Recent zbMATH articles in MSC 15B https://www.zbmath.org/atom/cc/15B 2021-04-16T16:22:00+00:00 Werkzeug A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications. https://www.zbmath.org/1456.15026 2021-04-16T16:22:00+00:00 "Zimmermann, Ralf" https://www.zbmath.org/authors/?q=ai:zimmermann.ralf Summary: For a matrix $$X \in \mathbb{R}^{n \times p}$$, we provide an analytic formula that keeps track of an orthonormal basis for the range of $$X$$ under rank-one modifications. More precisely, we consider rank-one adaptations $$X_{new} = X + ab^T$$ of a given $$X$$ with known matrix factorization $$X = UW$$, where $$U \in \mathbb{R}^{n \times p}$$ is column-orthogonal and $$W \in \mathbb{R}^{p \times p}$$ is invertible. Arguably, the most important methods that produce such factorizations are the singular value decomposition (SVD), where $$X = UW = U (\Sigma V^T)$$, and the QR-decomposition, where $$X = UW = QR$$. We give a geometric description of rank-one adaptations and derive a closed-form expression for the geodesic line that travels from the subspace $$\mathcal{S} = \mathrm{ran}(X)$$ to the subspace $$\mathcal{S}_{new} =\mathrm{ran}(X_{new}) = \mathrm{ran} (U_{new} W_{new})$$. This leads to update formulas for orthogonal matrix decompositions, where both $$U_{new}$$ and $$W_{new}$$ are obtained via elementary rank-one matrix updates in $$\mathcal{O}(np)$$ time for $$n \gg p$$. Moreover, this allows us to determine the subspace distance and the Riemannian midpoint between the subspaces $$\mathcal{S}$$ and $$\mathcal{S}_{new}$$ without additional computational effort. Alternating sign hypermatrix decompositions of Latin-like squares. https://www.zbmath.org/1456.05026 2021-04-16T16:22:00+00:00 "O'Brien, Cian" https://www.zbmath.org/authors/?q=ai:obrien.cian Summary: To any $$n\times n$$ Latin square $$L$$, we may associate a unique sequence of mutually orthogonal permutation matrices $$P=P_1,P_2,\dots,P_n$$ such that $$L=L(P)=\sum kP_k$$. \textit{R. A. Brualdi} and \textit{G. Dahl} [ibid. 95, 116--151 (2018; Zbl 1379.05020)] described a generalisation of a Latin square, called an alternating sign hypermatrix Latin-like square (ASHL), by replacing $$P$$ with an alternating sign hypermatrix (ASHM). An ASHM is an $$n\times n\times n$$ $$(0,1,-1)$$-hypermatrix in which the non-zero elements in each row, column, and vertical line alternate in sign, beginning and ending with 1. Since every sequence of $$n$$ mutually orthogonal permutation matrices forms the planes of a unique $$n\times n\times n$$ ASHM, this generalisation of Latin squares follows very naturally, with an ASHM $$A$$ having corresponding ASHL $$L=L(A)=\sum k A_k$$, where $$A_k$$ is the $$k$$th plane of $$A$$. This paper addresses open problems posed in Brualdi and Dahl's article, firstly by characterising how pairs of ASHMs with the same corresponding ASHL relate to one another and identifying the smallest dimension for which this can happen, and secondly by exploring the maximum number of times a particular integer may occur as an entry of an $$n\times n$$ ASHL. A construction is given for an $$n\times n$$ ASHL with the same entry occurring $$\lfloor\frac{n^2+4n-19}{2} \rfloor$$ times, improving on the previous best of $$2n$$. Equivalence classes of e-matrices and associated eigenvalue localization regions. https://www.zbmath.org/1456.15009 2021-04-16T16:22:00+00:00 "Marsli, Rachid" https://www.zbmath.org/authors/?q=ai:marsli.rachid "Hall, Frank J." https://www.zbmath.org/authors/?q=ai:hall.frank-j The authors continue their study on square e-matrices (real constant row-sum matrices) by looking at their partition into equivalence classes (known as e-similarity classes). They study the relationships between the spectra of different matrices belonging to the same equivalence class, as well as their left and generalized left eigenspaces. The results obtained are applied to improve the location of eigenvalues of e-matrices. Associated localization Gershgorin regions of the so-called second type are obtained. Many examples are provided to illustrate the theoretical results. Reviewer: George