Linear Algebra and its Applications最新文献

{"title":"The double almost-Riordan group","authors":"Tian-Xiao He","doi":"10.1016/j.laa.2024.10.027","DOIUrl":"10.1016/j.laa.2024.10.027","url":null,"abstract":"<div><div>In this paper, we define double almost-Riordan arrays and find that the set of all double almost-Riordan arrays forms a group, called the double almost-Riordan group. We also obtain the sequence characteristics of double almost-Riordan arrays and give the production matrices of two types for double almost-Riordan arrays. In addition, we discuss the algebraic properties of the double almost-Riordan group, and finally give the compression of double almost-Riordan arrays and their sequence characteristics.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 50-88"},"PeriodicalIF":1.0,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653401","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":"On the Colin de Verdière graph number and penny graphs","authors":"A.Y. Alfakih","doi":"10.1016/j.laa.2024.10.026","DOIUrl":"10.1016/j.laa.2024.10.026","url":null,"abstract":"<div><div>The Colin de Verdière number of graph <em>G</em>, denoted by <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is a spectral invariant of <em>G</em> that is related to some of its topological properties. For example, <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>3</mn></math></span> iff <em>G</em> is planar. A <em>penny graph</em> is the contact graph of equal-radii disks with disjoint interiors in the plane. In this note, we prove lower bounds on <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> when the complement <span><math><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math></span> is a penny graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 17-25"},"PeriodicalIF":1.0,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592670","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":"Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order","authors":"Gregor Dolinar ,&nbsp;Dijana Ilišević ,&nbsp;Bojan Kuzma ,&nbsp;Janko Marovt","doi":"10.1016/j.laa.2024.10.025","DOIUrl":"10.1016/j.laa.2024.10.025","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> be the cone of all positive semidefinite <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> real matrices. We describe the form of all surjective maps on <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, that preserve the minus partial order in both directions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 26-49"},"PeriodicalIF":1.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653400","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":"Spectral radius, odd [1,b]-factor and spanning k-tree of 1-binding graphs","authors":"Ao Fan ,&nbsp;Ruifang Liu ,&nbsp;Guoyan Ao","doi":"10.1016/j.laa.2024.10.023","DOIUrl":"10.1016/j.laa.2024.10.023","url":null,"abstract":"<div><div>The <em>binding number</em> <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the minimum value of <span><math><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>|</mo><mo>/</mo><mo>|</mo><mi>X</mi><mo>|</mo></math></span> taken over all non-empty subsets <em>X</em> of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>≠</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. A graph <em>G</em> is called 1<em>-binding</em> if <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math></span>. Let <em>b</em> be a positive integer. An <em>odd</em> <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span><em>-factor</em> of a graph <em>G</em> is a spanning subgraph <em>F</em> such that for each <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is odd and <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>b</mi></math></span>. Motivated by the result of Fan, Lin and Lu (2022) <span><span>[10]</span></span> on the existence of an odd <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor in connected graphs, we first present a tight sufficient condition in terms of the spectral radius for a connected 1-binding graph to contain an odd <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor, which generalizes the result of Fan and Lin (2024) <span><span>[8]</span></span> on the existence of a 1-factor in 1-binding graphs.</div><div>A spanning <em>k</em>-tree is a spanning tree with the degree of every vertex at most <em>k</em>, which is considered as a connected <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>]</mo></math></span>-factor. Inspired by the result of Fan, Goryainov, Huang and Lin (2022) <span><span>[9]</span></span> on the existence of a spanning <em>k</em>-tree in connected graphs, we in this paper provide a tight sufficient condition based on the spectral radius for a connected 1-binding graph to contain a spanning <em>k</em>-tree.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586752","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":"Around strongly operator convex functions","authors":"Nahid Gharakhanlu ,&nbsp;Mohammad Sal Moslehian","doi":"10.1016/j.laa.2024.10.021","DOIUrl":"10.1016/j.laa.2024.10.021","url":null,"abstract":"<div><div>We establish the subadditivity of strongly operator convex functions on <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>. Moreover, we study strongly operator convex functions on <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> with <span><math><mo>−</mo><mo>∞</mo><mo>&lt;</mo><mi>a</mi></math></span> and on the left half-line <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>b</mi><mo>)</mo></math></span> with <span><math><mi>b</mi><mo>&lt;</mo><mo>∞</mo></math></span>. We demonstrate that any nonconstant strongly operator convex function on <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> is strictly operator decreasing, and any nonconstant