Linear Algebra and its Applications最新文献

{"title":"Keldysh's theorem revisited","authors":"Johannes M. Schumacher","doi":"10.1016/j.laa.2025.10.020","DOIUrl":"10.1016/j.laa.2025.10.020","url":null,"abstract":"<div><div>In a variety of applications, the problem comes up of describing the principal part of the inverse of a holomorphic operator at an eigenvalue in terms of left and right root functions associated to the eigenvalue. Such a description was given by Keldysh in 1951. His theorem, the proof of which was published only in 1971, is a fundamental result in the local spectral theory of operator-valued functions. Here we present a streamlined derivation in the matrix case, and we extend Keldysh's theorem by means of a new principal part formula. Special attention is given to the semisimple case (first-order poles).</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 358-386"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418519","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":"New upper and lower bounds on the smallest singular values of nonsingular lower triangular (0,1)-matrices","authors":"Vesa Kaarnioja ,&nbsp;André-Alexander Zepernick","doi":"10.1016/j.laa.2025.10.033","DOIUrl":"10.1016/j.laa.2025.10.033","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the set of all nonsingular <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> lower triangular <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-matrices. Hong and Loewy (2004) introduced the number sequence<span><span><span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>λ</mi><mo>|</mo><mi>λ</mi><mspace></mspace><mtext>is an eigenvalue of</mtext><mspace></mspace><mi>X</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>X</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>n</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>.</mo></math></span></span></span> There have been a number of attempts in the literature to obtain bounds on the numbers <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by Mattila (2015), Altınışık et al. (2016), Kaarnioja (2021), Loewy (2021), and Altınışık (2021). In this paper, improved upper and lower bounds are derived for the numbers <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By considering the characteristic polynomial corresponding to the matrix <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfying <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>‖</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></math></span>, it is shown that the second largest eigenvalue of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is bounded from above by <span><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> leading to an improved upper bound on <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. On the other hand, Samuelson's inequality applied to the roots of the characteristic polynomial of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> yields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 483-497"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418521","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":"Maps preserving two small values of λ-th upper scrambling index","authors":"Y. Kulev,&nbsp;A. Maksaev,&nbsp;V.V. Promyslov","doi":"10.1016/j.laa.2025.10.006","DOIUrl":"10.1016/j.laa.2025.10.006","url":null,"abstract":"<div><div>The notion of <em>λ</em>-th upper scrambling index was introduced by Huang and Liu in 2010, as a generalization of a notion considered by Akelbek and Kirkland in 2009. For a primitive digraph <em>D</em>, it is defined as the smallest positive integer <em>k</em> such that for every <em>λ</em> vertices of <em>D</em> there exist directed paths of lengths <em>k</em> from these vertices to a common vertex. This concept can be reformulated in terms of Boolean matrices and extended to all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices, not only primitive. In this paper, for <span><math><mi>λ</mi><mo>&gt;</mo><mn>1</mn></math></span>, we completely characterize additive maps that preserve two different values of <em>λ</em>-th upper scrambling index belonging to <span><math><mo>[</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, or strongly preserve one fixed value.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 51-72"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323237","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":"Involutions and angles between subspaces","authors":"Jean-Christophe Bourin ,&nbsp;Eun-Young Lee","doi":"10.1016/j.laa.2025.10.017","DOIUrl":"10.1016/j.laa.2025.10.017","url":null,"abstract":"<div><div>If <em>S</em> is an involution matrix, which means <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>I</mi></math></span>, then <em>S</em> is unitarily equivalent to<span><span><span><math><mo>(</mo><mi>ε</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>⊕</mo><mrow><mo>{</mo><munderover><mo>⨁</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mrow><mo>(</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>}</mo></mrow></math></span></span></span> where <span><math><mi>ε</mi><mo>=</mo><mo>±</mo><mn>1</mn></math></span>, <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the identity of size <span><math><mi>k</mi><mo>=</mo><mo>|</mo><mrow><mi>Tr</mi><mspace></mspace></mrow><mi>S</mi><mo>|</mo></math></span>, and <span><math><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>1</mn></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span>. This can be used to introduce principal angles between subspaces and yields formulas for the angles between the eigenspaces of <em>S</em> such as <span><math><mi>sin</mi><mo>⁡</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>/</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 124-137"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364993","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":"Gershgorin type inclusion-exclusion sets for matrix polynomials","authors":"Christina Michailidou,&nbsp;Vasiliki Panagakou,&nbsp;Panayiotis Psarrakos","doi":"10.1016/j.laa.2025.10.012","DOIUrl":"10.1016/j.laa.2025.10.012","url":null,"abstract":"<div><div>In this paper, improvements of the spectrum estimation for matrix polynomials given by the Gershgorin set, the Brauer set, and the Dashnic-Zusmanovich set are derived by substracting regions of the complex plane which do not contain eigenvalues. Geometrical and topological properties of the exclusion sets are obtained, and illustrative examples are presented.