A non-zero value of the aforementioned parameter
+exists at which a new inflationary small-field scenario is successfully implemented. In this novel scenario
+most of the required number N0 of efolds (N0 ∼ 60) are collected in the vicinity of the minimum of the
+potential and the prediction for the tensor-to-scalar ratio density fluctuations in the early universe is
+r ≈ 4 × 10−4. Despite these successes of the model, yet the picture is not complete and a waterfall
+mechanism has to be realised in order to end inflation and bring the dS vacuum down to a lower value
+compatible with the cosmological constant. It has been shown [5] that when open string states appearing
+in the D7 brane intersections are included in the massless spectrum, appropriate magnetic fluxes and
+specific brane separations can be chosen in a way that, for V less than some critical value, a charged
+open string scalar becomes tachyonic. In [5, 6] it was shown that such a state can represent a waterfall
+field. In possible generalisations of this scenario, one may include several of such fields, which end
+inflation and provide a deeper vacuum in accordance with the present value of Λ.
+In the present work an alternative scenario is proposed where it is taken into account that, in addition
+to the moduli fields, the EFT model accommodates the ordinary fermion matter and the Higgs fields in
+appropriate representations on the gauge group stemming from the specific details of the compactifica-
+tion procedure. Then, a possible and interesting variation of the scenario described previously would be
+that the Higgs rolls down a potential hill towards a new lower minimum, where its initial condition is de-
+fined around the metastable vacuum of the moduli potential. In the present construction, the metastable
+vacuum is determined by the Kähler moduli and the associated compactification volume V . More pre-
+2
+
+cisely, we implement the standard hybrid inflation with a singlet scalar field acting as inflaton and the
+Higgses used as waterfall fields. In this scenario the vacuum energy is determined by the scalar field
+and waterfall fields. We consider radiative corrections which are essential in shaping the slope along the
+inflationary track. Since supersymmetry (SUSY) is broken during inflation, we also include SUSY soft
+terms as well. The soft terms play an important role in order to achieve spectral index (ns) values consis-
+tent with the current experimental bounds. We find small tensor-to-scalar values which can be probed in
+future designed experiments. We also discuss dark radiation and show that change in the effective degree
+of neutrinos satisfy the bound 0.95% confidence level with natural values of the involved couplings.
+The paper is organised as follows. In Section 2 we present the basic features of the model including
+the Kähler potential and the superpotential. An analysis of the effective potential is given in Section 3
+followed by the presentation of the inflation setup along with our inflationary numerical predictions. In
+Section 4 we discuss reheating and dark radiation predictions. Section 5 concludes the paper.
+2
+Description of the model and its constituents
+In this work we consider a type-IIB string framework in ten dimensions where six of them are compact-
+ified on a Calabi-Yau threefold X . We restrict our attention mainly to the moduli spectrum and use the
+following notation: φ represents the dilaton field, while Ti, and za denote the Kähler and the complex
+structure (CS) moduli respectively. Furthermore, we introduce the usual axion-dilaton combination
+τ = C0 +ie−φ ≡ C0 + i
+gs
+,
+(2.1)
+where gs is the string coupling and C0 a 0-form potential (RR-scalar). We assume a perturbative -flux
+induced- superpotential W0 of the form proposed in [7]. At the classical level, W0 is a holomorphic
+function which depends on the axion-dilaton modulus τ, and the CS moduli za 5. τ and za are stabilised
+in the standard supersymmetric way, by solving DτW0 = 0, DzaW0 = 0, where DI = ∂IW +W∂IK are the
+covariant derivatives.
+We consider a geometric configuration of three intersecting D7-brane stacks equipped with magnetic
+fluxes. Regarding the Kähler potential, we will take into account α′ corrections as well as the effects
+of a novel four-dimensional Einstein-Hilbert (EH) term (localised in the internal space) which is gener-
+ated from higher derivative terms in the ten-dimensional string effective action [3]. This setup induces
+logarithmic corrections to the scalar potential via loop effects. Taking into account these corrections the
+relevant part of the Kähler potential receives the form [3]
+K = −2log(V +ξ0 +η0 logV )+···
+(2.2)
+where the dots stand for terms depending on za and τi. The general form of the volume is given by
+V = 1
+6κi jktit jtk where ti are the two-cycle Kähler moduli fields and κi jk are triple intersection numbers
+5We dispense with the use of non-perturbative corrections which would also introduce the Kähler moduli through terms of
+the form WNP ∝ e−aTk. As explained in the subsequent analysis, Tk can be perturbatively stabilised through one-loop corrected
+Kähler potential.
+3
+
+on X . We will assume a particular CY manifold with three Kähler moduli fields where the simple
+relation τi = at jtk (where a a positive constant) holds between the two- and four-cycle moduli, and the
+volume is simply given by [8]
+V = at1t2t3 = τiti = 1
+√a
+√τ1τ2τ3
+(2.3)
+As will be shown subsequently, the quantum corrections break the no-scale structure of the effective
+theory and give a non-zero contribution to the F-part of the supergravity scalar potential.
+After the dimensional reduction, the effective field theory model is assumed to be either some Grand
+Unified Theory (GUT) or directly the MSSM model where the ordinary low energy (super)-fields appear
+in appropriate representations of the EFT gauge group. Thus, in addition to the quantum corrections
+considered above, we also include matter fields in the Kähler potential. These contributions to the Käh-
+ler potential are essential in studying soft supersymmetry breaking effects and cosmological inflation.
+Within the present context, in particular, we focus on the Higgs sector which plays a vital role in im-
+plementing the scenario of hybrid inflation and investigating the possible production of dark radiation.
+Thus, we consider a generic set of Higgs pairs Φi, ¯Φi which are assumed to break the gauge group at
+some GUT scale much lower that the Planck scale MP. We have also introduced a field S, representing
+a gauge singlet superfield which realises the trilinear superpotential couplings of the form SΦ ¯Φ. Such
+singlet fields are ubiquitous in effective string theory models. In the setup described so far, the relevant
+terms of the superpotential have the following generic form
+W = W0 +κS(ΦΦ−M2)+···
+(2.4)
+where κ is a coupling constant coefficient, M represents a high scale mass parameter, W0 the flux induced
+part introduced previously, and dots stand for possible terms irrelevant to our discussion.
+Including contributions of the above matter fields in the Kähler potential (2.2), while setting the
+Planck mass, MP = 1, we obtain the following generic form
+K = −2log
+�
+�
+�
+3
+∏
+k=1
+(Tk + ¯Tk)
+� 1
+2
++C
+�
+�+
+3
+∑
+k=1
+ak
+Tk + ¯Tk
+fk(Φi, ¯Φi...,S, ¯S)
+(2.5)
+where C is a function with a logarithmic dependence on the three moduli fields T1,2,3 defined as
+C = ξ0 +η0log
+�
+3
+∏
+k=1
+(Tk + ¯Tk)
+�
+.
+(2.6)
+In the above equation the parameter ξ0 stands for α′3 corrections [9] and is proportional to the Euler
+characteristic χCY of the Calabi-Yau manifold
+ξ0 = −ζ(3)
+4
+χCY ,
+(2.7)
+and η0 is an order one coefficient [3]. The functions fk in Eq (2.5) describe the visible Higgs sector. For
+simplicity we will assume that all fk are the same and have the following form
+f(Φ,Φ,S) = αΦΦ† +βΦΦ
+† +γSS† +λ(ΦΦ+h.c.)
+(2.8)
+4
+
+where α,β,γ and λ are dimensionless couplings. Furthermore, we will adopt the approach used in [10]
+and we will express the matter contribution term in (2.5) in terms of the compactification volume (note
+that T ∝ V 2/3), hence, the Kähler potential in Eq. (2.5) is reduced to the following form
+K = −2log[V +ξ0 +η0log(V )]+ 3a
+V 2/3
+�
+αΦΦ† +βΦΦ
+† +γSS† +λ(ΦΦ+h.c)
+�
+(2.9)
+where a is a constant and V is defined in (2.3).
+Up to this point we have presented the minimum number of moduli and matter fields which are
+necessary for our subsequent analysis. Next, we proceed with the computation of the scalar potential
+which is essential to investigate the properties of the model and compute the various cosmological and
+other phenomenological observables.
+3
+The effective potential
+The scalar potential of the effective field theory model receives various contributions. As we will see
+shortly, in the present construction there are F- and D-terms associated with the moduli sector, and
+contributions from the EFT matter fields, as well as supersymmetry breaking terms. We start with the
+F-term potential which is given by the generic formula
+VF = eG �
+GiG−1
+i j∗ G j∗ −3
+�
+,
+(3.1)
+where
+G = K +log|W|2 ≡ K +logW +logW ∗
+and the indices i, j in (3.1) denote the derivatives with respect to the various moduli and other fields.
+Computing the derivatives, and substituting in Eq. (3.1) while keeping only the leading order terms,
+the F-term potential receives the following simplified form
+VF
+≃
+κ2αβ
+�
+M2 −ϕϕ
+�2 +γκ2S2 �
+αϕ2 +βϕ2�
+3aαβγV 4/3
++ 3W 2
+0 (2η0 logV −8η0 +ξ0)
+2V 3
+,
+(3.2)
+where ϕ and ϕ are the bosonic components of the superfields Φ and Φ.
+At the extrema of the F-term potential the fields take the following values 6
+So = 0,
+ϕoϕo = M2,
+Vo = e
+13
+3 − ξ0
+2η0 .
+(3.3)
+Substituting the solution (3.3) into (3.2) we obtain
+V extr.
+F
+=
+3|W0|2η0(2logVo −8+ ξ0
+η0 )
+2V 3
+o
+≡ η0
+|W0|2
+V 3
+o
+.
+(3.4)
+6In more general EFT backgrounds it is possible that minimisation with respect to the fields S,Φi leads to a potential
+of the form V ∼
+a
+5V 4/3 + 1
+3
+b+η logV
+V 3
+. In this case it is possible to have a dS minimum with the volume acquiring a value
+V 5/3
+0
+=
+� 9n
+4a
+�
+W
+�
+4a
+9n
+�
+e
+1
+3 − b
+n
+�5/3�
+, where W is the product-log (Lambert) function.
+5
+
+20000
+30000
+40000
+50000
+60000
+-2.×10-13
+-1.5×10-13
+-1.×10-13
+-5.×10-14
+x
+V (x)
+25000 30000 35000 40000 45000 50000 55000 60000
+1.×10-15
+2.×10-15
+3.×10-15
+4.×10-15
+5.×10-15
+6.×10-15
+7.×10-15
+x
+V (x)
+Figure 1: Plots of the potential along the volume direction. The left panel shows the F-term potential, while in
+the right panel the D-term potential has also been included. We choose ξ0 = 10, η0 = −0.92, S = 0, ϕ = ϕ = M,
+κ = 0.1 and γ = 1. Here x represents the volume, x ≡ V .
+A simple analysis shows that this is a minimum of the potential as long as η0 < 0. However, since
+|W0|2
+V 3
+0
+> 0, the potential (3.4) at the minimum aquires a negative value, hence it turns out that the F-term
+potential predicts an anti-de Sitter (AdS) vacuum. In Fig. 1, left panel, the F-term potential with AdS
+minimum is plotted for a choice of the parameters ξ0,η0,W0.
+Despite the forgoing negative F-term contribution, the potential can be lifted to a de Sitter minimum,
+once D-term contributions [4, 11] associated with the U(1) symmetries of the D7-branes are taken into
+account [2]. More precisely, in the present geometric setting, there are D-term contributions due to the
+universal U(1) factors associated with the D7 brane stacks. These terms have the general form [2, 11]
+VD =
+3
+∑
+i=1
+g2
+D7i
+2
+�
+Qi∂TiK +∑
+j
+qj|Θj|2
+�2
+,
+(3.5)
+where gD7i = (ReTi)−1 and Qi,qj are “charges” while Θj represent possible gauge singlets of the effective
+field theory model. For ⟨Θj⟩ = 0 (see [11] for an extended discussion related to D-terms) the second term
+in the parenthesis vanishes and each component of the D-term aquires a simple -model independent- form
+VDi ≈ Q2
+i /τ3
+i . Then, the total D-term potential, being the sum of three components, is approximated by [2]
+VD =
+3
+∑
+i=1
+di
+τi
+�∂K
+∂τi
+�2
+≈
+3
+∑
+i=1
+di
+τ3
+i
+≡ d1
+τ3
+1
++ d3
+τ3
+3
++ d2τ3
+1τ3
+3
+V 6
+,
+(3.6)
+Here, di are positive constants related to the charges di ∼ Q2
+i > 0. Furthermore, using the volume formula
+V 2 = τ1τ2τ3, we have substituted the modulus τ2 with its equivalent τ2 = V 2/(τ1τ3) . Then, the total
+effective potential is the sum Veff = VF +VD which we must minimise it with respect to V and the two
+remaining Kähler moduli τ1,2. We have assumed that the F-part of the potential depends on the total
+volume, hence the explicit dependence of Veff on τ1,τ3 only comes through the VD part. It is found
+that the two minimisation conditions with respect to τ1,3 determine the ratios between the moduli, i.e.,
+�
+τi
+τ j
+�3
+= di
+d j [12]. Expressed in terms of the stabilised total volume V , the conditions for the two τi can
+6
+
+Figure 2: The shape of the effective potential ϕ − S plane for the choice of parameters ξ0 = 10, η0 = −0.92,
+V0 = 32000, κ = 0.1, γ = 1. Here x represents the compactification volume, x ≡ V .
+be written as
+τ3
+i =
+� d2
+i
+dkdj
+� 1
+3
+V 2 ,
+where i = 1,3. In this case the D-term potential receives the simple form
+VD ≈ d
+V 2 ,
+with
+d = 3(d1d2d3)
+1
+3 .
+(3.7)
+By choosing ϕ = ϕ and α = β, the D-term contribution from the Higgs fields vanishes. Then the effective
+potential has the following form
+Veff
+≃
+κ2αβ
+�
+M2 −ϕϕ
+�2 +γκ2S2 �
+αϕ2 +βϕ2�
+3aαβγV 4/3
++ 3W 2
+0 (2η0 log(V )−8η0 +ξ0)
+2V 3
++ d
+V 2 . (3.8)
+The shape of the potential along the volume modulus V when both F- and D-terms are included is shown
+in the right panel of Fig. 1. As can be seen, a positive D-term is sufficient to uplift the potential along the
+volume direction so that we achieve a de Sitter minimum. In order to find the extrema of the potential
+along the ϕ and S directions, we require the vanishing of its corresponding derivatives. Thus, for ϕ we
+impose the condition
+dVeff
+dϕ
+=
+0 ⇒ κ2 �
+4α2ϕ3 −4α2M2ϕ +4αγS2ϕ
+�
+3aα2γV 4/3
+= 0
+(3.9)
+which is solved for the following three ϕ-values
+ϕ = 0,
+ϕ± = ±
+�
+M2 − γ
+α S2.
+(3.10)
+7
+
+Similarly along the S direction
+dVeff
+dS
+=
+0 ⇒ 4κ2Sϕ2
+3aαV 4/3 = 0
+(3.11)
+which yields
+S = 0.
+(3.12)
+Combining Equations (3.10) and (3.12), in the large volume limit we obtain the following solutions
+(S = 0,ϕ = 0),
+(S = 0,ϕ = ±M).
+(3.13)
+We have already dealt with the minimisation of VF with respect to the volume modulus. However, in
+the presence of D-terms the minima along the volume direction are shifted. Thus, requiring the vanishing
+of the derivative of (3.8) with respect to V , we obtain the equation
+dVef f
+dV
+= 0
+⇒
+−4κ2 �
+α2ϕ4 +α2M4 −2α2M2ϕ2 +2αγS2ϕ2�
+9aα2γV 7/3
+− 2d
+V 3
+−
+9
+�
+2η0W 2
+0 log(V )−8η0W 2
+0 +ξ0W 2
+0
+�
+2V 4
++ 3η0W 2
+0
+V 4
+= 0.
+(3.14)
+In the large volume limit, to a good approximation, the above equation gives the solution
+V0 ≈ 9η0W 2
+0
+2d
+W
+�
+�2de
+13
+3 − ξ0
+2η0
+9η0W 2
+0
+�
+�,
+(3.15)
+where W represents the product-log (Lambert) function. The shape of the scalar potential Ve f f in the
+ϕ-S plane is displayed in Fig. 2. As discussed earlier the D-term contribution in the effective potential is
+essential in order to ensure de Sitter vacua when S approaches zero.
+3.1
+Inflationary phase
+Up to this point we have analysed in detail the scalar potential of the effective theory and the role of the
+various fields on its final shape. Therefore we are now fully equipped with all the tools and the necessary
+ingredients to examine whether cosmological inflation is realised in the present model. In a previous
+approach, within the same type-IIB framework and the geometric setup of intersecting D7 brane stacks,
+the role of the inflaton field was associated with the logarithm of the compactification volume modulus.
+The accumulation of the required 60 efolds to realise slow-roll inflation was converted to a lower bound
+on the minimum vacuum energy [12], yet much bigger than the cosmological constant. Subsequently,
+a new “waterfall” field was introduced which adds a new direction of the potential. The waterfall field
+rolls down towards the new lower minimum while at the same time it ends inflation. It was shown [5]
+that this role can be realised by open string states oscillating near the intersections of the D7 stacks.
+In the present case where the physical states from the effective theory model have been taken into
+account, new possibilities have emerged. At the minimum along the compactification volume modulus
+8
+
+V , the Higgs field ϕ, and the singlet S add new directions (transverse to that of V ) providing new
+lower minima for the scalar potential. As such, they are potential candidates for waterfall fields. In the
+present scenario inflation proceeds along the local minimum ϕ = 0 (the inflationary track), beginning at
+large S values. An instability occurs at the waterfall point S2
+c = M2 , which is the value of S, such that
+Sc = ∂ 2V
+∂S2 |S= 0. At this point the field falls naturally into one of the two SUSY minima at ϕ = ±M . At
+large S, the scalar potential is approximately quadratic in ϕ, whereas at S = 0, equation (3.8) becomes a
+Higgs potential. Along the inflationary track the constant term
+V vol
+0
+= κ2M4
+3aV 4/3
+0
++ 3W 2
+0 (2η0 log(V0)−8η0 +ξ0)
+2V 3
+0
+,
+is present at tree level, thus SUSY is broken during inflation. When SUSY breaks, splitting between
+fermionic and bosonic mass multiplets is created and contributions to radiative corrections occurs. Fol-
+lowing [13, 14], the soft terms are
+∆Vsoft = (m2
+3/2 +V0)M2y2 +O(1/V0)
+(3.16)
+where in above equation y defines the ratio y = S/M. The first extremum (S = 0,ϕ = 0) is the maxi-
+mum of the potential. The ϕ = 0 corresponds to the trajectory of the standard hybrid inflation for which
+{ϕ = 0, S > M}. When the inflaton reaches at S = M, the waterfall field takes over and the inflaton
+moves towards the SUSY minimum at ϕ = ±M. Since the potential is non-zero along the inflationary
+track, SUSY is broken along the latter, and the radiative corrections along with the soft SUSY-breaking
+potential Vsoft can lift its flatness, while also providing the necessary slope for driving inflation. The ef-
+fective contribution of the one-loop radiative corrections can be calculated using the Coleman-Weinberg
+formula [15] as
+∆V1-loop =
+κ4M4y4
+48π2a2α2γV 8/3
+0
+�F(y)
+3γ − κ2
+2
+�
+,
+(3.17)
+where
+F(y) = log
+�
+κ2M2y2
+3aαγQ2V 4/3
+0
+�
+− 9a2α2γ2
+V 4/3
+0
+log
+�κ2M2y2
+Q2V 2
+0
+�
+.
+(3.18)
+Including the various contributions computed above, we may write the scalar potential along the infla-
+tionary trajectory (i.e. ϕ = ϕ = 0) as follows,
+V
+≃
+VF +VD +∆V1-loop +∆Vsoft,
+≃
+κ2M4
+�
+V vol
+0
+κ2M4 +
+κ2y4F(y)
+144π2a2α2γ2V 8/3
+0
+−
+κ2y4
+96π2a2α2γ2V 8/3
+0
++ M2
+Sy2
+κ2M2
+�
+.
+(3.19)
+The prediction for the various inflationary observables are estimated using the standard slow-roll
+parameters defined as,
+ε = 1
+4
+� 1
+M
+�2 �V ′
+V
+�2
+, η = 1
+2
+� 1
+M
+�2 �V ′′
+V
+�
+,
+ξ 2 = 1
+4
+� 1
+M
+�4 �V ′V ′′′
+V 2
+�
+,
+(3.20)
+9
+
+where prime denotes the derivative with respect to y. Note that the extra factor of 1/2 is due to the
+relation between the canonically normalised real inflaton field, σ ≡ |S|/
+√
+2, and the complex field, S. In
+the slow-roll approximation, the scalar spectral index ns, the tensor-to-scalar ratio r and the running of
+the scalar spectral index αs ≡ dns/d lnk are given by,
+ns ≃ 1+2η −6ε,
+r ≃ 16ε,
+αs ≃ 16ε η −24ε2 −2ξ 2.
+(3.21)
+The value of the scalar spectral index ns in the ΛCDM model is ns = 0.9665±0.0038 [16].
+The amplitude of the scalar power spectrum is given by,
+As(k0) =
+1
+24π2
+�V(y0)
+ε(y0)
+�
+,
+(3.22)
+where As(k0) = 2.137×10−9 at the pivot scale k0 = 0.05Mpc−1 as measured by Planck 2018 [16]. The
+relevant number of e-folds, N0, before the end of inflation is,
+N0 = 2M2
+� y0
+ye
+� V
+V ′
+�
+dy,
+(3.23)
+where y0 ≡ y(k0) is the field value at the pivot scale k0, and ye is the field value at the end of inflation.
+As the case may be, the value of ye is fixed either by the breakdown of the slow-roll approximation
+(η(ye) = −1), or by a ‘waterfall’ destabilisation occurring at the value ye = 1.
+3.2
+Numerical results
+The results of our numerical calculations are displayed in Fig. 3, which show the ranges of r, m3/2 and a
+in the M −κ plane. We have used up to second-order approximation on the slow-roll parameters, and for
+simplicity we have set γ = 1, α = 0.25, V0 = 32000 and ye = M. Moreover, we have fixed the spectral
+index ns to the central value (ns = 0.96655) from Planck’s data.
+We further require a ≤ 1, m3/2 ≲ 6 × 1011 GeV, Higgs mass parameter M ≤ Mstring ∼ 1/V01/2 =
+5.5×10−10Mp = 1.36×1016 GeV and FT which is defined as the difference of field value at pivot scale
+and end of inflation FT ≡ y0 − ye ≲ 0.01. These constraints appear in Fig. 3 as the boundaries of the
+allowed region in the M −κ plane. In our analysis the soft SUSY contributions along with the radiative
+corrections play the dominant role in order to obtain a parametric space consistent with experimental
+bounds, parametrised by m3/2, a and the logarithmic term. It is useful to analytically examine some
+approximate equations to understand the behaviour depicted in Fig. 3. The spectral index ns and tensor-
+to-scalar ratio r in the leading order slow-roll approximation are given by
+ns ≃ 1+ 2M2
+s
+κ2M4 +
+κ2y2
+0
+36π2a2α2γ2M2V 8/3
+0
+�
+3log
+�
+κ2M2y2
+0
+3aαγQ2V 4/3
+0
+�
+−1
+�
+,
+(3.24)
+r ≃ 4
+M2
+�
+�
+�
+�
+2M2
+s y0
+κ2M2 +
+κ2y3
+0
+�
+log
+�
+κ2M2y2
+0
+3aαγQ2V 4/3
+0
+�
+−1
+�
+36π2a2α2γ2V 8/3
+0
+�
+�
+�
+�
+2
+.
+(3.25)
+10
+
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+1012
+1013
+1014
+1015
+1016
+κ
+M (GeV)
+M=Mstring=1.36×1016(GeV)
+Ms=6×1011(GeV)
+κ=1
+=1
+=0.1688
+FT=y0-ye=0.01%
+DR Area
+r=5×10-7
+r=10-10
+r=10-13
+r=10-16
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+1012
+1013
+1014
+1015
+1016
+κ
+M (GeV)
+M=Mstring=1.36×1016(GeV)
+Ms=6×1011(GeV)
+κ=1
+=1
+=0.1688
+FT=y0-ye=0.01%
+DR Area
+=10-2
+=10-4
+=10-6
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+1012
+1013
+1014
+1015
+1016
+κ
+M (GeV)
+M=Mstring=1.36×1016(GeV)
+Ms=6×1011(GeV)
+κ=1
+=1
+=0.2117
+FT=y0-ye=0.01%
+DR Area
+Ms=5×1010GeV
+Ms=109GeV
+Ms=107GeV
+Ms=105GeV
+Figure 3: Variations of r, m3/2 and a in M − κ plane. The solution between a = 0.1688 to a = 1 shows the
+parametric space consistent with dark radiation.
+Solving these two equations simultaneously, we obtain
+ns ≃ 0.96655, r ≃ 1.44×10−5
+for the following values of the parameters
+y0 ≃ 11.89, a ∼ 2.02×10−4, Ms ∼ 5.76×1011GeV, M ∼ 1016GeV, γ ∼ 1, α ∼ 0.25.
+These approximate values are very close to the actual ones obtained in the numerical calculations.
+The above analytical equations therefore give a valid approximation of our numerical results displayed
+in Fig. 3.
+For the scalar spectral index ns fixed at Planck’s central value, ns = 0.96655, according to our nu-
+11
+
+merical analysis the exact range of parameters where acceptable solutions occurs is presented below
+1.6×10−5 ≲ κ ≲ 1,
+(2.19×1012 ≲ M ≲ 1.3×1016) GeV,
+(1.8×104 ≲ MS ≲ 6×1011) GeV,
+9.1×10−20 ≲ r ≲ 2×10−5,
+1.6×10−8 ≲ a ≲ 1.
+From the exact ranges of solutions we can see that our calculations predict a low tensor-to-scalar
+ratio (r) compared to the current experimental bounds. However, ongoing and future gravity waves
+experiments are expected to reach much smaller ranges of tensor-to-scalar ratio values comparable to
+our numerical predictions.
+In all plots of Fig. 3 the solution between a = 0.1688 (blue curve) to a = 1 shows the parametric
+space consistent with dark radiation conditions which we discuss next.
+4
+Reheating and dark radiation
+After the end of inflation, the lightest moduli will begin to oscillate around their minima, acquiring a
+large energy density in the process. The decay products of the modulus fall into two categories. The
+first are decays that go to the visible sector that is, particles of the SM, or its extensions such as the
+MSSM. The decays to visible matter induce reheating, after which the standard hot Big Bang cosmology
+follows. In addition, there may also be decays to hidden sector states. The hidden sector contains several
+candidates for dark radiation, such as massless axions or light hidden gauge bosons. Let us consider the
+three Kähler moduli case, Tk = τk +iak,V = √τ1τ2τ3 where ak is the RR-axion. Decay to the light axion
+ak takes place primarily through the supergravity kinetic terms for the Kähler moduli which read as,
+L ⊃ Ki ¯j∂µT i∂ µT ¯j.
+(4.1)
+The tree level Kähler potential is,
+K = −2log
+�
+(T1 + ¯T1)(T2 + ¯T2)(T3 + ¯T3) = −log(τ1τ2τ3)+···
+(4.2)
+where the dots represent constant terms which are ignored. Then, the Kähler matrix is found to be
+Ki ¯j = 1
+4diag
+� 1
+τ2
+1
+, 1
+τ2
+3
+, 1
+τ2
+3
+�
+.
+(4.3)
+Therefore, Eq (4.1) can be rewritten as,
+L ⊃ 1
+4
+1
+τ2
+i
+∂µτi∂ µτi
+Note here that we have set the reduced Planck mass Mp = 1. To put this into canonical form we need to
+find the transformation τi(ui) so that
+L ⊃ 1
+2 ∑
+i
+∂µui∂ µui.
+12
+
+This implies
+τi = e
+√
+2ui
+. The moduli fields for canonical kinetic terms take the form
+uk = 1
+√
+2
+logτk
+. The corresponding volume modulus is
+t = u1 +u2 +u3
+√
+3
+= 1
+√
+3
+1
+√
+2 ∑
+k
+logτk =
+�
+2
+3 logV
+. The transverse directions are
+u = u1 −u2
+√
+2
+= 1
+2 log τ1
+τ2
+, v = u1 +u2 −2u3
+√
+6
+= 1
+√
+3 log τ1τ2
+τ2
+3
+. We reverse the relations
+�
+�
+�
+u1
+u2
+u3
+�
+�
+� =
+�
+�
+�
+�
+1
+√
+3
+1
+√
+2
+1
+√
+6
+1
+√
+3
+− 1
+√
+2
+1
+√
+6
+1
+√
+3
+0
+−
+�
+2
+3
+�
+�
+�
+�
+�
+�
+�
+t
+u
+v
+�
+�
+�
+while substituting to the Lagrangian part for axions we have
+L
+⊃
+1
+√
+3t
+�
+∂µa1∂ µa1 +∂µa2∂ µa2 +∂µa3∂ µa3
+�
++ 1
+√
+2
+u
+�
+∂µa1∂ µa1 −∂µa2∂ µa2
+�
++ 1
+√
+6
+v
+�
+∂µa1∂ µa1 +∂µa2∂ µa2 −2∂µa3∂ µa3
+�
+.
+(4.4)
+The decay rate of the lightest modulus u to axions is given by,
+Γ(u → a1a1) =
+1
+64π m3
+u
+(4.5)
+where mu is the modulus mass. In the large volume scenario a distinct hierarchy of mass scales is
+generated, (for details see Ref [17, 18]). The mass eigenstates after diagonalisation in a unit of Mp = 1
+can be written as,
+mt = mu = mv ∼
+1
+V 3/2
+0
+,
+mai ∼ e−2πV 2/3
+0
+,
+mso ft ∼ 1
+V 2
+0
+,
+m3/2 ∼ 1
+V0
+,
+Mstring ∼
+1
+V 1/2
+0
+.
+Similarly the dominant visible-sector decay channel is the decay to Higgs bosons. Specialising to
+the MSSM case, we can make the identifications Φ = Hu and Φ = Hd. Each of these is a two-component
+complex field, so there are eight degrees of freedom. Then the decay rate can be derived by including
+the matter contribution to the Kähler potential
+L ⊃ 3aλ
+√
+2
+HuHd□u+h.c.+··· .
+(4.6)
+13
+
+0.2
+0.4
+0.6
+0.8
+1.0
+2×107
+4×107
+6×107
+8×107
+
+Tr(GeV)
+Figure 4: Variations of the reheating temperature (Tr) with respect to coefficient a consistent with dark radiation
+constraint (∆Neff ≲ 0.95) at 95% confidence level.
+The dominant contribution to the decay of light moduli u comes from the Giudice-Masiero coupling [19]
+3aλHuHd□u as all other couplings are mass-suppressed [20]. Each field is a complex doublet, hence we
+include the partial widths from each of the four decay channels. This yields
+Γ(u −→ HuHd) = 9a2λ 2
+8π
+m3
+u.
+(4.7)
+The present-day radiation content of the Universe can be described in terms of the energy density asso-
+ciated with each relativistic particle species at present. This radiation consists of photons and neutrinos
+plus any additional hidden components, which we call dark radiation (DR):
+ρradiation = ρphoton +ρneutrino +ρDR
+(4.8)
+We can express this in terms of an effective number of neutrino species, Neff, as follows
+ρradiation = ρphoton
+�
+1+ 7
+8
+� 4
+11
+�4/3
+Neff
+�
+.
+(4.9)
+Any excess can be accounted for by the presence of DR which is expressed as
+ρDR = ρphoton
+�
+7
+8
+� 4
+11
+�4/3
+∆Neff
+�
+,
+(4.10)
+where ∆Neff = Neff − 3.046 is the change in the effective number of neutrino species. If there were no
+dark radiation we would expect to find Neff = 3.046 which is slightly greater than 3 to account for partial
+14
+
+reheating due to e+e− annihilation. ∆Neff can also be written in terms of decay rate channels
+∆Neff = 43
+7
+� 10.75
+g∗(Tr)
+� 1
+3 Γτ→DR
+Γτ→SM
+= 43
+7
+� 10.75
+g∗(Tr)
+� 1
+3
+1
+72a2λ 2 ,
+(4.11)
+where g∗(TRh) is the effective degree of freedom at the time of reheating the Universe. The measured
+values of Neff require ∆Neff ≲ 0.95 at the 95% confidence level, which translates into a bound on the pa-
+rameters of the model, a·λ ≳ 0.1688. In Fig. 3 the parametric space between 0.1688 ≤ a ≤ 1 represents
+the solutions consistent with dark radiation. In this parametric space we receive tensor-to-scalar ratio
+values r ≤ 1.5×10−9, M ≲ 2.54×1012 GeV and soft mass parameter Ms ≤ 4.98×109 GeV.
+The isotropy of the Cosmic microwave background (CMB) over large scales can be explained by
+inflation, which is followed by a period of reheating. During that, the expansion slows and energy is
+transferred to the SM particles, which enter local thermal equilibrium. We consider the scenario where
+the Universe is reheated by a modulus u decaying into SM particles. In this case the reheating temperature
+Tr, using Eq.(4.5) and Eq.(4.11) is defined as
+Tr =
+�
+Γu =
+�
+63
+344π ∆Neff
+�g∗(Tr)
+10.75
+�1/2
+a2λ 2m3u.
+(4.12)
+For a SUSY scale at the TeV regime, the constraint on reheating temperature is Tr ≲ 1 GeV, which
+corresponds to g∗(Tr) = 224/7, see Ref [21, 22]. For the present model we have V0 ∼ 3.2 × 104 which
+corresponds to split scale SUSY: msoft ∼ V −2
+0
+= 9.7656 × 10−10 = 2.3 × 109 GeV. As a result, the
+constraint on the reheating temperature is relaxed in this scenario. For split or high scale SUSY we have
+g∗(Tr) = 106.75, so the reheating temperature is Tr ∼ 107 GeV as shown in Fig. 4.
