Pub Date : 2024-11-05DOI: 10.1016/j.jco.2024.101907
Harmandeep Singh , Janak Raj Sharma
A two-step Newton-like method is proposed to efficiently solve the systems of nonlinear equations. Extending Newton scheme to a next step as weighted-Newton iteration, the proposed iteration scheme shows optimal fourth order of convergence. The primary objective in formulating the method is to keep the computational efficiency as high as possible. In this context, the efficiency analysis is thoroughly examined using a systematic approach, wherein the efficiency index of the new method is compared with those of existing methods of comparable complexity. Numerical experimentation is performed to investigate the computational efficacy of the developed method. Results indicate higher efficiency and numerical precision in comparison to the existing counterparts.
{"title":"A two-point Newton-like method of optimal fourth order convergence for systems of nonlinear equations","authors":"Harmandeep Singh , Janak Raj Sharma","doi":"10.1016/j.jco.2024.101907","DOIUrl":"10.1016/j.jco.2024.101907","url":null,"abstract":"<div><div>A two-step Newton-like method is proposed to efficiently solve the systems of nonlinear equations. Extending Newton scheme to a next step as weighted-Newton iteration, the proposed iteration scheme shows optimal fourth order of convergence. The primary objective in formulating the method is to keep the computational efficiency as high as possible. In this context, the efficiency analysis is thoroughly examined using a systematic approach, wherein the efficiency index of the new method is compared with those of existing methods of comparable complexity. Numerical experimentation is performed to investigate the computational efficacy of the developed method. Results indicate higher efficiency and numerical precision in comparison to the existing counterparts.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101907"},"PeriodicalIF":1.8,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.jco.2024.101906
Marcel Fernández , John Livieratos , Sebastià Martín
This paper presents an algorithmic approach to the construction of separating codes. In the first part of the work, the Lovász Local Lemma is used to obtain a lower bound on the code rate. This lower bound matches the previously best-known lower bound. In the second part, it is shown how the technique used in proving the lower bound leads to an algorithm that outputs an instance of a separating code. Moreover, the implications of the algorithm regarding computational complexity are considered. The discussion ends by presenting explicit separating codes with polynomial computational complexity in the length of the code, with rate that improves previously known constructions.
本文提出了一种构建分离码的算法方法。在工作的第一部分，利用 Lovász Local Lemma 获得了码率下限。该下限与之前最著名的下限相吻合。第二部分展示了证明下限时使用的技术如何导致一种算法输出分离代码实例。此外，还考虑了该算法对计算复杂性的影响。讨论的最后，提出了计算复杂度与代码长度成多项式关系的显式分离代码，其速率改进了之前已知的构造。
{"title":"Combinatorial constructions of separating codes","authors":"Marcel Fernández , John Livieratos , Sebastià Martín","doi":"10.1016/j.jco.2024.101906","DOIUrl":"10.1016/j.jco.2024.101906","url":null,"abstract":"<div><div>This paper presents an algorithmic approach to the construction of separating codes. In the first part of the work, the Lovász Local Lemma is used to obtain a lower bound on the code rate. This lower bound matches the previously best-known lower bound. In the second part, it is shown how the technique used in proving the lower bound leads to an algorithm that outputs an instance of a separating code. Moreover, the implications of the algorithm regarding computational complexity are considered. The discussion ends by presenting explicit separating codes with polynomial computational complexity in the length of the code, with rate that improves previously known constructions.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101906"},"PeriodicalIF":1.8,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-22DOI: 10.1016/j.jco.2024.101902
Erich Novak, Kateryna Pozharska, Mathias Sonnleitner, Michaela Szölgyenyi, Henryk Woźniakowski
{"title":"Matthieu Dolbeault is the winner of the 2024 Joseph F. Traub Information-Based Complexity Young Researcher Award","authors":"Erich Novak, Kateryna Pozharska, Mathias Sonnleitner, Michaela Szölgyenyi, Henryk Woźniakowski","doi":"10.1016/j.jco.2024.101902","DOIUrl":"10.1016/j.jco.2024.101902","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101902"},"PeriodicalIF":1.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142525769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jco.2024.101903
K.Yu. Osipenko
The paper concerns problems of the recovery of linear operators defined on sets of functions from information of these functions given with stochastic errors. The constructed optimal recovery methods, in general, do not use all the available information. As a consequence, optimal methods are obtained for recovering derivatives of functions from Sobolev classes by the information of their Fourier transforms given with stochastic errors. A similar problem is considered for solutions of the heat equation.
