Linear Algebra and its Applications最新文献

{"title":"On the local and global minimizers of the smooth stress function in Euclidean distance matrix problems","authors":"Mengmeng Song ,&nbsp;Douglas S. Gonçalves ,&nbsp;Woosuk L. Jung ,&nbsp;Carlile Lavor ,&nbsp;Antonio Mucherino ,&nbsp;Henry Wolkowicz","doi":"10.1016/j.laa.2025.08.012","DOIUrl":"10.1016/j.laa.2025.08.012","url":null,"abstract":"<div><div>We consider the nonconvex minimization problem, with quartic objective function, that arises in the exact recovery of a configuration matrix <span><math><mi>P</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>d</mi></mrow></msup></math></span> of <em>n</em> points when a Euclidean distance matrix, <strong>EDM</strong>, is given with embedding dimension <em>d</em>. It is an open question in the literature whether there are conditions such that the minimization problem admits a local nonglobal minimizer, <strong>lngm</strong>. We prove that all second-order stationary points are global minimizers whenever <span><math><mi>n</mi><mo>≤</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. And, for <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>7</mn><mo>&gt;</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>, we present an example where we can analytically exhibit a local nonglobal minimizer. For more general cases, we numerically find a second-order stationary point and then prove that there indeed exists a nearby <strong>lngm</strong> for the quartic nonconvex minimization problem. Thus, we answer the previously open question about their existence in the affirmative. Our approach to finding the <strong>lngm</strong> is novel in that we first exploit the translation and rotation invariance to remove the singularities of the Hessian, and reduce the size of the problem from <em>nd</em> variables in <em>P</em> to <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>d</mi><mo>−</mo><mi>d</mi><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> variables. This allows for stabilizing Newton's method, and for finding examples that satisfy the strict second order sufficient optimality conditions.</div><div>The motivation for being able to find global minima is to obtain <em>exact recovery</em> of the configuration matrix, even in the cases where the data is noisy and/or incomplete, without resorting to approximating solutions from convex (semidefinite programming) relaxations. In the process of our work we present new insights into when <strong>lngm</strong>s of the smooth stress function do and do not exist.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 234-267"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Irredundant generating sets for matrix algebras","authors":"Yonatan Blumenthal,&nbsp;Uriya A. First","doi":"10.1016/j.laa.2025.08.008","DOIUrl":"10.1016/j.laa.2025.08.008","url":null,"abstract":"<div><div>Let <em>F</em> be a field. We show that the largest irredundant generating sets for the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <em>F</em> have <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> elements when <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span>. (A result of Laffey states that the answer is <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span> when <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math></span>, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span> and <em>F</em> is algebraically closed. We use this description to compute the dimension of the variety of <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-tuples of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices which form an irredundant generating set when <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span>, and draw some consequences to <em>locally redundant</em> generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets <em>S</em> of subspaces of <span><math><msup><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with the property that every <span><math><mi>V</mi><mo>∈</mo><mi>S</mi></math></span> admits a matrix stabilizing every subspace in <span><math><mi>S</mi><mo>−</mo><mo>{</mo><mi>V</mi><mo>}</mo></math></span> and not stabilizing <em>V</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 308-335"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limit points of Aα-matrices of graphs","authors":"Elismar R. Oliveira,&nbsp;Vilmar Trevisan","doi":"10.1016/j.laa.2025.08.014","DOIUrl":"10.1016/j.laa.2025.08.014","url":null,"abstract":"<div><div>We study limit points of the spectral radii of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrices of graphs. Adapting a method used by J. B. Shearer in 1989, we prove a density property of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-limit points of caterpillars for <em>α</em> close to zero. Precisely, we show that for <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span> there exists a positive number <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>α</mi><mo>)</mo><mo>&gt;</mo><mn>2</mn></math></span> such that any value <span><math><mi>λ</mi><mo>&gt;</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>α</mi><mo>)</mo></math></span> is an <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-limit point. We also determine other intervals whose numbers are all <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-limit points.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 1-25"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tridiagonal generalized inverses of singular matrices possessing the triangle property","authors":"A.M. Encinas ,&nbsp;K. Kranthi Priya ,&nbsp;K.C. Sivakumar","doi":"10.1016/j.laa.2025.08.011","DOIUrl":"10.1016/j.laa.2025.08.011","url":null,"abstract":"<div><div>It is known that an invertible real square matrix has the triangle property if and only if the inverse is a tridiagonal matrix. This result has an implicit importance due to the fact that nonsingular tridiagonal matrices arise in a variety of problems in pure and applied mathematics and for this reason they have been extensively studied in the literature. However, the singular case has received comparatively much lesser attention. In particular, there has been little focus on the generalized inverses of such matrices. In this paper, we provide a complete description of those singular matrices possessing the triangle property to have the tridiagonal Moore-Penrose inverse or group inverse. The converse statements are also completely answered.