strongly operator convex function on <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>b</mi><mo>)</mo></math></span> is strictly operator monotone. Consequently, for a strongly operator convex function <em>g</em> on <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> or <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, we provide lower bounds for <span><math><mo>|</mo><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><mi>B</mi><mo>)</mo><mo>|</mo></math></span> whenever <span><math><mi>A</mi><mo>−</mo><mi>B</mi><mo>&gt;</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 231-248"},"PeriodicalIF":1.0,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554408","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":"Laplacian {−1,0,1}- and {−1,1}-diagonalizable graphs","authors":"Nathaniel Johnston ,&nbsp;Sarah Plosker","doi":"10.1016/j.laa.2024.10.016","DOIUrl":"10.1016/j.laa.2024.10.016","url":null,"abstract":"<div><div>A graph is called <em>Laplacian integral</em> if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> or <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. Such graphs include as special cases the recently-investigated families of <em>Hadamard-diagonalizable</em> and <em>weakly Hadamard-diagonalizable</em> graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call <em>balanced</em>, which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>- and <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>- or <span><math><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-diagonalizable.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 309-339"},"PeriodicalIF":1.0,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On generalized Sidon spaces","authors":"Chiara Castello","doi":"10.1016/j.laa.2024.10.015","DOIUrl":"10.1016/j.laa.2024.10.015","url":null,"abstract":"<div><div>Sidon spaces have been introduced by Bachoc, Serra and Zémor as the <em>q</em>-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of <em>r</em>-Sidon spaces, as an extension of Sidon spaces, which may be seen as the <em>q</em>-analogue of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and <em>r</em>-Sidon spaces, providing some upper and lower bounds on the possible dimension of their <em>r-span</em> and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of <em>r</em>-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>-sets constructed by means of them.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 270-308"},"PeriodicalIF":1.0,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Kursov's theorem for matrices over division rings","authors":"Truong Huu Dung ,&nbsp;Tran Nam Son","doi":"10.1016/j.laa.2024.10.018","DOIUrl":"10.1016/j.laa.2024.10.018","url":null,"abstract":"<div><div>Let <em>D</em> be a division ring with center <em>F</em> and multiplicative group <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span>, where each element of the commutator subgroup of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> can be expressed as a product of at most <em>s</em> commutators. A known theorem of Kursov states that if <em>D</em> is finite-dimensional over <em>F</em>, then every element of the commutator subgroup of the general linear group over <em>D</em> can be expressed as a product of at most <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> commutators. We show that this result remains valid when <em>F</em> has a sufficiently large number of elements, without requiring <em>D</em> to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 218-230"},"PeriodicalIF":1.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533210","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":"On the trace-zero doubly stochastic matrices of order 5","authors":"Amrita Mandal ,&nbsp;Bibhas Adhikari","doi":"10.1016/j.laa.2024.10.020","DOIUrl":"10.1016/j.laa.2024.10.020","url":null,"abstract":"<div><div>We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose <em>k</em>-th power is also a trace-zero DS matrix for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span>. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 340-360"},"PeriodicalIF":1.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579121","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":"A proof of the Paz conjecture for 6 × 6 matrices","authors":"M.A. Khrystik ,&nbsp;A.M. Maksaev","doi":"10.1016/j.laa.2024.10.019","DOIUrl":"10.1016/j.laa.2024.10.019","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span> and let <span><math><mi>S</mi></math></span> be its generating set (as an <span><math><mi>F</mi></math></span>-algebra). The length of <span><math><mi>S</mi></math></span> is the smallest number <em>k</em> such that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> equals the <span><math><mi>F</mi></math></span>-linear span of all products of the length at most <em>k</em> of matrices from <span><math><mi>S</mi></math></span>. The length of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span>, denoted by <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo></math></span>, is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span>, for any field <span><math><mi>F</mi></math></span>. This conjecture has been verified only for <span><math><mi>n</mi><mo>⩽</mo><mn>5</mn></math></span>. In this paper, we prove Paz's conjecture for <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span>, meaning that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>10</mn></math></span>. We also prove that <span><math><mn>12</mn><mo>⩽</mo><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>⩽</mo><mn>13</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 249-269"},"PeriodicalIF":1.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554409","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}