+5
+Conclusions
+We investigated the cosmological and low energy supersymmetry implications of an effective model [2]
+stemming from a geometric configuration of intersecting three D7-branes stacks, within the framework
+of type-IIB string theory. In this model perturbative string loop corrections which depend logarithmically
+on the compactification volume V , and D-terms associated with the universal U(1) factors of D7-brane
+stacks generate an effective scalar potential with de Sitter vacuum, and stabilise all three Kähler moduli
+fields of the specific geometric setting. In the present work we took into account the effects of ordinary
+matter contributions in the Kähler potential of the effective model and, in particular, we focused on
+the role of a generic pair of Higgs fields Φ, ¯Φ (related to the gauge group of the effective theory) on
+low energy phenomenology predictions and various cosmological observables. We included matter field
+content and soft-term contributions as well as Coleman-Weinberg corrections to the previously derived
+potential and studied the implementation of the standard hybrid inflationary scenario where a singlet
+gauge field sharing common couplings with the Higgs fields in the superpotential plays the role of the
+inflaton whilst the Higgs states act as waterfall fields. Fixing the spectral intex at central value, ns =
+0.96655, we provided predictions for the remaining cosmological observables in accordance with the
+latest Planck data. In particular, we predicted the value of the tensor-to-scalar ratio to be r ∼ .2×10−4
+15
+
+which is much smaller than the current experimental bounds, but within the reach of future designed
+experiments. Next, we discussed the decay of the lighter Kähler moduli after the end of inflation, which
+includes modes to visible as well as invisible particles. In particular, in the context of an MSSM effective
+theory and in the presence of a Giudice-Masiero coupling, the dominant decay of the lightest modulus
+is to the Higgs fields, in accordance with previous computations [21]. Furthermore, we investigated
+predictions of the model to dark radiation production and we found ∆Ne f f ≤ 0.95 at 2σ confidence
+level. This requires the model parameters a and λ (associated with the couplings ∝ aλ(Φ ¯Φ + h.c) in the
+Kähler potential) to satisfy the bound aλ ≳ 0.1688, which for λ ≈ 1 translates into a bound for a in the
+perturbative regime. Regarding other vital low energy predictions for a volume fixed at V0 ∼ 3.2 × 104
+which ensures a dS minimum, we fould that the (split)-SUSY scale is around mso ft ∼ V −2
+0
+MP ≈ 2.3×109
+GeV. As a result, the constraint imposed on the reheating temperature is relaxed in this scenario. For split
+or high scale SUSY we have g∗(Tr) = 106.75 [21], and the reheating temperature is Tr ∼ 107 GeV .
+Acknowledgements
+The work of GKL was supported by the “Hellenic Foundation for Research and Innovation (H.F.R.I.)
+under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and
+the procurement of high-cost research equipment grant” (Project Number: 2251)”
+16
+
+References
+[1] V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, “Systematics of moduli stabilisation
+in Calabi-Yau flux compactifications,” JHEP 03 (2005), 007 [arXiv:hep-th/0502058 [hep-th]].
+[2] I. Antoniadis, Y. Chen and G. K. Leontaris, “Perturbative moduli stabilisation in type IIB/F-theory
+framework,” Eur. Phys. J. C 78 (2018) no.9, 766 [arXiv:1803.08941 [hep-th]].
+[3] I. Antoniadis, Y. Chen and G. K. Leontaris, “Logarithmic loop corrections, moduli stabilisation and
+de Sitter vacua in string theory,” JHEP 01, 149 (2020) [arXiv:1909.10525 [hep-th]].
+[4] C. P. Burgess, R. Kallosh and F. Quevedo, “De Sitter string vacua from supersymmetric D terms,”
+JHEP 10, 056 (2003) [arXiv:hep-th/0309187 [hep-th]].
+[5] I. Antoniadis, O. Lacombe and G. K. Leontaris, “Hybrid inflation and waterfall field in string theory
+from D7-branes,” JHEP 01 (2022), 011 [arXiv:2109.03243 [hep-th]].
+[6] Ignatios Antoniadis, Osmin Lacombe and George K. Leontaris, “Type IIB moduli stabilization,
+inflation and waterfall fields”, International Journal of Modern Physics A, September 2022,
+[7] S. Gukov, C. Vafa and E. Witten, “CFT’s from Calabi-Yau four folds,” Nucl. Phys. B 584 (2000),
+69-108 [arXiv:hep-th/9906070 [hep-th]].
+[8] G. K. Leontaris and P. Shukla, “Stabilising all Kähler moduli in perturbative LVS,” JHEP 07 (2022),
+047 [arXiv:2203.03362 [hep-th]].
+[9] K. Becker, M. Becker, M. Haack and J. Louis, “Supersymmetry breaking and alpha-prime correc-
+tions to flux induced potentials,” JHEP 06, 060 (2002) [arXiv:hep-th/0204254 [hep-th]].
+[10] R. Blumenhagen, J. P. Conlon, S. Krippendorf, S. Moster and F. Quevedo, “SUSY Breaking in
+Local String/F-Theory Models,” JHEP 09 (2009), 007 [arXiv:0906.3297 [hep-th]].
+[11] M. Haack, D. Krefl, D. Lust, A. Van Proeyen and M. Zagermann, “Gaugino Condensates and D-
+terms from D7-branes,” JHEP 01 (2007), 078 [arXiv:hep-th/0609211 [hep-th]].
+[12] I. Antoniadis, O. Lacombe and G. K. Leontaris, “Inflation near a metastable de Sitter vacuum from
+moduli stabilisation,” Eur. Phys. J. C 80, no.11, 1014 (2020) [arXiv:2007.10362 [hep-th]].
+[13] A. Brignole, L. E. Ibanez and C. Munoz, “Towards a theory of soft terms for the supersymmetric
+Standard Model,” Nucl. Phys. B 422, 125-171 (1994) [erratum: Nucl. Phys. B 436, 747-748 (1995)]
+[arXiv:hep-ph/9308271 [hep-ph]].
+[14] J. P. Conlon, S. S. Abdussalam, F. Quevedo and K. Suruliz, “Soft SUSY Breaking Terms for Chiral
+Matter in IIB String Compactifications,” JHEP 01, 032 (2007) [arXiv:hep-th/0610129 [hep-th]].
+[15] S.R. Coleman and E.J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry
+Breaking,” Phys. Rev. D 7, 1888 (1973) [arXiv:hep-ph/].
+17
+
+[16] Y. Akrami et al. [Planck], “Planck 2018 results. X. Constraints on inflation,” Astron. Astrophys.
+641 (2020), A10 [arXiv:1807.06211 [astro-ph.CO]].
+[17] J. P. Conlon, F. Quevedo and K. Suruliz, “Large-volume flux compactifications: Moduli spectrum
+and D3/D7 soft supersymmetry breaking,” JHEP 08, 007 (2005) [arXiv:hep-th/0505076 [hep-th]].
+[18] M. Cicoli, C. P. Burgess and F. Quevedo, “Anisotropic Modulus Stabilisation: Strings at LHC
+Scales with Micron-sized Extra Dimensions,” JHEP 10, 119 (2011) [arXiv:1105.2107 [hep-th]].
+[19] G. F. Giudice and A. Masiero, “A Natural Solution to the mu Problem in Supergravity Theories,”
+Phys. Lett. B 206 (1988), 480-484
+[20] S. Angus, J. P. Conlon, U. Haisch and A. J. Powell, “Loop corrections to ∆Ne f f in large volume
+models,” JHEP 12 (2013), 061 [arXiv:1305.4128 [hep-ph]].
+[21] A. Hebecker, P. Mangat, F. Rompineve and L. T. Witkowski, “Dark Radiation predictions from
+general Large Volume Scenarios,” JHEP 09, 140 (2014) [arXiv:1403.6810 [hep-ph]].
+[22] M. Cicoli, K. Sinha and R. Wiley Deal, “The dark universe after reheating in string inflation,” JHEP
+12 (2022), 068 [arXiv:2208.01017 [hep-th]].
+18
+

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@@ -0,0 +1,1438 @@
+HETEROGENEOUS DOMAIN ADAPTATION AND EQUIPMENT
+MATCHING: DANN-BASED ALIGNMENT WITH CYCLIC
+SUPERVISION (DBACS)
+A PREPRINT
+Natalie Gentner∗
+Infineon Technologies AG
+Am Campeon 1-15,
+85579 Neubiberg, Germany
+Gian Antonio Stusto †
+Department of Information Engineering
+University of Padova
+Via Gradenigo 6/B,
+35131 Padova, Italy
+January 4, 2023
+ABSTRACT
+Process monitoring and control are essential in modern industries for ensuring high quality standards
+and optimizing production performance. These technologies have a long history of application in
+production and have had numerous positive impacts, but also hold great potential when integrated
+with Industry 4.0 and advanced machine learning, particularly deep learning, solutions. However, in
+order to implement these solutions in production and enable widespread adoption, the scalability and
+transferability of deep learning methods have become a focus of research. While transfer learning has
+proven successful in many cases, particularly with computer vision and homogenous data inputs, it
+can be challenging to apply to heterogeneous data.
+Motivated by the need to transfer and standardize established processes to different, non-identical
+environments and by the challenge of adapting to heterogeneous data representations, this work
+introduces the Domain Adaptation Neural Network with Cyclic Supervision (DBACS) approach.
+DBACS addresses the issue of model generalization through domain adaptation, specifically for
+heterogeneous data, and enables the transfer and scalability of deep learning-based statistical control
+methods in a general manner. Additionally, the cyclic interactions between the different parts of
+the model enable DBACS to not only adapt to the domains, but also match them. To the best of
+our knowledge, DBACS is the first deep learning approach to combine adaptation and matching
+for heterogeneous data settings. For comparison, this work also describes and analyzes subspace
+alignment and a multi-view learning method that deals with heterogeneous representations, called
+views, by mapping data into correlated latent feature spaces. Finally, the DBACS method, with its
+ability to adapt and match, is applied to a virtual metrology use case for an etching process run on
+different machine types in semiconductor manufacturing.
+Keywords deep learning · equipment matching · heterogeneous domain adaptation · multi-view learning · semiconductor
+manufacturing · virtual metrology
+1
+Introduction
+Process control and monitoring are essential elements in any automated production setting. Both have a long history
+of use, particularly in specialized and demanding manufacturing environments. In recent years, the complexity of
+these systems has made them the focus of ongoing research, particularly in the context of Industry 4.0 and due to
+∗natalie.gentner@infineon.com (Natalie Gentner)
+†[email protected] (Gian Antonio Susto)
+arXiv:2301.01038v1  [cs.LG]  3 Jan 2023
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+the increasing usage of sophisticated artificial intelligence-based solutions. While various machine learning and deep
+learning techniques have been applied successfully to a wide range of data types, the current focus is on scalability and
+the generalization of models, particularly for non-standardized environments.
+Despite the potential of machine learning-based technologies to improve automation in production, there are several
+issues that continue to limit their widespread success. These include limited data availability, small data sets, lack of
+standardization, low or inconsistent data quality, and complex, fragmented data. These challenges can make it difficult
+to transfer and generalize models, hindering progress towards higher levels of fab automation and overall digitalization.
+As a result, the focus is now on standardization and scalability, particularly for application-driven research. This is
+important due to the financial and technological investments required for method and model development, as well as the
+need for 24/7 support for critical production infrastructure and maintenance in highly automated environments.
+In non-standardized environments, there are two main approaches to supporting the scalability of methods and model
+transfer, as discussed in the semiconductor literature: (i) matching and (ii) transfer learning, with a focus on domain
+adaptation. The goal of matching is to harmonize environments and processes by using data and expert knowledge, with
+the aim of eliminating differences. Transfer learning, on the other hand, uses a purely data-driven approach to change
+the data representation (but not the data itself or any equipment or process properties) in order to bring corresponding
+data sets closer together, or in the best case, make them indistinguishable.
+However, most machine learning-based transfer learning methods, which are driven by computer vision and naturally
+homogeneous data input, are not designed to handle heterogeneous data. While this is not a common issue when
+modeling tasks use images as the main input, it becomes a significant challenge in semiconductor scenarios where an
+established process must be transferred to a different, non-identical equipment due to availability or utilization. This
+raises the question of how to match non-identical equipment and use knowledge gained from one tool to optimize the
+same process on a different, non-identical tool in order to improve output quality.
+To address the research gap related to heterogeneous domain adaptation (DA), this paper introduces an extended version
+of DBAM called DANN-based Alignment with Cyclic Supervision (DBACS). This method, which was previously
+applied to a homogeneous VM modeling task in previous studies Gentner et al. [2020, 2021], has the ability to map
+unpaired samples in their original feature spaces, enabling the functionality of matching. This capability allows DBACS
+to naturally enrich the existing method.
+This contribution methodically extends the work presented in Gentner et al. [2021] by demonstrating an extension
+suitable for heterogeneous domain adaptation (DA) and matching. The main contributions of the proposed DBACS
+method are as follows:
+• DBACS is able to handle high data complexity caused by heterogeneous systems in production and is applicable
+to various data types, such as time series data;
+• DBACS is able to tackle both supervised and unsupervised adaptation for heterogeneous input data using the
+original input feature spaces;
+• DBACS enables model scaling by allowing the use of a well-trained model for another data set with no
+assumption of the same feature representation, but only identical underlying physical information;
+• DBACS ensures interpretability and comparability of all parts of the model and allows unpaired feature
+matching on top of the adaptation in both directions.
+To evaluate the performance of DBACS in the context of heterogeneous data, the method is compared to selected
+benchmark models, including subspace alignment (SA) using principle component analysis (PCA) with and without
+correlation alignment (CORAL) and canonical correlation analysis (CCA), a method well known in the field of
+multi-view learning.
+Virtual metrology (VM), a representative of standard process control mechanisms, is chosen as a real-world application
+showcase for this study. VM, also known as a soft sensor, is a statistical model that predicts inline wafer properties based
+on process information and sensor measurements. Since its introduction to the semiconductor industry in 2005 Chen
+et al. [2005], VM has a long research history and has benefited greatly from the adoption of new modeling techniques
+driven by Industry 4.0 and the use of artificial intelligence. In addition to being useful for predictive maintenance,
+fault detection and classification, and defect classification, VM is a key mechanism for direct/early fault detection and
+enabling quality improvements by increasing monitoring capacity, control through real-time process corrections in
+combination with a Run-to-Run system, and smart capacity usage by preparing input for smart sampling strategies and
+improved decision making.
+The rest of the paper is organized into six more sections: Section 2 introduces related literature, Section 3 formalizes the
+problem and presents the main model DBACS and selected benchmarks. Section 4 gives details on virtual metrology,
+2
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+etching process, data, preprocessing and the experimental design, while in Section 5 implementation details including
+hyperparameter and architectures as well as results are reported. Finally in Section 6 DBACS suitability for matching is
+discussed. Section 7 closes with conclusive remarks and future research directions are envisioned.
+2
+Literature and Background
+In this section we summarize literature related to both relevant methodological approaches as well as application works.
+One of the main issue in adopting ML-based solutions in complex production is the need for scalability. With a large
+number of machines, products, and recipes (e.g., in semiconductor production), it is often infeasible to build ad-hoc
+analytics solutions for each scenario. In this context, scalability in learning frameworks is critical. In this context,
+scalability in learning framework is of fundamental importance; one of this is matching, which has Matching has a
+long history in non-standardized manufacturing environments and can be implemented using both classical methods
+Chouichi et al. [2020] and DL techniques Heng et al. [2021].
+With the rise of Domain Adversarial networks (DANNs) Ganin et al. [2016] and the related concept of domain
+adaptation, a theory made to deal with occurrence of different data distributions for one modeling task, there has
+been a surge in the number of publications focusing on transfer learning for semiconductor applications Kang [2017],
+Tsutsui and Matsuzawa [2019] and Chien et al. [2022a]. Domain adaptation also enables semi-supervised learning,
+as demonstrated in Farahani et al. [2020] and Li et al. [2020]. Unsupervised domain adaptation for semiconductor
+applications has also been explored using DANNs Shim and Kang [2022]. A variety of metrics and losses can be
+used to measure distribution distances in domain adaptation settings, such as maximum mean discrepancy (MMD)
+Azamfar et al. [2020]. For a broader overview of domain adaptation, see Wang and Deng [2018] for a computer
+vision survey and Courty et al. [2017] for an example using optimal transport. Generative models have also been
+widely used in production-related research. For example, Lu et al. [2019] presents a generative adversarial network
+(GAN)-inspired approach using pseudo labeling to address class imbalance in defect inspection in industrial settings.
+While the literature on homogeneous domain adaptation for semiconductor applications is growing, there is still a
+lack of research on heterogeneous domain adaptation for specific semiconductor use cases. However, literature from
+other industry sectors shows promising results for heterogeneous domain adaptation tasks, such as classification of
+heterogeneous information networks Yang et al. [2020], image-to-text transfer Fang et al. [2022], Tsai et al. [2016] and
+the combination of distribution alignment via subspace mapping with pseudo-labeling Alipour and Tahmoresnezhad
+[2022].
+Another approach for handling heterogeneous data distributions is multi-view learning (MVL) Perry et al. [2021]. A
+systematic overview of MVL can be found in Sun [2013] and in Xu et al. [2013]. There are few examples of MVL
+applied to fault detection and performance systems in manufacturing environments, such as Chen et al. [2016] and Yu
+et al. [2021], which use correlation and Canonical Correlation Analysis (CCA) Hardoon et al. [2004] for fault detection
+and performance evaluation. A review of MVL in the deep learning (DL) context is provided by Yan et al. [2021], while
+a range of CCA approaches is discussed in Chapman and Wang [2021].
+Metrology and its relationship to process control have been discussed in the literature, such as in early works such as
+Chen et al. [2005] and Su et al. [2007]. While metrology is essential for quality and control, it can be costly in terms
+of productivity, which has led to the development of numerous virtual metrology (VM) approaches in the literature.
+VM is still an active research topic, with state-of-the-art methods like isolation forest being used in a decision-based
+model framework (e.g., Chien et al. [2022b]). VM tools are often developed based on data from fault detection
+and classification (FDC) systems, which are monitoring software used to overview different types of equipment in
+semiconductor manufacturing. FDC data typically consists of descriptive statistics computed from raw, time-dependent
+physical sensor measurements installed on the equipment, making the VM problem a classic tabular data regression
+task. Given the high dimensionality of FDC data, feature selection is an important step in the context of tabular data
+VM and has been widely discussed in the literature ; see Saeys et al. [2007] for a general review of selection techniques
+and Kang et al. [2016] or Lynn et al. [2009] and Fan et al. [2020] for more sophisticated VM specific preprocessing and
+selection techniques. Other notable regression methods for VM prediction include those presented in Lynn et al. [2009],
+Susto and Beghi [2012] and Park and Kim [2016]. Chen et al. [2020] compares tree-based methods for VM modeling
+to other regression methods and neural networks.
+Another set of approaches in the VM literature aims to solve the regression task using time series data collected
+from equipment sensor data. These approaches include those presented in Park and Kim [2016] and add Susto et al.
+[2015], which introduced the Supervised Aggregative Feature Extraction framework for feature selection. DL-based
+approaches have also been successfully employed for modeling with time series input data, such as in Lee and Kim
+[2020], Maggipinto et al. [2018, 2019] and Lee and Kim [2020].
+3
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+3
+Proposed Approaches
+In this section, a general description and mathematical formalization of a modeling task with heterogeneous input are
+given, including the necessary assumptions. We also provide a formal description of the methods and algorithms used
+to solve the task under exam.
+The modeling task at hand is formulated as regression, with the goal of scaling a selected model to make it usable
+for two data sets with different distributions and heterogeneous data representations. Since the input spaces do not
+have a common subspace sufficient for the task, the modeling must be done in a domain-specific manner. To address
+scalability, the goal is to use a trained statistical model for both data sets in parallel while minimizing the prediction
+error and maximizing the accuracy of a dedicated model. To achieve this, we compare methods from the field of domain
+adaptation and multi-view learning.
+First, we mathematically formalize the regression task followed by selected methods. Let fS define a modeling task,
+let a hypothesis class H be a set of all possible modeling functions h ∈ H that are considered for a specific task. Let
+XS ⊂ RnS be defined as the first input space and Y as the output space. The output space Y is defined as Y ⊂ R in
+case of a regression task. A distribution over XS × Y is called source domain. For time series data, let XS ⊂ T × RnS
+where T describes the set of considered points in time and xt
+S ∈ RnS a sample from the source feature space taken
+at a fixed point in time t ∈ T . A learning algorithm is provided with a source data set S drawn i.i.d. from the source
+domain DS with XS × YS, XS ⊂ X, YS ⊂ Y . In the SSL setting, it is distinguished between labeled and unlabeled
+data and define S := SL ∪ SU where SU stands for the unlabeled source sample subset and SL for the labeled one.
+Without loss of generality for UDA and SSL, SU = ∅ sine the source domain is assumed to be labeled. Hence
+S = {XS, YS} = {XSL, YSL} = {xi
+S, yi
+S}nS
+i=1 ∼ {DS}ns,
+(1)
+with nS being the number of drawn samples (all labeled) and therefore XS = XSL ⊂ RnS, YS = YSL ⊂ Y .
+A learning algorithm is provided with a second set T drawn i.i.d. from the target domain DT with different data
+distribution, representation and feature space. Hence, let T = TL ∪ TU be the second called target data set T drawn
+i.i.d. from a target domain DT with a distribution over XT × YT , XT ⊂ RnT , YT ⊂ Y , and consisting of unlabeled
+TU and/or labeled TL samples.
+TL = {XT L, YT L} = {xj
+T , yj
+T }nT −l
+j=1
+∼ {DT }nt−l;
+(2)
+TU = {XT U} = {xj
+T }nT
+j=nT −l+1 ∼ {DX
+T }nt;
+(3)
+with nT being the number of drawn target samples, therefore XT L ⊂ XT ⊂ RnT , YT L ⊂ YT ⊂ Y and XT U ⊂ XT ⊂
+RnT . For time series data, let XT ⊂ T × RnT where T describes the set of considered points in time and xt
+T ∈ RnT a
+sample from the target feature space taken at a fixed point in time t ∈ T .
+3.1
+DANN-based Alignment with Cyclic Supervision (DBACS)
+In this work, we present a new framework, called DANN-based Alignment with Cyclic Supervision (DBACS). The
+DBACS approach (illustrated in Figure 1) is an extented version of DBAM Gentner et al. [2021] and is designed for
+binary heterogeneous domain adaptation using source and target domain. DBACS consists of five main parts:
+• the baseline or reference prediction model P;
+• an encoder/alignment model called aligner F used to map the target domain to the source domain (the output
+of the aligner is called aligned);
+• F is connected to a second encoder/alignment model aligner G that maps the source domain to the target
+domain. By combining both aligners, it is possible to introduce cycle-consistency by comparing source samples
+with its cycled sample and target samples with its cycled samples;
+• a domain discriminator A for classification of source domain versus aligned target domain;
+• Adversarial training in both directions is enabled by adding a second domain discriminator (called discriminator
+B) for target versus aligned source comparison.
+The various components of the DBACS architecture are discussed in the following.
+4
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Figure 1: Graphical representation of the proposed DBACS system exploiting input data from two non-identical
+domains. The arrows represent the data flows. An autoencoder shaped aligner can be used for noise reduction especially
+for homogeneous DA but is not mandatory.
+Prediction loss
+Let hP : XS → Y be a dedicated statistical model trained on a data set S, let S be a labeled source
+sample set drawn i.i.d. from a domain DS with nS = |S| being the number of drawn samples. The neural network
+representation is parameterized by θP and P(xS, θP ) where P is the model function with parameters θP that outputs
+the prediction for xS ∈ XS. The loss LP used for training and minimization is defined as:
+min
+hP ∈H LP (XS) = min
+hP ∈H LDS(hP (XS)) = min
+θP LDS(XS, θP ) = min
+θP L(x,y)∈S (P (x, θP ) , y) .
+(4)
+where L is selected based on the modeling task at hand; for VM regression task we choose mean absolute error (MAE).
+Cycle consistency loss
+Let hF : XT → XS be a statistical model function aligning target to source and F(xT , θF )
+be its parameterized representation where F is the model function with parameters θF that outputs the prediction
+for xT ∈ XT . Let hG : XS → XT be a statistical model function aligning source to target and G(xS, θG) be its
+parameterized representation where G is the model function with parameters θG that outputs the prediction for xS ∈ XS.
+Let xS ∼ DS the data distribution according to DS and xT ∼ DT according to DT . Then, the cycle-consistency loss
+is defined as:
+LcycleS(XS) := LG,F,DS(XS) = LxS∼DS (F(G (xS, θG), θF )) = LxS∼DS(F(G(xS)), xS);
+(5)
+LcycleT (XT ) := LF,G,DT (XT ) = LxT ∼DT (G(F (xT , θF ), θG)) = LxT ∼DT (G(F(xT )), xT ).
+(6)
+To give an example we follow the description in Zhu et al. [2017] where the L1 norm is used as cycle loss function:
+LcycleS(XS) = LxS∼DS(F(G(xS)), xS) = ExS∼DS [∥F(G(xS)) − xS∥1]
+(7)
+LcycleT (XT ) = LxT ∼DT (G(F(xT )), xT ) = ExT ∼DT [∥G(F(xT )) − xT ∥1]
+(8)
+In short, the cycle consistency loss is defined as
+Lcyc(F, G, XS, XT ) = LcycleS(XS) + LcycleT (XT ).
+(9)
+Here, the optimization goal is to reproduce a bijective mapping so that each xS ∈ XS is mapped to XT and back to XS
+with F(G(xS)) ≈ xS. The same goes for xT ∈ XT with G(F(xT )) ≈ xT .
+5
+
+source
+Aligner F
+Classifierf
+aligned
+target
+target
+IC
+Discriminator A
+aligned
+source
+source
+Aligner G
+DiscriminatorBHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Adversarial loss
+Let hDA : XS → I, hDB : XT → I with I = [0, 1] be two statistical model functions describing
+respectively the distance of source versus aligned target and target versus aligned source. Let hDA be parameterized by
+θDA and let DA(xS, θDA) be the parameter representation of discriminator A where DA is the model function with
+parameters θDA that outputs the prediction for xS ∈ XS. Let hDB be parameterized by θDB and let DB(xT , θDB) be
+the parameter representation of discriminator B where DB is the model function with parameters θDB that outputs the
+prediction for xT ∈ XT . Then the adversarial loss first for source LadvS and second for target LadvT is defined based
+on a selected loss function L:
+LadvS(XS, XT ) := LDA,DS(XS) − LDA,DT (XT )
+= LxS∼DX
+S (hDA (xS)) − LxT ∼DX
+T (hDA (hF (xT )))
+= LxS∼DX
+S (DA (xS, θDA)) − LxT ∼DX
+T (DA (F(xT , θF ), θDA))
+(10)
+LadvT (XS, XT ) := LDB,DT (XT ) − LDB,DS(XS)
+= LxT ∼DX
+T (hDB (xT )) − LxS∼DX
+S (hDB (hG(xS)))
+= LxT ∈DX
+T (DB (xT , θDB)) − LxS∼DX
+S (DB (G(xS, θG), θDB))
+(11)
+The adversarial loss is applied to the output of each discriminator and enables an adversarial training approach (see
+Gentner et al. [2021], Gulrajani et al. [2017]). For a regression modeling task: Let I ⊂ R or I = R and the discriminator
+a regression model with linear output activation function. Then the adversarial loss is defined as
+LadvS(XS, XT ) = ExS∼DS [DA (xS, θDA)] − ExT ∼DT [(DA(F(xT , θF ), θDA))],
+(12)
+LadvT (XS, XT ) = ExT ∼DT [DB (xT , θDB)] − ExS∼DS[(DB(G(xS, θG), θDB))],
+(13)
+where E defines the expected value and the loss is an approximation of the Wasserstein distance of two sampled
+distributions, for details see Gulrajani et al. [2017].
+Remark
+It is recommended by Zhu et al. [2017] based on Taigman et al. [2017] to add one more additional loss term
+namely the identity loss. The idea is that if almost identical samples in the other domain occur, the aligner should per-
+form close to an identity function. Since source and target are heterogeneous in our case we do not apply this kind of loss.
+The training itself happens in an adversarial setting with a two-player game approach. The adversarial training routine
+includes parallel training of both aligner and both discriminator using adversarial loss plus inclusion of the additional
+loss terms. For training, two fixed training data sets S and T (training samples are drawn i.i.d from DS and DT ) are
+used. During the aligners training phase, the adversarial loss is minimized, during discriminator training phase it is
+maximized (or its negative value minimized).:
+• The first competitor of the adversarial training is the discriminator DA trained to distinguish between source
+and aligned target data meaning optimizing the adversarial source loss. In parallel the discriminator DB is
+trained to distinguish between target and aligned source data also meaning optimizing the adversarial target
+loss. The optimization of the discriminator A and discriminator B loss LDtotal is defined as
+max
+θDA,θDB
+LDtotal(XS, XT ) = max
+θDA
+LadvS(XS, XT ) + max
+θDB
+LadvT (XS, XT )
+= max
+θDA
+LDA,DS(XS, θDA) − LDA,DT (F(XT , θF ), θDA)
++ max
+θDB
+LDB,DT (XT , θDB) − LDB,DS(G(XS, θG), θDB)
+= max
+θDA
+LxS∈S (DA (xS, θDA)) − LxT ∈T (DA (F(xT , θF ), θDA))
++ max
+θDB
+LxT ∈T (DB (xT , θDB)) − LxS∈S (DB (G(xS, θG), θDB))
+(14)
+• The second competitor in the adversarial training is the aligner cycle. We define LAtotal using the adversarial
+loss for both aligners and the cycle consistency loss. In case of labeled target data the aligner F is also updated
+in order to optimize prediction loss LP for aligned target data. The adversarial part of the aligner losses is set
+in opposite direction compared to the ones used to update the two discriminator:
+min
+θF ,θG LAtotal(XS, XT ) = min
+θF ,θG λadvSLadvS (XS, XT ) + λP LP (F(XT )) + λadvT LadvT (XS, XT )
+= min
+θF ,θG[−LxT ∈T (DA (F(xT , θF ), θDA))
++ λP L(xT ,y)∈T L(P (F(xT , θF ), θP )) − LxS∈S (DB (G(xS, θG), θDB))]
+(15)
+6
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+where λ(·) represents the weight assigned to each corresponding loss term. For λP = 0 the training happens in
+an unsupervised setting where no target labels are available. A gradient penalty regularization term is added
+when updating both aligners following the recommendations of Gulrajani et al. [2017].
+3.2
+Subspace Alignment using Principle Component Analysis
+Subspace Alignment (SA), presented by Fernando et al. [2013], linearly aligns subspaces generated by Principle
+Component Analysis (PCA).
+SA was introduced as unsupervised DA method for classification task. Overall benefits of SA lies in the simplicity and
+in the speed of the method while still presenting high accuracy. For heterogeneous domain adaptation, we slightly adapt
+here the SA approach by first applying PCA separately to source and target and then align the corresponding subspaces
+using CORrelation ALignment (CORAL). CORAL by Sun et al. [2016] is an unsupervised domain adaptation method
+that aligns second order statistics of source and target domain.
+PCA
+Principle Component Analysis (PCA) is a linear transformation of a vector space with respect to its points/vectors.
+The projection is created in a way that highest occurring variance is represented by the first latent dimension (the
+so-called first principle component), the second highest variance by the second principle component and so on. Let X
+be a vector space, let ψ : X → X′ define the nonlinear principle component transformation to be computed. Then PCA
+is formalized via
+x′ = ψ(x) = ΓT x
+(16)
+where x′ ∈ X′ describes the transformed input, Γ consists of the eigenvectors and is computed via Λ = ΓT ΣΓ where
+Λ is a diagonal matrix defined by the eigenvalues and Σ is the covariance matrix. PCA is applied to XS and XT
+accordingly resulting in S′ and T ′ as projected input sets. Since PCA is a very well-known method, we refer to Jolliffe
+[2010] for a more detailed description.
+CORAL
+Let S′ = {x′
+Si}, T ′ = {x′
+Ti} be the PCA projected input sets from the source and target domains. Let
+Υ : X′
+S → X′
+T with Υ(X′
+S) = X′
+S ∗ A describe the feature transformation of the source space to the target space. Let
+µS′, µT ′ be the feature mean of S′, T ′ and CS′, CT ′ the corresponding covariance matrices. Then, the distance between
+the covariance matrices (assuming normalized features with zero mean) is minimized by:
+min
+A
+��C ˆ
+S′ − CT ′��2
+F = min
+A
+��AT CS′A − C′
+T
+��2
+F
+where A is the matrix used in linear transformation that is applied to the source, C ˆ
+S′ describes the covariance of the
+transformed source features S′∗A and ∥·∥2
+F denoting the squared Frobenius norm selected as distance metric. It is
+called CORAL loss. In order to solve this equation, we follow Algorithm 1 in Sun et al. [2016] and compute first the
+covariance matrices followed by whitening the source and then recoloring it with the target covariance.
+3.3
+Canonical Correlation Analysis (CCA)
+Canonical Correlation Analysis (CCA) defines linear transformation for each set of variables such that after the
+transformation the projected features are maximal correlated. A summary of the descriptions is taken from Hardoon
+et al. [2004].
+Let S = {xS}, T = {xT } be two sample sets wanted to be projected into direction wS, wT . Let ΦS : XS → X′
+S,
+ΦT : XT → X′
+T define the linear transformation for each domain. Then:
+ΦS(S) = S
+′ = SxS,wS = ⟨wS, xS⟩,
+ΦT (T) = T
+′ = TxT ,wT = ⟨wT , xT ⟩.
+(17)
+Specifically, it is looked for wS, wT such that the correlation between the projected vectors is maximised, hence:
+ρ = max
+wS,wT corr (SxS,wS, TxT ,wT ) = max
+wS,wT
+⟨SxS,wS, TxT ,wT ⟩
+∥SxS,wS∥ · ∥TxT ,wT ∥.
+(18)
+The previous equation can be reformulated as
+ρ = max
+wS,wT
+w′
+sE[xSx′
+T ]wT
+�
+w′
+SE[xSx′
+S]wSw′
+T E[xT x′
+T ]wT
+(19)
+7
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+with E denoting the discrete empirical expectation, ′ denotes the transpose of a vector or a matrix and properties of the
+inner product are used. Using the covariance matrix with
+C = C(xS, xT ) = E[xSxT ] =
+�CxSxS
+CxT xS
+CxSxT
+CxT xT
+�
+(20)
+where C is a block matrix with the within-covariance CxSxS, CxT xT and between-covariance matrices CxSxT , CxT xS
+as entries. Finally the optimization problem can be formulated in the following way:
+ρ = max
+wS,wT
+w′
+sCxSxT wT
+�
+w′
+SCxSxSwSw′
+T CxT xT wT
+(21)
+By checking that rescaling of wS, wT does not change the problem, it can be maximized subject to
+w′
+SCxSxSwS = 1,
+w′
+T CxT xT wT = 1.
+(22)
+The formulation of the dual problem is used, hence computing the corresponding Lagrangian L leads to
+L(λ, wS, wT ) = w′
+sCxSxT wT − λS
+2 (w′
+SCxSxSwS − 1) − λT
+2 (w′
+T CxT xT wT − 1).
+(23)
+The partial derivatives in the direction of wS, wT are:
+∂L
+wS
+= CxSxT wT − λSCxSxSwS = 0,
+(24)
+∂L
+wT
+= CxT xSwS − λT CxT xT wT = 0.
