{"title":"Davis-Wielandt shells of 4 by 4 matrices","authors":"Mao-Ting Chien , Hiroshi Nakazato","doi":"10.1016/j.laa.2025.06.006","DOIUrl":"10.1016/j.laa.2025.06.006","url":null,"abstract":"<div><div>In this paper, we study possible degrees of the boundary generating surfaces of the Davis-Wielandt shells of 4-by-4 upper triangular unitarily irreducible matrices. The degree can be any even number between 6 and 36 except 14,26 and 30.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 182-200"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291206","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":"Orthogonalisability of joins of graphs","authors":"Rupert H. Levene , Polona Oblak , Helena Šmigoc","doi":"10.1016/j.laa.2025.06.001","DOIUrl":"10.1016/j.laa.2025.06.001","url":null,"abstract":"<div><div>A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the connected components of two graphs for their join to be orthogonalisable. In some cases, those conditions coincide, and we present several families of joins of graphs that are orthogonalisable.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 162-181"},"PeriodicalIF":1.0,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270437","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 indefinite-inner-product spaces induced by non-zero-scaled hypercomplex numbers","authors":"Daniel Alpay , Ilwoo Cho","doi":"10.1016/j.laa.2025.06.002","DOIUrl":"10.1016/j.laa.2025.06.002","url":null,"abstract":"<div><div>In this paper, we consider a new type of adjoint <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span> on the algebra <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> of all <em>t</em>-scaled hypercomplex numbers over the real field <span><math><mi>R</mi></math></span>, for all “non-zero” scales <span><math><mi>t</mi><mo>∈</mo><mi>R</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></math></span>. We show that such a <span><math><mi>R</mi></math></span>-adjoint <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span> generates a well-defined indefinite inner product <span><math><msub><mrow><mo>[</mo><mo>,</mo><mo>]</mo></mrow><mrow><mi>t</mi></mrow></msub></math></span> on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, inducing a complete indefinite inner product space <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mo>[</mo><mo>,</mo><mo>]</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></math></span> over <span><math><mi>R</mi></math></span>. Analysis and operator theory on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> is considered up to this adjoint <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span>. As application, by regarding <em>t</em>-scaled hypercomplex numbers of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> as embedded subset <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>, the corresponding (usual operator-theoretic) spectral theory on <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> is studied (over the complex field <span><math><mi>C</mi></math></span>). And we study relations between these usual spectral-theoretic results and the operator-theoretic results obtained from the <span><math><mo>[</mo><mo>⁎</mo><mo>]</mo></math></span>-depending structures; and then the free distributions of self-adjoint matrices of <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> are characterized up to the normalized trace <em>τ</em> on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 99-161"},"PeriodicalIF":1.0,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255189","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":"Maximal determinants of matrices over the roots of unity","authors":"Guillermo Nuñez Ponasso","doi":"10.1016/j.laa.2025.05.024","DOIUrl":"10.1016/j.laa.2025.05.024","url":null,"abstract":"<div><div>We study the maximum absolute value of the determinant of matrices with entries in the set of <em>ℓ</em>-th roots of unity — this is a generalization of <em>D</em>-optimal designs and Hadamard's maximal determinant problem, which involves ±1 matrices. For general values of <em>ℓ</em>, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span>, <span><math><mi>ℓ</mi><mo>=</mo><mn>4</mn></math></span> is similar to the classical Hadamard maximal determinant problem for matrices with entries ±1, and many techniques can be generalized. For <span><math><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span> we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant of matrices with entries in the third-roots of unity for all orders <span><math><mi>n</mi><mo><</mo><mn>14</mn></math></span>. Additionally, we survey the case <span><math><mi>ℓ</mi><mo>=</mo><mn>4</mn></math></span> and exhibit an infinite family of maximal determinant matrices of odd order over the fourth roots of unity.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 201-243"},"PeriodicalIF":1.0,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291012","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}
Charles R. Johnson , Benjamin Mudrak , Carlos M. Saiago
{"title":"Multiplicity lists by diameter: All trees of diameter <7","authors":"Charles R. Johnson , Benjamin Mudrak , Carlos M. Saiago","doi":"10.1016/j.laa.2025.05.023","DOIUrl":"10.1016/j.laa.2025.05.023","url":null,"abstract":"<div><div>This paper considers the problem of determining all multiplicity lists occurring among Hermitian matrices whose graph is a given tree from a new perspective. For all trees of diameter <7, it is shown how to generate all possible lists. For diameter 5 and 6, this includes many nonlinear trees. In the process, for diameter 4 (double stars), the first succinct and direct description of all ordered lists, for each instance, is given. Observations of topological relationships are helpful.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 33-57"},"PeriodicalIF":1.0,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144242091","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":"Algebraic notes on testing sets for lower and upper grids","authors":"Eduardo Marques de Sá","doi":"10.1016/j.laa.2025.05.022","DOIUrl":"10.1016/j.laa.2025.05.022","url":null,"abstract":"<div><div>For a given finite dimensional subspace <span><math><mi>P</mi></math></span> of <span><math><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, where <em>k</em> is a field, a subset <span><math><mi>N</mi><mo>⊆</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a <span><math><mi>P</mi></math></span><em>-testing set</em> if any member of <span><math><mi>P</mi></math></span> that vanishes at all points of <span><math><mi>N</mi></math></span>, vanishes all over <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>; and we say <span><math><mi>N</mi></math></span> is <em>optimal</em> if it has the smallest cardinality among all <span><math><mi>P</mi></math></span>-testing sets. This is related to Lagrangian interpolation of data on a set <span><math><mi>N</mi></math></span> of