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 198-230"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364997","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":"Solutions for underdetermined generalized absolute value equations","authors":"Cairong Chen ,&nbsp;Xuehua Li ,&nbsp;Ren-Cang Li","doi":"10.1016/j.laa.2025.10.027","DOIUrl":"10.1016/j.laa.2025.10.027","url":null,"abstract":"<div><div>An underdetermined generalized absolute value equation (GAVE) may have no solution, one solution, finitely many or infinitely many solutions. This paper is concerned with sufficient conditions that guarantee the existence of solutions to an underdetermined GAVE. Particularly, sufficient conditions are established for an underdetermined GAVE to have infinitely many solutions with no zero entry that possess a particular or any given sign pattern. Some existing results for square GAVE are also extended. It is noted that some of the proofs are constructive and lead to iterative schemes that can solve the underdetermined GAVE in question.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 461-482"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418517","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 numerical radius of fractional powers of matrices","authors":"Eman Aldabbas ,&nbsp;Mohammad Sababheh","doi":"10.1016/j.laa.2025.10.031","DOIUrl":"10.1016/j.laa.2025.10.031","url":null,"abstract":"<div><div>Using integral representations of the fractional power of matrices, and the geometric intuition of sectorial matrices, we show that for any accretive-dissipative matrix <em>A</em> and any <span><math><mi>t</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, the matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span> is accretive-dissipative, and that<span><span><span><math><mi>ω</mi><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>ω</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> is the numerical radius. This inequality complements the well-known power inequality <span><math><mi>ω</mi><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo><mo>≤</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, valid for any square matrix and positive integer power <em>k</em>. As an application, we prove that if <em>A</em> is accretive, then the above fractional inequality holds if <span><math><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≤</mo><mi>t</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Other consequences will be given too.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 450-460"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418518","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":"Two-sided preconditioned CGLS for the solution of factorized linear systems","authors":"Yurui Jiang ,&nbsp;Junfeng Yin","doi":"10.1016/j.laa.2025.10.018","DOIUrl":"10.1016/j.laa.2025.10.018","url":null,"abstract":"<div><div>Two-sided preconditioned CGLS method is proposed for solving large-scale rank-deficient factorized linear systems, where the original system is reformulated into an augmented linear system with a scaling parameter. A class of structured two-sided preconditioners based on sketching-and-QR strategy is studied in detail. The convergence theory of the proposed method is established and the properties of the preconditioned system are analyzed. Numerical experiments demonstrate that the proposed method with sketch-based preconditioners is efficient and outperforms existing approaches.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 387-415"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418534","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 conditions for the maximum subgraph edge-connectivity of graphs","authors":"Jia Wei ,&nbsp;Muhuo Liu ,&nbsp;Zhifu You ,&nbsp;Hong-Jian Lai","doi":"10.1016/j.laa.2025.10.023","DOIUrl":"10.1016/j.laa.2025.10.023","url":null,"abstract":"<div><div>Let <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the edge connectivity of a graph <em>G</em>. The strength of <em>G</em>, denoted by <span><math><msup><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is the maximum value of <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>H</mi><mo>)</mo></math></span> where <em>H</em> runs over all subgraphs of <em>G</em>. For a positive integer <em>k</em>, Mader in 1971 initiated the study of a simple graph <em>G</em> that does not have a subgraph with edge connectivity exceeding <em>k</em> but the addition of any edge to <em>G</em> will create a subgraph of edge connectivity at least <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. For any simple graph <em>G</em> on <span><math><mi>n</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>2</mn></math></span> vertices with the spectral radius of <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, we will show the followings:</div><div>(i) If the minimum degree of <em>G</em> is at least <em>k</em>, and<span><span><span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>n</mi><mi>k</mi><mo>−</mo><mn>3</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo></math></span></span></span> then <span><math><msup><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>&gt;</mo><mi>k</mi></math></span> unless <em>G</em> belongs to a well classified family of graphs.</div><div>(ii) If <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>, then there exists an edge <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></math></span> such that <span><math><msup><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>+</mo><mi>e</mi><mo>)</mo><mo>≤</mo><mi>k</mi></math></span> unless <em>G</em> belongs to a well classified family of graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 416-429"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418535","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":"Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics","authors":"Simon Mataigne ,&nbsp;P.-A. Absil ,&nbsp;Nina Miolane","doi":"10.1016/j.laa.2025.10.003","DOIUrl":"10.1016/j.laa.2025.10.003","url":null,"abstract":"<div><div>We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 1-34"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323235","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}