+(25)
+Multiplying (25) with wS∗ and multiplying (24) with wT ∗ and subtracting the one from the other, define λ = λS = λT ,
+assuming CxT xT is invertible, rearrange the equation and use the partial derivative leaves to
+CxSxT C−1
+xT xT CxT xSwS = λ2CxSxSwS
+(26)
+which is equivalent to a generalised eigenproblem of the form Ax = λBx. Using Cholesky decomposition, the previous
+can be even more simplified to a symmetric eigenvalue problem Ax = λx. For visualization see Figure 2.
+Figure 2: Visualization of Canonical Correlation Analysis (CCA). The canonical components of source and target
+are a weighted combination of corresponding input features. The correlation of the canonical components within the
+red box is maximized. Similarly to PCA, the number of canonical components can be tuned.
+4
+Case Study: Dataset Description and Experimental Settings
+4.1
+Semiconductor Manufacturing: Etching process and Virtual Metrology
+Wafers are the basis for every semiconductor manufacturing process. A wafer consists of pure (99.9999%) silicon, has a
+disc shape and houses several thousand chips (the end product) on average. The specific technology structure of a chip
+is built up layer by layer on the wafer during a couple of hundred process steps. Each wafer is considered a separate
+sample in this work.
+8
+
+Source
+TargetHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Etching is a common process in semiconductor manufacturing and is frequently studied and discussed in semiconductor
+research and literature, along with chemical vapor deposition and implantation. The etching process removes material
+from a surface or transfers a structure created during the lithography step to the layer below Hilleringmann [1996],
+May and Spanos [2006]. Reactive-ion etching uses a high-frequency alternating energy field applied to the cathode on
+which the wafer is placed. Positively charged ions in the plasma are accelerated towards the wafer and collide with its
+surface at high kinetic energy, causing atoms from the wafer’s surface to be dislodged from the crystal lattice, resulting
+in partial physical etching. In addition, a partial chemical reaction occurs due to the highly reactive free radicals.
+The plasma etching process includes up to ten sub-steps during which input sensors must be adjusted to the target values
+specified in a recipe. Sensors that measure properties such as chamber pressure, applied high frequency voltage, gas
+type, gas flow, and wafer temperature, as well as electrode temperature and bias, play a crucial role in achieving the
+desired wafer properties. End point detection, or the etching time, is one of the most critical aspects of the process, as it
+is highly sensitive and closely related to other variables such as gases, pressure, current, and temperature. Incorrect
+etching times, inadequate end point detection, uncontrolled reactions, and interference in the chamber can negatively
+affect the layer thickness and overall quality and functionality of the wafer.
+Process monitoring and control are essential for reliable, standardized, and repeatable production processes that produce
+high-quality products. In this work, we focus on a process control method called virtual metrology (VM) and analyze it
+through a case study involving an etching process. In general, control quantities are typically measured in metrology
+stations or tools after the process is completed, using multiple measurements on a sample of wafers. Traditional
+metrology is a univariate or multivariate control system that uses control charts with defined upper and lower control
+limits to monitor process performance. However, due to cost and time constraints, not all wafers can be physically
+measured after the process.
+Virtual metrology (VM) or soft sensing modules utilize data collected by process equipment to model the relationship
+between wafer properties and process input and feedback sensor measurements. VM techniques allow for the inclusion
+of non-measured but predicted control measures in order to enhance analysis. VM technologies offer several benefits,
+including:
+• costs and time savings due to reduced mandatory measurements;
+• quality assurance through enhanced and comprehensive monitoring;
+• real-time control, assessment and process updates in conjunction with Run-to-Run controllers Su et al. [2007];
+• data-driven process optimization including fault detection, root cause analysis and improved sample selection
+Feng et al. [2019].
+4.2
+Data Preparation
+The data used in this work is collected from two different etching equipment types from the same vendor. The data set
+is restricted to a specific etching recipe that was transferred from one equipment to the other and now runs regularly
+on both equipment. Raw sensor measurements in form of time series data and their corresponding metrology/inline
+measurements over a period of 3 years are considered.
+• The equipment type 1 with 35 activated sensors - older equipment type hence higher number of samples (∼10
+000) and original tool to run the specific recipe- is selected as source;
+• the equipment type 2 with 55 activated sensors - newer equipment type with ∼6000 data samples - is defined
+as target.
+The following preprocessing steps were applied to the collected time series sensor data, with each equipment treated
+separately due to its heterogeneous nature:
+1. removal of constant features;
+2. removal of features that show small fluctuation that can be detect as noise (variations smaller than 0.01) and a
+constant behavior underneath the noise;
+3. removal of samples showing label outliers based on interquantile range;
+4. removal of samples where the length of the time series lies below or above 25 percent respective 75 percent
+quantile of time series length;
+5. equal-distributed upsampling of timestamps and feature values to generate time series with equal length.
+33 features for equipment type 1 respective 49 for equipment type 2 are finally selected as input features. No significant
+label shift is detected, see Figure 3.
+9
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Figure 3: Boxplot of normalized layer thickness from two equipment. Boxplot graphs of normalized metrol-
+ogy/inline measurements from both equipment types considered in the analysis.
+4.3
+Experimental Design
+Virtual metrology is modeled as prediction task with sensor data as input mapped to a single continuous metrology
+value. Due to the heterogeneous nature of the data representation from equipment type 1 and 2, no common model can
+be used without additional transfer. We present the analysis in the following order:
+1. DBACS is trained and tested as domain adaptation model using autoencoder to align the original input features
+to enable usage of a dedicated pretrained model;
+2. PCA and CCA are selected as benchmark models; for the heterogeneous VM task the alignment happens by
+creating a common (latent) feature space that is then used to train a common model. CORAL on the latent
+features is tested as final combination for both PCA and CCA.
+For training, distribution comparison and alignment evaluation the following metrics are considered:
+• Mean absolute error (MAE) is used as performance based loss; Adam optimizer is applied for training;
+• To test inner versus outer domain distance - the divergence between data selected from source and data selected
+from target domain - we use Frechet inception distance (FID);
+• 5-fold cross validation is applied, hence split both data sets into 5 subsets each and using 4 merged sets as
+train and 1 as test set per fold. Architectures of all models stay fixed for all 5 folds;
+• pearson correlation of features is tested after alignment.
+For the correlation analysis we use the function implementation available in python module numpy Harris et al. [2020]
+and for PCA and CCA we use existing function implementation in the python module scikit-learn Pedregosa et al.
+[2011]. For CORAL we use the implementations from the python module transfertools Vincent et al. [2020]. DBACS
+is trained using the described adversarial training approach. PCA and CCA expect a two dimensional input, hence
+we keep the original data and reshape the 3 dimensional sample into a two dimensional one by treating each value at
+each time step as separate sample. The selected number of latent features are based on the variation coverage of both
+domains. In the following, model details and hyperparameters choices are reported.
+DBCAS
+1DCNN is chosen since it is simple but proven to be well performing for time series data Gentner et al.
+[2021].
+• The predictor consists of 3 convolutional layers (dimension 32, 16, 8 and kernel size 53, 33 and 33), followed
+by one max pooling layer, a flattening layer and two dense layers (dimension 16 and 1, Leaky ReLU activation
+except sigmoid output).
+• The domain discriminators both have the same architecture besides the respective input shape: 3 convolutional
+layers (dimension 24, 16 and 8, kernel size 17), causal padding and leaky ReLU activation function, max
+10
+
+Normalized metrology measurementsforboth equipment types
+1.0
+0.8
+0.6
+0.4 
+0.2
+0.0
+equipment type 1 - train data
+equipment type 1 -test data
+equipment type 2 - train data
+equipment type 2 -test dataHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+pooling of size 4 and 2 times 2, followed by a flattening layer and 6 dense layers (dimension 512, 256, 128, 64,
+32, 1, Leaky ReLU activation and linear output).
+• Both aligners consist of 6 convolutional layers, first 5 followed by Leaky ReLU activation function, the final
+output is kept linear. The aligner that maps target domain to source domain has filter size 48, 42, 36, 32, 32
+and final filter size is set to number of features of the source domain; kernel size is 37, 37, 37, 37, 57 and 7.
+Upsampling with size 3 and 2 is done after 4th and 5th layer block. The aligner that maps source domain to
+target domain has filter size 32, 36, 42, 46, 48 and final filter size is set to number of features of the target
+domain; kernel size is also 37, 37, 37, 37, 57 and 7. Upsampling with size 3 and 2 is also done after 4th and
+5th layer block.
+For an improved initialization both aligners are pretrained separately using SSIM loss Wang et al. [2004]: therefore, we
+select sample pairs from source and target based on closest label value.
+PCA and CORAL
+The 3 dimensional input is reshaped into a two dimensional one by treating each value at each
+time step as separate sample. For each equipment type, we select the first 10 principal components in order to cover
+around 95% variance and to create same dimensional input space. For equipment type 1 we cover 97% of the variance
+and for equipment type 2 we cover 94%. 1DCNN prediction model with reduced number of features based on PCA
+(reshaped back to 3 dimensions) of source and target domain is used as prediction model.
+CCA
+27 canonical components (CCs) are kept since it shows the most stable results in our experiments. The original
+data is reshaped from 3 dimensional input into a two dimensional one by treating each value at each time step as separate
+sample. 1DCNN prediction model with reduced number of features based on CCA (reshaped back to 3 dimensions) of
+source and target domain is used as prediction model.
+5
+Experimental Results
+Table 1 shows the average 5 fold CV results for DBACS compared to the dedicated lower bound values meaning perfor-
+mance errors for dedicated models trained only on source and only on target data. The numbers given in Table 1 confirm
+DBACS MAE for source and aligned target
+Source domain
+Target domain
+Train
+Test
+Train
+Test
+Lower Bound
+0.084
+0.094
+0.102
+0.128
+DBACS
+0.084
+0.094
+0.102
+0.131
+Table 1: DBACS performance errors for source and aligned target. Source and aligned target data DBACS training
+and test scores average over 5 fold CV. Target data is mapped to source domain using trained aligner F from DBACS
+and evaluated after the mapping using the VM prediction model trained on source. Lower bound prediction models are
+dedicated meaning trained only on source train data and evaluated only on source test data respective trained only on
+target train data and evaluated only on target test data.
+the visual convergence seen in the t-SNE plot in Figure 4. This is supported by frechet inception distance (FID) 0.01 for
+outer domain distance after alignment compared to FID inner domain distance close to 0 for equipment type 1 as well as
+for equipment type 2. Next, Figure 5 shows true versus predicted values of different alignment states - randomly initial-
+ized aligner, after the pretraining of the aligner and after DA training with DBACS). Again, the visualization supports the
+results presented in Table 1: Enabeling usage of a dedicated source model to mapped target data for high accuracy pre-
+dictions. A visualization of both aligners output is presented in Figure 6 and compared to original domain sensor signals.
+Next, we present results for PCA analysis in Table 2. Optional DA with CORAL on top shows slightly improved results
+if model is trained on data from both domains. The FID score for outer domain distance after PCA + CORAL on the
+latent features generated by PCA is significant lower than before with 0.0001 for train and 0.001 for test. Only the first
+two principal components show a correlation higher than r = 0.5.
+For CCA, the performance is presented in Table 3. Optional DA with CORAL on top shows improved results since
+model training for both domains is enabled and can be executed using CCs. The FID score for outer domain distance
+after CCA + CORAL on the latent features generated by CCA is again significant lower than before with very close to 0
+11
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Figure 4: T-SNE visualization before and after alignment with DBACS. Graphical t-SNE representation of source
+and target domain in different stages of the alignment process: (a) shows features mapped by a randomly initialized
+aligner, (b) after the pretraining of the aligner and (c) after DA with DBACS is done. The source is colored in blue
+and contains data from equipment type 1, the target is colored red and contains data from the equipment type 2. The
+axes are dimensionless. The effect of the adaptation of the input features after DBACS is applied during training. The
+adaptation brings the distributions of the target domain closer and finally target overlaps source domain.
+Figure 5: True versus predicted scatter plot for DBACS before and after alignment. The graph shows predictions of
+aligned target data after mapped to source space by a randomly initialized aligner, after the pretraining of the aligner
+and predictions of aligned target data after DA training with DBACS is done. Only test data is presented, the test source
+data is colored in blue, the the aligned target test data is colored in red.
+for train and test. The first five CCs have a correlation higher than r = 0.5.
+6
+Equipment Matching Experiments
+Having with DBACS a methodology that allows parallel training and transfer in both directions - source to target
+but also target to source - mis- or abnormal behavior detected for aligned data can be compared to normal as well as
+abnormal data from source. These kind of comparisons enables equipment matching for nonidentical equipment with
+heterogeneous data representations.
+12
+
+20
+0
+-20
+-20
+0
+20
+4020
+10
+0
+-10
+20
+-30
+-20
+-10
+0
+10
+20
+3040
+20
+0
+-20
+20
+20sourcetestset
+alignedtargettest
+0.8
+0.6
+value
+0.4
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+true values1.0
+source test set
+aligned target test set
+0.8
+0.6
+edicted
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+true values1.0
+sourcetestset
+aligned target test set
+0.8
+E0.6
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+true valuesHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Figure 6: Aligner F and G visualizations of 2 times 3 raw sensor measurements of both equipment types before and
+after the corresponding alignment. The graph shows results for trained aligner F and mapped sensor signals from
+target to source domain in red and compares it to corresponding original source sensor signals plotted in black. It also
+shows results from trained aligner G and mapped sensor signals from source to target domain in blue and compares it to
+corresponding original target sensor signals plotted in black. A good alignment is visible as well. The x axis shows the
+timestamps of the sensor signals, y axis the sensor measurement values.
+VM prediction model performance for PCA based principle components
+Source domain
+Target domain
+Train MAE
+Test MAE
+Train MAE
+Test MAE
+PCA(source)
+0.09
+0.09
+0.32
+0.33
+PCA(target)
+0.47
+0.47
+0.12
+0.13
+PCA(both)
+0.10
+0.14
+0.09
+0.14
+PCA+CORAL(both)
+0.08
+0.09
+0.12
+0.13
+Table 2: VM prediction model performance for PCA based principle components. Results for VM prediction
+models that are trained with reduced number of latent features that are created via PCA.
+First, we compare source signals with its cycled signals on the signal shape itself as well target signals with its cycled
+target signals. Examples from both are presented in Figure 7.
+Next, we check differences within source domain of samples having a high, middle and low prediction value. This
+helps to better understand univariate feature behavior for source. The middle prediction is the preferred and targeted
+one. Figure 8 shows euclidean barycenter averages of tree example signals from source domain for low, middle and
+high label values. Sensor offsets for deviating metrology measurements are clearly visible for some of the signals.
+For final equipment matching, we compare preferred shape of signals from the source domain meaning signals
+with metrology measurements close to target 0.5 (see Figure 8 to corresponding as well as deviating signals from
+target domain. Therefore, we use the DBACS to map selected source signals into the target domain. Different
+sensors measurements and their euclidean barycenter averages of groups according to low, middle and high metrology
+measurements are shown in Figure 9 and compared to mapped sensor signals (source to target) corresponding to the
+middle meaning preferred metrology group in the source domain.
+13
+
+1.0
+1.0
+1.0
+source-domain
+target domain aligned
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+0.0
+0
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+time
+time
+time1.0
+1.0
+1.0
+targetdomain
+source domain aligned
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+time
+time
+timeHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+VM prediction model performance for CCA based canonical components
+Source domain
+Target domain
+Train MAE
+Test MAE
+Train MAE
+Test MAE
+CCA(source)
+0.12
+0.13
+0.29
+0.29
+CCA(target)
+0.29
+0.29
+0.10
+0.14
+CCA(both)
+0.10
+0.13
+0.12
+0.14
+CCA+CORAL(both)
+0.07
+0.08
+0.13
+0.13
+Table 3: VM prediction model performance for CCA based canonical components. Results for VM prediction
+models that are trained with latent features created via CCA.
+Figure 7: Aligner F and G visualizations of 2 times 3 cycled raw sensor measurements of both equipment types
+in its original form as well as after its bijective mapping. The first graph shows results for source signals and cycled
+source signals from source to target to source domain. The cycled signals are plotted in red and compared to its original
+source sensor signals plotted in black. The second graph shows results for target signals and cycled target signals from
+target to source back to target domain. The cycled signals are plotted in blue and compared to its original target sensor
+signals plotted in black. The x axis shows the timestamps of the sensor signals, y axis the sensor measurement values.
+7
+Conclusion and Future Work
+The paper presents DBACS, a Deep Learning approach that is able to deal with heterogeneous domain adaptation while
+allowing comparison of aligned signals for a VM use case in semiconductor manufacturing. Linear transformation
+methods from subspace alignment and multi-view learning are selected as benchmarks and show comparable results
+when training with data from both domains is possible. Especially for classification tasks, the correlation within
+CCA can be further exploited for cross-modal or mate-based retrieval. A big advantage of DBACS is the presented
+combination of domain adaptation with matching, two of the main approaches for standardization and scalability in the
+semiconductor field.
+Envisioned future work could go in the direction of root cause analysis based on the matching results. Another important
+step could to enrich the data with more equipment for multi-source or multi-target alignment. Other applications from
+semiconductor manufacturing like predictive maintenance and defect classification could be involved and tested for
+example against computer vision inspired state-of-the-art transfer learning benchmark models like pseudo-labeling.
+Since only offline model training is executed (training time is not a critical aspect of VM here), online model training
+could also be explored in that context.
+14
+
+1.0
+1.0
+1.0
+source_domain
+source domain cycled
+0.8
+0.8
+0.8
+value
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+time
+time
+time1.0
+1.0
+1.0
+target domain
+target domain cycled
+0.8
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+400
+600
+800
+1000
+0.0
+200
+400
+600
+800
+0.0
+200
+0
+1000
+0
+200
+400
+600
+800
+1000
+time
+time
+timeHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Figure 8: Comparison of raw source sensor measurements via barycenter average grouped into low, middle
+high label values. Graphical representation of euclidean barycenter averages for 3 example sensors of the source
+domain. The x axis shows the timestamps of the sensor signals, y axis the sensor measurement values. Example sensor
+measurements of samples corresponding to low label values - meaning values smaller 0.1 - are plotted in green, example
+sensor measurements of samples corresponding to middle therefore preferred label values - meaning values around 0.5 -
+are plotted in grey and example sensor measurements of samples corresponding to high label values - meaning values
+higher than 0.9 - are plotted in orange. Sensor offsets for deviating metrology measurements are clearly visible.
+Acknowledgment
+Infineon Technologies AG is gratefully acknowledged for the financial support of this research. The Italian Government
+PNRR iniatiatives ’Partenariato 11: Made in Italy circolare e sostenibile’ and ’Ecosistema dell’Innovazione - iNest’ are
+also gratefully acknowledged for partially financing this research activity.
+References
+N. Alipour and J. Tahmoresnezhad. Heterogeneous domain adaptation with statistical distribution alignment and
+progressive pseudo label selection. Appl. Intell., 52:8038–8055, 2022. doi:https://doi.org/10.1007/s10489-021-02756-
+x.
+M. Azamfar, X. Li, and J. Lee.
+Deep learning-based domain adaptation method for fault diagnosis in
+semiconductor manufacturing.
+IEEE Transactions on Semiconductor Manufacturing, 33(3):445–453, 2020.
+doi:https://doi.org/10.1109/TSM.2020.2995548.
+J. Chapman and H.-T. Wang.
+Cca-zoo: A collection of regularized, deep learning based, kernel, and proba-
+bilistic cca methods in a scikit-learn style framework.
+Journal of Open Source Software, 6(68):3823, 2021.
+doi:https://doi.org/10.21105/joss.03823.
+C.-H. Chen,
+W.-D. Zhao,
+T. Pang,
+and Y.-Z. Lin.
+Virtual metrology of semiconductor pvd pro-
+cess based on combination of tree-based ensemble model.
+ISA Transactions, 103:192 – 202, 2020.
+doi:https://doi.org/10.1016/j.isatra.2020.03.031.
+P. Chen, S. Wu, J. Lin, F. Ko, H. Lo, J. Wang, C. Yu, and M. Liang. Virtual metrology: a solution for wafer to wafer
+advanced process control. In ISSM 2005, IEEE International Symposium on Semiconductor Manufacturing, 2005.,
+pages 155–157, 2005. doi:https://doi.org/10.1109/ISSM.2005.1513322.
+Z. Chen, K. Zhang, S. X. Ding, Y. A. Shardt, and Z. Hu.
+Improved canonical correlation analysis-
+based fault detection methods for industrial processes.
+Journal of Process Control, 41:26–34, 2016.
+doi:https://doi.org/10.1016/j.jprocont.2016.02.006.
+C.-F. Chien, W.-T. Hung, and E. T.-Y. Liao. Redefining monitoring rules for intelligent fault detection and classification
+via cnn transfer learning for smart manufacturing. IEEE Transactions on Semiconductor Manufacturing, 35(2):
+158–165, 2022a. doi:https://doi.org/10.1109/TSM.2022.3164904.
+C.-F. Chien, W.-T. Hung, C.-W. Pan, and T. H. Van Nguyen. Decision-based virtual metrology for advanced process
+control to empower smart production and an empirical study for semiconductor manufacturing. Computers &
+Industrial Engineering, page 108245, 2022b.
+A. Chouichi,
+J. Blue,
+C. Yugma,
+and F. Pasqualini.
+Chamber-to-chamber discrepancy detection in
+semiconductor manufacturing.
+IEEE Transactions on Semiconductor Manufacturing, 33(1):86–95, 2020.
+doi:https://doi.org/10.1109/TSM.2020.2965288.
+15
+
+1.0
+1.0
+1.0
+low
+middle
+high
+0.8
+0.8 
+0.8
+I sensor value
+value
+value
+0.6
+0.6
+sensor
+0.6
+0.4
+0.2
+0.2 
+0.2 
+0.0 +
+400
+0.0 +
+400
+600
+800
+0.0 +
+0
+200
+600
+800
+1000
+200
+1000
+0
+200
+400
+600
+800
+1000
+time
+time
+timeHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Figure 9: Visualization of equipment matching: mapped source sensors vs. target sensors grouped into low,
+middle high label values. Graphical representation of 6 sensor signals. The plots show euclidean barycenter averages
+of signals. Groups are defined by their metrology values and plotted in different colors. The x axis shows the
+timestamps of the sensor signals, y axis the sensor measurement values. Selected target sensor measurements of
+samples corresponding to low label values - meaning values smaller 0.1 - are plotted in green, example target sensor
+measurements of samples corresponding to middle therefore preferred label values - meaning values around 0.5 - are
+plotted in black and example sensor measurements of samples corresponding to high label values - meaning values
+higher than 0.9 - are plotted in orange. Mapped sensor signals corresponding to source samples with middle label
+values are colored black. Sensor offsets for deviating metrology measurements are clearly visible from middle target as
+well as mapped middle source signals.
+N. Courty,
+R. Flamary,
+D. Tuia,
+and A. Rakotomamonjy.
+Optimal transport for domain adapta-
+tion.
+IEEE Transactions on Pattern Analysis and Machine Intelligence,
+39(9):1853–1865,
+2017.
+doi:https://doi.org/10.1109/TPAMI.2016.2615921.
+S.-K. S. Fan, C.-Y. Hsu, D.-M. Tsai, F. He, and C.-C. Cheng. Data-driven approach for fault detection and diagnostic in
+semiconductor manufacturing. IEEE Transactions on Automation Science and Engineering, 17(4):1925–1936, 2020.
+doi:https://doi.org/10.1109/TASE.2020.2983061.
+Z. Fang, J. Lu, F. Liu, and G. Zhang.
+Semi-supervised heterogeneous domain adaptation:
+Theory and
+algorithms.
+IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(1):1087–1105, 2022.
+doi:https://doi.org/10.1109/TPAMI.2022.3146234.
+H. S. Farahani, A. Fatehi, A. Nadali, and M. A. Shoorehdeli. A novel method for designing transferable soft sensors
+and its application. arXiv, 2020. doi:https://doi.org/10.48550/arxiv.2008.02186.
+J. Feng, X. Jia, F. Zhu, J. Moyne, J. Iskandar, and J. Lee. An online virtual metrology model with sample selection for the
+tracking of dynamic manufacturing processes with slow drift. IEEE Transactions on Semiconductor Manufacturing,
+32(4):574–582, 2019. doi:https://doi.org/10.1109/TSM.2019.2942768.
+B. Fernando, A. Habrard, M. Sebban, and T. Tuytelaars.
+Unsupervised visual domain adaptation using sub-
+space alignment.
+In 2013 IEEE International Conference on Computer Vision, pages 2960–2967, 2013.
+doi:https://doi.org/10.1109/ICCV.2013.368.
+Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempit-
+sky.
+Domain-adversarial training of neural networks.
+J. Mach. Learn. Res., 17(1):2096–2030, Jan. 2016.
+doi:https://doi.org/10.5555/2946645.2946704.
+N. Gentner, M. Carletti, A. Kyek, G. A. Susto, and Y. Yang. Enhancing scalability of virtual metrology: A deep
+learning-based approach for domain adaptation. In Proceedings of the 2020 Winter Simulation Conference, pages
+1898–1909, 2020. doi:https://doi.org/10.1109/WSC48552.2020.9383945.
+16
+
+1.0
+1.0
+1.0
+low
+mapped middle
+middle
+0.8
+0.8
+0.8
+high
+0.6
+0.6 
+0.6 
+0.4
+0.4 
+0.4 
+0.2 
+0.2
+0.2 
+0.0 +
+0.0
+0.0 +
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+time
+time
+time
+1.0
+1.0
+1.0
+0.8
+0.8 
+0.8
+value
+physical sensor value
+physical sensor 
+0.6 
+0.6
+ 0.6 
+0.4
+0.4
+0.4 
+0.2 
+0.2 
+0.2 
+0.0
+0
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+U
+200
+400
+600
+800
+1000
+time
+time
+timeHeterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+N. Gentner, M. Carletti, A. Kyek, G. A. Susto, and Y. Yang.
+Dbam: Making virtual metrology/soft sens-
+ing with time series data scalable through deep learning.
+Control Engineering Practice, 116:104914, 2021.
+doi:https://doi.org/10.1016/j.conengprac.2021.104914.
+I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, and A. Courville. Improved training of wasserstein gans. In
+Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page
+5769–5779, Red Hook, NY, USA, 2017. Curran Associates Inc. doi:https://doi.org/10.48550/arxiv.1704.00028.
+D. R. Hardoon, S. Szedmak, and J. Shawe-Taylor. Canonical correlation analysis: An overview with application to
+learning methods. Neural Computation, 16(12):2639–2664, 2004. doi:https://doi.org/10.1162/0899766042321814.
+C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg,
+N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe,
+P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant.
+Array programming with NumPy. Nature, 585(7825):357–362, Sept. 2020. doi:https://doi.org/10.1038/s41586-020-
+2649-2.
+H. Heng, T. Liao, S. Didari, and H. Rajagopal. Chamber matching with neural networks in semiconductor equipment
+tools. Applied Materials, Inc., US Patent 11,133,204, 2021.
+U. Hilleringmann. Silizium-Halbleitertechnologie. Springer, 1996.
+I. Jolliffe. Principal Component Analysis. Springer Series in Statistics. Springer New York, 2010. ISBN 9781441929990.
+S. Kang.
+On effectiveness of transfer learning approach for neural network-based virtual metrol-
+ogy
+modeling.
+IEEE
+Transactions
+on
+Semiconductor
+Manufacturing,
+31(1):149–155,
+2017.
+doi:https://doi.org/10.1109/TSM.2017.2787550.
+S. Kang, D. Kim, and S. Cho.
+Efficient feature selection-based on random forward search for vir-
+tual metrology modeling.
+IEEE Transactions on Semiconductor Manufacturing, 29(4):391–398, 2016.
+doi:https://doi.org/10.1109/TSM.2016.2594033.
+K. B. Lee and C. O. Kim.
+Recurrent feature-incorporated convolutional neural network for virtual metrology
+of the chemical mechanical planarization process.
+Journal of Intelligent Manufacturing, 31(1):73–86, 2020.
+doi:https://doi.org/10.1007/s10845-018-1437-4.
+B. Li, Y. Wang, S. Zhang, D. Li, T. Darrell, K. Keutzer, and H. Zhao. Learning invariant representations and risks for
+semi-supervised domain adaptation. arXiv, 2020. doi:https://doi.org/10.48550/arxiv.2010.04647.
+Y.-W. Lu, K.-L. Liu, and C.-Y. Hsu. Conditional generative adversarial network for defect classification with class
+imbalance. In 2019 IEEE International Conference on Smart Manufacturing, Industrial & Logistics Engineering
+(SMILE), pages 146–149, 2019. doi:https://doi.org/10.1109/SMILE45626.2019.8965320.
+S. Lynn, J. Ringwood, E. Ragnoli, S. McLoone, and N. MacGearailty. Virtual metrology for plasma etch using tool
+variables. In 2009 IEEE/SEMI Advanced Semiconductor Manufacturing Conference, pages 143–148. IEEE, 2009.
+doi:https://doi.org/10.1109/ASMC.2009.5155972.
+M. Maggipinto, C. Masiero, A. Beghi, and G. A. Susto. A convolutional autoencoder approach for feature extraction in
+virtual metrology. Procedia Manufacturing, 17:126–133, 2018. doi:https://doi.org/10.1016/j.promfg.2018.10.023.
+28th International Conference on Flexible Automation and Intelligent Manufacturing (FAIM2018), June 11-14, 2018,
+Columbus, OH, USAGlobal Integration of Intelligent Manufacturing and Smart Industry for Good of Humanity.
+M. Maggipinto, A. Beghi, S. McLoone, and G. A. Susto.
+Deepvm: A deep learning-based approach with au-
+tomatic feature extraction for 2d input data virtual metrology.
+Journal of Process Control, 84:24–34, 2019.
+doi:https://doi.org/10.1016/j.jprocont.2019.08.006.
+G. May and C. Spanos. Fundamentals of Semiconductor Manufacturing and Process Control. IEEE Press. John Wiley
+& Sons, 2006.
+C. Park and S. B. Kim. Virtual metrology modeling of time-dependent spectroscopic signals by a fused lasso algorithm.
+Journal of Process Control, 42:51–58, 2016. doi:https://doi.org/10.1016/j.jprocont.2016.04.002.
+F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer,
+R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duches-
+nay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.
+doi:https://doi.org/10.5555/1953048.2078195.
+R. Perry, G. Mischler, R. Guo, T. Lee, A. Chang, A. Koul, C. Franz, H. Richard, I. Carmichael, P. Ablin, A. Gramfort,
+and J. T. Vogelstein. mvlearn: Multiview machine learning in python. Journal of Machine Learning Research, 22
+(109):1–7, 2021. doi:https://doi.org/10.5555/3546258.3546367.
+17
+
+Heterogeneous Domain Adaptation and Equipment Matching: DANN-based Alignment with Cyclic Supervision
+(DBACS)
+A PREPRINT
+Y. Saeys, I. Inza, and P. Larrañaga. A review of feature selection techniques in bioinformatics. Bioinformatics (Oxford,
+England), 23(19):2507–2517, 08 2007. doi:https://doi.org/10.1093/bioinformatics/btm344.
+J. Shim and S. Kang. Domain-adaptive active learning for cost-effective virtual metrology modeling. Computers in
+Industry, 135, 2022. doi:https://doi.org/10.1016/j.compind.2021.103572.
+A.-J. Su, J.-C. Jeng, H.-P. Huang, C.-C. Yu, S.-Y. Hung, and C.-K. Chao. Control relevant issues in semiconduc-
+tor manufacturing: Overview with some new results. Control Engineering Practice, 15(10):1268–1279, 2007.
+doi:https://doi.org/10.1016/j.conengprac.2006.11.003. Special Issue - International Symposium on Advanced Control
+of Chemical Processes (ADCHEM).
+B. Sun, J. Feng, and K. Saenko. Return of frustratingly easy domain adaptation. In Proceedings of the AAAI Conference
+on Artificial Intelligence, volume 30, 2016. doi:https://doi.org/10.5555/3016100.3016186.
+S. Sun.
+A survey of multi-view machine learning.
+Neural Computing and Applications, 23, 12 2013.
+doi:https://doi.org/10.1007/s00521-013-1362-6.
+G.
+A.
+Susto
+and
+A.
+Beghi.
+Least
+angle
+regression
+for
+semiconductor
+manufacturing
+modeling.
+In
+2012
+IEEE
+International
+Conference
+on
+Control
+Applications,
+pages
+658–663.
+IEEE,
+2012.
+doi:https://doi.org/10.1109/CCA.2012.6402409.
+G. A. Susto, A. Schirru, S. Pampuri, S. McLoone, and A. Beghi.
+Machine learning for predictive mainte-
+nance: A multiple classifier approach.
+IEEE Transactions on Industrial Informatics, 11(3):812–820, 2015.
+doi:https://doi.org/10.1109/TII.2014.2349359.
+Y. Taigman, A. Polyak, and L. Wolf. Unsupervised cross-domain image generation. ArXiv, abs/1611.02200, 2017.
+doi:https://doi.org/10.48550/arxiv.1611.02200.
+Y.-H. H. Tsai, Y.-R. Yeh, and Y.-C. F. Wang. Heterogeneous domain adaptation with label and structure consistency. In
+2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 2842–2846, 2016.
+doi:https://doi.org/10.1007/s11042-020-08731-x.
+T. Tsutsui and T. Matsuzawa.
+Virtual metrology model robustness against chamber condition varia-
+tion using deep learning.
+IEEE Transactions on Semiconductor Manufacturing, 32(4):428–433, 2019.
+doi:https://doi.org/10.1109/ISSM.2018.8651170.
+V. Vincent, M. Wannes, and D. Jesse. Transfer learning for anomaly detection through localized and unsupervised
+instance selection. Proceedings of the AAAI Conference on Artificial Intelligence, 34(04):6054–6061, Apr. 2020.
+doi:https://doi.org/10.1609/aaai.v34i04.6068.
+M. Wang and W. Deng.
+Deep visual domain adaptation: A survey.
+Neurocomputing, 312:135–153, 2018.
+doi:https://doi.org/10.1016/j.neucom.2018.05.083.
+Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli. Image quality assessment: from error visibility to structural similarity.
+IEEE Transactions on Image Processing, 13(4):600–612, 2004. doi:https://doi.org/10.1109/TIP.2003.819861.
+C.
+Xu,
+D.
+Tao,
+and
+C.
+Xu.
+A
+survey
+on
+multi-view
+learning.
+arXiv,
+2013.
+doi:https://doi.org/10.48550/ARXIV.1304.5634.
+X. Yan, S. Hu, Y. Mao, Y. Ye, and H. Yu. Deep multi-view learning methods: A review. Neurocomputing, 448:106–129,
+2021. doi:https://doi.org/10.1016/j.neucom.2021.03.090.