{"title":"Optimal recovery of linear operators from information of random functions","authors":"K.Yu. Osipenko","doi":"10.1016/j.jco.2024.101903","DOIUrl":"10.1016/j.jco.2024.101903","url":null,"abstract":"<div><div>The paper concerns problems of the recovery of linear operators defined on sets of functions from information of these functions given with stochastic errors. The constructed optimal recovery methods, in general, do not use all the available information. As a consequence, optimal methods are obtained for recovering derivatives of functions from Sobolev classes by the information of their Fourier transforms given with stochastic errors. A similar problem is considered for solutions of the heat equation.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101903"},"PeriodicalIF":1.8,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.jco.2024.101901
Erich Novak , Friedrich Pillichshammer
We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number N of function evaluations that is necessary such that the N-th minimal error is less than a factor ε times the initial error, i.e., the error for , where ε belongs to . We are interested to which extent the information complexity depends on the number d of variables of the integrands. If the information complexity grows exponentially fast in d, then the integration problem is said to suffer from the curse of dimensionality.
Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.
我们证明了张量乘空间中数值积分最坏情况误差的下限。信息复杂度是函数求值的最小次数 N，即 N 次最小误差小于初始误差的系数 ε 倍，即 N=0 时的误差，其中 ε 属于 (0,1)。我们感兴趣的是，信息复杂度在多大程度上取决于积分变量的数量 d。如果信息复杂度在 d 的范围内呈指数增长，那么积分问题就会受到维度诅咒的影响。在单变量问题存在最坏情况函数的假设下，我们提出了两种提供信息复杂度下限的方法。第一种方法基于对最坏情况函数的适当分解，可视为可分解再现核方法的一般化。第二种方法虽然只适用于正二次函数规则，但不需要对最坏情况函数进行适当分解。相反，它以最坏情况函数的样条近似为基础，可用于解析函数。本文介绍了这两种方法的几种应用。
{"title":"Intractability results for integration in tensor product spaces","authors":"Erich Novak , Friedrich Pillichshammer","doi":"10.1016/j.jco.2024.101901","DOIUrl":"10.1016/j.jco.2024.101901","url":null,"abstract":"<div><div>We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number <em>N</em> of function evaluations that is necessary such that the <em>N</em>-th minimal error is less than a factor <em>ε</em> times the initial error, i.e., the error for <span><math><mi>N</mi><mo>=</mo><mn>0</mn></math></span>, where <em>ε</em> belongs to <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We are interested to which extent the information complexity depends on the number <em>d</em> of variables of the integrands. If the information complexity grows exponentially fast in <em>d</em>, then the integration problem is said to suffer from the curse of dimensionality.</div><div>Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101901"},"PeriodicalIF":1.8,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.jco.2024.101900
Thomas Hangelbroek , Christian Rieger
Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a -cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.
{"title":"Kernel multigrid on manifolds","authors":"Thomas Hangelbroek , Christian Rieger","doi":"10.1016/j.jco.2024.101900","DOIUrl":"10.1016/j.jco.2024.101900","url":null,"abstract":"<div><div>Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a <span><math><mi>τ</mi><mo>≥</mo><mn>2</mn></math></span>-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101900"},"PeriodicalIF":1.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-24DOI: 10.1016/j.jco.2024.101899
Mingyao Ai , Yunfan Yang , Xiangshun Kong
This paper proposes a new approach to generating space-filling designs over Riemannian manifolds by using a Hilbert curve. Different from ordinary Euclidean spaces, a novel transformation is constructed to link the uniform distribution over a Riemannian manifold and that over its parameter space. Using this transformation, the uniformity of the design points in the sense of Riemannian volume measure can be guaranteed by the intrinsic measure preserving property of the Hilbert curve. It is proved that these generated designs are not only asymptotically optimal under minimax and maximin distance criteria, but also perform well in minimizing the Wasserstein distance from the target distribution and controlling the estimation error in numerical integration. Furthermore, an efficient algorithm is developed for numerical generation of these space-filling designs. The advantages of the new approach are verified through numerical simulations.
{"title":"Space-filling designs on Riemannian manifolds","authors":"Mingyao Ai , Yunfan Yang , Xiangshun Kong","doi":"10.1016/j.jco.2024.101899","DOIUrl":"10.1016/j.jco.2024.101899","url":null,"abstract":"<div><div>This paper proposes a new approach to generating space-filling designs over Riemannian manifolds by using a Hilbert curve. Different from ordinary Euclidean spaces, a novel transformation is constructed to link the uniform distribution over a Riemannian manifold and that over its parameter space. Using this transformation, the uniformity of the design points in the sense of Riemannian volume measure can be guaranteed by the intrinsic measure preserving property of the Hilbert curve. It is proved that these generated designs are not only asymptotically optimal under minimax and maximin distance criteria, but also perform well in minimizing the Wasserstein distance from the target distribution and controlling the estimation error in numerical integration. Furthermore, an efficient algorithm is developed for numerical generation of these space-filling designs. The advantages of the new approach are verified through numerical simulations.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101899"},"PeriodicalIF":1.8,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-24DOI: 10.1016/j.jco.2024.101898
Carlo Sanna
The Permuted Kernel Problem (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an matrix A and an vector b over a finite field of q elements , the PKP asks to find an permutation matrix π such that belongs to the kernel of A. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters . Previous works have heuristically estimated the expected number of solutions to be .
We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related Inhomogeneous Permuted Kernel Problem (IPKP), considering two natural ways of generating random instances.