nodes using functions from <span><math><mi>P</mi></math></span>. We consider a <em>generic version</em> of this interpolation problem, when <span><math><mi>P</mi></math></span> has a monomial basis <span><math><mi>B</mi></math></span> that we identify with a <em>grid</em> (i.e. a finite subset of <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math></span>), each node is an <em>n</em>-tuple of independent variables and the set of nodes is identified with a grid <span><math><mi>C</mi><mo>⊆</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math></span>. A corollary to our main result offers an explicit formula for the determinant of the linear system corresponding to the generic interpolation problem in case <span><math><mi>B</mi><mo>=</mo><mi>C</mi></math></span> is a <em>σ</em>-lower (or <em>σ</em>-upper) grid, where we say <span><math><mi>B</mi></math></span> is a <em>σ-lower</em> (resp., <em>σ-upper</em>) <em>grid</em> if it is a union of intervals of <span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mspace></mspace><mi>n</mi></mrow></msubsup></math></span> having <em>σ</em> as common origin (resp., endpoint). We give explicit (optimal) <span><math><mi>P</mi></math></span>-testing sets for spaces having monomial bases determined by <em>σ</em>-lower (or <em>σ</em>-upper) grids. The corollaries at the end, for the finite field case, have potential use in Number Theory and Coding Theory.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 78-98"},"PeriodicalIF":1.0,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255188","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":"Grothendieck group of the Leavitt path algebra over power graphs of prime-power cyclic groups","authors":"Aslı Güçlükan İlhan , Müge Kanuni , Ekrem Şimşek","doi":"10.1016/j.laa.2025.05.021","DOIUrl":"10.1016/j.laa.2025.05.021","url":null,"abstract":"<div><div>In this paper, the Grothendieck group of the Leavitt path algebra over the power graphs of all prime-power cyclic groups is studied by using a well-known computation from linear algebra. More precisely, the Smith normal form of the matrix derived from the adjacency matrix associated with the power graph of prime-power cyclic group is calculated.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 58-77"},"PeriodicalIF":1.0,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255187","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}
Shuaiwei Zhai , Yuanyuan Chen , Dan Li , Yinfen Zhu
{"title":"Characterizations of spectral radius and toughness of graphs","authors":"Shuaiwei Zhai , Yuanyuan Chen , Dan Li , Yinfen Zhu","doi":"10.1016/j.laa.2025.05.020","DOIUrl":"10.1016/j.laa.2025.05.020","url":null,"abstract":"<div><div>The <em>toughness</em> of graph <em>G</em> and the <em>bipartite toughness</em> of bipartite graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> are defined by Chvátal <span><span>[9]</span></span> and Bian <span><span>[5]</span></span>, respectively. In this paper, we obtained the maximum spectral radius of connected graph with given toughness, and characterize the extremal graph. The extremal balanced bipartite graph with maximal spectral radius with fixed toughness is also characterized. Furthermore, we provide spectral radius conditions for balanced bipartite graph to be <em>r</em>-tough (<span><math><mi>r</mi><mo>⩾</mo><mn>2</mn></math></span> is an integer) and to be <em>r</em>-tough (<span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>≥</mo><mn>1</mn></math></span> is a positive integer), respectively.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 1-14"},"PeriodicalIF":1.0,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190160","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":"Decomposable numerical ranges of normal matrices","authors":"Pan-Shun Lau , Chi-Kwong Li , Nung-Sing Sze","doi":"10.1016/j.laa.2025.05.016","DOIUrl":"10.1016/j.laa.2025.05.016","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> (<span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) be the set of <span><math><mi>n</mi><mo>×</mo><mi>k</mi></math></span> (<span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span>) complex matrices, and <span><math><mrow><mi>per</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the permanent of a square matrix <em>X</em>. We study the three types of generalized numerical ranges associated with generalized matrix functions<span><span><span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></munderover><msub><mrow><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mi>V</mi><mo>)</mo></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>:</mo><mi>V</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>V</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>,</mo></math></span></span></span><span><span><span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mi>det</mi><mo></mo><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mi>V</mi><mo>)</mo><mo>:</mo><mi>V</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>V</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>,</mo></math></span></span></span> and<span><span><span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mi>per</mi></mrow><mo>(</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>A</mi><mi>V</mi><mo>)</mo><mo>:</mo><mi>V</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>,</mo><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>V</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>.</mo></math></span></span></span> We give complete descriptions of the set <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for essentially hermitian matrices <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"722 ","pages":"Pages 237-254"},"PeriodicalIF":1.0,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144139034","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}
Martin Plávala , Laurens T. Ligthart , David Gross
{"title":"The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank","authors":"Martin Plávala , Laurens T. Ligthart , David Gross","doi":"10.1016/j.laa.2025.05.019","DOIUrl":"10.1016/j.laa.2025.05.019","url":null,"abstract":"<div><div>We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the <em>polarization hierarchy</em>, which has previously been introduced for the study of polynomial optimization problems over state spaces of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain <em>de Finetti theorem</em> for <em>general probabilistic theories</em>, which are studied as possible generalizations of quantum mechanics. We apply the method to the problem of nonnegative matrix factorization, and in particular to the <em>nested rectangles problem</em>. A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"723 ","pages":"Pages 15-32"},"PeriodicalIF":1.0,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144212748","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}
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