+S. Yang, G. Song, Y. Jin, and L. Du. Domain adaptive classification on heterogeneous information networks. In
+C. Bessiere, editor, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence,
+IJCAI-20, pages 1410–1416. International Joint Conferences on Artificial Intelligence Organization, 7 2020.
+doi:https://doi.org/10.5555/3491440.3491636.
+Q. Yu, L. Li, H. Zhao, Y. Liu, and K.-Y. Lin. Evaluation system and correlation analysis for determining the
+performance of a semiconductor manufacturing system. Complex System Modeling and Simulation, 1(3):218–231,
+2021. doi:https://doi.org/10.23919/CSMS.2021.0015.
+J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros. Unpaired image-to-image translation using cycle-consistent adver-
+sarial networks. In 2017 IEEE International Conference on Computer Vision (ICCV), pages 2242–2251, 2017.
+doi:10.1109/ICCV.2017.244.
+18
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+arXiv:2301.00331v1  [math.AG]  1 Jan 2023
+Optimality of Curtiss Bound on
+Poincare Multiplier for
+Positive Univariate Polynomials
+Hoon Hong and Brittany Riggs
+December 31 2022
+Abstract
+Let f be a monic univariate polynomial with non-zero constant term. We say that f is positive if
+f(x) is positive over all x ≥ 0. If all the coefficients of f are non-negative, then f is trivially positive.
+In 1888, Poincar´e proved thatf is positive if and only if there exists a monic polynomial g such that all
+the coefficients of gf are non-negative. Such polynomial g is called a Poincar´e multiplier for the positive
+polynomial f. Of course one hopes to find a multiplier with smallest degree. This naturally raised a
+challenge: find an upper bound on the smallest degree of multipliers. In 1918, Curtiss provided such a
+bound. Curtiss also showed that the bound is optimal (smallest) when degree of f is 1 or 2. It is easy to
+show that the bound is not optimal when degree of f is higher. The Curtiss bound is a simple expression
+that depends only on the angle (argument) of non-real roots of f. In this paper, we show that the Curtiss
+bound is optimal among all the bounds that depends only on the angles.
+1
+Introduction
+Let f be a monic univariate polynomial with non-zero constant term. We say that f is positive if f(x) is
+positive over all x ≥ 0. Consider the following small (toy) examples,
+• f1 = x4 + x3 + 10x2 + 2x + 10
+• f2 = x4 − x3 − 10x2 − 2x + 10
+• f3 = x4 − x3 + 10x2 − 2x + 10
+Note that all the coefficients of f1 are non-negative. Thus it is trivial to see that f1 is positive. However f2
+and f3 have non-negative coefficients, and thus it is not obvious whether they are positive or not. It turns
+out that f2 is not positive and f3 is positive.
+In 1888, Poincar´e [5] proved a general result that implies the following specific claim: f is positive if
+and only if there exists a monic polynomial g such that all the coefficients of gf are non-negative. Such
+polynomial g is called a Poincar´e multiplier for the positive polynomial f. For the above examples, we have
+• Since f2 is not positive, there is no a Poincar´e multiplier for f2.
+• Since f3 is positive, there is a Poincar´e multiplier for f3. For instance, let g = x + 1. Then
+gf3 = (x + 1)
+�
+x4 − x3 + 10x2 − 2x + 10
+�
+= x5 + 9x3 + 8x2 + 8x + 10
+Note that all the coefficients of gf3 are non-negative.
+1
+
+Note that there are (infinitely) many Poincar´e multipliers for f3. For instance, let g = x2 + x + 1. Then
+gf3 =
+�
+x2 + x + 1
+� �
+x4 − x3 + 10x2 − 2x + 10
+�
+= x6 + 10x4 + 7x3 + 18x2 + 8x + 10
+Note that all the coefficients of gf3 are again non-negative. Of course one hopes to find a Poincar´e multiplier
+with optimal (smallest) degree. It turns out that the smallest degree for Poincar´e multipliers for f3 is 1.
+This naturally raised a challenge: find an upper bound on the smallest degree of multipliers.
+In 1918, Curtiss [3] provided such a bound (See Theorem 1). Curtiss also showed that the bound is
+optimal when degree of f
+is 1 or 2. It is easy to show that the bound is not optimal when degree of f is
+higher. For example, the Curtiss bound for f3 is 2, which is bigger than the optimal degree which is 1.
+It seems that Curtiss bound was forgotten for almost a century. In 2010 Avenda˜no [1] (see Lemma 2.1),
+evidently not informed of the Curtiss bound, derived another bound implicitly. Its derivation is very elegant
+and short, but it turns out that the implied bound is roughly twice the Curtiss bound.
+Boththe Curtiss bound and Avenda˜no bound depend only on the angles (arguments) of non-real roots of
+f. The main contribution of this paper is to show that the Curtiss bound is optimal among all the bounds
+that depends only on the angles of the non-real roots (see Theorem 2).
+2
+Main Result
+Let f ∈ R [x] be monic such that ∀
+x≥0f (x) > 0, that is, f does not have any non-negative real root.
+Definition 1 (Optimal Bound) The optimal degree for f, written as opt (f), is defined by
+opt (f) =
+min
+g∈R[x]\0
+coeff(gf)≥0
+deg(g)
+Theorem 1 (Curtiss Bound 1918[3]) Let r1e±iθ1, . . . , rme±iθm be the non-real roots of f where multiple
+roots are repeated. Let
+b (f) =
+m
+�
+i=1
+�� π
+θi
+�
+− 2
+�
+Then opt (f) ≤ b (f) and the equality holds if deg f ≤ 2.
+Theorem 2 (Main Result: Angle-Based Optimality of Curtiss Bound) We have
+∀
+θ1,...,θm∈(0,π)
+∃
+p1,...,pt>0
+∃
+r1,...,rm>0 opt (f) = b (f)
+3
+Proof of Curtiss Bound (Theorem 1)
+In this section, we will provide an alternative proof for Curtiss’ theorem (Theorem 1). This alternative proof
+will be based on a new proof strategy that will be found crucial for proving our main result (Theorem 2).
+Let f ∈ R [x] be monic without losing generality. Assume that f does not have any non-negative real roots.
+The problem is to find non-zero g ∈ R [x] such that coeffs (gf) ≥ 0. We will reduce the problem to that of
+linear algebra. Let
+f = anxn + · · · + a0x0
+g = bsxs + · · · + b0x0
+where an = 1 and bs = 1. We first rewrite them using vectors. Let
+a =
+�
+a0
+· · ·
+an
+�
+b =
+�
+b0
+· · ·
+bs
+�
+2
+
+and let
+xk =
+
+
+x0
+...
+xk
+
+
+Then we can write f and g compactly as
+f = axn
+g = bxs
+Let
+As =
+
+
+a0
+· · ·
+· · ·
+an
+...
+...
+a0
+· · ·
+· · ·
+an
+
+ ∈ R(s+1)×(s+n+1)
+Lemma 3 coeffs (gf) = bAs
+Proof: Note
+gf = (bxs) (axn)
+= b (xsaxn)
+= b
+
+
+x0
+...
+xs
+
+
+�
+a0
+· · ·
+an
+�
+
+
+x0
+...
+xn
+
+
+= b
+
+
+a0
+· · ·
+· · ·
+an
+...
+...
+a0
+· · ·
+· · ·
+an
+
+
+
+
+x0
+...
+xs+n
+
+
+= bAsxs+n
+Hence coeffs (gf) = bAs. □
+We partition As into two submatrices as As = [Ls|Rs] where
+Ls =
+
+
+a0
+· · ·
+an−1
+...
+...
+a0
+
+
+∈ R(s+1)×n
+and Rs =
+
+
+an
+...
+...
+...
+...
+a0
+...
+...
+...
+a0
+· · ·
+· · ·
+an
+
+
+∈ R(s+1)×(s+1)
+Let
+c = bRs
+∈ R1×(s+1)
+Ts = R−1
+s Ls
+∈ R(s+1)×n
+Lemma 4 coeffs (gf) = c [Ts|I]
+3
+
+Proof: Note
+bAs = b
+�
+RsR−1
+s
+�
+As
+= (bRs)
+�
+R−1
+s As
+�
+= (bRs)
+�
+R−1
+s
+[Ls|Rs]
+�
+= (bRs)
+�
+R−1
+s Ls|R−1
+s Rs
+�
+= (bRs)
+�
+R−1
+s Ls|I
+�
+= c [Ts|I]
+where c = bRs
+and Ts = R−1
+s Ls
+□
+Lemma 5 We have
+∃
+g̸=0, deg(g)≤s coeffs (gf) ≥ 0
+⇐⇒
+ConvexHull (Ts) ∩ Rn
+≥0 ̸= ∅
+where Ts is a viewed as a set of row vectors.
+Proof: Note
+∃
+g̸=0, deg(g)≤s coeffs (gf) ≥ 0
+⇐⇒
+∃
+c̸=0 c [Ts|I] ≥ 0
+(from Lemma 4)
+⇐⇒
+∃
+c̸=0 cTs ≥ 0 and c ≥ 0
+⇐⇒
+∃
+c≥0, c̸=0 cTs ≥ 0
+⇐⇒
+∃
+c≥0, c0+···+cs=1 cTs ≥ 0
+⇐⇒
+ConvexHull (Ts) ∩ Rn
+≥0 ̸= ∅
+□
+Let
+f = an
+n
+�
+i=1
+(x − αi)
+Lemma 6 The entries of Ts are given by
+Tskℓ = −|N|
+|D|
+where
+D =
+
+
+α0
+1
+· · ·
+α0
+n
+...
+...
+αn−1
+1
+· · ·
+αn−1
+n
+
+
+and N is obtained from D by replacing the ℓ-th row with
+�
+αk+n
+1
+· · ·
+αk+n
+n
+�
+.
+Proof: Note
+xsf = Asxs+n
+4
+
+= RsR−1
+s Asxs+n
+= RsR−1
+s
+[Ls|Rs] xs+n
+= Rs
+�
+R−1
+s Ls|R−1
+s Rs
+�
+xs+n
+= Rs [Ts|I] xs+n
+= Rs
+
+
+
+Ts
+
+
+x0
+...
+xn−1
+
+ +
+
+
+xn
+...
+xs+n
+
+
+
+
+
+
+By evaluating the above on each root, we have
+
+
+α0
+i
+...
+αs
+i
+
+ f (αi) = Rs
+
+
+
+Ts
+
+
+α0
+i
+...
+αn−1
+i
+
+ +
+
+
+αn
+i
+...
+αs+n
+i
+
+
+
+
+
+
+Since f (αi) = 0, we have
+0 = Rs
+
+
+
+Ts
+
+
+α0
+i
+...
+αn−1
+i
+
+ +
+
+
+αn
+i
+...
+αs+n
+i
+
+
+
+
+
+
+Since Rs is an invertible matrix, we have
+0 = Ts
+
+
+α0
+i
+α1
+i
+...
+αn−1
+i
+
+
++
+
+
+αn
+i
+...
+αs+n
+i
+
+
+Rearranging,
+Ts
+
+
+α0
+i
+α1
+i
+...
+αn−1
+i
+
+
+= −
+
+
+αn
+i
+...
+αn+s
+i
+
+
+Combining the above equations for all the roots, we have
+Ts
+
+
+α0
+1
+· · ·
+α0
+n
+...
+...
+αn−1
+1
+· · ·
+αn−1
+n
+
+ = −
+
+
+αn
+1
+· · ·
+αn
+n
+...
+αs+n
+1
+· · ·
+αs+n
+n
+
+
+By applying Cramer’s rule, we have
+Tskℓ = −|N|
+|D|
+where
+D =
+
+
+α0
+1
+· · ·
+α0
+n
+...
+...
+αn−1
+1
+· · ·
+αn−1
+n
+
+
+and N is obtained from D by replacing the ℓ-th row with
+�
+αk+n
+1
+· · ·
+αk+n
+n
+�
+. □
+5
+
+Remark 7 Note that Tskℓ does not depend on s. Thus we will often write it as Tkℓ.
+Lemma 8 Let f ∈ R[x] be such that deg(f) = 2 without real roots. Let the roots be α1 = reiθ and α2 = re−iθ.
+Then we have
+Tk0
+=
++rk+2 sin (k + 1) θ
+sin θ
+=
+r2 Im(αk+1
+i
+)
+Im(αi)
+Tk1
+=
+−rk+1 sin(k + 2)θ
+sin θ
+=
+− Im(αk+2
+i
+)
+Im(αi)
+Proof: From Lemma 6 we have
+Tk0
+= −
+�����
+αk+2
+1
+αk+2
+2
+α1
+1
+α1
+2
+�����
+�����
+α0
+1
+α0
+2
+α1
+1
+α1
+2
+�����
+= −αk+2
+1
+α2 − α1αk+2
+2
+α2 − α1
+= −rk+2 +2i sin(k + 1) θ
+−2i sinθ
+= +rk+2 sin (k + 1) θ
+sin θ
+= r2 Im(αk+1
+i
+)
+Im(αi)
+Tk1
+= −
+�������
+α0
+1
+α0
+2
+αk+2
+1
+αk+2
+2
+�������
+�������
+α0
+1
+α0
+2
+α1
+1
+α1
+2
+�������
+= −αk+2
+2
+− αk+2
+1
+α2 − α1
+= −rk+1 −2i sin(k + 2)θ
+−2i sinθ
+= −rk+1 sin(k + 2)θ
+sin θ
+= − Im(αk+2
+i
+)
+Im(αi)
+□
+Lemma 9 Let f ∈ R[x] be such that deg(f) = 2 without real roots. Let α1 = reiθ and α2 = re−iθ be the
+roots of f.
+s =
+�π
+θ
+�
+− 2.
+Then
+∃
+g̸=0, deg(g)≤s coe��s (gf) ≥ 0.
+Proof: Note
+∃
+g̸=0, deg(g)≤s coeffs (gf) ≥ 0
+⇐⇒
+ConvexHull (Ts) ∩ Rn
+≥0 ̸= ∅
+(from Lemma 5)
+⇐=
+Ts0, Ts1 ≥ 0
+⇐⇒
+sin (s + 1) θ ≥ 0 ∧ sin (s + 2) θ ≤ 0 (from Lemma 8)
+⇐=
+0 < (s + 1)θ ≤ π ∧ π ≤ (s + 2)θ < 2π
+⇐⇒
+s ≤ π
+θ − 1 ∧ s ≥ π
+θ − 2
+⇐⇒
+π
+θ − 2 ≤ s ≤ π
+θ − 1
+⇐=
+s =
+�π
+θ
+�
+− 2
+□
+Proof: [Proof of Theorem 1]
+6
+
+1. Let f be quadratic with non-real roots re±iθ. Note that bc(f) is optimal for any f with π
+2 ≤ θ < π
+since bc(f) = 0. Let 0 < θ < π
+2 . Let g be such that g ̸= 0 and deg(g) = v < s, where s = bc(f). Thus
+k
+≤ v
+=⇒
+k
+< s
+=⇒
+k
+<
+�π
+θ
+�
+− 2
+=⇒
+k
+< π
+θ − 2
+(since k ≤
+�π
+θ
+�
+− 3)
+=⇒
+(k + 2)θ
+< π
+=⇒
+sin(k + 2)θ
+> 0
+(since (k + 2)θ > 0)
+=⇒
+−rk+1 sin(k + 2)θ
+sin θ
+< 0
+⇐⇒
+Tvk1
+< 0
+⇐⇒
+ConvexHull (Tv) ∩ R2
+≥0
+= ∅
+⇐⇒
+coeffs (gf)
+< 0
+(by Lemma 5)
+Hence, opt(f) ̸< s. By Lemma 9, opt(f) = s.
+2. Consider the factorization of f over R into linear and irreducible quadratic factors as follows.
+f = (x + p1) · · · (x + pt)
+�
+x2 − 2r1 cos θ1x + r2
+1
+�
+· · ·
+�
+x2 − 2rm cos θmx + r2
+m
+�
+f = l1 · · · lt q1 · · · qm
+where
+li = x + pi
+qi = x2 − 2ri cos θix + r2
+i
+where again −pi stand for negative real roots and ri (cos θi ± i sin θi) stand for the complex conjugate
+root pairs. The coefficients of each li and qi with π
+2 ≤ θi < π are non-negative. From Lemma 8, for those
+qi with 0 < θi < π
+2 , there exists non-zero gi ∈ R[x] such that coeffs (gi qi) ≥ 0 and deg(gi) =
+� π
+θi
+�
+− 2.
+Let g = g1 · · · gm. Then coeffs (gf) ≥ 0 and deg(g) =
+m
+�
+i=1
+�� π
+θi
+�
+− 2
+�
+= b(f). Hence we have
+opt (f) ≤ b (f) .
+□
+4
+Proof of Angle-Based Optimality (Theorem 2)
+Let
+f = fπ,p fφ,rφ fθ,rθ
+fπ,p =
+�
+1≤i≤t
+(x + pi) where pi > 0
+fφ,rφ =
+�
+1≤i≤k
+π
+2 ≤φi<π
+�
+x2 − 2rφi cos φi x + r2
+φi
+�
+where rφi > 0 and π > φ1 ≥ · · · ≥ φk ≥ π
+2
+7
+
+fθ,rθ =
+�
+1≤i≤ℓ
+0<θi< π
+2
+�
+x2 − 2rθi cos θi x + r2
+θi
+�
+where rθi > 0 and π
+2 > θ1 ≥ · · · ≥ θℓ > 0
+where k + ℓ = m (the number of complex root pairs of f) and 2m + t = n = deg(f).
+Proof: [Proof of Theorem 2]
+1. We need to show
+∀
+π>φ1≥···≥φk≥ π
+2
+∀
+π
+2 >θ1≥···≥θℓ>0
+∃
+p1,...,pt>0
+∃
+rφ1,...,rφk>0
+∃
+rθ1 ,...,rθℓ>0
+opt (f) = b (f)
+2. Let π > φ1 ≥ · · · ≥ φk ≥ π
+2 and π
+2 > θ1 ≥ · · · ≥ θℓ > 0 be arbitrary but fixed. We need to show
+∃
+p1,...,pt>0
+∃
+rφ1,...,rφk>0
+∃
+rθ1 ,...,rθℓ>0
+opt (f) = b (f) .
+We need to find a witness for p, rφ, rθ such that opt (f) = b (f).
+3. We propose a witness candidate as follows.
+(a) From Lemma 10, for the fixed θ, we have
+∃
+rθ1 ,...,rθℓ>0
+opt (fθ,rθ) = b (fθ,rθ)
+(1)
+(b) From Lemma 15, for the fixed φ and θ, we have
+∃
+p1,...,pt>0
+∃
+rφ1 ,...,rφk>0
+∀
+rθ1 ,...,rθℓ>0
+opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+(2)
+(c) We propose p, rφ, rθ appearing in the above two facts as a witness candidate.
+4. We verify that the proposed candidate is indeed a witness, that is, opt (f) = b (f).
+Note
+opt (f) = opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+by (2)
+= b (fθ,rθ) by (1)
+= b (fπ,p) + b
+�
+fφ,rφ
+�
++ b (fθ,rθ) since b (fπ,p) = b
+�
+fφ,rφ
+�
+= 0
+= b
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= b (f)
+□
+5
+Supporting Lemmas for Proof of Theorem 2
+5.1
+Concerning Irreducible Quadratic Factors with 0 < θ < π
+2
+Let
+αi = rieiθi
+for 1 ≤ i ≤ ℓ
+8
+
+αm+i = rie−iθi
+for 1 ≤ i ≤ ℓ
+ti = cos θi
+f =
+ℓ
+�
+i=1
+�
+x2 − 2ritix + r2
+i
+�
+=
+ℓ
+�
+i=1
+(x − αi)(x − αℓ+i) =
+2ℓ
+�
+i=0
+aixi
+s = b(f)
+g = xs−1 + bs−2xs−2 + · · · + b1x + b0
+ck = coeff(gf, xk)
+Note that
+1. ai = (−1)2ℓ−ie2ℓ−i (α1, . . . , α2ℓ) where ek (α1, . . . , α2ℓ) is the elementary symmetric polynomial of
+degree k in the roots α1, . . . , α2ℓ. When ℓ = 0, we define e0 = 1.
+2. ti > 0 since 0 < θi < π
+2 .
+Lemma 10
+∀
+ℓ≥0
+∀
+π
+2 >θ1≥...≥θℓ>0
+∃
+rθ1,...,rθℓ>0 opt (fθ,rθ) = b (fθ,rθ)
+Proof: We need to prove the following claim for every ℓ ≥ 0.
+∀
+π
+2 >θ1≥...≥θℓ>0
+∃
+rθ1 ,...,rθℓ>0 opt (fθ,rθ) = s
+By Lemma 11, it suffices to show
+∀
+π
+2 >θ1≥...≥θℓ>0
+∃
+rθ1 ,...,rθℓ>0
+∀
+g∈R[x]
+deg(g)=s−1
+∃
+0≤k≤2ℓ+s−1 ck < 0
+⇐⇒
+∀
+0<t1≤···≤tℓ<1
+∃
+rθ1 ,...,rθℓ>0 ∀
+b
+�
+0≤k≤2ℓ+s−1
+ck < 0
+⇐⇒
+∀
+0<t1≤···≤tℓ<1
+∃
+rθ1 ,...,rθℓ>0 ∀
+b
+�
+0≤k≤2ℓ+s−2
+ck < 0
+(since the leading coefficient bs−1 = 1)
+1. Case 1: ℓ = 0
+Here, f = 1. The claim is trivially true since opt (fθ,rθ) = b (fθ,rθ) = 0.
+2. Case 2: ℓ = 1
+Immediate from Remark ??.
+3. Case 3: ℓ = 2
+Immediate from Lemma 12.
+4. Case 3: ℓ ≥ 3
+Immediate from Lemma 13.
+□
+Lemma 11
+∄
+g, deg(g)=s ∀
+k ck ≥ 0
+=⇒
+∄
+g, deg(g)<s ∀
+k ck ≥ 0
+Proof: We will prove via the contrapositive:
+∃
+g, deg(g)<s ∀
+k ck ≥ 0
+=⇒
+∃
+g, deg(g)=s ∀
+k ck ≥ 0
+9
+
+1. Assume
+∃
+g, deg(g)<s
+∀
+k
+ck ≥ 0.
+Then there exists a g with deg(g) = t < s such that gf has all
+non-negative coefficients. Let u = s − t.
+2. Consider the multiplier xu g. Note that xu gf = xu(gf) must have all non-negative coefficients and
+deg(xu g) = u + t = s.
+3. Then there exists a multiplier, xu g with degree equal to s such that the product has all non-negative
+coefficients.
+4. Hence we have
+∃
+g, deg(g)<s ∀
+k ck ≥ 0
+=⇒
+∃
+g, deg(g)=s ∀
+k ck ≥ 0
+□
+Lemma 12 For ℓ = 2,
+∀
+0<t1≤t2<1
+∃
+r1,r2>0 ∀
+b
+�
+0≤k≤2ℓ+s−2
+ck < 0
+Proof: We will prove the following stronger statement.
+∀
+0<t1≤t2<1
+∃
+r1,r2>0 ∀
+b
+�
+k∈{s−2,s−1, 2ℓ+s−3, 2ℓ+s−2}
+ck < 0.
+1. Let t1, t2 be such that 0 < t1 ≤ t2 < 1. Let r1 > 0.
+2. Note
+cs−2 = a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + a4bs−6
+= a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + bs−6
+cs−1 = a0bs−1 + a1bs−2 + a2bs−3 + a3bs−4 + a4bs−5
+= a0 + a1bs−2 + a2bs−3 + a3bs−4 + bs−5
+c2ℓ+s−3 = a2bs−1 + a3bs−2 + a4bs−3
+= a2 + a3bs−2 + bs−3
+c2ℓ+s−2 = a3bs−1 + a4bs−2
+= a3 + bs−2
+(a) Note that coeff(gf, xk) =
+�
+i+j=k
+aibj =
+k
+�
+i=0
+aibk−i for 0 ≤ k ≤ 2ℓ + s − 1.
+Recall from Lemma 3 for 0 ≤ i ≤ 2ℓ and 0 ≤ j ≤ s − 1
+coeffs(gf) =
+�
+b0
+· · ·
+bs−1
+�
+
+
+a0
+· · ·
+· · ·
+a2ℓ
+...
+...
+a0
+· · ·
+· · ·
+a2ℓ
+
+
+Then
+coeff(gf, xk) =
+�
+i+j=k
+aibj
+10
+
+(b) Then
+cs−2 =
+�
+i+j=s−2
+aibj
+= a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + a4bs−6
+= a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + bs−6
+since a4 = 1
+cs−1 =
+�
+i+j=s−1
+aibj
+= a0bs−1 + a1bs−2 + a2bs−3 + a3bs−4 + a4bs−5
+= a0 + a1bs−2 + a2bs−3 + a3bs−4 + bs−5
+since a4 = bs−1 = 1
+c2ℓ+s−3 =
+�
+i+j=2ℓ+s−3
+aibj =
+�
+i+j=s+1
+aibj
+= a2bs−1 + a3bs−2 + a4bs−3
+= a2 + a3bs−2 + bs−3
+since a4 = bs−1 = 1
+c2ℓ+s−2 =
+�
+i+j=2ℓ+s−2
+aibj =
+�
+i+j=s+2
+aibj
+= a3bs−1 + a4bs−2
+= a3 + bs−2
+since a4 = bs−1 = 1
+3. Claim 1: When s = 2,
+∃
+r(1)
+2
+>0
+∀
+r2≥r(1)
+2
+∀
+b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.
+(a) Note
+cs−1 = a0 + a1b0
+c2ℓ+s−2 = a3 + b0
+(b) We have
+cs−1 < 0
+⇐⇒
+b0 > −a0
+a1
+⇐⇒
+b0 > e4(α1, α2, α3, α4)
+e3(α1, α2, α3, α4)
+⇐⇒
+b0 >
+r1r2
+2r2t1 + 2r1t2
+c2ℓ+s−2 < 0
+⇐⇒
+b0 < −a3
+⇐⇒
+b0 < e1(α1, α2, α3, α4)
+⇐⇒
+b0 < 2r1t1 + 2r2t2
+(c) Note
+∃
+r2>0 ∀
+b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0. Note
+∀
+b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0
+⇐=
+r1r2
+2r2t1 + 2r1t2
+< 2r1t1 + 2r2t2
+⇐⇒
+r1r2
+(2r2t1 + 2r1t2)(2r1t1 + 2r2t2)
+< 1
+Note
+∀
+b0
+lim
+r2→∞
+r1r2
+(2r2t1 + 2r1t2)(2r1t1 + 2r2t2) = 0
+since
+degr2 (r1r2) = 1
+degr2 ((2r2t1 + 2r1t2)(2r1t1 + 2r2t2)) = 2
+(d) Hence
+∃
+r2>0 ∀
+b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.
+11
+
+4. Let s ≥ 3.
+5. Note
+cs−2 = 0 ⇐⇒ bs−3 = −a0
+a1
+bs−2 − a2bs−4 + a3bs−5 + bs−6
+a1
+c2ℓ+s−3 = 0 ⇐⇒ bs−3 = −a3bs−2 − a2
+Let b0, . . . , bs−4 be arbitrary but fixed.
+Let Lk be the line given by ck = 0. Let µk be the slope of Lk. Then µs−2 = −a0
+a1
+and µ2ℓ+s−3 = −a3.
+6. Since ai = (−1)4−ie4−i (α1, α2, α3, α4), we have
+a0 = e4 (α1, α2, α3, α4)
+a1 = −e3 (α1, α2, α3, α4)
+a3 = −e1 (α1, α2, α3, α4)
+Then
+µs−2
+= e4 (α1, α2, α3, α4)
+e3 (α1, α2, α3, α4)
+=
+r1r2
+2r2t1 + 2r1t2
+µ2ℓ+s−3
+= e1 (α1, α2, α3, α4)
+= 2r1t1 + 2r2t2
+7. Claim 2:
+∃
+r(2)
+2
+>0
+∀
+r2≥r(2)
+2
+µ2ℓ+s−3 > µs−2.
+(a) Note
+µ2ℓ+s−3
+> µs−2
+⇐⇒
+2r1t1 + 2r2t2
+>
+r1r2
+2r2t1 + 2r1t2
+⇐⇒
+1
+>
+r1r2
+(2r2t1 + 2r1t2)(2r1t1 + 2r2t2)
+(b) Note
+lim
+r2→∞
+r1r2
+(2r2t1 + 2r1t2)(2r1t1 + 2r2t2) = 0
+since
+degr2 (r1r2) = 1
+degr2 ((2r2t1 + 2r1t2)(2r1t1 + 2r2t2)) = 2
+8. Let r(2)
+2
+be such that
+∀
+r2≥r(2)
+2
+µ2ℓ+s−3 > µs−2. Such r(2)
+2
+exists due to the previous claim. Let r2 be
+arbitrary but fixed such that r2 ≥ r(2)
+2 .
+9. Over the space (bs−2, bs−3), there exists a unique intersection point of Ls−2 and L2ℓ+s−3. Let (p, q) be
+the intersection point.
+10. We have p = a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)
+a0a4 − a1a3
+and q = −a0a3 − a3(−a2bs−4 − a3bs−5 − bs−6)
+a0a4 − a1a3
+.
+12
+
+Note
+cs−2 = 0
+∧
+c2ℓ+s−3 = 0
+⇐⇒
+a0bs−2 + a1bs−3 + a2bs−4 + a3bs−5 + bs−6 = 0
+∧
+a2 + a3bs−2 + a4bs−3 = 0
+⇐⇒
+a0bs−2 + a1bs−3 = −a2bs−4 − a3bs−5 − bs−6
+∧
+a3bs−2 + a4bs−3 = −a2
+⇐⇒
+�
+a0
+a1
+a3
+a4
+� �
+bs−2
+bs−3
+�
+=
+�
+−a2bs−4 − a3bs−5 − bs−6
+−a3
+�
+⇐⇒
+bs−2 =
+�������
+−a2bs−4 − a3bs−5 − bs−6
+a1
+−a3
+a4
+�������
+�������
+a0
+a1
+a3
+a4
+�������
+∧
+bs−3 =
+�������
+a0
+−a2bs−4 − a3bs−5 − bs−6
+a3
+−a3
+�������
+�������
+a0
+a1
+a3
+a4
+�������
+⇐⇒
+bs−2 = a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)
+a0a4 − a1a3
+∧
+bs−3 = −a0a3 − a3(−a2bs−4 − a3bs−5 − bs−6)
+a0a4 − a1a3
+11. Claim 3:
+∃
+r(2)
+2
+>0
+∀
+r2≥r(2)
+2
+∀
+b0,...,bs−2
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−3 < 0.
+Let r(2)
+2
+be such that
+∀
+r2≥r(2)
+2
+µ2ℓ+s−3 > µs−2. In the above, we have shown that such r(2)
+2
+exists.
+Let r2 ≥ r(2)
+2
+be arbitrary but fixed.
+Let b0, . . . , bs−4 be arbitrary but fixed. We need to show
+∀
+bs−2,bs−3
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−3 < 0.
+(a) Over the space (bs−2, bs−3) where bs−2 > p, we have
+i. c2ℓ+s−3 = 0 line and cs−2 = 0 line do not intersect
+ii. c2ℓ+s−3 = 0 line is above cs−2 = 0 line.
+(b) Let (bs−2, bs−3) be an arbitrary but fixed point such that bs−2 > p. Then (bs−2, bs−3) is above
+Ls−2 or below L2ℓ+s−3.
+(c) Note
+(bs−2, bs−3) is above Ls−2
+⇐⇒
+bs−3 > − a0
+a1 bs−2 − a2bs−4+a3bs−5+bs−6
+a1
+⇐⇒
+cs−2 < 0
+(bs−2, bs−3) is below L2ℓ+s−3
+⇐⇒
+bs−3 < −a3bs−2 − a2
+⇐⇒
+c2ℓ+s−3 < 0
+(d) Thus cs−2 < 0 or c2ℓ+s−3 < 0.
+12. Claim 4:
+∃
+r(3)
+2
+>0
+∀
+r2≥r(3)
+2
+∀
+b0,...,bs−2
+bs−2≤p
+c2ℓ+s−2 < 0.
+(a) Note
+∀
+r2>0
+∀
+b0,...,bs−2
+bs−2≤p
+a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)
+e1 (α1, α2, α3, α4) (a0a4 − a1a3)
+< 1
+=⇒
+c2ℓ+s−2 < 0
+13
+
+since
+c2ℓ+s−2
+< 0
+⇐⇒
+bs−2
+< −a3
+⇐=
+p
+< −a3
+since bs−2 ≤ p
+⇐⇒
+a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)
+a0a4 − a1a3
+< e1 (α1, α2, α3, α4)
+⇐⇒
+a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)
+e1 (α1, α2, α3, α4) (a0a4 − a1a3)
+< 1
+(b) Let ek = ek (α1, α2, α3, α4). Note
+a1a3 − a4(a2bs−4 + a3bs−5 + bs−6)
+e1 (α1, α2, α3, α4) (a0a4 − a1a3)
+= e3e1 − e0(e2bs−4 − e1bs−5 + bs−6)
+e1 (e4e0 − e3e1)
+(c) Note
+∀
+b0,...,bs−2
+lim
+r2→∞
+e3e1 − e0(e2bs−4 − e1bs−5 + bs−6)
+e1 (e4e0 − e3e1)
+= 0
+since
+degr2
+�e3e1 − e0(e2bs−4 − e1bs−5 + bs−6)
+e1 (e4e0 − e3e1)
+�
+≤ 3
+degr2 (e1 (e4e0 − e3e1)) = 4
+13. Claim 5:
+∃
+r2>0
+∀
+b0,...,bs−2
+�
+0≤k≤2ℓ+s−2
+ck < 0
+From Claim 1, when s = 2, for some r(1)
+2
+> 0 we have
+∃
+r2>0 ∀
+b0 cs−1 < 0 ∨ c2ℓ+s−2 < 0.
+From Claim 3, when s ≥ 3, for some r(2)
+2
+> 0 we have
+∀
+r2>r(1)
+2
+∀
+b0,...,bs−2
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−3 < 0.
+From Claim 4, when s ≥ 3, for some r(3)
+2
+> 0 we have
+∀
+r2≥r(2)
+2
+∀
+b0,...,bs−2
+bs−2≤p
+c2ℓ+s−2 < 0.
+Then for r∗
+2 = max{r(1)
+2 , r(2)
+2 , r(3)
+2 }, we have
+∀
+r2≥r∗
+2
+
+
+∀
+b0,...,bs−2
+bs−2>p
+cs−1 < 0 ∨ cs−2 < 0 ∨ c2ℓ+s−3 < 0
+∧
+∀
+b0,...,bs−2
+bs−2≤p
+cs−1 < 0 ∨ c2ℓ+s−2 < 0
+
+
+Hence
+∃
+r2>0
+∀
+b0,...,bs−2
+�
+k∈{s−2,s−1,2ℓ+s−3,2+s−2}
+ck < 0.