{"title":"On the number of solutions to a random instance of the permuted kernel problem","authors":"Carlo Sanna","doi":"10.1016/j.jco.2024.101898","DOIUrl":"10.1016/j.jco.2024.101898","url":null,"abstract":"<div><div>The <em>Permuted Kernel Problem</em> (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an <span><math><mi>ℓ</mi><mo>×</mo><mi>m</mi></math></span> matrix <strong><em>A</em></strong> and an <span><math><mi>m</mi><mo>×</mo><mn>1</mn></math></span> vector <strong><em>b</em></strong> over a finite field of <em>q</em> elements <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, the PKP asks to find an <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> permutation matrix <strong><em>π</em></strong> such that <span><math><mi>π</mi><mi>b</mi></math></span> belongs to the kernel of <strong><em>A</em></strong>. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters <span><math><mi>q</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>m</mi></math></span>. Previous works have heuristically estimated the expected number of solutions to be <span><math><mi>m</mi><mo>!</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>ℓ</mi></mrow></msup></math></span>.</div><div>We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related <em>Inhomogeneous Permuted Kernel Problem</em> (IPKP), considering two natural ways of generating random instances.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101898"},"PeriodicalIF":1.8,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X2400075X/pdfft?md5=939873f4b51043507214927d47f2bb37&pid=1-s2.0-S0885064X2400075X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-13DOI: 10.1016/j.jco.2024.101897
Gaurav Mittal , Harshit Bajpai , Ankik Kumar Giri
In Banach spaces, the convergence analysis of iteratively regularized Landweber iteration (IRLI) is recently studied via conditional stability estimates. But the formulation of IRLI does not include general non-smooth convex penalty functionals, which is essential to capture special characteristics of the sought solution. In this paper, we formulate a generalized form of IRLI so that its formulation includes general non-smooth uniformly convex penalty functionals. We study the convergence analysis and derive the convergence rates of the generalized method solely via conditional stability estimates in Banach spaces for both the perturbed and unperturbed data. We also discuss few examples of inverse problems on which our method is applicable.
{"title":"Convergence analysis of iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces","authors":"Gaurav Mittal , Harshit Bajpai , Ankik Kumar Giri","doi":"10.1016/j.jco.2024.101897","DOIUrl":"10.1016/j.jco.2024.101897","url":null,"abstract":"<div><p>In Banach spaces, the convergence analysis of iteratively regularized Landweber iteration (IRLI) is recently studied via conditional stability estimates. But the formulation of IRLI does not include general non-smooth convex penalty functionals, which is essential to capture special characteristics of the sought solution. In this paper, we formulate a generalized form of IRLI so that its formulation includes general non-smooth uniformly convex penalty functionals. We study the convergence analysis and derive the convergence rates of the generalized method solely via conditional stability estimates in Banach spaces for both the perturbed and unperturbed data. We also discuss few examples of inverse problems on which our method is applicable.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101897"},"PeriodicalIF":1.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000748/pdfft?md5=5ae8eeac0a143f493ee150c18db69cf1&pid=1-s2.0-S0885064X24000748-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1016/j.jco.2024.101896
Raziyeh Erfanifar, Masoud Hajarian
Many practical problems, such as the Malthusian population growth model, eigenvalue computations for matrices, and solving the Van der Waals' ideal gas equation, inherently involve nonlinearities. This paper initially introduces a two-parameter iterative scheme with a convergence order of two. Building on this, a three-parameter scheme with a convergence order of four is proposed. Then we extend these schemes into higher-order schemes with memory using Newton's interpolation, achieving an upper bound for the efficiency index of . Finally, we validate the new schemes by solving various numerical and practical examples, demonstrating their superior efficiency in terms of computational cost, CPU time, and accuracy compared to existing methods.
{"title":"High-efficiency parametric iterative schemes for solving nonlinear equations with and without memory","authors":"Raziyeh Erfanifar, Masoud Hajarian","doi":"10.1016/j.jco.2024.101896","DOIUrl":"10.1016/j.jco.2024.101896","url":null,"abstract":"<div><p>Many practical problems, such as the Malthusian population growth model, eigenvalue computations for matrices, and solving the Van der Waals' ideal gas equation, inherently involve nonlinearities. This paper initially introduces a two-parameter iterative scheme with a convergence order of two. Building on this, a three-parameter scheme with a convergence order of four is proposed. Then we extend these schemes into higher-order schemes with memory using Newton's interpolation, achieving an upper bound for the efficiency index of <span><math><msup><mrow><mn>7.88748</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>≈</mo><mn>1.99057</mn></math></span>. Finally, we validate the new schemes by solving various numerical and practical examples, demonstrating their superior efficiency in terms of computational cost, CPU time, and accuracy compared to existing methods.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101896"},"PeriodicalIF":1.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000736/pdfft?md5=6dc221bf6c1e2ffc085c2b830768b4e4&pid=1-s2.0-S0885064X24000736-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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