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 268-307"},"PeriodicalIF":1.1,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Construction of exceptional copositive matrices","authors":"Tea Štrekelj ,&nbsp;Aljaž Zalar","doi":"10.1016/j.laa.2025.08.010","DOIUrl":"10.1016/j.laa.2025.08.010","url":null,"abstract":"<div><div>An <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrix <em>A</em> is copositive if the quadratic form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>A</mi><mi>x</mi></math></span> is nonnegative on the nonnegative orthant <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. The cone of copositive matrices contains the cone of matrices which are the sum of a positive semidefinite matrix and a nonnegative one and the latter contains the cone of completely positive matrices. These are the matrices of the form <span><math><mi>B</mi><msup><mrow><mi>B</mi></mrow><mrow><mi>T</mi></mrow></msup></math></span> for some <span><math><mi>n</mi><mo>×</mo><mi>r</mi></math></span> matrix <em>B</em> with nonnegative entries. The above inclusions are strict for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes ≥5, i.e., copositive matrices that are not the sum of a positive semidefinite matrix and a nonnegative one. The second contribution of this paper addresses the asymptotic ratio of the volume radii of compact sections of the cones of copositive and completely positive matrices. In a previous work by Klep and the authors, it was shown that, by identifying symmetric matrices naturally with quartic even forms, and equipping them with the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> inner product and the Lebesgue measure, the ratio of the volume radii of sections with a suitably chosen hyperplane is bounded below by a constant independent of <em>n</em> as <em>n</em> tends to infinity. In this paper, we complement this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 368-384"},"PeriodicalIF":1.1,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Lawrence–Krammer representation is a quantization of the symmetric square of the Burau representation","authors":"Alexandr V. Kosyak","doi":"10.1016/j.laa.2025.08.009","DOIUrl":"10.1016/j.laa.2025.08.009","url":null,"abstract":"<div><div>We show that the Lawrence–Krammer representation is a quantization of the symmetric square of the Burau representation. Here, by quantization we mean changing the natural numbers by <em>q</em>-natural numbers and the corresponding quantization of the Pascal triangle, appearing naturally in representations of the braid groups. This connection allows us to construct new representations of the braid groups.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 203-233"},"PeriodicalIF":1.1,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144894874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Graded representations of current Lie superalgebra sl(1|2)[t]","authors":"Shushma Rani ,&nbsp;Divya Setia","doi":"10.1016/j.laa.2025.08.007","DOIUrl":"10.1016/j.laa.2025.08.007","url":null,"abstract":"<div><div>In this paper, we study finite-dimensional graded representations of the current Lie superalgebra <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. We define the notion of super POPs, a combinatorial tool to provide another parametrization of the basis of the local Weyl module given by Brito, Calixto and Macedo. We derive the graded character formula of the local Weyl module for <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. Furthermore, we construct a short exact sequence of Chari-Venkatesh modules for <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. As a consequence, we prove that Chari-Venkatesh modules are isomorphic to the fusion product of generalized Kac modules.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 178-202"},"PeriodicalIF":1.1,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Kreĭn-Šmul'jan theorem revisited","authors":"Santiago Gonzalez Zerbo ,&nbsp;Alejandra Maestripieri ,&nbsp;Francisco Martínez Pería","doi":"10.1016/j.laa.2025.08.006","DOIUrl":"10.1016/j.laa.2025.08.006","url":null,"abstract":"<div><div>We present a generalization of the Kreĭn-Šmul'jan theorem involving several operators: Given bounded selfadjoint operators <span><math><mi>A</mi><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> acting on a Hilbert space <span><math><mi>H</mi></math></span>, we provide sufficient conditions to determine whether there are <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∈</mo><mi>R</mi></math></span> such that <span><math><mi>A</mi><mo>+</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a positive semidefinite operator.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 163-177"},"PeriodicalIF":1.1,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Metric Frobenius norms and inner products of matrices and linear maps","authors":"Roland Herzog ,&nbsp;Frederik Köhne ,&nbsp;Leonie Kreis ,&nbsp;Anton Schiela","doi":"10.1016/j.laa.2025.08.005","DOIUrl":"10.1016/j.laa.2025.08.005","url":null,"abstract":"<div><div>The Frobenius norm is a frequent choice of norm for matrices. We provide a broader view on the Frobenius norm and Frobenius inner product for linear maps or matrices, and establish their dependence on inner products in the domain and co-domain spaces. These new concepts are termed the metric Frobenius norm and metric Frobenius inner product. We demonstrate that the classical Frobenius norm is merely one particular element of the family of metric Frobenius norms. We also show that the metric Frobenius norm has an interpretation similar to an operator norm of a linear map. While the usual operator norm is defined as the maximal norm response of the map w.r.t. inputs in the unit sphere, the Frobenius norm turns out to measure the average norm response.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 112-128"},"PeriodicalIF":1.1,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homomorphisms from the Coxeter graph","authors":"Marko Orel ,&nbsp;Draženka Višnjić","doi":"10.1016/j.laa.2025.08.003","DOIUrl":"10.1016/j.laa.2025.08.003","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the set of all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> symmetric matrices with coefficients in the binary field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>, and let <span><math><mi>S</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be its subset formed by invertible matrices. Let <span><math><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the graph with the vertex set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> where a pair of vertices <span><math><mo>{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>}</mo></math></span> form an edge if and only if <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Similarly, let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the subgraph in <span><math><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which is induced by the set <span><math><mi>S</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. Graph <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> generalizes the well-known Coxeter graph, which is isomorphic to <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Motivated by research topics in coding theory, matrix theory, and graph theory, this paper represents the first step towards the characterization of all graph homomorphisms <span><math><mi>Φ</mi><mo>:</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>m</mi></mrow></msub></math></span> where <span><math><mi>n</mi><mo>,</mo><mi>m</mi></math></span> are positive integers. Here, the case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> is solved.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 129-162"},"PeriodicalIF":1.1,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}