+□
+Lemma 13 For ℓ ≥ 3,
+∀
+0<t1≤···≤tℓ<1
+∃
+r1,...,rℓ>0 ∀
+b
+�
+0≤k≤2ℓ+s−2
+ck < 0
+14
+
+Proof: We will prove the following stronger statement.
+∀
+0<t1≤···≤tℓ<1
+∃
+r1,...,rℓ−1>0
+C(r)<1
+∃
+rℓ>0 ∀
+b
+�
+k∈{s−2, 2ℓ+s−5, 2ℓ+s−2}
+ck < 0
+where C(r) = e0 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) e2ℓ−2 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e1 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) e2ℓ−3 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1).
+1. Let t1, . . . , tℓ be such that 0 < t1 ≤ · · · ≤ tℓ < 1.
+2. Claim 1:
+∀
+0<t1≤···≤tℓ−1<1
+∃
+r1,...,rℓ−1>0 C(r) < 1
+(a) Note
+e0 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) = 1
+e1 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) = α1 + · · · + αℓ−1 + αℓ+1 + · · · + α2ℓ−1
+= (α1 + αℓ+1) + · · · + (αℓ−1 + α2ℓ−1)
+= 2r1t1 + · · · + 2rℓ−1tℓ−1
+e2ℓ−3 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) =
+�
+i∈{1,...,ℓ−1,ℓ+1,...,2ℓ−1}
+
+�
+j̸=i
+αj
+
+
+=
+�
+1≤i≤ℓ−1
+
+�
+j̸=i
+αj +
+�
+j̸=ℓ+i
+αj
+
+
+=
+�
+1≤i≤ℓ−1
+
+(αℓ+i + αi)
+�
+j̸=i,ℓ+1
+αj
+
+
+=
+�
+1≤i≤ℓ−1
+
+(2riti)
+�
+j̸=i
+r2
+j
+
+
+e2ℓ−2 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) = α1 · · · αℓ−1αℓ+1 · · · α2ℓ−1
+= α1αℓ+1 · · · αℓ−1α2ℓ−1
+= r2
+1 · · · r2
+ℓ−1
+Then
+C(r) = e0 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) e2ℓ−2 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e1 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1) e2ℓ−3 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+=
+r2
+1 · · · r2
+ℓ−1
+(2r1t1 + · · · + 2rℓ−1tℓ−1)
+��
+1≤i≤ℓ−1
+�
+(2riti) �
+j̸=i r2
+j
+��
+=
+r1 · · · rℓ−1
+(2r1t1 + · · · + 2rℓ−1tℓ−1)
+��
+1≤i≤ℓ−1
+�
+(2ti) �
+j̸=i rj
+��
+=
+1
+(2r1t1 + · · · + 2rℓ−1tℓ−1)
+�2t1
+r1
++ · · · + 2tℓ−1
+rℓ−1
+�
+(b) Consider the following witness candidates for the existentially quantified variables r1, . . . , rℓ−1:
+r1 = 2t1
+15
+
+ri = 1
+2ti
+for 2 ≤ t ≤ ℓ − 1
+(c) Obviously, r1, . . . , rℓ−1 > 0. Note
+1
+(2r1t1 + 2r2t2 + · · · + 2rℓ−1tℓ−1)
+�2t1
+r1
++ 2t2
+r2
++ · · · + 2tℓ−1
+rℓ−1
+�
+=
+1
+�
+2 (2t1) t1 + 2
+� 1
+2t2
+�
+t2 + · · · + 2
+�
+1
+2tℓ−1
+�
+tℓ−1
+� �
+2t1
+2t1
++ 2t2
+1
+2t2
++ · · · + 2tℓ−1
+1
+2tℓ−1
+�
+=
+1
+(4t2
+1 + 1 + · · · + 1)
+�
+1 + 4t2
+2 + · · · + 4t2
+ℓ−1
+�
+< 1
+Hence the candidates are witnesses.
+3. Let r1, . . . , rℓ−1 > 0 be such that C(r) < 1.
+4. Note
+cs−2 = a0bs−2 + a1bs−3 +
+s−2
+�
+i=2
+aib(s−2)−i
+c2ℓ+s−5 = a2ℓ−4bs−1 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + a2ℓbs−5
+= a2ℓ−4 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + bs−5
+c2ℓ+s−2 = a2ℓ−1bs−1 + a2ℓbs−2
+= a2ℓ−1 + bs−2
+(a) Note that coeff(gf, xk) =
+�
+i+j=k
+aibj =
+k
+�
+i=0
+aibk−i for 0 ≤ k ≤ 2ℓ + s − 1.
+Recall from Lemma 3 for 0 ≤ i ≤ 2ℓ and 0 ≤ j ≤ s − 1
+coeffs(gf) =
+�
+b0
+· · ·
+bs−1
+�
+
+
+a0
+· · ·
+· · ·
+a2ℓ
+...
+...
+a0
+· · ·
+· · ·
+a2ℓ
+
+
+Then
+coeff(gf, xk) =
+�
+i+j=k
+aibj
+=
+k
+�
+i=0
+aibk−i
+(b) Then
+cs−2 =
+s−2
+�
+i=0
+aib(s−2)−i
+16
+
+= a0bs−2 + a1bs−3 +
+s−2
+�
+i=2
+aib(s−2)−i
+c2ℓ+s−5 =
+�
+i+j=2ℓ+s−5
+aibj
+= a2ℓ−4bs−1 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + a2ℓbs−5
+= a2ℓ−4 + a2m−3bs−2 + a2m−2bs−3 + a2m−1bs−4 + bs−5
+since a2ℓ = bs−1 = 1
+c2ℓ+s−2 =
+�
+i+j=2ℓ+s−2
+aibj
+= a2ℓ−1bs−1 + a2ℓbs−2
+= a2ℓ−1 + bs−2
+since a2ℓ = bs−1 = 1
+5. Note
+cs−2 = 0 ⇐⇒ bs−3 = −a0
+a1
+bs−2 −
+s−2
+�
+i=2
+ai
+a1
+b(s−2)−i
+c2ℓ+s−5 = 0 ⇐⇒ bs−3 = −a2ℓ−3
+a2ℓ−2
+bs−2 − a2ℓ−1bs−4 + bs−5 + a2ℓ−4
+a2ℓ−2
+Let b0, . . . , bs−4 be arbitrary but fixed.
+Let Lk be the line given by ck = 0. Let µk be the slope of Lk. Then µs−2 = −a0
+a1
+and µ2ℓ+s−5 = −a2ℓ−3
+a2ℓ−2
+.
+6. Since ai = (−1)2ℓ−ie2ℓ−i (α1, . . . , α2ℓ), we have
+a0 = e2ℓ (α1, . . . , α2ℓ)
+a1 = −e2ℓ−1 (α1, . . . , α2ℓ)
+a2ℓ−3 = −e3 (α1, . . . , α2ℓ)
+a2ℓ−2 = e2 (α1, . . . , α2ℓ)
+Then µs−2 =
+e2ℓ (α1, . . . , α2ℓ)
+e2ℓ−1 (α1, . . . , α2ℓ) and µ2ℓ+s−5 = e3 (α1, . . . , α2ℓ)
+e2 (α1, . . . , α2ℓ).
+7. Claim 2:
+∃
+r(1)
+ℓ
+>0
+∀
+rℓ≥r(1)
+ℓ
+µ2ℓ+s−5 > µs−2.
+(a) Note as rℓ → ∞
+e2ℓ (α1, . . . , α2ℓ) → αℓα2ℓ e2ℓ−2 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e2ℓ−1 (α1, . . . , α2ℓ) → αℓα2ℓ e2ℓ−3 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e3 (α1, . . . , α2ℓ) → αℓα2ℓ e1 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e2 (α1, . . . , α2ℓ) → αℓα2ℓ e0 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+Then
+µs−2 → e2ℓ−2 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e2ℓ−3 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+µ2ℓ+s−5 → e1 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e0 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+17
+
+(b) Then for sufficiently large rℓ,
+µ2ℓ+s−5
+> µs−2
+⇐⇒
+e1 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e0 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+> e2ℓ−2 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+e2ℓ−3 (α1, . . . , αℓ−1, αℓ+1, . . . , α2ℓ−1)
+⇐⇒
+1
+> C(r)
+which is true for the r1, . . . , rℓ−1 we have chosen based on Claim 1.
+8. Let r(1)
+ℓ
+be such that
+∀
+rℓ≥r(1)
+ℓ
+µ2ℓ+s−5 > µs−2. Such r(1)
+ℓ
+exists due to the previous claim. Let rℓ be
+arbitrary but fixed such that rℓ ≥ r(1)
+ℓ .
+9. Over the space (bs−2, bs−3), there exists a unique intersection point of Ls−2 and L2ℓ+s−5. Let (p, q) be
+the intersection point.
+10. We have p =
+a2ℓ−2d0 − a1d1
+a0a2ℓ−2 − a1a2ℓ−3
+and q =
+a0d1 − a2ℓ−3d0
+a0a2ℓ−2 − a1a2ℓ−3
+where d0 = −
+s−2
+�
+i=2
+aib(s−2)−i and d1 =
+−a2ℓ−1bs−4 − bs−5 − a2ℓ−4.
+Note
+cs−2 = 0
+∧
+c2ℓ+s−5 = 0
+⇐⇒
+a0bs−2 + a1bs−3 + �s−2
+i=2 aib(s−2)−i = 0
+∧
+a2ℓ−4 + a2ℓ−3bs−2 + a2ℓ−2bs−3 + a2ℓ−1bs−4 + bs−5 = 0
+⇐⇒
+a0bs−2 + a1bs−3 = − �s−2
+i=2 aib(s−2)−i
+∧
+a2ℓ−3bs−2 + a2ℓ−2bs−3 = −a2ℓ−1bs−4 − bs−5 − a2ℓ−4
+⇐⇒
+a0bs−2 + a1bs−3 = d0
+where d0 = − �s−2
+i=2 aib(s−2)−i
+∧
+a2ℓ−3bs−2 + a2ℓ−2bs−3 = d1
+where d1 = −a2ℓ−1bs−4 − bs−5 − a2ℓ−4
+⇐⇒
+�
+a0
+a1
+a2ℓ−3
+a2ℓ−2
+� �
+bs−2
+bs−3
+�
+=
+�
+d0
+d1
+�
+⇐⇒
+bs−2 =
+�������
+d0
+a1
+d1
+a2ℓ−2
+�������
+�������
+a0
+a1
+a2ℓ−3
+a2ℓ−2
+�������
+∧
+bs−3 =
+�������
+a0
+d0
+a2ℓ−3
+d1
+�������
+�������
+a0
+a1
+a2ℓ−3
+a2ℓ−2
+�������
+⇐⇒
+bs−2 =
+a2ℓ−2d0 − a1d1
+a0a2ℓ−2 − a1a2ℓ−3
+∧
+bs−3 =
+a0d1 − a2ℓ−3d0
+a0a2ℓ−2 − a1a2ℓ−3
+11. Claim 3:
+∃
+r(1)
+ℓ
+>0
+∀
+rℓ≥r(1)
+ℓ
+∀
+b0,...,bs−2
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−5 < 0.
+Let r(1)
+ℓ
+be such that
+∀
+rℓ≥r(1)
+ℓ
+µ2ℓ+s−5 > µs−2. In the above, we have shown that such r(1)
+ℓ
+exists.
+Let rℓ ≥ r(1)
+ℓ
+be arbitrary but fixed.
+Let b0, . . . , bs−4 be arbitrary but fixed. We need to show
+∀
+bs−2,bs−3
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−5 < 0.
+(a) Over the space (bs−2, bs−3) where bs−2 > p, we have
+18
+
+i. c2ℓ+s−5 = 0 line and cs−2 = 0 line do not intersect
+ii. c2ℓ+s−5 = 0 line is above cs−2 = 0 line.
+(b) Let (bs−2, bs−3) be an arbitrary but fixed point such that bs−2 > p. Then (bs−2, bs−3) is above
+Ls−2 or below L2ℓ+s−5.
+(c) Note
+(bs−2, bs−3) is above Ls−2
+⇐⇒
+bs−3 > − a0
+a1 bs−2 − d0
+a1
+⇐⇒
+cs−2 < 0
+(bs−2, bs−3) is below L2ℓ+s−5
+⇐⇒
+bs−3 < − a2ℓ−3
+a2ℓ−2 bs−2 −
+d1
+a2ℓ−2
+⇐⇒
+c2ℓ+s−5 < 0
+since a1 = (−1)2ℓ−1e2ℓ−1 (α1, . . . , α2ℓ) < 0 and a2ℓ−2 = (−1)2e2 (α1, . . . , α2ℓ) > 0.
+(d) Thus cs−2 < 0 or c2ℓ+s−5 < 0.
+12. Claim 4:
+∃
+r(2)
+ℓ
+>0
+∀
+rℓ≥r(2)
+ℓ
+∀
+b0,...,bs−2
+bs−2≤p
+c2ℓ+s−2 < 0.
+(a) Note
+∀
+rℓ>0
+∀
+b0,...,bs−2
+bs−2≤p
+a2ℓ−2d0 − a1d1
+e1 (α1, . . . , α2ℓ) (a0a2ℓ−2 − a1a2ℓ−3) < 1
+=⇒
+c2+s−2 < 0
+since
+c2ℓ+s−2
+< 0
+⇐⇒
+bs−2
+< −a2ℓ−1
+⇐=
+p
+< −a2ℓ−1
+since bs−2 ≤ p
+⇐⇒
+a2ℓ−2d0 − a1d1
+a0a2ℓ−2 − a1a2ℓ−3
+< e1 (α1, . . . , α2ℓ)
+⇐⇒
+a2ℓ−2d0 − a1d1
+e1 (α1, . . . , α2ℓ) (a0a2ℓ−2 − a1a2ℓ−3)
+< 1
+(b) Let ek = ek (α1, . . . , α2ℓ). Note
+a2ℓ−2d0 − a1d1
+e1 (α1, . . . , α2ℓ) (a0a2ℓ−2 − a1a2ℓ−3)
+= −e2
+�s−2
+i=2 (−1)2ℓ−ie2ℓ−ib(s−2)−i + e2ℓ−1 (e1bs−4 − bs−5 − e4)
+e1 (e2ℓe2 − e2ℓ−1e3)
+(c) Note
+∀
+b0,...,bs−2
+lim
+rℓ→∞
+−e2
+�s−2
+i=2 (−1)2ℓ−ie2ℓ−ib(s−2)−i + e2ℓ−1 (e1bs−4 − bs−5 − e4)
+e1 (e2ℓe2 − e2ℓ−1e3)
+= 0
+since
+degrℓ
+�
+−e2
+s−2
+�
+i=2
+(−1)2ℓ−ie2ℓ−ib(s−2)−i + e2ℓ−1 (e1bs−4 − bs−5 − e4)
+�
+≤ 4
+degrℓ (e1 (e2ℓe2 − e2ℓ−1e3)) = 5
+13. Claim 5:
+∃
+rℓ>0
+∀
+b0,...,bs−2
+�
+0≤k≤2ℓ+s−2
+ck < 0
+From Claim 3, for some r(1)
+ℓ
+> 0 we have
+∀
+rℓ>r(1)
+ℓ
+∀
+b0,...,bs−2
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−5 < 0.
+19
+
+From Claim 4, for some r(2)
+ℓ
+> 0 we have
+∀
+rℓ≥r(2)
+ℓ
+∀
+b0,...,bs−2
+bs−2≤p
+c2ℓ+s−2 < 0.
+Then for r∗
+ℓ = max{r(1)
+ℓ , r(2)
+ℓ }, we have
+∀
+rℓ≥r∗
+ℓ
+
+
+∀
+b0,...,bs−2
+bs−2>p
+cs−2 < 0 ∨ c2ℓ+s−5 < 0
+∧
+∀
+b0,...,bs−2
+bs−2≤p
+c2ℓ+s−2 < 0
+
+
+Hence
+∃
+rℓ>0
+∀
+b0,...,bs−2
+�
+k∈{s−2,2ℓ+s−5,2ℓ+s−2}
+ck < 0.
+□
+Lemma 14 If arg(αi) < π
+2 , then ek (α1, . . . , α2ℓ) > 0 for k = 0, . . . , 2ℓ.
+Proof: We will induct on ℓ, the number of quadratic factors of f with non-real roots.
+1. Base Case: ℓ = 0
+Note that e0 = 1 > 0.
+2. Hypothesis: Assume ek (α1, . . . , α2ℓ) > 0 for ℓ ≥ 1 and 0 ≤ k ≤ 2ℓ.
+3. Induction Step: Prove ek
+�
+α1, . . . , α2(ℓ+1)
+�
+> 0 for 0 ≤ k ≤ 2(ℓ + 1).
+Let fℓ =
+ℓ
+�
+i=1
+�
+x2 − 2ritix + r2
+i
+�
+and aℓ,k = coeff(fℓ, xk). Note that
+fℓ+1
+=
+�
+x2 − 2rℓ+1tℓ+1x + r2
+ℓ+1
+�
+fℓ
+aℓ+1,k
+= aℓ,k−2 − 2rℓ+1,t+1aℓ,k−1 + r2
+ℓ+1aℓ,k
+Then
+aℓ+1,k = (−1)2(ℓ+1)−ke2(ℓ+1)−k
+�
+α1, . . . , α2(ℓ+1)
+�
+= (−1)2ℓ−(k−2)e2ℓ−(k−2) (α1, . . . , α2ℓ)
+− 2rℓ+1tℓ+1(−1)2ℓ−(k−1)e2ℓ−(k−1) (α1, . . . , α2ℓ)
++ r2
+ℓ+1(−1)2ℓ−ke2ℓ−k (α1, . . . , α2ℓ)
+Hence
+e2(ℓ+1)−k
+�
+α1, . . . , α2(ℓ+1)
+�
+= e2ℓ−(k−2) (α1, . . . , α2ℓ)
+− 2rℓ+1tℓ+1(−1)−1e2ℓ−(k−1) (α1, . . . , α2ℓ)
++ r2
+ℓ+1(−1)−2e2ℓ−k (α1, . . . , α2ℓ)
+= e2ℓ−(k−2) (α1, . . . , α2ℓ)
++ 2rℓ+1tℓ+1e2ℓ−(k−1) (α1, . . . , α2ℓ)
++ r2
+ℓ+1e2ℓ−k (α1, . . . , α2ℓ)
+> 0
+by the inductive hypothesis and the fact that rℓ+1, tℓ+1 > 0.
+□
+20
+
+5.2
+Concerning Linear and Irreducible Quadratic Factors with π
+2 < θ < π
+Lemma 15 We have
+∀
+π>φ1≥···≥φk≥ π
+2
+∀
+π
+2 >θ1≥···≥θℓ>0
+∃
+p1,...,pt>0
+∃
+rφ1,...,rφk>0
+∀
+rθ1,...,θℓ>0
+opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+Proof: We will first induct on k, the number of quadratic factors with π
+2 ≤ φi < π in fφ,rφ, to show that
+opt
+�
+fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ).
+1. Base: The claim holds for k = 0.
+Trivially true.
+2. Hypo: Assume the claim holds for k quadratic factors with π
+2 ≤ φi < π.
+∃
+rφ1,...,rφk>0
+∀
+rθ1 ,...,rθℓ>0
+opt
+�
+fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+3. Induction Step: Consider k + 1 quadratic factors with π
+2 ≤ φi < π.
+Assume rφ1, . . . , rφk > 0 are such that the induction hypothesis holds. By Lemma 19,
+∃
+rφk+1 >0
+∀
+rθ1,...,rθℓ>0
+opt
+��
+x2 − 2rφk+1 cos φk+1 x + r2
+φk+1
+�
+fφ,rφ fθ,rθ
+�
+= opt
+�
+x2 − 2rφk+1 cos φk+1 x + r2
+φk+1
+�
++ opt
+�
+fφ,rφ fθ,rθ
+�
+By Remark ??, opt
+�
+x2 − 2rφk+1 cos φk+1 x + r2
+φk+1
+�
+= 0, so we have
+opt
+��
+x2 − 2rφk+1 cos φk+1 x + r2
+φk+1
+�
+fφ,rφ fθ,rθ
+�
+= opt
+�
+fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+Hence, we have
+∃
+rφ1,...,rφk>0
+∀
+rθ1 ,...,rθℓ>0
+opt
+�
+fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ).
+Now we will induct on t, the number of linear factors in fπ,p, to show that
+∃
+p1,...,pt>0
+∃
+rφ1,...,rφk>0
+∀
+rθ1,...,rθℓ>0
+opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+1. Base: The claim holds for t = 0.
+Trivially true.
+2. Hypo: Assume the claim holds for t linear factors.
+∃
+p1,...,pt>0
+∃
+rφ1,...,rφk>0
+∀
+rθ1 ,...,rθℓ>0
+opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+3. Induction Step: Consider t + 1 linear factors.
+Assume p1, . . . , pt, rφ1, . . . , rφk > 0 are such that the induction hypothesis holds. By Lemma 18,
+∃
+pt+1>0
+∀
+rθ1,...,rθℓ>0
+opt ((x + pt+1) fπ,p fφ,σ fθ,τ) = opt (x + pt+1) + opt (fπ,p fφ,σ fθ,τ)
+By Lemma 16, opt (x + pt+1) = 0, so we have
+opt ((x + pt+1) fπ,p fφ,σ fθ,τ) = opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ)
+Hence, we have
+∃
+p1,...,pt>0
+∃
+rφ1 ,...,rφk>0
+∀
+rθ1,...,rθℓ>0
+opt
+�
+fπ,p fφ,rφ fθ,rθ
+�
+= opt (fθ,rθ).
+21
+
+□
+Notation 2 Note
+• hr = x + r
+• hθ,r = x2 − 2r cos θ x + r2
+• P = {f ∈ R[x] : f(x) > 0 for x ≥ 0}
+Lemma 16
+∀
+f∈P, deg(f)=1 opt (f) = b (f)
+Proof: Note that b (f) = 0. □
+Lemma 17
+∀
+f∈P, deg(f)=2 opt (f) = b (f)
+Proof: By Theorem 1. □
+Lemma 18
+∀
+f∈P, deg(f)≥1 ∃
+r>0 opt (hr f) = opt (hr) + opt (f)
+Proof: Let f ∈ P. Let h = hr f. We need to show ∃
+r>0 opt(h) = opt(f). We will divide the proof into several
+claims.
+C1:
+∃
+r>0 opt(h) = opt(f)
+⇐=
+∃
+r>0 CH(T ∗
+h,0, . . . , T ∗
+h,s−1) ∩ Rn
+≥0 = ∅
+where
+• n = deg (f)
+• s = opt(f)
+• The Th,i are the rows of Ts−1 for h.
+• T ∗
+h,i is obtained from Th,i by deleting the first element.
+Proof: Note
+opt(h) = opt(f)
+⇐⇒
+opt(h) ≥ s
+since
+opt(h) ≤ opt(f) + opt(x + r) = opt(f) = s
+⇐⇒
+¬
+∃
+g̸=0, deg(g)<s coeffs(gh) ≥ 0
+⇐⇒
+¬
+�
+CH(Th,0, . . . , Th,s−1) ∩ Rn+1
+≥0
+̸= ∅
+�
+⇐⇒
+CH(Th,0, . . . , Th,s−1) ∩ Rn+1
+≥0
+= ∅
+⇐=
+CH(T ∗
+h,0, . . . , T ∗
+h,s−1) ∩ Rn
+≥0 = ∅.
+C2:
+∃
+r>0 opt(h) = opt(f)
+⇐=
+∃
+r>0εh (r) > 0
+where εh (r) stands for the minimum Euclidean distance between CH(T ∗
+h,0, . . . , T ∗
+h,s−1) and Rn
+≥0, that
+is,
+εh(r) :=
+min
+x∈CH(T ∗
+h,0,...,T ∗
+h,s−1)
+y∈Rn
+≥0
+∥x − y∥
+Proof: Immediate from the above claim.
+22
+
+C3: εh (r) is continuous at r = 0.
+Proof: We will divide it into several steps.
+(a) Note
+εh(r) =
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+s−1
+�
+i=0
+ciT ∗
+h,i − y
+�����
+=
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+�s−1
+�
+i=0
+ciT ∗
+h,i,1 − y1 , . . . ,
+s−1
+�
+i=0
+ciT ∗
+h,i,n − yn
+������
+=
+min
+z∈Rs+n
+≥0
+z0+···+zs−1=1
+�����
+�s−1
+�
+i=0
+ziT ∗
+h,i,1 − zs , . . . ,
+s−1
+�
+i=0
+ziT ∗
+h,i,n − zs+n−1
+������
+where z = (c, y)
+= min
+z∈C p (z, r)
+where p(z, r) is Euclidean distance and
+C =
+�
+z ∈ Rs+n
+≥0 : z1 + · · · + zs = 1
+�
+.
+(b) Note, since p(z, r) is the distance between two closed sets, the minimum distance is realized by a
+point in each set. Hence,
+εh(r) = min
+z∈C p (z, r) = inf
+z∈C p (z, r) .
+(c) Note the following:
+i. By Section 3.1.5 of [2], p(z, r) is a convex function since p(z, r) is a norm.
+ii. By Section 3.2.5 of [2], εh(r) = inf
+z∈C p (z, r) is a convex function, since C is convex and p(z, r)
+is bounded below.
+iii. By Corollary 3.5.3 in [4], since εh(r) is a convex function defined on a convex set R, εh(r) is
+continuous on the relative interior of R, ri (R) = R.
+iv. Hence, εh(r) is continuous at r = 0.
+C4: εh (0) > 0.
+Proof: Let r = 0. Then h = x · f.
+(a) Subclaim: T ∗
+h,i,j = Tf,i,j. Note that these are the entries of T ∗
+h,s−1 and Tf,s−1. We will prove the
+claim using the fact that Ts−1 = R−1
+s−1Ls−1 for any f.
+i. Note Rf,s−1 = Rh,s−1. This is clear from the definition of Rs−1. Hence R−1
+f,s−1 = R−1
+h,s−1.
+ii. Note that Lf,i,j = Lh,i,j+1. This is clear from the definition of Ls−1, since Lh,s−1 is composed
+of Lf,s−1 with a column of zeroes added on the left.
+iii. Note that for i = 0, . . . , s − 1 and j = 0, . . . , n − 1,
+T ∗
+h,i,j = Th,i,j+1
+=
+s−1
+�
+k=0
+R−1
+h,i,kLh,k,j+1
+23
+
+=
+s−1
+�
+k=0
+R−1
+f,i,kLf,k,j
+= Tf,i,j
+Hence T ∗
+h,i,j = Tf,i,j.
+(b) Subclaim: εh(0) = εf
+where εf stands for the minimum Euclidean distance between CH(Tf,0, . . . , Tf,s−1) and Rn
+≥0, that
+is,
+εf :=
+min
+x∈CH(Tf,0,...,Tf,s−1)
+y∈Rn
+≥0
+∥x − y∥.
+To see this, note
+εh(0) =
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+s−1
+�
+i=0
+ciT ∗
+h,i − y
+�����
+=
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+s−1
+�
+i=0
+ciTf,i − y
+�����
+=
+min
+x∈CH(Tf,0,...,Tf,s−1)
+y∈Rn
+≥0
+∥x − y∥
+= εf
+(c) Subclaim: εf > 0.
+Note since opt (f) = s, we have CH(Tf,0, . . . , Tf,s−1) ∩ Rn
+≥0 = ∅. Thus εf > 0.
+(d) From the above two subclaims, we have εh (0) > 0.
+From the above four claims, we immediately have
+∃
+r>0 opt(h) = opt(f).
+□
+Lemma 19
+∀
+f∈P, deg(f)≥1
+∀
+π
+2 ≤θ< π
+1
+∃
+r>0 opt (hθ,r f) = opt (hθ,r) + opt (f)
+Proof: Let f ∈ P and h = hθ,r f. We need to show ∃
+r>0 opt(h) = opt(f) since opt(hθ,r) = 0. We will divide
+the proof into several claims.
+C1:
+∃
+r>0 opt(h) = opt(f)
+⇐=
+∃
+r>0 CH(T ∗
+h,0, . . . , T ∗
+h,s−1) ∩ Rn
+≥0 = ∅
+where
+• n = deg (f)
+• s = opt(f)
+• The Th,i are the rows of Ts−1 for h.
+• T ∗
+h,i is obtained from Th,i by deleting the first two elements.
+24
+
+Proof: Note
+opt(h) = opt(f)
+⇐⇒
+opt(h) ≥ s
+since
+opt(h) ≤ opt(f) + opt(hr,θ) = opt(f) = s
+⇐⇒
+¬
+∃
+g̸=0, deg(g)<s coeffs(gh) ≥ 0
+⇐⇒
+¬
+�
+CH(Th,0, . . . , Th,s−1) ∩ Rn+2
+≥0
+̸= ∅
+�
+⇐⇒
+CH(Th,0, . . . , Th,s−1) ∩ Rn+2
+≥0
+= ∅
+⇐=
+CH(T ∗
+h,0, . . . , T ∗
+h,s−1) ∩ Rn
+≥0 = ∅.
+C2:
+∃
+r>0 opt(h) = opt(f)
+⇐=
+∃
+r>0εh (r) > 0
+where εh (r) stands for the minimum Euclidean distance between CH(T ∗
+h,0, . . . , T ∗
+h,s−1) and Rn
+≥0, that
+is,
+εh(r) :=
+min
+x∈CH(T ∗
+h,0,...,T ∗
+h,s−1)
+y∈Rn
+≥0
+∥x − y∥
+Proof: Immediate from the above claim.
+C3: εh (r) is continuous at r = 0.
+Proof: We will divide it into several steps.
+(a) Note
+εh(r) =
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+s−1
+�
+i=0
+ciT ∗
+h,i − y
+�����
+=
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+�s−1
+�
+i=0
+ciT ∗
+h,i,1 − y1 , . . . ,
+s−1
+�
+i=0
+ciT ∗
+h,i,n − yn
+������
+=
+min
+z∈Rs+n
+≥0
+z0+···+zs−1=1
+�����
+�s−1
+�
+i=0
+ziT ∗
+h,i,1 − zs , . . . ,
+s−1
+�
+i=0
+ziT ∗
+h,i,n − zs+n−1
+������
+where z = (c, y)
+= min
+z∈C p (z, r)
+where p(z, r) is Euclidean distance and
+C =
+�
+z ∈ Rs+n
+≥0 : z1 + · · · + zs = 1
+�
+.
+(b) Note, since p(z, r) is the distance between two closed sets, the minimum distance is realized by a
+point in each set. Hence,
+εh(r) = min
+z∈C p (z, r) = inf
+z∈C p (z, r) .
+(c) Note the following:
+i. By Section 3.1.5 of [2], p(z, r) is a convex function since p(z, r) is a norm.
+ii. By Section 3.2.5 of [2], εh(r) = inf
+z∈C p (z, r) is a convex function, since C is convex and p(z, r)
+is bounded below.
+25
+
+iii. By Corollary 3.5.3 in [4], since εh(r) is a convex function defined on a convex set R, εh(r) is
+continuous on the relative interior of R, ri (R) = R.
+iv. Hence, εh(r) is continuous at r = 0.
+C4: εh (0) > 0.
+Proof: Let r = 0. Then h = x2 · f.
+(a) Subclaim: T ∗
+h,i,j = Tf,i,j. Note that these are the entries of T ∗
+h,s−1 and Tf,s−1. We will prove the
+claim using the fact that Ts−1 = R−1
+s−1Ls−1 for any f.
+i. Note Rf,s−1 = Rh,s−1. This is clear from the definition of Rs−1. Hence R−1
+f,s−1 = R−1
+h,s−1.
+ii. Note that Lf,i,j = Lh,i,j+2. This is clear from the definition of Ls−1, since Lh,s−1 is composed
+of Lf,s−1 with two columns of zeroes added on the left.
+iii. Note that for i = 0, . . . , s − 1 and j = 0, . . . , n − 1,
+T ∗
+h,i,j = Th,i,j+2
+=
+s−1
+�
+k=0
+R−1
+h,i,kLh,k,j+2
+=
+s−1
+�
+k=0
+R−1
+f,i,kLf,k,j
+= Tf,i,j
+Hence T ∗
+h,i,j = Tf,i,j.
+(b) Subclaim: εh(0) = εf
+where εf stands for the minimum Euclidean distance between CH(Tf,0, . . . , Tf,s−1) and Rn
+≥0, that
+is,
+εf :=
+min
+x∈CH(Tf,0,...,Tf,s−1)
+y∈Rn
+≥0
+∥x − y∥.
+To see this, note
+εh(0) =
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+s−1
+�
+i=0
+ciT ���
+h,i − y
+�����
+=
+min
+c∈Rs
+≥0
+c0+···+cs−1=1
+y∈Rn
+≥0
+�����
+s−1
+�
+i=0
+ciTf,i − y
+�����
+=
+min
+x∈CH(Tf,0,...,Tf,s−1)
+y∈Rn
+≥0
+∥x − y∥
+= εf
+(c) Subclaim: εf > 0.
+Note since opt (f) = s, we have CH(Tf,0, . . . , Tf,s−1) ∩ Rn
+≥0 = ∅. Thus εf > 0.
+(d) From the above two subclaims, we have εh (0) > 0.
+From the above four claims, we immediately have
+∃
+r>0 opt(h) = opt(f).
+□
+26
+
+References
+[1] M. Avendano. Descartes’ rule of signs is exact! Journal of Algebra - J ALGEBRA, 324:2884–2892, 11
+2010.
+[2] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004.
+[3] D. Curtiss. Recent extensions of descartes rule of signs. Annals of Mathematics, 19(4):251–278, 6 1918.
+[4] C. Niculescu and L. Persson. Convex functions and their applications. Springer Verlag New York, 2006.
+[5] H. Poincar´e. Sur les ´equations alg´ebriques. Comptes Rendus, 97:1418–1419, 1888.
+27
+

FdE2T4oBgHgl3EQf-QlB/content/tmp_files/2301.04236v1.pdf.txt

@@ -0,0 +1,593 @@
+arXiv:2301.04236v1  [math.AP]  10 Jan 2023
+Infinitely many solutions for Kirchhoff equations with
+indefinite potential
+Shuai Jiang a, Shibo Liu b
+aSchool of Mathematical Sciences, Xiamen University
+Xiamen 361006, P.R. China
+bDepartment of Mathematical Sciences, Florida Institute of Technology
+Melbourne, FL 32901, USA
+Abstract. We obtain a sequence of solutions converging to zero for the Kirchhoff
+equation
+−
+�
+1 +
+ˆ
+Ω
+|∇u|2
+�
+∆u + V(x)u = f(u),
+u ∈ H1
+0(Ω)
+via truncating technique and a variant of Clark’s theorem due to Liu–Wang, where Ω
+is a bounded smooth domain Ω ⊂ RN. Similar result for Schr¨odinger-Poisson system
+on a bounded smooth domain Ω ⊂ R3 is also presented.
