Linear Algebra and its Applications最新文献

{"title":"New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties","authors":"Peter Šepitka,&nbsp;Roman Šimon Hilscher","doi":"10.1016/j.laa.2024.11.017","DOIUrl":"10.1016/j.laa.2024.11.017","url":null,"abstract":"<div><div>In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 187-224"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748261","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":"Stabilization of associated prime ideals of monomial ideals – Bounding the copersistence index","authors":"Clemens Heuberger,&nbsp;Jutta Rath,&nbsp;Roswitha Rissner","doi":"10.1016/j.laa.2024.11.020","DOIUrl":"10.1016/j.laa.2024.11.020","url":null,"abstract":"<div><div>The sequence <span><math><msub><mrow><mo>(</mo><mi>Ass</mi><mo>(</mo><mi>R</mi><mo>/</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> of associated primes of powers of a monomial ideal <em>I</em> in a polynomial ring <em>R</em> eventually stabilizes by a known result by Markus Brodmann. Lê Tuân Hoa gives an upper bound for the index where the stabilization occurs. This bound depends on the generators of the ideal and is obtained by separately bounding the powers of <em>I</em> after which said sequence is non-decreasing and non-increasing, respectively. In this paper, we focus on the latter and call the smallest such number the copersistence index. We take up the proof idea of Lê Tuân Hoa, who exploits a certain system of inequalities whose solution sets store information about the associated primes of powers of <em>I</em>. However, these proofs are entangled with a specific choice for the system of inequalities. In contrast to that, we present a generic ansatz to obtain an upper bound for the copersistence index that is uncoupled from this choice of the system. We establish properties for a system of inequalities to be eligible for this approach to work. We construct two suitable inequality systems to demonstrate how this ansatz yields upper bounds for the copersistence index and compare them with Hoa's. One of the two systems leads to an improvement of the bound by an exponential factor.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 162-186"},"PeriodicalIF":1.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722539","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":"Characterization of almost-Riordan arrays with row sums","authors":"Yasemin Alp ,&nbsp;E. Gokcen Kocer","doi":"10.1016/j.laa.2024.11.019","DOIUrl":"10.1016/j.laa.2024.11.019","url":null,"abstract":"<div><div>The almost-Riordan arrays and their inverses are investigating by the generating functions of the row sum, the alternating row sum, and the weighted row sum. The <em>A</em>, <em>Z</em>, and <em>ω</em>-sequences of the almost-Riordan arrays are characterized by the generating functions of these row sums. Additionally, using the generating functions of these row sums, the product of two almost-Riordan arrays is obtained.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 101-123"},"PeriodicalIF":1.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722518","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 symplectic solvmanifolds satisfying the hard-Lefschetz condition","authors":"Adrián Andrada,&nbsp;Agustín Garrone","doi":"10.1016/j.laa.2024.11.018","DOIUrl":"10.1016/j.laa.2024.11.018","url":null,"abstract":"<div><div>A compact symplectic manifold <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span> is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>. This loosely means that there is a notion of harmonicity of differential forms in <em>M</em>, depending on <em>ω</em> alone, such that every de Rham cohomology class in has a <em>ω</em>-harmonic representative. In this article, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-Kähler structure and compute their cohomology explicitly. We show that they satisfy the hard-Lefschetz condition with respect to any left-invariant symplectic structure by exploiting an unforeseen connection with Kneser graphs. We also show that for some choice of parameters their associated simply connected, completely solvable Lie groups admit lattices, thereby constructing examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition, in such a way that their de Rham cohomology is fully known.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 70-100"},"PeriodicalIF":1.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722517","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 energies of vertices","authors":"J. Guerrero","doi":"10.1016/j.laa.2024.11.016","DOIUrl":"10.1016/j.laa.2024.11.016","url":null,"abstract":"<div><div>In this work, we define the Laplacian and Normalized Laplacian energies of vertices in a graph, we derive some of its properties and relate them to combinatorial, spectral and geometric quantities of the graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 124-143"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722519","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":"Determinants of Seidel tournament matrices","authors":"Sarah Klanderman ,&nbsp;MurphyKate Montee ,&nbsp;Andrzej Piotrowski ,&nbsp;Alex Rice ,&nbsp;Bryan Shader","doi":"10.1016/j.laa.2024.11.011","DOIUrl":"10.1016/j.laa.2024.11.011","url":null,"abstract":"<div><div>The Seidel matrix of a tournament on <em>n</em> players is an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> skew-symmetric matrix with entries in <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span> that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Seidel matrix is 0 if <em>n</em> is odd, and is an odd perfect square if <em>n</em> is even. This leads to the study of the set, <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, of square roots of determinants of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Seidel matrices. It is shown that <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is a proper subset of <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span> for every positive even integer, and every odd integer in the interval <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>]</mo></math></span> is in <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for <em>n</em> even. The expected value and variance of <span><math><mi>det</mi><mo>⁡</mo><mi>S</mi></math></span> over the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Seidel matrices chosen uniformly at random is determined, and upper bounds on <span><math><mi>max</mi><mo>⁡</mo><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many <em>n</em>, <span><math><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contains a gap (that is, there are odd integers <span><math><mi>k</mi><mo>&lt;</mo><mi>ℓ</mi><mo>&lt;</mo><mi>m</mi></math></span> such that <span><math><mi>k</mi><mo>,</mo><mi>m</mi><mo>∈</mo><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> but <span><math><mi>ℓ</mi><mo>∉</mo><mi>D</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>) and several properties of the characteristic polynomials of Seidel matrices are established.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 126-151"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722543","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":"Strongly real adjoint orbits of complex symplectic Lie group","authors":"Tejbir Lohan ,&nbsp;Chandan Maity","doi":"10.1016/j.laa.2024.11.015","DOIUrl":"10.1016/j.laa.2024.11.015","url":null,"abstract":"<div><div>We consider the adjoint action of the symplectic Lie group <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> on its Lie algebra <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. An element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span> for some <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. Moreover, if <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>h</mi><mo>)</mo><mi>X</mi></math></span> for some involution <span><math><mi>h</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, then <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is called strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real. In this paper, we prove that for every element <span><math><mi>X</mi><mo>∈</mo><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, there exists a skew-involution <span><math><mi>g</mi><mo>∈</mo><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> such that <span><math><mo>−</mo><mi>X</mi><mo>=</mo><mrow><mi>Ad</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo><mi>X</mi></math></span>. Furthermore, we classify the strongly <span><math><msub><mrow><mi>Ad</mi></mrow><mrow><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></msub></math></span>-real elements in <span><math><mrow><mi>sp</mi></mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"706 ","pages":"Pages 144-156"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746666","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 note on a conjecture from distillability of quantum entanglement","authors":"Weng Kun Sio,&nbsp;Che-Man Cheng","doi":"10.1016/j.laa.2024.11.012","DOIUrl":"10.1016/j.laa.2024.11.012","url":null,"abstract":"<div><div>A conjecture from the distillability of quantum entanglement is that when <em>A</em> and <em>B</em> are <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> trace zero complex matrices and <span><math><msup><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>‖</mo><mi>B</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>/</mo><mn>4</mn></math></span> (where <span><math><mo>‖</mo><mo>⋅</mo><mo>‖</mo></math></span> is the Frobenius norm), the sum of squares of the largest two singular values of <span><math><mi>A</mi><mo>⊗</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>⊗</mo><mi>B</mi></math></span> does not exceed 1/2. In this paper, the conjecture is proved when<ul><li><span>(i)</span><span><div><em>A</em> or <em>B</em> is unitarily similar to a direct sum of <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> trace zero matrices;</div></span></li><li><span>(ii)</span><span><div><em>A</em> and <em>B</em> are unitarily similar to matrices, when partitioned into <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> blocks, having zero diagonal blocks.</div></span></li></ul></div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 152-161"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722542","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":"Tight frames generated by a graph short-time Fourier transform","authors":"Martin Buck,&nbsp;Kasso A. Okoudjou","doi":"10.1016/j.laa.2024.11.014","DOIUrl":"10.1016/j.laa.2024.11.014","url":null,"abstract":"<div><div>A <em>graph short-time Fourier transform</em> is defined using the eigenvectors of the graph Laplacian and a graph heat kernel as a window parametrized by a nonnegative time parameter <em>t</em>. We show that the corresponding Gabor-like system forms a frame for <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and gives a description of the spectrum of the corresponding frame operator in terms of the graph heat kernel and the spectrum of the underlying graph Laplacian. For two classes of algebraic graphs, we prove the frame is tight and independent of the window parameter <em>t</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 107-125"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707118","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":"Upper bounds for the rank of powers of quadrics","authors":"Cosimo Flavi","doi":"10.1016/j.laa.2024.11.009","DOIUrl":"10.1016/j.laa.2024.11.009","url":null,"abstract":"<div><div>We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any <span><math><mi>s</mi><mo>∈</mo><mi>N</mi></math></span>, we prove that the <em>s</em>-th power of a quadratic form of rank <em>n</em> grows as <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>. Furthermore, we demonstrate that its rank is subgeneric for all <span><math><mi>n</mi><mo>&gt;</mo><msup><mrow><mo>(</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 49-79"},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706749","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}