+1. Introduction
+In a recent paper [8], He and Wu studied the following elliptic boundary value
+problem
+−∆u + V(x)u = f (x, u),
+u ∈ H1
+0(Ω)
+with indefinite linear part −∆ + V, where Ω ⊂ RN is a bounded smooth domain and the
+odd nonlinearity f : Ω × R → R is sublinear at zero:
+lim
+|t|→0
+1
+t2
+ˆ t
+0
+f (x, s) ds = +∞.
+Using truncating technique and Liu–Wang’s variant of Clark’s theorem [9, Theorem
+1.1], they obtained a sequence of solutions conversing to zero in H1
+0(Ω).
+Motivated by [8], in this note we consider the following Kirchhoff equation on a
+bounded smooth domain Ω ⊂ RN,
+−
+�
+1 +
+ˆ
+Ω
+|∇u|2
+�
+∆u + V(x)u = f (x, u),
+u ∈ H1
+0(Ω).
+(1.1)
+We impose the following conditions on the potential V and the nonlinearity f ,
+(V) V ∈ C(Ω) is bounded;
+1
+
+2
+S. JIANG AND S. LIU
+( f1) f ∈ C(Ω × R) is subcritical, that is
+lim
+|t|→∞
+f (x, t)t
+|t|2∗
+= 0,
+where 2∗ =
+2N
+N − 2 is the critical exponent;
+( f2) f (x, ·) is odd for all x ∈ Ω, f (x, 0) = 0, and is sublinear at zero:
+lim
+|t|→0
+F(x, t)
+t2
+= +∞,
+where F(x, t) =
+ˆ t
+0
+f (x, s) ds.
+(1.2)
+We will prove the following theorem.
+Theorem 1.1. Suppose (V), (f1) and (f2) hold, then the problem (1.1) possesses a
+sequence of nontrivial solutions converging to zero.
+Boundary value problems of the form (1.1) are closely related to the wave equation
+ψtt −
+�
+a + b
+ˆ
+Ω
+|∇ψ|2
+�
+∆ψ = g(x, ψ),
+(t, x) ∈ (0, T) × Ω,
+which was used by G. Kirchhoff to investigate vibrations of elastic strings with chang-
+ing length. Starting from Alves et al. [1], where a variational approach is developed
+to solve (1.1), many existence results for (1.1) appear. For example, Cheng et al. [4]
+considered the case that V(x) = 0 and the nonlinearity is of the form
+f (x, t) = α(x) |t|q−2 t + g(x, t),
+(1.3)
+where q ∈ (1, 2), g(x, t) = o(|t|) as t → 0. Obviously such f satisfies our assumption
+( f2). Since they need H1
+0(Ω) ֒→ Lr(Ω) for r > 4, it is assumed in [4] that N ≤ 3.
+Furtado and Zanata [7] also considered (1.1) with V(x) = 0 and f as in (1.3); but they
+only imposed local conditions to g(x, t) for |t| small (g needs not be odd and subcritical
+for |t| large). Using some idea from Wang [11], they got a sequence of solutions {uk} for
+the truncated problem with an odd and subcritical ˜g in place of g, ˜g(x, t) = g(x, t) for |t|
+small; then applied L∞-estimate to show that |uk|∞ → 0 and concluded that for k large
+uk are solutions of the original problem. Since our problem (1.1) may be indefinite,
+such L∞-estimate seems not applicable, this is why we need f to be globally odd and
+subcritical. For more recent papers on Kirchhoff equations, the reader is referred to
+[5,6,10].
+When N = 3, for the following Schr¨odinger-Poisson system on a bounded smooth
+domain Ω
+
+−∆u + V(x)u + φu = f (x, u)
+in Ω,
+−∆φ = u2
+in Ω,
+u = φ = 0
+on ∂Ω,
+(1.4)
+we have similar result.
+Theorem 1.2. Suppose (V), (f1) and (f2) hold, then the problem (1.4) possesses a
+sequence of nontrivial solutions (un, φn) → (0, 0) in H1
+0(Ω) × H1
+0(Ω).
+Since the seminar work or Benci et al. [3], Schr¨odinger-Poisson system has been
+an active field of research, for recent work on Schr¨odinger-Poisson system on bounded
+domain we mention [2,12,13].
+
+INFINITELY MANY SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIAL
+3
+2. Proof of Theorem 1.1
+The dependence on x in f (x, u) is not essential in our discussion of (1.1) and (1.4).
+Therefore in what follows we write f (u) for f (x, u) for simplicity.
+It is well known that to find weak solutions of (1.1), it suffices to find critical points
+of the C1-functional Φ : H1
+0(Ω) → R defined by
+Φ(u) = 1
+2
+ˆ �
+|∇u|2 + V(x)u
+�
++ 1
+4
+�ˆ
+|∇u|2
+�2
+−
+ˆ
+F(u),
+(2.1)
+here and below the integrals are taken over Ω. Let E−, E0, and E+ be the negative
+space, null space, and positive space of the quadratic form (the first term) in (2.1). For
+u ∈ E := H1
+0(Ω), we always denote by u± and u0 the orthogonal projections of u on E±
+and E0. Because of the condition (V), there is an equivalent norm ∥ · ∥ on E such that
+Φ(u) = 1
+2
+�
+∥u+∥2 −
+���u−���
+2�
++ 1
+4
+�ˆ
+|∇u|2
+�2
+−
+ˆ
+F(u).
+We denote by (·, ·) the corresponding inner product.
+To prove Theorem 1.1 it suffices to find a sequence of critical points of Φ. For this
+purpose, we need the following variant of the Clark’s theorem due to Liu–Wang [9].
+Theorem 2.1 ([9, Theorem 1.1]). Let E be a Banach space and Φ ∈ C1(E, R) be an
+even coercive functional satisfying the (PS ) condition and Φ(0) = 0. If for any k ∈ N,
+there is an k-dimensional subspace Xk and ρk > 0 such that
+sup
+Xk∩S ρk
+Φ < 0,
+(2.2)
+where S r = {u ∈ E| ∥u∥ = r}, then Φ has a sequence of critical points uk � 0 such that
+Φ(uk) ≤ 0, uk → 0.
+As pointed out in He–Wu [8, Remark 2.5], in Theorem 2.1, instead of (PS ) con-
+dition, it suffices to assume (PS )c for c ≤ 0. That is, any sequence {un} such that
+Φ′(un) → 0 and Φ(un) → c ≤ 0, possesses a convergent subsequence.
+We need the following lemma.
+Lemma 2.2. If un ⇀ u in E, then
+lim
+n→∞
+��ˆ
+|∇un|2
+� ˆ
+∇un · ∇(un − u) −
+�ˆ
+|∇u|2
+� ˆ
+∇u · ∇(un − u)
+�
+≥ 0,
+(2.3)
+Proof. By direct computation we have
+�ˆ
+|∇un|2
+� ˆ
+∇un · ∇(un − u) −
+�ˆ
+|∇u|2
+� ˆ
+∇u · ∇(un − u)
+=
+�ˆ
+|∇un|2
+� ˆ
+|∇(un − u)|2 +
+�ˆ
+|∇un|2 −
+ˆ
+|∇u|2
+� ˆ
+∇u · ∇(un − u)
+≥
+�ˆ
+|∇un|2 −
+ˆ
+|∇u|2
+� ˆ
+∇u · ∇(un − u).
+
+4
+S. JIANG AND S. LIU
+Since un ⇀ u in E, the right hand side goes to zero. The desired result follows from
+taking lower limit on both sides of the above inequality.
+Now, we are ready to prove Theorem 1.1.
+Proof (
+Proof of Theorem 1.1). Let φ : [0, ∞) → R be a decreasing C∞-function
+such that |φ′(t)| ≤ 2,
+φ(t) = 1
+for t ∈ [0, 1] ,
+φ(t) = 0
+for t ≥ 2.
+We consider the following truncated functional I : E → R, which is a modification of
+the truncated functional used in [8],
+I(u) = 1
+2 ∥u∥2 − 1
+2
+�
+∥u∗∥2 + 2
+ˆ
+F(u)
+�
+φ(∥u∥2) + 1
+4
+�ˆ
+|∇u|2
+�2
+,
+(2.4)
+where u∗ = u− + u0 ∈ E− ⊕ E0. The derivative I′ is given by
+⟨I′(u), v⟩ =
+�
+1 −
+�
+∥u∗∥2 + 2
+ˆ
+F(u)
+�
+φ′(∥u∥2)
+�
+(u, v)
+−
+�
+(u∗, v∗) +
+ˆ
+f (u)v
+�
+φ(∥u∥2) +
+�ˆ
+|∇u|2
+� ˆ
+∇u · ∇v
+(2.5)
+for u, v ∈ E.
+We will apply Theorem 2.1 to I to get a sequence of critical points {uk} for I such
+that
+I(uk) ≤ 0,
+uk → 0.
+Since I(u) = Φ(u) for ∥u∥ ≤ 1, we see that for large k all the uk are critical points of Φ
+and Theorem 1.1 is proved.
+Obviously I is even. If ∥u∥ ≥ 2, then φ(∥u∥2) = 0. Hence
+I(u) = 1
+2 ∥u∥2 + 1
+4
+�ˆ
+|∇u|2
+�2
+≥ 1
+2 ∥u∥2 → +∞,
+as ∥u∥ → ∞.
+This means that I is coercive.
+To verify (PS )c for c ≤ 0, let {un} be a sequence in E such that I(un) → c ≤ 0,
+I′(un) → 0. Since I is coercive, {un} is bounded in E. Up to a subsequence, we may
+assume that un ⇀ u in E. Then
+−
+����u∗
+n
+���
+2 + 2
+ˆ
+F(un)
+�
+φ(∥un∥2) = 2I(un) − ∥un∥2 − 1
+2
+�ˆ
+|∇un|2
+�2
+≤ 0.
+Hence
+���u∗
+n
+���
+2 + 2
+ˆ
+F(un) ≥ 0.
+(2.6)
+Because φ′(∥un∥2) ≤ 0 and
+lim
+n→∞
+(un, un − u) = lim
+n→∞
+∥un∥2 − ∥u∥2 ≥ 0,
+
+INFINITELY MANY SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIAL
+5
+up to a further subsequence we may assume
+����u∗
+n
+���
+2 + 2
+ˆ
+F(un)
+�
+φ′(∥un∥2) (un, un − u) −→ α ≤ 0,
+(2.7)
+note here that by the boundedness of {un}, the coefficient of (un, un − u) is bounded.
+Thanks to Lemma 2.2, we may also assume
+�ˆ
+|∇un|2
+� ˆ
+∇un · ∇(un − u) −
+�ˆ
+|∇u|2
+� ˆ
+∇u · ∇(un − u) −→ β ≥ 0.
+(2.8)
+From the subcritical assumption (f1) and the compact embedding E ֒→ L2(Ω), it is well
+known that
+ˆ
+f (un) (un − u) → 0,
+ˆ
+f (u) (un − u) → 0.
+(2.9)
+Finally, because dim(E− ⊕ E0) < ∞, we also have
+�u∗
+n, u∗
+n − u∗� → 0,
+�u∗, u∗
+n − u∗� → 0.
+(2.10)
+Computing ⟨I′(un), un − u⟩ and ⟨I′(u), un − u⟩ via (2.5) then subtracting the results, we
+deduce from (2.7), (2.8), (2.9) and (2.10) that
+∥un − u∥2 = ⟨I′(un) − I′(u), un − u⟩
++
+����u∗
+n
+���
+2 + 2
+ˆ
+F(un)
+�
+φ′(∥un∥2) (un, un − u)
+−
+�
+∥u∗∥2 + 2
+ˆ
+F(u)
+�
+φ′(∥u∥2) (u, un − u)
++
+��u∗
+n, u∗
+n − u∗� +
+ˆ
+f (un) (un − u)
+�
+φ(∥un∥2)
+−
+��u∗, u∗
+n − u∗� +
+ˆ
+f (u) (un − u)
+�
+φ(∥u∥2)
+−
+�ˆ
+|∇un|2
+� ˆ
+∇un · ∇(un − u) +
+�ˆ
+|∇u|2
+� ˆ
+∇u · ∇(un − u)
+= �o(1) + α − β� → (α − β) ≤ 0.
+(2.11)
+It follows that un → u in E and I satisfies (PS )c for c ≤ 0.
+Finally, for k ∈ N, let Xk be an arbitrary k-dimensional subspace of E. There is
+Λk > 0 such that
+|u|2
+2 ≥ Λk ∥u∥2
+for u ∈ Xk.
+There is also a constant η > 0 such that for all u ∈ E we have
+ˆ
+|∇u|2 ≤ η ∥u∥2 .
+From (f2), there is δ > 0 such that
+F(t) ≥ 1 + η2
+Λk
+t2
+for t ∈ (−δ, δ) .
+(2.12)
+
+6
+S. JIANG AND S. LIU
+Take ρk ∈ (0, 1) such that if u ∈ Xk, ∥u∥ = ρk, then |u|∞ < δ. For u ∈ Xk ∩ S ρk we have
+|u(x)| ≤ δ for all x ∈ Ω. Hence by (2.12),
+I(u) = Φ(u) = 1
+2
+�
+∥u+∥2 −
+���u−���
+2�
++ 1
+4
+�ˆ
+|∇u|2
+�2
+−
+ˆ
+F(u)
+≤ ∥u∥2 + η2
+4 ∥u∥4 − 1 + η2
+Λk
+ˆ
+u2
+≤ η2
+4 ρ4
+k − η2ρ2
+k ≤ −3η2
+4 ρ2
+k.
+Thus
+sup
+Xk∩S ρk
+I ≤ −3η2
+4 ρ2
+k < 0.
+Now, by Theorem 2.1, I has a sequence of critical points {uk} such that uk → 0 in E.
+For some k0, if k ≥ k0 then ∥uk∥ < 1 and uk is a critical point of Φ. Hence Φ has a
+sequence of critical points {uk}k≥k0 converging to zero.
+3. Proof of Theorem 1.2
+Given u ∈ E, let φu be the solution of the second equation in the system (1.4). It is
+well known that if u ∈ E is a critical point of Φ : E → R,
+Φ(u) = 1
+2
+ˆ �
+|∇u|2 + V(x)u2�
++ 1
+4
+ˆ
+φuu2 −
+ˆ
+F(u)
+= 1
+2
+�
+∥u+∥2 −
+���u−���
+2�
++ 1
+4
+ˆ
+φuu2 −
+ˆ
+F(u),
+then (u, φu) is a solution of (1.4), this idea was initiated from Benci et al. [3]. Similar
+to (2.4) we consider a truncated functional I : E → R
+I(u) = 1
+2 ∥u∥2 − 1
+2
+�
+∥u∗∥2 + 2
+ˆ
+F(u)
+�
+φ(∥u∥2) + 1
+4
+ˆ
+φuu2.
+Then I is an even coercive functional with I(0) = 0. Similar to the last section, using
+( f2), for k-dimensional subspace Xk there is ρk > 0 such that (2.2) holds.
+To verify the (PS )c condition with c ≤ 0 for I, we need the following analogue of
+Lemma 2.2.
+Lemma 3.1. If un ⇀ u in E, then
+lim
+n→∞
+�ˆ
+φunun (un − u) −
+ˆ
+φuu (un − u)
+�
+= 0.
+(3.1)
+Proof. It is well known that φu is obtained from applying Riesz lemma to the func-
+tional ℓu : v �→
+´
+u2v on E. Thus
+∥φu∥ = ∥ℓu∥ = sup
+∥v∥=1
+�����
+ˆ
+u2v
+�����
+≤ sup
+∥v∥=1
+�
+|u2|3 |v|3/2
+�
+= |u|2
+6 sup
+∥v∥=1
+|v|3/2 ≤ C ∥u∥2 .
+(3.2)
+
+INFINITELY MANY SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIAL
+7
+Since {un} is bounded, we know that �φun
+� is also bounded in E. By the compactness of
+the embedding E ֒→ L12/5(Ω), up to a subsequence we have un → u in L12/5(Ω). Hence
+�����
+ˆ
+φunun (un − u)
+����� ≤
+���φun
+���6 |un|12/5 |un − u|12/5 → 0,
+because �φun
+� and {un} are bounded in L6(Ω) and L12/5(Ω), respectively. Similarly, the
+second integral in (3.1) vanishes as n → ∞.
+Let {un} be a (PS )c sequence of Φ with c ≤ 0. It is easy to see that (2.6) still holds
+in current situation, thus we have (2.7). Using (2.7), (2.9), (2.10), and Lemma 3.1 we
+have an analogue of (2.11)
+∥un − u∥2 → α ≤ 0.
+Thus un → u in E and (PS )c is verified. Applying Theorem 2.1, I has a sequence of
+critical points uk → 0. Since I(u) = Φ(u) for ∥u∥ ≤ 1, for large k, uk is critical point of
+Φ. Thus Φ has a sequence of critical points uk → 0 in E. From (3.2) we have φuk → 0
+in E. Thus (1.4) has a sequence of solutions (uk, φuk) → (0, 0) in E × E.
+References
+[1] C. O. Alves, F. J. S. A. Corrˆea, and T. F. Ma, Positive solutions for a quasilinear elliptic equation
+of Kirchhoff type, Comput. Math. Appl., 49 (2005), pp. 85–93.
+[2] C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schr¨odinger-Poisson
+system in bounded domains, Z. Angew. Math. Phys., 65 (2014), pp. 1153–1166.
+[3] V. Benci and D. Fortunato, An eigenvalue problem for the Schr¨odinger-Maxwell equations, Topol.
+Methods Nonlinear Anal., 11 (1998), pp. 283–293.
+[4] B. Cheng, X. Wu, and J. Liu, Multiple solutions for a class of Kirchhoff type problems with concave
+nonlinearity, NoDEA Nonlinear Differential Equations Appl., 19 (2012), pp. 521–537.
+[5] F. Faraci and K. Silva, On the Brezis-Nirenberg problem for a Kirchhoff type equation in high
+dimension, Calc. Var. Partial Differential Equations, 60 (2021), pp. Paper No. 22, 33.
+[6] M. C. Ferreira and P. Ubilla, A critical concave-convex Kirchhoff-type equation in R4 involving
+potentials which may vanish at infinity, Ann. Henri Poincar´e, 23 (2022), pp. 25–47.
+[7] M. F. Furtado and H. R. Zanata, Multiple solutions for a Kirchhoff equation with nonlinearity
+having arbitrary growth, Bull. Aust. Math. Soc., 96 (2017), pp. 98–109.
+[8] W. He and Q. Wu, Multiplicity results for sublinear elliptic equations with sign-changing potential
+and general nonlinearity, Bound. Value Probl., (2020), pp. Paper No. 159, 9.
+[9] Z. Liu and Z.-Q. Wang, On Clark’s theorem and its applications to partially sublinear problems,
+Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 32 (2015), pp. 1015–1037.
+[10] R. Pei and C. Ma, Multiple solutions for a Kirchhoff-type equation, Mediterr. J. Math., 17 (2020),
+pp. Paper No. 78, 16.
+[11] Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,
+NoDEA Nonlinear Differential Equations Appl., 8 (2001), pp. 15–33.
+[12] Z.-L. Yang and Z.-Q. Ou, Nodal solutions for Schr¨odinger-Poisson systems with concave-convex
+nonlinearities, J. Math. Anal. Appl., 499 (2021), pp. Paper No. 125006, 15.
+[13] S. Yu and Z. Zhang, Sufficient and necessary conditions for ground state sign-changing solutions
+to the Schr¨odinger-Poisson system with cubic nonlinearity on bounded domains, Appl. Math. Lett.,
+123 (2022), pp. Paper No. 107570, 5.
+

FdE2T4oBgHgl3EQf-QlB/content/tmp_files/load_file.txt

@@ -0,0 +1,300 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf,len=299
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='04236v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='AP]  10 Jan 2023  Infinitely many solutions for Kirchhoff equations with  indefinite potential  Shuai Jiang a, Shibo Liu b  aSchool of Mathematical Sciences, Xiamen University  Xiamen 361006, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' China  bDepartment of Mathematical Sciences, Florida Institute of Technology  Melbourne, FL 32901, USA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' We obtain a sequence of solutions converging to zero for the Kirchhoff  equation  −  �  1 +  ˆ  Ω  |∇u|2  �  ∆u + V(x)u = f(u),  u ∈ H1  0(Ω)  via truncating technique and a variant of Clark’s theorem due to Liu–Wang, where Ω  is a bounded smooth domain Ω ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Similar result for Schr¨odinger-Poisson system  on a bounded smooth domain Ω ⊂ R3 is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Introduction  In a recent paper [8], He and Wu studied the following elliptic boundary value  problem  −∆u + V(x)u = f (x, u),  u ∈ H1  0(Ω)  with indefinite linear part −∆ + V, where Ω ⊂ RN is a bounded smooth domain and the  odd nonlinearity f : Ω × R → R is sublinear at zero:  lim  |t|→0  1  t2  ˆ t  0  f (x, s) ds = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Using truncating technique and Liu–Wang’s variant of Clark’s theorem [9, Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1], they obtained a sequence of solutions conversing to zero in H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Motivated by [8], in this note we consider the following Kirchhoff equation on a  bounded smooth domain Ω ⊂ RN,  −  �  1 +  ˆ  Ω  |∇u|2  �  ∆u + V(x)u = f (x, u),  u ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1)  We impose the following conditions on the potential V and the nonlinearity f ,  (V) V ∈ C(Ω) is bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  1  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' JIANG AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' LIU  ( f1) f ∈ C(Ω × R) is subcritical, that is  lim  |t|→∞  f (x, t)t  |t|2∗  = 0,  where 2∗ =  2N  N − 2 is the critical exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  ( f2) f (x, ·) is odd for all x ∈ Ω, f (x, 0) = 0, and is sublinear at zero:  lim  |t|→0  F(x, t)  t2  = +∞,  where F(x, t) =  ˆ t  0  f (x, s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2)  We will prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Suppose (V), (f1) and (f2) hold, then the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) possesses a  sequence of nontrivial solutions converging to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Boundary value problems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) are closely related to the wave equation  ψtt −  �  a + b  ˆ  Ω  |∇ψ|2  �  ∆ψ = g(x, ψ),  (t, x) ∈ (0, T) × Ω,  which was used by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Kirchhoff to investigate vibrations of elastic strings with chang-  ing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Starting from Alves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' [1], where a variational approach is developed  to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1), many existence results for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' For example, Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' [4]  considered the case that V(x) = 0 and the nonlinearity is of the form  f (x, t) = α(x) |t|q−2 t + g(x, t),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='3)  where q ∈ (1, 2), g(x, t) = o(|t|) as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Obviously such f satisfies our assumption  ( f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Since they need H1  0(Ω) ֒→ Lr(Ω) for r > 4, it is assumed in [4] that N ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Furtado and Zanata [7] also considered (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) with V(x) = 0 and f as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' but they  only imposed local conditions to g(x, t) for |t| small (g needs not be odd and subcritical  for |t| large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Using some idea from Wang [11], they got a sequence of solutions {uk} for  the truncated problem with an odd and subcritical ˜g in place of g, ˜g(x, t) = g(x, t) for |t|  small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' then applied L∞-estimate to show that |uk|∞ → 0 and concluded that for k large  uk are solutions of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Since our problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) may be indefinite,  such L∞-estimate seems not applicable, this is why we need f to be globally odd and  subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' For more recent papers on Kirchhoff equations, the reader is referred to  [5,6,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  When N = 3, for the following Schr¨odinger-Poisson system on a bounded smooth  domain Ω  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  −∆u + V(x)u + φu = f (x, u)  in Ω,  −∆φ = u2  in Ω,  u = φ = 0  on ∂Ω,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4)  we have similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Suppose (V), (f1) and (f2) hold, then the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4) possesses a  sequence of nontrivial solutions (un, φn) → (0, 0) in H1  0(Ω) × H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Since the seminar work or Benci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' [3], Schr¨odinger-Poisson system has been  an active field of research, for recent work on Schr¨odinger-Poisson system on bounded  domain we mention [2,12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  INFINITELY MANY SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIAL  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1  The dependence on x in f (x, u) is not essential in our discussion of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Therefore in what follows we write f (u) for f (x, u) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  It is well known that to find weak solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1), it suffices to find critical points  of the C1-functional Φ : H1  0(Ω) → R defined by  Φ(u) = 1  2  ˆ �  |∇u|2 + V(x)u  �  + 1  4  �ˆ  |∇u|2  �2  −  ˆ  F(u),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1)  here and below the integrals are taken over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Let E−, E0, and E+ be the negative  space, null space, and positive space of the quadratic form (the first term) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' For  u ∈ E := H1  0(Ω), we always denote by u± and u0 the orthogonal projections of u on E±  and E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Because of the condition (V), there is an equivalent norm ∥ · ∥ on E such that  Φ(u) = 1  2  �  ∥u+∥2 −  ���u−���  2�  + 1  4  �ˆ  |∇u|2  �2  −  ˆ  F(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  We denote by (·, ·) the corresponding inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1 it suffices to find a sequence of critical points of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' For this  purpose, we need the following variant of the Clark’s theorem due to Liu–Wang [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1 ([9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Let E be a Banach space and Φ ∈ C1(E, R) be an  even coercive functional satisfying the (PS ) condition and Φ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' If for any k ∈ N,  there is an k-dimensional subspace Xk and ρk > 0 such that  sup  Xk∩S ρk  Φ < 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2)  where S r = {u ∈ E| ∥u∥ = r}, then Φ has a sequence of critical points uk � 0 such that  Φ(uk) ≤ 0, uk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  As pointed out in He–Wu [8, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='5], in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1, instead of (PS ) con-  dition, it suffices to assume (PS )c for c ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' That is, any sequence {un} such that  Φ′(un) → 0 and Φ(un) → c ≤ 0, possesses a convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  We need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' If un ⇀ u in E, then  lim  n→∞  ��ˆ  |∇un|2  � ˆ  ∇un · ∇(un − u) −  �ˆ  |∇u|2  � ˆ  ∇u · ∇(un − u)  �  ≥ 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' By direct computation we have  �ˆ  |∇un|2  � ˆ  ∇un · ∇(un − u) −  �ˆ  |∇u|2  � ˆ  ∇u · ∇(un − u)  =  �ˆ  |∇un|2  � ˆ  |∇(un − u)|2 +  �ˆ  |∇un|2 −  ˆ  |∇u|2  � ˆ  ∇u · ∇(un − u)  ≥  �ˆ  |∇un|2 −  ˆ  |∇u|2  � ˆ  ∇u · ∇(un − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' JIANG AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' LIU  Since un ⇀ u in E, the right hand side goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' The desired result follows from  taking lower limit on both sides of the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Now, we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Proof (  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Let φ : [0, ∞) → R be a decreasing C∞-function  such that |φ′(t)| ≤ 2,  φ(t) = 1  for t ∈ [0, 1] ,  φ(t) = 0  for t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  We consider the following truncated functional I : E → R, which is a modification of  the truncated functional used in [8],  I(u) = 1  2 ∥u∥2 − 1  2  �  ∥u∗∥2 + 2  ˆ  F(u)  �  φ(∥u∥2) + 1  4  �ˆ  |∇u|2  �2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4)  where u∗ = u− + u0 ∈ E− ⊕ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' The derivative I′ is given by  ⟨I′(u), v⟩ =  �  1 −  �  ∥u∗∥2 + 2  ˆ  F(u)  �  φ′(∥u∥2)  �  (u, v)  −  �  (u∗, v∗) +  ˆ  f (u)v  �  φ(∥u∥2) +  �ˆ  |∇u|2  � ˆ  ∇u · ∇v  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='5)  for u, v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  We will apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1 to I to get a sequence of critical points {uk} for I such  that  I(uk) ≤ 0,  uk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Since I(u) = Φ(u) for ∥u∥ ≤ 1, we see that for large k all the uk are critical points of Φ  and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Obviously I is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' If ∥u∥ ≥ 2, then φ(∥u∥2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Hence  I(u) = 1  2 ∥u∥2 + 1  4  �ˆ  |∇u|2  �2  ≥ 1  2 ∥u∥2 → +∞,  as ∥u∥ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  This means that I is coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  To verify (PS )c for c ≤ 0, let {un} be a sequence in E such that I(un) → c ≤ 0,  I′(un) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Since I is coercive, {un} is bounded in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Up to a subsequence, we may  assume that un ⇀ u in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Then  −  ����u∗  n  ���  2 + 2  ˆ  F(un)  �  φ(∥un∥2) = 2I(un) − ∥un∥2 − 1  2  �ˆ  |∇un|2  �2  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Hence  ���u∗  n  ���  2 + 2  ˆ  F(un) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='6)  Because φ′(∥un∥2) ≤ 0 and  lim  n→∞  (un, un − u) = lim  n→∞  ∥un∥2 − ∥u∥2 ≥ 0,  INFINITELY MANY SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIAL  5  up to a further subsequence we may assume  ����u∗  n  ���  2 + 2  ˆ  F(un)  �  φ′(∥un∥2) (un, un − u) −→ α ≤ 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='7)  note here that by the boundedness of {un}, the coefficient of (un, un − u) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2, we may also assume  �ˆ  |∇un|2  � ˆ  ∇un · ∇(un − u) −  �ˆ  |∇u|2  � ˆ  ∇u · ∇(un − u) −→ β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='8)  From the subcritical assumption (f1) and the compact embedding E ֒→ L2(Ω), it is well  known that  ˆ  f (un) (un − u) → 0,  ˆ  f (u) (un − u) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='9)  Finally, because dim(E− ⊕ E0) < ∞, we also have  �u∗  n, u∗  n − u∗� → 0,  �u∗, u∗  n − u∗� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='10)  Computing ⟨I′(un), un − u⟩ and ⟨I′(u), un − u⟩ via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='5) then subtracting the results, we  deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='10) that  ∥un − u∥2 = ⟨I′(un) − I′(u), un − u⟩  +  ����u∗  n  ���  2 + 2  ˆ  F(un)  �  φ′(∥un∥2) (un, un − u)  −  �  ∥u∗∥2 + 2  ˆ  F(u)  �  φ′(∥u∥2) (u, un − u)  +  ��u∗  n, u∗  n − u∗� +  ˆ  f (un) (un − u)  �  φ(∥un∥2)  −  ��u∗, u∗  n − u∗� +  ˆ  f (u) (un − u)  �  φ(∥u∥2)  −  �ˆ  |∇un|2  � ˆ  ∇un · ∇(un − u) +  �ˆ  |∇u|2  � ˆ  ∇u · ∇(un − u)  = �o(1) + α − β� → (α − β) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='11)  It follows that un → u in E and I satisfies (PS )c for c ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Finally, for k ∈ N, let Xk be an arbitrary k-dimensional subspace of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' There is  Λk > 0 such that  |u|2  2 ≥ Λk ∥u∥2  for u ∈ Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  There is also a constant η > 0 such that for all u ∈ E we have  ˆ  |∇u|2 ≤ η ∥u∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  From (f2), there is δ > 0 such that  F(t) ≥ 1 + η2  Λk  t2  for t ∈ (−δ, δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='12)  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' JIANG AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' LIU  Take ρk ∈ (0, 1) such that if u ∈ Xk, ∥u∥ = ρk, then |u|∞ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' For u ∈ Xk ∩ S ρk we have  |u(x)| ≤ δ for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Hence by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='12),  I(u) = Φ(u) = 1  2  �  ∥u+∥2 −  ���u−���  2�  + 1  4  �ˆ  |∇u|2  �2  −  ˆ  F(u)  ≤ ∥u∥2 + η2  4 ∥u∥4 − 1 + η2  Λk  ˆ  u2  ≤ η2  4 ρ4  k − η2ρ2  k ≤ −3η2  4 ρ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Thus  sup  Xk∩S ρk  I ≤ −3η2  4 ρ2  k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Now, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1, I has a sequence of critical points {uk} such that uk → 0 in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  For some k0, if k ≥ k0 then ∥uk∥ < 1 and uk is a critical point of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Hence Φ has a  sequence of critical points {uk}k≥k0 converging to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2  Given u ∈ E, let φu be the solution of the second equation in the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' It is  well known that if u ∈ E is a critical point of Φ : E → R,  Φ(u) = 1  2  ˆ �  |∇u|2 + V(x)u2�  + 1  4  ˆ  φuu2 −  ˆ  F(u)  = 1  2  �  ∥u+∥2 −  ���u−���  2�  + 1  4  ˆ  φuu2 −  ˆ  F(u),  then (u, φu) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4), this idea was initiated from Benci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Similar  to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4) we consider a truncated functional I : E → R  I(u) = 1  2 ∥u∥2 − 1  2  �  ∥u∗∥2 + 2  ˆ  F(u)  �  φ(∥u∥2) + 1  4  ˆ  φuu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Then I is an even coercive functional with I(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Similar to the last section, using  ( f2), for k-dimensional subspace Xk there is ρk > 0 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  To verify the (PS )c condition with c ≤ 0 for I, we need the following analogue of  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' If un ⇀ u in E, then  lim  n→∞  �ˆ  φunun (un − u) −  ˆ  φuu (un − u)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' It is well known that φu is obtained from applying Riesz lemma to the func-  tional ℓu : v �→  ´  u2v on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Thus  ∥φu∥ = ∥ℓu∥ = sup  ∥v∥=1  �����  ˆ  u2v  �����  ≤ sup  ∥v∥=1  �  |u2|3 |v|3/2  �  = |u|2  6 sup  ∥v∥=1  |v|3/2 ≤ C ∥u∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2)  INFINITELY MANY SOLUTIONS FOR KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIAL  7  Since {un} is bounded, we know that �φun  � is also bounded in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' By the compactness of  the embedding E ֒→ L12/5(Ω), up to a subsequence we have un → u in L12/5(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Hence  �����  ˆ  φunun (un − u)  ����� ≤  ���φun  ���6 |un|12/5 |un − u|12/5 → 0,  because �φun  � and {un} are bounded in L6(Ω) and L12/5(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Similarly, the  second integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1) vanishes as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Let {un} be a (PS )c sequence of Φ with c ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' It is easy to see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='6) still holds  in current situation, thus we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='10), and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1 we  have an analogue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='11)  ∥un − u∥2 → α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Thus un → u in E and (PS )c is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='1, I has a sequence of  critical points uk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Since I(u) = Φ(u) for ∥u∥ ≤ 1, for large k, uk is critical point of  Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Thus Φ has a sequence of critical points uk → 0 in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='2) we have φuk → 0  in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Thus (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='4) has a sequence of solutions (uk, φuk) → (0, 0) in E × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Alves, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Corrˆea, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Ma, Positive solutions for a quasilinear elliptic equation  of Kirchhoff type, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 49 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 85–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Alves and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Souto, Existence of least energy nodal solution for a Schr¨odinger-Poisson  system in bounded domains, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 65 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 1153–1166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Benci and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Fortunato, An eigenvalue problem for the Schr¨odinger-Maxwell equations, Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  Methods Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 11 (1998), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 283–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Wu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Liu, Multiple solutions for a class of Kirchhoff type problems with concave  nonlinearity, NoDEA Nonlinear Differential Equations Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 19 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 521–537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Faraci and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Silva, On the Brezis-Nirenberg problem for a Kirchhoff type equation in high  dimension, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Partial Differential Equations, 60 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 22, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Ferreira and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Ubilla, A critical concave-convex Kirchhoff-type equation in R4 involving  potentials which may vanish at infinity, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Henri Poincar´e, 23 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 25–47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Furtado and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Zanata, Multiple solutions for a Kirchhoff equation with nonlinearity  having arbitrary growth, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 96 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 98–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' He and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Wu, Multiplicity results for sublinear elliptic equations with sign-changing potential  and general nonlinearity, Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Value Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 159, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [9] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Liu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Wang, On Clark’s theorem and its applications to partially sublinear problems,  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Poincar´e Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Non Lin´eaire, 32 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 1015–1037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Pei and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Ma, Multiple solutions for a Kirchhoff-type equation, Mediterr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 17 (2020),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 78, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [11] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,  NoDEA Nonlinear Differential Equations Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 8 (2001), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 15–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Yang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Ou, Nodal solutions for Schr¨odinger-Poisson systems with concave-convex  nonlinearities, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=', 499 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 125006, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Yu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Zhang, Sufficient and necessary conditions for ground state sign-changing solutions  to the Schr¨odinger-Poisson system with cubic nonlinearity on bounded domains, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=',  123 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}
+page_content=' 107570, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQf-QlB/content/2301.04236v1.pdf'}

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+SUPERCONCENTRATION FOR MINIMAL SURFACES IN
+FIRST PASSAGE PERCOLATION AND
+DISORDERED ISING FERROMAGNETS
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Abstract. We consider the standard first passage percolation model on Zd
+with a distribution G taking two values 0 < a < b. We study the maximal
+flow through the cylinder [0, n]d−1 × [0, hn] between its top and bottom as
+well as its associated minimal surface(s). We prove that the variance of the
+maximal flow is superconcentrated, i.e. in O( nd−1
+log n ), for h ≥ h0 (for a large
+enough constant h0 = h0(a, b)).
+Equivalently, we obtain that the ground state energy of a disordered Ising
+ferromagnet in a cylinder [0, n]d−1 × [0, hn] is superconcentrated when opposite
+boundary conditions are applied at the top and bottom faces and for a large
+enough constant h ≥ h0 (which depends on the law of the coupling constants).
+Our proof is inspired by the proof of Benjamini–Kalai–Schramm [3]. Yet,
+one major difficulty in this setting is to control the influence of the edges since
+the averaging trick used in [3] fails for surfaces.
+Of independent interest, we prove that minimal surfaces (in the present
+discrete setting) cannot have long thin chimneys.
+1. Introduction
+1.1. Context and main results. We focus in this paper on the fluctuations of the
+maximal flow (or equivalently of the minimal surface of the dual problem) through
+a cylinder in Zd of the form [0, n]d−1 × [0, H], where the vertical height H will
+be through most of this text of order hn. It is defined informally as follows (see
+Subsection 1.3 below for a more formal definition). Each non-oriented edge e inside
+[0, n]d−1 × [0, hn] carries an i.i.d capacity t(e) whose distribution takes two values
+0 < a < b. Without much loss of generality, one can think of t(e) ∈ {1, 2} with
+equal probability. The (vertical) maximum flow through this cylinder is informally
+the maximum amount of water which can be injected at the bottom, say, of the
+cylinder so that it can flow upwards in such a way that the amount of water flowing
+through any given edge e is less or equal than t(e). Let us denote this maximal flow
+by Φ = Φ([0, n]d−1 × {0}, H). By max-flow/min-cut principle, it is well-known that
+this maximal flow can be computed by minimizing the capacity over all possible
+cut-sets. I.e,
+Φ = min
+E
+��
+e∈E
+t(e)
+�
+,
+where the mimimum is taken over all cut-sets E which separate the bottom [0, n]d−1×
+{0} from the top [0, n]d−1 × {H}. There may be several such minimizing cut-sets E
+and by duality each of those correspond to a minimal surface embedded in Rd (see
+Figure 1).
+In dimension d = 2, the minimal cut-sets in [0, n] × [0, H] correspond to geodesics
+on the dual graph (Z2)∗ = Z2 + ( 1
+2, 1
+2) which connect the left and right boundaries
+of the rectangle. The maximal flow can then be studied as a random metric problem
+1
+arXiv:2301.11248v1  [math.PR]  26 Jan 2023
+
+2
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+in this special case and much is known about fluctuations, large-deviations etc. in
+this case. Let us mention in particular the breakthrough work by Benjamini-Kalai-
+Schramm [3] which implies in the present setting that Var[Φ([0, n] × {0}, H)] =
+O(
+n
+log n) as long as H = Ω(nϵ). Furthermore, in this d = 2 case, the fluctuations
+are believed to be described as n → ∞ by the KPZ universality class (in particular
+it is conjectured that Var[Φ] ≍ n2/3, see for example [19] where this is proved for
+directed last-passage percolation).
+In higher dimensions d ≥ 3, the problem may no longer be formulated in terms of
+geodesics and is expressed instead in terms of minimal surfaces (of co-dimension 1).
+The analysis of such maximal flows/minimal surfaces in d ≥ 3 was first considered in
+the seminal paper by Kesten for d = 3: Surfaces with minimal random weights and
+maximal flows: a higher dimensional version of first-passage percolation ([20]) where
+he obtained a law of large numbers for Φ as well as some large deviations estimates.
+Since the work [20], there has been a lot of activity on the analysis of the maximal
+flow Φ: Kesten’s results were extended by Zhang [27] to any dimensions, and by
+Rossignol–Théret in [24] to any dimensions for tilted flat cylinders (with height
+H = o(n)). Cerf–Théret proved a law of large number for more general domains
+in [5]. They later studied the speed of upper and lower large deviations in [6, 7].
+Interestingly, upper large deviations are in nd while lower large deviations are in
+nd−1. In [15, 14], Dembin–Théret proved upper and lower large deviations principle
+for the maximal flow in general domains.
+Let us now introduce another setting where minimal surfaces appear in the same
+fashion. Consider the disordered Ising ferromagnet in [0, n]d−1×[0, hn] with opposite
+boundary conditions applied at the top and the bottom. Each non-oriented edge
+e inside [0, n]d−1 × [0, hn] carries an i.i.d coupling constant Je whose distribution
+takes two values 0 < a < b. For a configuration σ ∈ {−1, 1}[0,n]d−1×[0,hn]∩Zd, its
+associated energy is
+H(σ) = −
+�
+e={x,y}
+Jeσxσy.
+One can check that the ground state energy (i.e. the minimal energy) corresponds
+to Φ and the corresponding minimal surface corresponds to the interface of a ground
+state (i.e. a configuration achieving the minimal energy). This connection was
+mentioned for example in Licea–Newman [21].
+To our knowledge, prior to this work, nothing was known on the fluctuations of
+Φ = Φ([0, n]d−1 × [0, H]) (besides the easy upper bound Var[Φ] = O(nd−1)). As
+we shall explain further in the next subsection, this may be due to the following
+reason. A crucial step in the proof of Benjamini-Kalai-Schramm in [3] is based on a
+beautiful averaging trick which no longer works with minimal surfaces.
+Our main result can be stated as follows.
+Theorem 1.1. For any d ≥ 2 and any distribution G on 0 < a < b, there exist
+C > 0 and h0 > 0, such that for any n ≥ 1 and H ≥ h0n, we have
+Var(Φ([0, n]d−1 × {0}, H) ≤ C nd−1
+log n .
+As it has been identified in the seminal work by Chatterjee [8], a variance of
+order O( nd−1
+log n ) versus a variance of order Ω(nd−1) induces a completely different
+behaviour of minimal cut-sets under small random perturbations of the capacities
+{t(e)}e. Indeed, a variance negligible w.r.t nd−1 corresponds to the phenomenon of
+superconcentration ([8]) and it implies a certain chaoticity property for the minimal
+cut-sets. We shall illustrate this in Corollary 6.1 where we will rely on a mild
+extension of a very useful identity from [26]. See also the recent work of Chatterjee [9]
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+3
+which analyzed the groundstate of an Ising model with non-ferromagnetic disordered
+coupling constants.
+We complete our analysis of the fluctuations of Φ = Φ([0, n]d−1 × {0}, H) by the
+following easier lower bound on the variance. Its proof in Section 5 will rely on the
+martingale decomposition method from Newman–Piza [22].
+Theorem 1.2. Let G be a distribution on {a, b} such that G({b}) > pc, where pc
+is the critical parameter for Bernoulli bond percolation on (Zd, Ed). There exists a
+constant c = c(G) > 0 such that for all n, H ≥ 1, we have
+Var(Φ([0, n]d−1 × {0}, H)) ≥ cnd−1
+H
+.
+We now introduce a slightly different model for which a greatly simplified version
+of our proof also implies superconcentration (see Remark 1 below). In the same
+cylinder [0, n]d−1 × [0, H], we now assign i.i.d weights {t(x)} to the vertices of the
+cylinder, again with a distribution G on 0 < a < b. We consider the following
+minimal weight
+ΨLip = ΨLip([0, n]d−1 × {0}, H) := min
+ψ
+�
+�
+�
+�
+u∈[0,n]d−1
+t(u, ψ(u))
+�
+�
+� ,
+where the minimum is taken over all 1-Lipschitz functions ψ : [0, n]d−1 → {0, 1, . . . , H}
+(i.e. such that |ψi − ψj| ≤ 1 for any i ∼ j in [0, n]d−1). We obtain in this setting
+the analog of Theorem 1.1.
+Theorem 1.3. There exist C, c > 0 and h0 > 0, both depending on 0 < a < b, such
+that for any n ≥ 1 and H ≥ h0n, we have
+�
+cnd−1
+H
+≤
+�
+Var(ΨLip([0, n]d−1 × {0}, H) ≤ C nd−1
+log n .
+To conclude this introduction, we wish to emphasise that if minimal surfaces
+happen to be anchored at some deterministic curve along the boundary of the
+cylinder, then we expect a completely different scenario for their fluctuations in
+large enough dimensions d. We discuss two possible such situations:
+(1) Instead of considering the maximum flow Φ from the bottom [0, n]d−1×{0} to
+the top [0, n]d−1×{H}, let us consider the maximal flow τ([0, n]d−1×{0}, H)
+between the bottom half and the top half of the cylinder, (i.e. between
+∂([0, n]d−1 × [0, H]) ∩ {x ∈ Rd, x · ed < H
+2 ]} and ∂([0, n]d−1 × [0, H]) ∩ {x ∈
+Rd, x·ed > H
+2 ]}). Then, the associated minimal surfaces are anchored in the
+boundary of the meridian plane of the cylinder [0, n]d−1 ×{ H
+2 }. For a formal
+definition, we refer to [24]. In high dimensions, by analogy to other models
+of surface (see in particular [23]), we expect that the anchored surface is
+localized, that is, there exists a constant C > 0 such that for any n, almost
+all the surface is within distance C of the meridian plane [0, n]d−1 × { H
+2 }.
+In that case, by a similar proof as Theorem 1.2, we can prove that there
+exists c > 0 depending on G such that for all n, H ≥ 1
+Var(τ([0, n]d−1 × {0}, H)) ≥ cnd−1 .
+This implies that in high dimensions, we don’t expect the variance of the
+anchored surface to be superconcentrated. This is another hint that minimal
+surfaces behave very differently as geodesics (of codimension d − 1) in
+standard first percolation theory.
+
+4
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+(2) In the spirit of the easier Theorem 1.3, we may further restrict the 1-Lipschitz
+functions ψ to be equal to H
+2 along ∂[0, n]d−1. The localisation result for
+uniform such 1-Lipschitz functions proved by Peled in [23] highly suggests
+that in high enough dimension, the variance of the associated minimal weight
+Ψanchored
+Lip
+will be ≥ cnd−1.
+We shall discuss this expected different behaviour further in Proposition 5.1 as well
+as in open question 1.
+1.2. Idea of proof.
+Benjamini-Kalai-Schramm and Talagrand. As we mentioned above, a similar
+theorem was first proved for the study of passage times in first passage percolation
+by Benjamini–Kalai–Schramm [3]. A key ingredient of [3] which we will also use
+is Talagrand’s inequality [25] (see Theorem 1.4). To obtain a “sub-surface” (i.e.
+o(nd−1)) upper-bound using Talagrand’s inequality, one needs to prove that most
+edges have a low influence on the maximal flow Φ. In [3], the influence of an edge is
+related to the probability that the geodesic goes through that edge. In our setting,
+it will be related to the probability that the minimal surfaces goes through the
+plaquette dual to that edge. We refer to [17, 16] for background on the interplay
+between Boolean functions and statistical physics.
+The main difficulty of this approach, already in [3], is that it happens to be very
+challenging to upper-bound the influence of any fixed given edge. In fact, for the
+passage times in first passage percolation, proving that the maximum influence in
+the bulk goes to zero (this is known as the BKS midpoint problem) was only proved
+a few years ago by Damron–Hanson [10], Ahlberg–Hoffman [1] and was recently
+solved quantitatively by Dembin–Elboim–Peled in [13].
+To circumvent this, Benjamini–Kalai–Schramm relied in [3] on a very nice av-
+eraging trick by randomizing the endpoints of the desired passage times. Since
+the randomized endpoints remain close to the original endpoints of the geodesic,
+it follows that the difference of passage times between the new geodesic and the
+original geodesic is negligible compared to the upper bound on standard deviation
+√n.
+No averaging trick for surfaces. We now explain why this averaging trick fails
+for surfaces. Indeed, consider two surfaces anchored respectively in the boundary of
+[0, n]d−1 × {0} and [0, n]d−1 × {1}, the best control we can get on the difference of
+capacity is of order nd−2. When d ≥ 3, we have nd−2 ≥ n(d−1)/2 where n(d−1)/2 is
+the order of the upper bound for the standard deviation for surfaces (obtained for
+example via Efron-Stein). This shows that as soon as d ≥ 3, we need to proceed
+differently as in [3] and a close inspection of influences will be needed.
+Idea and structure of the proof. We start by noting that if we were considering
+a maximal flow in a transitive graph, for example the maximal flow with non-trivial
+homology along the dth direction in a torus Td−1
+n
+× TH, then a direct application of
+Talagrand’s inequality (Theorem 1.4) would readily imply fluctuations of order at
+most n
+d−1
+2 /√log n for any H ≥ Ω(nϵ) just by using the fact that all edges have the
+same influence by transitivity of the graph.
+In our present case, despite the lack of transitive action acting on the cylinder
+[0, n]d−1 × [0, H], the rough idea is that if the minimal surface En (chosen among all
+possible minimal surfaces in any deterministic way, say) happens to be with high
+probability at distance at least 1 from the top and bottom boundary, then if we shift
+vertically by one the set of capacities {t(e)} (and also replace the missing bottom
+capacities by the top capacities that went off the cylinder), one may guess that,
+again with high probability, the new minimal surface En(tshifted) will be nothing but
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+5
+the vertical shift of En(t). Of course what could prevent this to happen comes from
+the effect of shuffling the top and bottom capacities. If one could prove that these
+two claims indeed happen with high enough probability, then it would imply that
+all edges in a vertical column have a very close influence which would allow us to
+conclude using Talagrand’s inequality 1.4.
+In the end, we do not quite succeed making this intuition rigorous but our proof
+is strongly influenced by analysing the effect of such vertical shifts. The proof of
+Theorem 1.1 will be based on the following three main steps which are of independent
+interest and do not have an analog in the analysis of Benjamini-Kalai-Schramm in
+[3]:
+(1) First, we shall prove that minimal surfaces cannot wiggle too much vertically.
+This will be achieved in Proposition 2.1. A similar phenomenon is known to
+arise in the analysis of minimal surfaces, see [12]. Our proof in the discrete
+setting will rely on the isoperimetric bounds in Zd obtained in [4]. This
+proposition is the technical step which is causing the restriction h ≥ h0 in
+our main theorem. Its proof will be given in Section 3.
+(2) Second, we need to know that minimal surfaces are unlikely to stay too
+close to the top and bottom boundaries. We will not prove this for the
+true minimal surfaces which lead to the maximal flow Φ([0, n]d−1 × {0}, H)
+but rather for a slightly modified notion of maximal flow in which minimal
+surfaces too close to the top and bottom boundaries receive a penalisation.
+This modified notion of maximal flow is called �Φ (see (5)) and is introduced
+in Section 2. For this modified maximal flow �Φ, we can show that the
+associated minimal surfaces are typically away from the top and bottom
+boundaries. This is the purpose of Proposition 4.1.
+(3) Finally, the last difficulty we are facing is the possibility that the minimal
+surface (for the modified �Φ) may often produce a high vertical cliff at certain
+locations. This would make the influence profile too inhomogeneous to
+allow us to control the magnitude of influences. Using a deep estimate from
+Zhang’s work [27] (inspired by the original work by Kesten [20]), we will
+prove Proposition 2.2 which shows that there are only few edges that may
+carry a large influence (we believe such edges do not exist but we cannot
+rule this out rigorously). Its proof will be the purpose of Section 4.
+Remark 1. We claim that one can prove Theorem 1.2 using the same proof, except
+there are several drastic simplifications. First, the absence of long thin chimneys
+(Proposition 2.1) is obvious in this case. Also, vertical cliffs do not exist by definition
+(thanks to the 1-Lipschitz condition) and as such Proposition 2.2 is much easier to
+prove in this case. We leave the details to the reader.
+1.3. Background.
+Definition of maximal flow. We now provide a more formal definition of maximal
+flows/minimal surfaces. We consider a first passage percolation on the graph (Zd, Ed)
+where Ed is the set of edges that link all the nearest neighbors for the Euclidean norm
+in Zd. Write (e1, . . . , ed) for the canonical basis of Rd. We consider a distribution
+G on R+. For each edge e in Ed we assign a random variable te of distribution G
+such that the family (te)e∈Ed is independent.
+Let A ⊂ Rd−1 × {0}. Let h > 0, we denote by cyl(A, h) the cylinder of basis A
+and height h defined by
+cyl(A, h) := {x + ted : x ∈ A, t ∈ [0, h]} .
+
+6
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Define the discretized versions B(A, h) and T(A, h) of the bottom and the top of
+the cylinder cyl(A, h)
+B(A, h) :=
+�
+x ∈ Zd ∩ cyl(A, h) :
+∃y /∈ cyl(A, h), ⟨x, y⟩ ∈ Ed
+and ⟨x, y⟩ intersects A
+�
+and
+T(A, h) :=
+�
+x ∈ Zd ∩ cyl(A, h) :
+∃y /∈ cyl(A, h), ⟨x, y⟩ ∈ Ed
+and ⟨x, y⟩ intersects A + hed
+�
+.
+Let E ⊂ Ed be a set of edges. We say that E cuts B(A, h) from T(A, h) in
+cyl(A, h) (or is a cutset, for short) if any path from B(A, h) to T(A, h) in cyl(A, h)
+intersects E.
+We associate with any set of edges E ⊂ Ed its capacity T(E) defined by
+T(E) :=
+�
+e∈E
+te .
+We define the maximal flow from the top to the bottom of the cylinder cyl(A, h)
+Φ(A, h) := min{T(E) : E cuts T(A, h) from B(A, h) in cyl(A, h)} .
+(1)
+As already mentioned in the introduction, we use the terminology maximal flow as
+by max-flow min-cut theorem, the dual problem of finding minimal surface boils
+down to computing the maximal flow.
+From now on, we assume that G can only take two values 0 < a < b. See Open
+Question 3 for a discussion of possible extensions to more general distributions using
+for example [2, 11].
+Dual representation of cutsets. Let E ⊂ Ed be a cutset separating T(A, h)
+from B(A, h) in cyl(A, h). The set E is a (d − 1)-dimensional object, that can be
+seen as a surface. To better understand this interpretation in term of surfaces,
+we can associate with each edge e ∈ E a small plaquette e∗. The plaquette e∗ is
+an hypersquare of dimension d − 1 whose sides have length one and are parallel
+to the edges of the graphs, such that e∗ is normal to e and cuts it in its middle.
+We also define the dual of a set of edge E by E∗ := {e∗, e ∈ E} (see Figure 1).
+Roughly speaking, if the set of edges E cuts T(A, h) from B(A, h) in cyl(A, h), the
+surface of plaquettes E∗ disconnects T(A, h) from B(A, h) in cyl(A, h). Note that,
+in dimension 2, a surface of plaquettes is very similar to a path in the dual graph of
+Z2 and thus the study of minimal cutsets is very similar to the study of geodesics.
+e
+e∗
+Figure 1. The dual of a cutset between the top and the bottom
+of a cylinder for d = 3.
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+7
+Concentration inequalities. Let J be a finite set of indices. For ω ∈ {a, b}J and
+j ∈ J denote σjω the function that switches the value in the j-th coordinate. For
+f : {a, b}J → R, denote
+∂jf := f − f ◦ σj
+2
+.
+For p ∈ (0, 1), consider µp the product measure on {a, b}J which gives a with
+probability p and b with probability 1 − p. We denote ∥f∥2
+2 =
+�
+f 2dµp.
+Theorem 1.4 (Talagrand’s inequality [25] Theorem 1.5). Let f : {a, b}J → R and
+p ∈ {0, 1}. We have
+Var(f) ≤ C log
+2
+p(1 − p)
+�
+j∈J
+∥∂jf∥2
+2
+1 + log(∥∂jf∥2/∥∂jf∥1)
+(2)
+where C is a universal constant.
+The following proposition is an upper bound on the variance using Efron–Stein
+inequality.
+Theorem 1.5 (Efron-Stein inequality). Let X = (X1, . . . , Xn) and X′ = (X′
+1, . . . , X′
+n)
+be two independent and identically distributed vectors taking values in a space X n.
+Let f : X n → R. We have
+Var(f(X)) ≤
+n
+�
+i=1
+E
+�
+(f(X) − E[f(X(i))|X])2�
+=
+n
+�
+i=1
+E
+�
+(f(X) − f(X(i)))2
+−
+�
+,
+where X(i) := (X1, . . . , Xi−1, X′
+i, Xi+1, . . . , Xn) and x− = max(−x, 0).
+2. Proof of the main theorem
+In this section, we state the main intermediate Propositions which were mentioned
+in the Section idea of proof and which will be proved in the next two Sections. We
+also implement the penalisation scheme used to “localize” the optimal surface away
+from the top and bottom boundaries. This will be the purpose of the re-weighting
+function Yi below. Finally, using these ingredients we give the proof of Theorem 1.1.
+Geometric control on minimal surfaces. The proposition stated below will be
+proved in Section 3.
+Proposition 2.1 (“Absence of long thin chimneys”). Fix 0 < a < b. There exists
+an even h0 > 0 depending only on 0 < a < b such that for any n ≥ 1, H ≥ 1
+2h0n and
+any configuration of capacities in {a, b} assigned to the edges of [0, n]d−1 ×[0, H], all
+minimal-cut sets E (i.e. that achieve the infimum in Φ([0, n]d−1 × {0}, H) defined
+in (1)) are contained in a cylinder of vertical height bounded by 1
+2h0n. I.e. for any
+minimal cut-set E, there exists some u ≥ 0 such that E ⊂ [0, n]d−1 × [u, u + 1
+2h0n].
+Fix H ≥ h0n. Write A = [0, n]d−1 × {0}. Define for i ≤ H − 1
+2h0n
+Xi := min
+�
+�
+�T(E) :
+E cuts B(A + ied, 1
+2h0n) from T(A + ied, 1
+2h0n)
+in cyl(A + ied, 1
+2h0n)
+and E ∩ (B(A + ied, 1
+2h0n) ∪ T(A + ied, 1
+2h0n)) ̸= ∅
+�
+�
+� .
+(3)
+
+8
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Penalisation scheme. Let 0 < ε < δ < 1/4. Set M := ⌊nε⌋ where ⌊x⌋ denotes
+the largest integer smaller than x. Let (Zi)1≤i≤M be a family of i.i.d. random
+variables that takes the value −1 with probability G({a}) and 1 with probability
+1 − G({a}) = G({b}). The reason for this choice is that to apply Talagrand formula
+(Theorem 1.4) the te and Zi must be parameterized by a Bernoulli random variable
+with the same parameter. Set
+SM :=
+M
+�
+k=1
+Zi.
+We define
+i0 :=
+�H
+2
+�
++ SM.
+In particular i0 is a random integer variable taking value in [⌊H/2⌋−M, ⌊H/2⌋+M].
+We define the family (Yi)1≤i≤H as follows
+∀1 ≤ i ≤ H
+Yi = Yi(i0) :=
+�
+0
+if |i0 − i| ≤ H
+2 − nδ
+n(d−1)/2
+nδ log n
+�
+|i0 − i| − H
+2 + nδ�
+otherwise.
+(4)
+Let j0 be such that
+Xj0 + Yj0 =
+min
+1≤i≤H− 1
+2 h0n Xi + Yi.
+If there are several possible choices, we pick the smallest. Let Emin(j0) be the surface
+achieving the minimum in the definition of Xj0. Again if there are several possible
+choices, we choose in a deterministic way (that is invariant by translation along the
+ed axis).
+Edges with large influence. The following proposition will be proved in Section
+4.
+Proposition 2.2. There exist n0 = n0(G) and ξ > 0 such that for all n ≥ n0
+���
+e ∈ cyl(A, H) : P(e ∈ Emin(j0)) ≥ n−ξ��� ≤ nd−1−ξ.
+We are now in position of proving Theorem 1.1.
+Proof of Theorem 1.1. Set E be the set of edges in cyl([0, n]d−1 × {0}, H). Let I be
+the set of indices that encode the choice of i0, in particular |I| = M. Set
+�Φ :=
+min
+1≤i≤H− 1
+2 h0n(Xi + Yi)
+(5)
+where (Xi)i was defined in (3) and (Yi)i in (4). Thanks to Proposition 2.1, we have
+Φ([0, n]d−1 × {0}, H) =
+min
+1≤i≤H− 1
+2 h0n Xi.
+It is easy to check that
+�����
+min
+1≤i≤H− 1
+2 h0n(Xi + Yi) −
+min
+1≤i≤H− 1
+2 h0n Xi
+����� ≤ n(d−1)/2
+log n
+.
+It follows that
+���E[�Φ] − E[Φ([0, n]d−1 × {0}, H)]
+��� ≤ n(d−1)/2
+log n
+.
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+9
+and
+Var(Φ([0, n]d−1 × {0}, H))
+= E((Φ([0, n]d−1 × {0}, H) − EΦ([0, n]d−1 × {0}, H))2)
+= E((Φ([0, n]d−1 × {0}, H) − �Φ + E�Φ − EΦ([0, n]d−1 × {0}, H) + �Φ − E�Φ)2)
+≤ 3
+�
+Var(�Φ) + 2nd−1
+log n
+�
+.
+(6)
+Let us compute the influence of the bits in I and E. For j ∈ I, we have |∂jSM| ≤ 2
+and it yields that
+|∂ji0| ≤ 2
+and
+|∂jYi0| ≤ 2n(d−1)/2
+nδ log n .
+As a result,
+∀j ∈ I
+|∂j �Φ|2 ≤
+4nd−1
+n2δ log2 n.
+Denote ∆e�Φ = �Φ ◦ σb
+e − �Φ ◦ σa
+e where σa
+e, σb
+e is the function that changes the value
+of the edge e to a, respectively b. We have
+P(∂e�Φ ̸= 0) = P(∆e�Φ ̸= 0) =
+1
+G({a})P(∆e�Φ ̸= 0, te = a) ≤
+1
+G({a})P(e ∈ Emin(j0)).
+Note that if ∆e�Φ ̸= 0 and te = a, then necessarily e has to belong to the minimal
+surface. For e ∈ E, thanks to the previous inequality, we have
+∥∂e�Φ∥2
+2 ≤ (b − a)2
+4
+P(∂e�Φ ̸= 0) ≤ (b − a)2
+4G({a})P(e ∈ Emin(j0)).
+Besides, we have by Cauchy–Schwarz inequality
+∥∂e�Φ∥1 = E
+����∂e�Φ
+���
+���
+≤
+�
+P(∂e�Φ ̸= 0) ∥∂e�Φ∥2 ≤
+�
+G({a})−1P(e ∈ Emin(j0)) ∥∂e�Φ∥2.
+Let n0 be as in the statement of Proposition 2.2. Finally, by applying Theorem 1.4
+and Proposition 2.2, we get for n ≥ n0
+Var(�Φ)
+≤ C
+�
+�
+�
+�
+�
+j∈I
+∥∂j �Φ∥2
+2 +
+�
+e∈E:
+P(e∈Emin(j0))≥n−ξ
+∥∂e�Φ∥2
+2 +
+�
+e∈E:
+P(e∈Emin(j0))<n−ξ
+∥∂e�Φ∥2
+2
+1 − log(G({a})−1P(e ∈ Emin(j0))/2
+�
+�
+�
+�
+≤ C
+�
+|I|
+nd−1
+n2δ log2 n + (b − a)2
+G({a}) nd−1−ξ +
+(b − a)2
+G({a})(1 + ξ
+4 log n)
+�
+e∈E
+P(e ∈ Emin(j0)))
+�
+.
+(7)
+Besides, note that the following set is a cutset from the top to the bottom of the
+cylinder cyl
+�
+A +
+� H
+2
+�
+ed, 1
+2h0n
+�
+F :=
+�
+{x, x + ed}, x ∈
+�
+[0, n]d−1 ×
+��H
+2
+���
+∩ Zd
+�
+.
+It follows that
+�Φ ≤ X⌊ H
+2 ⌋ ≤ b|F| = b(n + 1)d−1
+and
+a|Emin(j0)| ≤ b(n + 1)d−1.
+(8)
+We conclude by combining inequalities (6), (7) and (8).
+□
+
+10
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+3. Proof of Proposition 2.1 (absence of long chimneys)
+We shall need the following discrete isoperimetric inequality from [4] (N.B. the
+result in [4] is essentially sharp both in the side-length n and in the dimension d − 1,
+we only need the weaker statement given below).
+Theorem 3.1 (Corollary of Theorem 2 in [4]). For any d ≥ 2, there exists c =
+c(d) > 0 s.t. for any n ≥ 1 and any set A ⊂ [0, n]d−1,
+|∆A| ≥ c|A|1−
+1
+d−1 ∧ ((n + 1)d−1 − |A|)1−
+1
+d−1 ,
+where ∆A stands for the edge boundary of the set A in [0, n]d−1 (i.e.
+∆A :=
+�
+{i, j}, ∥i − j∥2 = 1, i ∈ A and j ∈ [0, n]d−1 \ A
+�
+).
+Proof of Proposition 2.1. Let h0 > 0 whose value will be chosen later depending on
+a and b. Let H ≥ 1
+2h0n and let E ⊂ Ed be a cut-set that achieves the infimum in
+Φ([0, n]d−1 × {0}, H).
+Let hmax be the maximum height in {0, . . . , H} of a vertex belonging to an edge in
+the minimal cut-set E. Define similarly hmin. Our goal is then to show that uniformly
+in the configuration of capacities {t(e)}, one necessarily has hmax − hmin ≤ h0
+2 .
+Scanning the upper horizontal slices. We start by scanning the upper horizon-
+tal layers of the cut-set E as follows. For any 1 ≤ i ≤ hmax, we call the ith upper
+layer, Ui := [0, n]d−1 × {hmax − i} and we define the following subset of Ui. Let
+A(i) ⊂ Ui be the set of all points x ∈ Ui such that any path γ connecting x to
+[0, n]d−1 × {H} inside the cylinder [0, n]d−1 × [hmax − i, H] necessarily intersects E.
+Let us start with the following two easy observations:
+• Since E is a minimal cut-set, it is easy to check that A(i) ̸= ∅ for all i ≥ 1.
+• Notice that the edge boundary ∆A(i) ⊂ E ∩ Ui (N.B. in general, there is
+no equality).
+We will need the following Lemma.
+Lemma 3.2. For each i ≥ 1, let Fi := E ∩ [0, n]d−1 × [0, hmax − i], i.e. the set
+of all edges in E that belong to the layer Ui or are below that layer. Then for any
+i ≥ 1, the set
+Ei := Fi ∪
+�
+{x, x − ed}, x ∈ A(i)
+�
+is a cut-set of the cylinder [0, n]d−1×[0, H]. (N.B. Its dual may no longer correspond
+to a simply connected surface. See Figure 2).
+Proof.
+Let γ = {x0, x1, . . . , xN} be any connected vertex-path connecting the
+bottom to the top of the cylinder. Let 1 ≤ m < N be the first time where the
+path reaches the layer Ui, i.e x0, . . . , xm−1 stays strictly below Li and xm ∈ Ui.
+We need to discuss the following two cases: First, if xm ∈ A(i), then we are
+done as the edge {xm−1, xm} belongs to
+�
+{x, x − ed}, x ∈ A(i)
+�
+. If, on the other
+hand, the point xm /∈ A(i), then we claim that the path {x0, . . . , xm} has necessarily
+intersected an edge of Fi. Indeed, if this was not the case then the path {x0, . . . , xm}
+would arrive at xm /∈ A(i) without ever crossing E and by definition of A(i), one
+could find a connected continuation of this path y1, . . . , yM such that the path
+x0, . . . , xm, y1, . . . , yM connects the bottom to the top of the cylinder without ever
+intersecting the cut-set E. This gives us a contradiction and thus concludes our
+proof.
+□
+The reason of this Lemma is that it immediately provides us with the following
+highly useful constraint: since E is a minimal cut-set and since Fi ∪
+�
+{x, x−ed}, x ∈
+A(i)
+�
+is a cut-set, we have for all i ≥ 1,
+a |E \ Fi| ≤ b|
+�
+{x, x − ed}, x ∈ A(i)
+�
+| = b |A(i)|
+(9)
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+11
+i
+Figure 2. Illustration in dimension d = 2(= 1 + 1) of the cut-set
+Ei defined in Lemma 3.2. It is made here of all the blue edges
+below level i as well as the additional green edges. By extrapolating
+such a picture in higher dimension d ≥ 3, one can easily produce
+situations where the set Ei splits into distant disconnected parts
+even though it arises from a minimal cut-set.
+We now define
+T := min{i ≥ 1 s.t. |A(i)| ≥ (1 − a
+10b)(n + 1)d−1} .
+(10)
+We shall prove the following Lemma.
+Lemma 3.3. For any 0 < a < b, there exists ϵ = ϵ(a, b) > 0 s.t. for any 1 ≤ k ≤
+T − 1,
+|∆A(k)| ≥ ϵ kd−2 ,
+The Lemma is easily proved by induction as follows. Unless T = 1, the lemma
+clearly holds for k = 1. (This is because in this case ∅ ⊊ A(1) ⊊ [0, n]d−1). Now,
+suppose the Lemma holds for a certain constant ϵ > 0 for all m < k ≤ T − 1.
+We shall use the above constraint (9) at the layer i = k. Notice that the set of
+edges E \ Fk is by definition the set of edges that are above the layer k (including
+some vertical edges pointing at that layer). In particular, this set is larger than the
+set of horizontal edges which lie above the kth layer Uk, namely,
+E \ Fk ⊃
+k−1
+�
+m=1
+E ∩ Um .
+Our next crucial point is the fact that for any m, as pointed out earlier, one has
+∆A(m) ⊂ E ∩ Um. As such, this gives us
+|E \ Fk| ≥
+k−1
+�
+m=1
+|E ∩ Um| ≥
+k−1
+�
+m=1
+|∆A(m)|
+≥ ϵ
+k−1
+�
+m=1
+md−2 ≥ ϵ C(d) kd−1 .
+Now plugging this into the constraint (9) gives us
+b|A(k)| ≥ a|E \ Fk| ≥ aϵ C(d) kd−1 .
+(11)
+
+12
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Now, using the fact that |A(k)| < (1 −
+a
+10b)(n + 1)d−1 (this is because k < T), we
+obtain from Theorem 3.1 that
+|∆A(k)| ≥ c(a, b)|A(k)|1−
+1
+d−1 .
+(Where for example c(a, b) = c( a
+20b)1−
+1
+d−1 ). Plugging this into (11) now gives us
+|∆A(k)| ≥ c(a, b)
+�aϵC(d)
+b
+�1−
+1
+d−1
+kd−2 .
+For 0 < a < b and the dimension d fixed, one can choose the constant ϵ small
+enough so that
+c(a, b)
+�aϵC(d)
+b
+�1−
+1
+d−1
+> ϵ ,
+which ends the proof of the Lemma.
+□
+Now using the Lemma 3.3 until k = T − 1, we extract the following estimate:
+C(d)ϵT d−1 ≤
+T −1
+�
+k=1
+|∆A(k)| ≤ |E \ FT | ≤ |E| ≤ b
+a(n + 1)d−1 .
+This implies the deterministic statement that the stopping time T is always bounded
+from above by ¯h0 n, where ¯h0 is a constant which only depends on 0 < a < b and
+the dimension d.
+The rest of the proof will proceed as follows: we will now scan horizontally the
+cut-set E from its bottom hmin and proceed upwards until we reach hmin + T ′. We
+will be left with showing that hmax − T cannot be much bigger than hmin + T ′. In
+order to keep a control on hmax − T versus hmin + T ′, it will be important to use
+exactly the same combinatorial definitions when scanning from below.
+Scanning the lower horizontal slices. We proceed in the same fashion. For
+any 1 ≤ i ≤ H − hmin, we call the ith lower layer, Li := [0, n]d−1 × {hmin + i}
+and we define the following subset of Li. Let ˆA(i) ⊂ Li be the set of all points
+x ∈ Li such that any path γ connecting x to [0, n]d−1 × {H} inside the cylinder
+[0, n]d−1 × [hmax − i, H] necessarily intersects E. (Notice and this is a key point
+that the set ˆA(i) is nothing but the previous set A(j) with j = hmax − hmin − i).
+We will need the following slight adaptation of Lemma 3.2 where we now add
+additional edges on the top of the complement of ˆA(i).
+Lemma 3.4. For each i ≥ 1, let Gi := E ∩ [0, n]d−1 × [hmin + i, H], i.e. the set
+of all edges in E that belong to the layer Li or are above that layer. Then for any
+i ≥ 1, the set
+ˆEi := Gi ∪
+�
+{x, x + ed}, x /∈ ˆA(i)
+�
+is a cut-set of the cylinder [0, n]d−1 × [0, H].
+Proof. Let γ = {x0, x1, . . . , xN} be any connected vertex-path connecting the bottom
+to the top of the cylinder. Let 1 ≤ m < N be the last passage time of this path
+through the layer Li. If xm ∈ ˆA(i), then by definition of this set, the rest of the
+connected path {xm, . . . , xN} will go through an edge in Gi. If on the other hand
+xm /∈ ˆA(i), then since xm is the last passage through Li, the next edge is necessarily
+a vertical edge {xm, xm+ed} which belongs to
+�
+{x, x + ed}, x /∈ ˆA(i)
+�
+, this ends the
+proof.
+□
+Similarly as for the upper layers, we define
+ˆT := min{i ≥ 1 s.t. |( ˆA(i))c| ≥ (1 − a
+10b)(n + 1)d−1} .
+(12)
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+13
+We claim that the exact same analysis as for the upper layers shows the following
+two facts:
+(1) for any 1 ≤ k ≤ ˆT − 1, |∆ ˆA(k)| = |∆( ˆA(k)c| ≥ ϵ kd−2.
+(2) ˆT ≤ ¯h0n.
+To conclude our proof, it remains to show that the upper layer where we stop the
+scanning from above, i.e. hmax − T cannot be much higher then the lower layer
+hmin + ˆT at which we stop the scanning from below. In fact, with our choices
+of stopping times T and ˆT, we will show more in the next Lemma, i.e. that up
+to a safety margin of 1, the top exploration necessarily stops below the bottom
+exploration.
+Lemma 3.5.
+hmin + ˆT + 1 ≥ hmax − T .
+To prove this Lemma, now that we have analyzed upper and lower horizontal
+slices, it remains to understand what would happen for the intermediate slices if
+they were to exist.
+Scanning the intermediate slices. Let us suppose by contradiction that hmin +
+ˆT + 1 < hmax − T. Introduce
+M := hmax − T − hmin − ˆT (M ≥ 2),
+the number of intermediate slices. Let us reparametrize the layers so that i = 0
+corresponds to the height hmin + ˆT while i = M corresponds to the top intermediate
+layer hmax − T. We shall denote by { ˜A(i)}1≤i≤M−1 the same sets as before (we
+use ˜A instead of A or ˆA just because of the reparametrization). Note that we have
+˜A(0) = ˆA( ˆT) and ˜A(M) = A(T).
+Lemma 3.6. For each 1 ≤ i ≤ M − 1, we have the following 2 constraints.
+(1) a| ˜A(i)c| ≤ b|A(T)c| (≤
+a
+10
+(n+1)d−1
+2
+)
+(2) a| ˜A(i)| ≤ b| ˆA( ˆT)| (≤
+a
+10
+(n+1)d−1
+2
+)
+For the inequalities in the parenthesis, we used the definitions of our stopping
+times T and ˆT (given in (10) and (12)). Conditions 1) and 2) are incompatible.
+Therefore this lemma implies that such intermediate layers cannot exist. This implies
+Lemma 3.5. To conclude the proof of Proposition 2.1, we are thus left with proving
+Lemma 3.6.
+Proof of Lemma 3.6. Let us start with item 1. Each point x in the intermediate
+layer i (i.e. at height hmin + ˆT + i) which belongs to the set ( ˜A(i))c has a path in
+its upper cylinder which connects it to [0, n]d−1 × {H} without intersecting E. By
+concatenating this path together with a vertical path pointing down all the way
+from x to the bottom face [0, n]d−1 × {0}, since E is a cut-set, it is necessary that
+at least one edges in this vertical path belongs to E. This implies in particular that
+we have at least |( ˜A(i))c| edges of E which are located below layer i. Finally, there
+cannot be too many such edges since E is a minimal cut-set. Using Lemma 3.4 for
+the layer at height hmax − T (or i = M), leads us precisely to the constraint 1).
+Item 2 is proved in a similar way. For any point x which belongs to ˜A(i), if we
+follow the vertical path above x until we reach the top layer [0, n]d−1 × {T}, then
+by definition of ˜A(i), the path will go through at least one edge of E. This implies
+in particular that there are at least | ˜A(i)| edges in E above (or touching) layer i.
+Now using Lemma 3.2 for the layer at height hmin + ˆT (or i = 0) together with the
+fact that E is minimal leads us to constraint 2.
+□
+
+14
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Remark 2. In the context of minimal surfaces in the continuum setting, a similar
+phenomenon of absence of “long thin chimneys" has been observed for example in
+[12].
+4. Proof of Proposition 2.2
+Let us first prove the following proposition which states that it is unlikely that
+the minimal surface Emin(j0) sticks to the bottom or the top of the cylinder.
+Proposition 4.1. There exists n0 = n0(G) ≥ 1 such that for all n ≥ n0, we have
+P (j0 ∈ {1, 2}) ≤
+2
+√n
+and
+P
+�
+j0 ∈
+�
+H − 1
+2h0n, H − 1
+2h0n − 1
+��
+≤
+2
+√n.
+To prove this proposition, we will need the following upper bound on the variance.
+Proposition 4.2 (Efron–Stein). There exists a constant β > 0 depending on G
+such that for all n ≥ 1 and H ≥ 1, we have
+Var(Φ([0, n]d−1 × {0}, H)) ≤ βnd−1 .
+Proof. The proof is a straightforward application of Theorem 1.5. Let e1, . . . , eN
+be a deterministic ordering of the edges of the cylinder cyl([0, n]d−1 × {0}, H)).
+Set X = (te1, . . . , ten) and f(X) = Φ([0, n]d−1 × {0}, H). Let Emin be a minimal
+surface for X (chosen according to a deterministic rule in case of ties). Recall that
+X(i) denotes the vector X where the i-th edge has been resampled. Note that if
+f(X) < f(X(i)) then ei belongs Emin. By similar reasoning as in (8), we have
+|Emin| ≤ b
+a(n + 1)d−1.
+By applying Theorem 1.5, it follows that
+Var(f(X)) ≤
+N
+�
+i=1
+(b − a)2P(ei ∈ Emin) ≤ (b − a)2 b
+a(n + 1)d−1.
+This concludes the proof.
+□
+Proof of Proposition 4.1. Thanks to Proposition 2.1, we have
+Φ([0, n]d−1 × {0}, H) =
+min
+1≤i≤H− h0
+2 n
+Xi .
+We will just prove the first inequality as the proof for the second inequality is similar.
+Let us assume by contradiction that
+P (j0 = 1) = P
+�
+min
+1≤i≤H− 1
+2 h0n Xi + Yi = X1 + Y1
+�
+≥
+1
+√n .
+We have for n large enough
+|i0 − 1| ≥ H
+2 − nε − 1 > H
+2 − nδ + nδ
+2
+and
+Y1 ≥ n(d−1)/2
+2 log n .
+For all i ∈ [2nδ, 3H/4], we have Yi = 0. On the event {min1≤i≤H− 1
+2 h0n Xi + Yi =
+X1 + Y1}, we have
+X1 ≤
+min
+i∈[2nδ,3H/4] Xi − n(d−1)/2
+2 log n .
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+15
+Hence,
+P
+�
+X1 ≤
+min
+i∈[2nδ,3H/4] Xi − n(d−1)/2
+2 log n
+�
+≥
+1
+√n .
+Set
+Ej :=
+�
+Xj ≤
+min
+i∈[j+2nδ,3H/4] Xi − n(d−1)/2
+2 log n
+�
+.
+Since the distribution of (Xi)1≤i≤3H/4 is the same as the distribution of (Xi)j≤i≤3H/4+j−1,
+we have
+P(Ej) ≥
+1
+√n .
+Set for 1 ≤ k ≤ H/4nδ
+Ik := [4knδ, 2(2k + 1)nδ]
+and
+Fk :=
+�
+j∈Ik
+Ej .
+Let N be the number of k ≤ H/4nδ such that Fk occurs, that is,
+N :=
+�
+1≤k≤H/4nδ
+1Fk .
+We have
+E[N] ≥
+�
+1≤k≤H/4nδ
+P(Fk) ≥ h0
+√n
+4nδ
+≥ h0
+2 n1/4
+(13)
+where we recall that H ≥ h0n. Let i1 < · · · < iN be integers such that they all
+belong to different intervals in (Ik, 1 ≤ k ≤ H/4nδ) and for all 1 ≤ j ≤ N, the
+event Eij occurs. Note that ij+1 − ij ≥ 2nδ since they belong to different intervals.
+Moreover, on the event Eij, we have
+Xij ≤ Xij+1 − n(d−1)/2
+2 log n .
+We can prove by induction that for 0 ≤ k ≤ N − 1
+XiN−k ≤
+min
+H/2+1≤i≤3H/4 Xi − (k + 1)n(d−1)/2
+2 log n .
+Hence,
+min
+1≤i≤H/2 Xi ≤
+min
+H/2+1≤i≤3H/4 Xi − Nn(d−1)/2
+2 log n .
+It follows that for t ≥ 0 using Bienaymé–Chebyshev’s inequality and Proposition 4.2
+P(N ≥ 2t log2 n) ≤ P
+�
+min
+H/2+1≤i≤3H/4 Xi −
+min
+1≤i≤H/2 Xi ≥ tn(d−1)/2
+�
+≤ 2Var(min1≤i≤H/2 Xi)
+t2nd−1
+≤ 2β
+t2 .
+It yields that
+E(N) ≤ 2(1 + 2β) log2 n .
+This contradicts inequality (13) for n large enough depending on G. By the same
+reasoning we can prove that
+P (j0 = 2) ≤
+1
+√n.
+This completes the proof.
+□
+To prove Proposition 2.2, we will also need the following lemma on the regularity
+of influences under translation by ed.
+
+16
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Lemma 4.3. There exists n0 = n0(G) such that for all n ≥ n0, H ≥ h0n the
+following holds. Let e be an edge of cyl(A, H) such that e + 2ed ⊂ cyl(A, H), we
+have
+|P(e ∈ Emin(j0)) − P(e + 2ed ∈ Emin(j0))| ≤
+2
+nε/2 .
+Proof of Lemma 4.3. Let (te)e∈cyl(A,h). We define t′
+e as follows
+t′
+e :=
+� te+2ed
+if e + 2ed ∈ cyl(A, H)
+t′′
+e
+otherwise
+where (t′′
+e)e∈cyl(A,h) is independent from (te). Let (Zi)1≤i≤M, (Z′
+i)1≤i≤M be two
+independent family of random variables that take the value −1 with probability
+G({a}) and 1 with probability 1 − G({a}) = G({b}). Set
+Sk :=
+k
+�
+k=1
+Zi
+and
+S′
+k :=
+k
+�
+k=1
+Z′
+i.
+Let
+τ := inf{k ∈ {1, . . . , M} : S′
+k ≥ Sk + 2}
+where we use the convention inf ∅ = +∞. Finally, we set
+i0 :=
+M
+�
+k=1
+Zk
+and
+i′
+0 :=
+min(τ,M)
+�
+k=1
+Z′
+k +
+M
+�
+k=min(τ,M)+1
+Zk.
+Denote by E′
+min(j′
+0) the minimal cutset corresponding to the family (t′
+e)e∈cyl(A,h)
+and i′
+0. It is easy to check that it has the same law as Emin(j0). Moreover, there
+exists a universal C > 0 s.t.
+P(i′
+0 − i0 ̸= 2) = P(τ = ∞) = P(∀k ∈ {1, . . . , M}
+Sk − S′
+k ≥ 0) ≤
+C
+√
+M
+.
+On the event {i′
+0 = i0 + 2} ∩ {j0 /∈ {H − 1
+2h0n, H − 1
+2h0n − 1}} ∩ {j′
+0 /∈ {1, 2}}, we
+have
+∀1 ≤ j ≤ H − 1
+2h0n − 2
+Xj(te) + Yj(i0) = Xj+2(t′
+e) + Yj+2(i′
+0)
+and Emin(j0) + 2ed = E′
+min(j′
+0). It yields
+|P(e ∈ Emin(j0)) − P(e + 2ed ∈ Emin(j0))|
+≤ P(i′
+0 − i0 ̸= 2) + P(j0 ∈ {1, 2}) + P
+�
+j0 ∈
+�
+H − 1
+2h0n, H − 1
+2h0n − 1
+��
+.
+Finally, by combining the two previous inequalities and using Proposition 4.1, it
+follows that for n ≥ n0 (where n0 is as in the statement of Proposition 4.1)
+|P(e ∈ Emin(j0)) − P(e + 2ed ∈ Emin(j0))| ≤
+2
+nε/2
+The result follows.
+□
+Proof of Proposition 2.2. Let n0 be as in the statement of Lemma 4.3. Let n ≥ n0.
+Let m ≥ 1 that we will choose later depending on n.
+Set k = ⌊n/m⌋.
+For
+i = (i1, . . . , id−1) ∈ {1, . . . , k}d−1, we define
+Ai :=
+d
+�
+j=1
+[(ij − 1)m, ijm) × {0} .
+We denote by J the set of cylinders that contain an edge such that P(e ∈ Emin(j0)) ≥
+n−ε/8, that is,
+J :=
+�
+i ∈ {1, . . . , k}d−1 : ∃e ∈ cyl(Ai, H)
+P(e ∈ Emin(j0)) ≥ n−ε/8�
+.
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+17
+Note that the set J is deterministic. By definition, the edges e ∈ cyl(Ai, H) for i /∈ J
+have a small influence. We need to make sure that there is a negligible number of
+edges with a large influence in cyl(Ai, H) for i ∈ J. In particular, we need to avoid
+that the minimal surface has a too large intersection with these cylinders.
+Let us first bound the size of J. Let us assume that there exists e ∈ cyl(Ai, H) such
+that P(e ∈ Emin(j0)) ≥ n−ε/8. Without loss of generality assume that e + √ned ∈
+cyl(Ai, H). By Proposition 4.3, we have
+|P(e ∈ Emin(j0)) − P(e + 2jed ∈ Emin(j0))| ≤
+2j
+nε/2 .
+Hence, for every j ≤ nε/4/4, we have
+P(e + 2jed ∈ Emin(j0)) ≥
+1
+nε/8 − 2j
+nε/2 ≥
+1
+2nε/8 .
+It yields that
+E[|Emin(j0) ∩ cyl(Ai, H)|] ≥ nε/4
+8nε/8 ≥ 1
+8nε/8 .
+Hence, we get using inequality (8)
+|J|nε/8
+8
+≤
+�
+i∈J
+E[|Emin(j0) ∩ cyl(Ai, H)|] ≤ E[|Emin(j0) ∩ cyl(A, H)|] ≤ b
+a(n + 1)d−1,
+it follows that for some positive constant β depending on a, b and d
+|J| ≤ βnd−1−ε/8.
+Next, we aim at upper bounding the total influence of edges in cyl(Ai, H) for i ∈ J,
+that is E [|Emin(j0) ∩ ∪i∈J cyl(Ai, H)|].
+Let E be a cutset in the cylinder cyl(A, h), one can check that E ∩ cyl(Ai, H) is
+also a cutset from the top to the bottom for the cylinder cyl(Ai, H). It follows that
+Φ(Ai, H) ≤ T(E ∩ cyl(Ai, H)).
+Hence, it yields
+�
+i∈{1,...,k}d−1\J
+Φ(Ai, H)+a
+�
+i∈J
+|Emin(j0)∩cyl(Ai, H)| ≤ T(Emin(j0)) ≤ Φ(A, H)+n(d−1)/2.
+Taking the expectation, we get
+aE
+��
+i∈J
+|Emin(j0) ∩ cyl(Ai, H)|
+�
+≤ E[Φ(A, H)]−
+�
+i∈{1,...,k}d−1\J
+E[Φ(Ai, H)]+n(d−1)/2 .
+(14)
+To control the right hand side, we will need a result of Zhang [27].
+Let K = ⌈n/(m − ⌊m5/6⌋)⌉. Set A′ := [0, K(m − ⌊m5/6⌋)]d−1 × {0} where K was
+chosen in such a way that A ⊂ A′. Thanks to the fine study of Zhang [27, inequality
+(10.22)], there exists C > 0 such that we have
+E[Φ(A′, H)] ≤
+�
+i∈{1,...,K}d−1
+E[Φ(Ai, H)] + C nd−1
+m1/16 .
+(15)
+Let us briefly explain how to prove this inequality. Let us assume we could
+prescribe in each cylinder cyl(Ai, H) a boundary condition for the minimal surface
+(that is the trace of the surface on the lateral side) in such a way that these boundary
+conditions match for adjacent cylinders. In other words, by taking the union of all
+minimal cutsets in cyl(Ai, H), i ∈ {1, . . . , k}d−1, one would get a cutset in the big
+cylinder cyl(A, H) and so Φ(A, H)] ≤ � Φ(Ai, H). The issue with this strategy is
+as follows: in order to prescribe a boundary condition without affecting too much
+the expectation E[Φ(Ai, H)], one needs that the trace of the minimal cutset on the
+
+18
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+lateral sides is negligible with nd−1. Since this fact is not known, Zhang overpasses
+this issue by slightly reducing the dimensions of the cylinder’s basis (it accounts for
+the m − ⌊m5/6⌋)): since the total size of the minimal surface is of order md−1, we
+can find a smaller cylinder where the trace of the minimal surface on the lateral
+sides is negligible. Once we can prescribe a given boundary condition, we use the
+symmetry to prescribe to adjacent cylinders some symmetric matching boundary
+conditions. The union of all these cutsets form a cutset in the big cylinder. Since
+the cylinders with prescribed boundary conditions are smaller than the original ones,
+we need to use a larger K ≥ k to be sure that A ⊂ A′.
+Let us now explain how we can control the right hand side of (14) using (15)
+from [27, inequality (10.22)]. The notation τmin(k1, . . . , kd−1, m) corresponds to
+Φ(�
+i=1...d[0, ki] × {0}, m). We apply the inequality with k1 = · · · = kd−1 = m,
+w1 = · · · = wd−1 = K, δ = 1/2. With these notations, the left hand side in (10.22)
+is equal to E[Φ(A′, H)]. Since A ⊂ A′, we have E[Φ(A, H)] ≤ E[Φ(A′, H)]. It follows
+that
+E[Φ(A, H)]−
+�
+i∈{1,...,k}d−1\J
+E[Φ(Ai, H)]
+≤ E[Φ(A, H)] −
+�
+i∈{1,...,K}d−1
+E[Φ(Ai, H)] + (|J| + (K − k)d−1)bmd−1
+≤ C nd−1
+m1/16 + bβnd−1−ε/8md−1 + b
+nd−1
+m(d−1)/6 .
+(16)
+Finally, combining (14) and (16), we get
+a E
+��
+i∈J
+|Emin(j0) ∩ cyl(Ai, H)|
+�
+≤ nd−1
+m1/16 + bβnd−1−ε/8md−1 + n(d−1)/2.
+Now choose m = nε/(16(d−1)). There exists ξ ≤ ε/16 depending on ε such that for n
+large enough
+E
+��
+i∈J
+|Emin(j0) ∩ cyl(Ai, H)|
+�
+≤ nd−1−ξ.
+We conclude that
+�����
+�
+e ∈
+�
+i∈J
+cyl(Ai, H) : P(e ∈ Emin(j0)) ≥ n−ξ/2
+������ ≤ nd−1−ξ/2.
+Since ξ ≤ ε/16, we have by definition of J
+���
+�
+e ∈ cyl(A, H) : P(e ∈ Emin(j0)) ≥ n−ξ/2���� ≤ nd−1−ξ/2
+(indeed, in the remaining cylinders, all edges have influence less than n−ε/8 which is
+smaller than n−ξ/2). As such, the result follows.
+□
+5. Proof of Theorem 1.2 and fluctuations of anchored
+surfaces
+We start with the proof of Theorem 1.2 which relies on the martingale decompo-
+sition method from Newman–Piza [22].
+Proof of Theorem 1.2.
+Let e1, . . . , eN be a deterministic ordering of the edges of the cylinder cyl([0, n]d−1×
+{0}, H)). Denote by Fk the σ-algebra generated by te1, . . . , tek. To simplify the
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+19
+notations, denote f(te1, . . . , teN ) = Φ([0, n]d−1 × {0}, H)). We have the following
+martingale decomposition
+Var(f) =
+N
+�
+k=1
+E[(E(f|Fk) − E(f|Fk−1))2).
+Let (t′
+e) be an independent family distributed as (te) and denote
+tk := (te1, . . . , tek, t′
+ek+1, . . . , t′
+eN ),
+tk
+a := (te1, . . . , tek−1, a, t′
+ek+1, . . . , t′
+eN )
+and
+tk
+b := (te1, . . . , tek−1, b, t′
+ek+1, . . . , t′
+eN ).
+In particular, we have
+f(tk) = (tek − a)1f(tk
+b )−f(tka)>0 + f(tk
+a).
+If f(tk
+b)−f(tk
+a) > 0 we say that the edge ek is pivotal. We can rewrite the expression
+as follows
+Var(f) =
+N
+�
+k=1
+E[E(f(tk) − f(tk−1)|(te)e)2) =
+N
+�
+k=1
+E(E((tek − t′
+ek)1f(tk
+b )−f(tk
+a)>0|(te)e)2)
+≥ Var(te)
+N
+�
+k=1
+P(f(tk
+b) − f(tk
+a) > 0)2
+≥ Var(te)
+N
+�
+k=1
+P(ek ∈ Emin, tek = b)2.
+When G({b}) > pc(d), there exists c > 0 such that the number of disjoint paths
+from the top to the bottom of the cylinder with only edges of time b is at least cnd−1
+with high probability (see for instance Theorem 7.68 in [18]). In particular, we have
+E[#{e ∈ Emin, te = b}] ≥ cnd−1.
+It follows that by Cauchy-Schwarz inequality
+Var(f) ≥ Var(te)
+N
+E[#{e ∈ Emin, te = b}]2 ≥ c0
+nd−1
+H
+where c0 depends on G and d.
+□
+The same proof allows us to show that fluctuations for anchored surfaces are
+not superconcentrated under the following hypothesis (H) of localisation. For any
+sequence (hn) such that hn goes to infinity with n, we have
+lim
+C→∞ lim sup
+n→∞
+1
+nd−1 E[#{e ∈ Emin : e /∈ {x ∈ Rd : |x · ed − hn
+2 | ≤ C}] = 0
+(H)
+where Emin is the minimal cutset for the anchored flow τ([0, n]d−1, hn).
+Proposition 5.1. Under the hypothesis (H), the variance of the anchored flow
+τ([0, n]d−1, H) (defined at the end of the introduction) is in Ω(nd−1).
+6. Chaoticity of the minimal surface
+Consider the notations of the previous section: f(te1, . . . , teN ) = Φ([0, n]d−1 ×
+{0}, H)). Set X := (te1, . . . , teN ). Let X′ be an independent vector distributed as
+X. Consider (U1, . . . , UN) an i.i.d. family of uniform random variables on [0, 1]. For
+any t ∈ [0, 1], we define
+∀ 1 ≤ i ≤ N
+Xt
+i :=
+� Xi
+if Ui ≥ t
+X′
+i
+otherwise.
+
+20
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+Denote by Pt the set of pivotal edges for f(Xt) and by It the set of edges that are
+in the intersection of all the minimal surfaces for f(Xt). It is easy to check that
+It ⊂ Pt. Following [8], we obtain the following Corollary of Theorem 1.1.
+Corollary 6.1. There exists a positive constant C such that for any n ≥ 1 and
+H ≥ h0n
+∀t ≥ 0
+E[|I0 ∩ It|] ≤ E[|P0 ∩ Pt|] ≤ C
+nd−1
+t log n Var(te).
+More precisely, this result follows from the following mild extension of Lemma
+3.3 from [26].
+Lemma 6.2 (Small extension of Lemma 3.3 in [26]). For any n ≥ 1 and H ≥ h0n,
+we have
+Var(Φ([0, n]d−1 × {0}, H)) = Var(te)
+� 1
+0
+E[|P0 ∩ Pt|]dt .
+Moreover, the function t → E[|P0 ∩ Pt|] is non-increasing.
+7. Open questions
+Open question 1. Prove that anchored maximal flow / minimal surfaces are not
+superconcentrated in high enough dimension d. (Thanks to Proposition 5.1, this
+boils down to showing that Hypothesis (H) holds).
+Open question 2. Prove superconcentration for maximal flows/minimal surfaces
+in more general domains, as considered for example in [5, 6, 7]. In fact, even
+extending Theorem 1.1 to the case of tilted cylinders with a rational slope appears to
+be challenging as Zhang’s inequality from [27] relies strongly on symmetry and does
+not adapt easily to rational directions.
+Open question 3. In this work, we focused on distributions G taking two values
+0 < a < b. It would be interesting to extend this analysis to more general distributions.
+The works [2, 11] by Benaïm–Rossignol and Damron–Hanson–Sosoe, where they
+extend the study of [3] to more general distributions are likely to play a key role
+here.
+Note that for a continuous distribution G, the chaoticity property proved in
+Corollary 6.1 would be more meaningful as the minimal surface would then be a.s.
+unique. In particular one would control the true intersection of minimal surfaces
+before and after noise.
+Open question 4. Our main result, Theorem 1.1, only works for thick enough
+cylinders (H ≥ h0n, for some large enough constant h0). This barrier h0 is there
+only for technical reasons (coming from Proposition 2.1). Show that the result still
+holds for any H ≥ Ω(nϵ).
+Open question 5. How do the fluctuations scale with n ? Is there an exponent
+α(d) ∈ (d − 2, d − 1) which describes the variance of Φ([0, n]d−1 × {0}, H) when H
+is, say, linear in n ?
+Acknowledgments. We wish to thank Itai Benjamini, Guy David, Simon Masnou,
+Ron Peled and Hugo Vanneuville for useful discussions. The research of B.D is
+supported by the European Research Council (ERC) under the European Union’s
+Horizon 2020 research and innovation program (grant agreement No 851565). The
+research of C.G. is supported by the Institut Universitaire de France (IUF) and the
+French ANR grant ANR-21-CE40-0003.
+
+SUPERCONCENTRATION FOR MINIMAL SURFACES
+21
+References
+[1] Daniel Ahlberg and Christopher Hoffman.
+Random coalescing geodesics in first-passage
+percolation, 2019.
+[2] Michel Benaïm and Raphaël Rossignol. Exponential concentration for first passage percolation
+through modified Poincaré inequalities. Annales de l’Institut Henri Poincaré, Probabilités et
+Statistiques, 44(3):544 – 573, 2008.
+[3] Itai Benjamini, Gil Kalai, and Oded Schramm. First passage percolation has sublinear distance
+variance. Ann. Probab., 31:1970–1978, January 2003.
+[4] Béla Bollobás and Imre Leader. Edge-isoperimetric inequalities in the grid. Combinatorica,
+11(4):299–314, 1991.
+[5] Raphaël Cerf and Marie Théret. Law of large numbers for the maximal flow through a domain
+of Rd in first passage percolation. Trans. Amer. Math. Soc., 363(7):3665–3702, 2011.
+[6] Raphaël Cerf and Marie Théret. Lower large deviations for the maximal flow through a
+domain of Rd in first passage percolation. Probability Theory and Related Fields, 150:635–661,
+2011.
+[7] Raphaël Cerf and Marie Théret. Upper large deviations for the maximal flow through a
+domain of Rd in first passage percolation. Annals of Applied Probability, 21(6):2075–2108,
+2011.
+[8] Sourav Chatterjee. Superconcentration and related topics, volume 15. Springer, 2014.
+[9] Sourav Chatterjee. Spin glass phase at zero temperature in the edwards-anderson model.
+arXiv preprint arXiv:2301.04112, 2023.
+[10] Michael Damron and Jack Hanson. Bigeodesics in first-passage percolation. Communications
+in Mathematical Physics, 349(2):753–776, 2017.
+[11] Michael Damron, Jack Hanson, and Philippe Sosoe.
+Sublinear variance in first-passage
+percolation for general distributions. Probability Theory and Related Fields, 163(1):223–258,
+Oct 2015.
+[12] Guy David and Stephen Semmes. Quasiminimal surfaces of codimension 1 and john domains.
+pacific journal of mathematics, 183(2):213–277, 1998.
+[13] Barbara Dembin, Dor Elboim, and Ron Peled. Coalescence of geodesics and the bks midpoint
+problem in planar first-passage percolation, 2022.
+[14] Barbara Dembin and Marie Théret. Large deviation principle for the streams and the maximal
+flow in first passage percolation, 2020.
+[15] Barbara Dembin and Marie Théret. Large deviation principle for the cutsets and lower large
+deviation principle for the maximal flow in first passage percolation, 2021.
+[16] Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion. Sharp phase transition for the
+random-cluster and potts models via decision trees. Annals of Mathematics, 189(1):75–99,
+2019.
+[17] Christophe Garban and Jeffrey E Steif. Noise sensitivity of Boolean functions and percolation,
+volume 5. Cambridge University Press, 2014.
+[18] Geoffrey Grimmett.
+Percolation, volume 321 of Grundlehren der Mathematischen Wis-
+senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin,
+second edition, 1999.
+[19] Kurt Johansson. Shape fluctuations and random matrices. Communications in mathematical
+physics, 209(2):437–476, 2000.
+[20] Harry Kesten. Surfaces with minimal random weights and maximal flows: a higher dimensional
+version of first-passage percolation. Illinois Journal of Mathematics, 31(1):99–166, 1987.
+[21] Cristina Licea and Charles M. Newman. Geodesics in two-dimensional first-passage percolation.
+The Annals of Probability, 24(1):399 – 410, 1996.
+[22] Charles M Newman and Marcelo ST Piza. Divergence of shape fluctuations in two dimensions.
+The Annals of Probability, pages 977–1005, 1995.
+[23] Ron Peled. High-dimensional lipschitz functions are typically flat. The Annals of Probability,
+45(3):1351–1447, 2017.
+[24] Raphaël Rossignol and Marie Théret. Lower large deviations and laws of large numbers for
+maximal flows through a box in first passage percolation. Ann. Inst. Henri Poincaré Probab.
+Stat., 46(4):1093–1131, 2010.
+[25] Michel Talagrand.
+On Russo’s Approximate Zero-One Law.
+The Annals of Probability,
+22(3):1576 – 1587, 1994.
+[26] Vincent Tassion and Hugo Vanneuville.
+Noise sensitivity of percolation via differential
+inequalities. arXiv preprint arXiv:2011.04572, 2020.
+[27] Yu Zhang. Limit theorems for maximum flows on a lattice. Probability Theory and Related
+Fields, May 2017.
+
+22
+BARBARA DEMBIN
+CHRISTOPHE GARBAN
+D-MATH, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
+Email address: [email protected]
+Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille
+Jordan, 69622 Villeurbanne, France , Institut Universitaire de France
+(IUF) and Université de Genève (Unige)
+Email address: [email protected]
+

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