Stoica (Saint John) A data-driven McMillan degree lower bound. https://www.zbmath.org/1456.37098 2021-04-16T16:22:00+00:00 "Hokanson, Jeffrey M." https://www.zbmath.org/authors/?q=ai:hokanson.jeffrey-m Classical adjoint commuting and determinant preserving linear maps on Kronecker products of Hermitian matrices. https://www.zbmath.org/1456.15002 2021-04-16T16:22:00+00:00 "Chooi, Wai-Leong" https://www.zbmath.org/authors/?q=ai:chooi.wai-leong "Kwa, Kiam-Heong" https://www.zbmath.org/authors/?q=ai:kwa.kiam-heong Let $$H_n$$ be a space of Hermitian $$n\times n$$ matrices and consider the space $$\otimes_{i=1}^{d} H_{n_i}$$. An automorphism $$\Psi$$ of this tensor product is called determinant-preserving if $\mathrm{det} (\Psi \otimes_{i=1}^{d} A_{n_i})= \mathrm{det} ( \otimes_{i=1}^{d} A_{n_i})$ for all $$A_{n_i}\in H_{n_i}$$. An automorphism $$\Psi$$ of the tensor product is called classical adjoint commuting if $\mathrm{adj} (\Psi \otimes_{i=1}^{d} A_{n_i})= \Psi(\mathrm{adj} \otimes_{i=1}^{d} A_{n_i})$ for all $$A_{n_i}\in H_{n_i}$$. The authors prove that nonzero classical adjoint commuting linear maps on Kronecker products of Hermitian matrices are also determinant-preserving. Conversely, they show that a determinant-preserving linear map may not be classical adjoint commuting unless $\Psi (\otimes_{i=1}^{d} A_{n_i})\Psi (\mathrm{adj}\otimes_{i=1}^{d} A_{n_i})= \mathrm{det}( \otimes_{i=1}^{d} A_{n_i})I_{n_1,...,n_d}.$ Reviewer: Dmitry Artamonov (Moskva) Image encryption algorithm for synchronously updating Boolean networks based on matrix semi-tensor product theory. https://www.zbmath.org/1456.68034 2021-04-16T16:22:00+00:00 "Wang, Xingyuan" https://www.zbmath.org/authors/?q=ai:wang.xingyuan "Gao, Suo" https://www.zbmath.org/authors/?q=ai:gao.suo Summary: This paper studies chaotic image encryption technology and an application of matrix semi-tensor product theory, and a Boolean network encryption algorithm for a synchronous update process is proposed. A 2D-LASM chaotic system is used to generate a random key stream. First, a Boolean network is coded, and a Boolean matrix is generated. If necessary, the Boolean network matrix is diffused in one round so that the Boolean matrix can be saved in the form of an image. Then, three random position scramblings are used to scramble the plaintext image. Finally, using a matrix semi-tensor product technique to generate an encrypted image in a second round of diffusion, a new Boolean network can be generated by encoding the encrypted image. In secure communications, users can choose to implement an image encryption transmission or a Boolean network encryption transmission according to their own needs. Compared with other algorithms, this algorithm exhibits good security characteristics. On a parametrization of non-compact wavelet matrices by Wiener-Hopf factorization. https://www.zbmath.org/1456.15038 2021-04-16T16:22:00+00:00 "Ephremidze, Lasha" https://www.zbmath.org/authors/?q=ai:ephremidze.lasha "Salia, Nika" https://www.zbmath.org/authors/?q=ai:salia.nika "Spitkovsky, Ilya" https://www.zbmath.org/authors/?q=ai:spitkovsky.ilya-matvey Summary: A complete parametrization (one-to-one and onto mapping) of a certain class of noncompact wavelet matrices is introduced in terms of coordinates of infinite-dimensional Euclidian space. The developed method relies on Wiener-Hopf factorization of corresponding unitary matrix functions. Solvability of new constrained quaternion matrix approximation problems based on core-EP inverses. https://www.zbmath.org/1456.15014 2021-04-16T16:22:00+00:00 "Kyrchei, Ivan" https://www.zbmath.org/authors/?q=ai:kyrchei.ivan "Mosić, Dijana" https://www.zbmath.org/authors/?q=ai:mosic.dijana "Stanimirović, Predrag S." https://www.zbmath.org/authors/?q=ai:stanimirovic.predrag-s Summary: Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: $$\min \Vert \mathbf{AXB}-\mathbf{C}\Vert_F$$ subject to the constraints imposed to the right column space of $$\mathbf{A}$$ and the left row space of $$\mathbf{B}$$. Unique solution to the considered Q-matrix problem is expressed in terms of the core inverse of $$\mathbf{A}$$ and/or the dual core-EP inverse of $$\mathbf{B}$$. Thus, we propose and solve problems which generalize a well-known constrained approximation problem for complex matrices with index one to quaternion matrices with arbitrary index. Determinantal representations for solutions of proposed constrained quaternion matrix approximation problems obtained. An example is given to justify obtained theoretical results. Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices. https://www.zbmath.org/1456.62113 2021-04-16T16:22:00+00:00 "Cai, T. Tony" https://www.zbmath.org/authors/?q=ai:cai.tianwen-tony|cai.tony-tony "Han, Xiao" https://www.zbmath.org/authors/?q=ai:han.xiao "Pan, Guangming" https://www.zbmath.org/authors/?q=ai:pan.guangming Let $$\mathbf{Y}=\mathbf{\Gamma X}$$ be the data matrix, where $$\mathbf{X}$$ be a $$(p+l)\times n$$ random matrix whose entries are independent with mean means and unit variances and $$\mathbf{\Gamma}$$ is a $$p\times(p+l)$$ deterministic matrix under condition $$l/p\rightarrow0$$. Let $$\mathbf{\Sigma}=\mathbf{\Gamma}\mathbf{\Gamma}^\intercal$$ be the population covariance matrix. The sample covariance matrix in such a case is $S_n=\frac{1}{n}\mathbf{Y}\mathbf{Y}^\intercal=\frac{1}{n}\mathbf{\Gamma X}\mathbf{X}^\intercal\mathbf{\Gamma}^\intercal.$ Let $$\mathbf{V}\mathbf{\Lambda}^{1/2}\mathbf{U}$$ denote the singular value decomposition of matrix $$\mathbf{\Gamma}$$, where $$\mathbf{V}$$ and $$\mathbf{U}$$ are orthogonal matrices and $$\mathbf{\Lambda}$$ is a diagonal matrix consisting in descending order eigenvalues $$\mu_1\geqslant\mu_2\geqslant\ldots\geqslant\mu_p$$ of matrix $$\mathbf{\Sigma}$$. Authors of the paper suppose that there are $$K$$ spiked eigenvalues that are separated from the rest. They assume that eigenvalues $$\mu_1\geqslant\ldots\geqslant\mu_K$$ tends to infinity, while the other eigenvalues $$\mu_{K+1}\geqslant\ldots\geqslant\mu_p$$ are bounded. In the paper, the asymptotic behaviour is considered of the spiked eigenvalues and the largest non-spiked eigenvalue. The limiting normal distribution for the spiked sample eigenvalues is established. The limiting \textit{Tracy-Widom} law for the largest non-spiked eigenvalues is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are considered. Reviewer: Jonas Šiaulys (Vilnius) Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without a fixed trace. https://www.zbmath.org/1456.60012 2021-04-16T16:22:00+00:00 "Akemann, Gernot" https://www.zbmath.org/authors/?q=ai:akemann.gernot "Vivo, Pierpaolo" https://www.zbmath.org/authors/?q=ai:vivo.pierpaolo Revisiting Horn's problem. https://www.zbmath.org/1456.15034 2021-04-16T16:22:00+00:00 "Coquereaux, Robert" https://www.zbmath.org/authors/?q=ai:coquereaux.robert "McSwiggen, Colin" https://www.zbmath.org/authors/?q=ai:mcswiggen.colin "Zuber, Jean-Bernard" https://www.zbmath.org/authors/?q=ai:zuber.jean-bernard A memory-based method to select the number of relevant components in principal component analysis. https://www.zbmath.org/1456.62116 2021-04-16T16:22:00+00:00 "Verma, Anshul" https://www.zbmath.org/authors/?q=ai:verma.anshul "Vivo, Pierpaolo" https://www.zbmath.org/authors/?q=ai:vivo.pierpaolo "Di Matteo, Tiziana" https://www.zbmath.org/authors/?q=ai:di-matteo.tiziana Averages of products and ratios of characteristic polynomials in polynomial ensembles. https://www.zbmath.org/1456.60011 2021-04-16T16:22:00+00:00 "Akemann, Gernot" https://www.zbmath.org/authors/?q=ai:akemann.gernot "Strahov, Eugene" https://www.zbmath.org/authors/?q=ai:strahov.eugene "Würfel, Tim R." https://www.zbmath.org/authors/?q=ai:wurfel.tim-r Summary: Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov [\textit{Y. V. Fyodorov} et al., J. Phys. A, Math. Theor. 51, No. 13, Article ID 134003, 30 p. (2018; Zbl 1388.60025)]. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble. Invariant sums of random matrices and the onset of level repulsion. https://www.zbmath.org/1456.82010 2021-04-16T16:22:00+00:00 "Burda, Zdzisław" https://www.zbmath.org/authors/?q=ai:burda.zdzislaw "Livan, Giacomo" https://www.zbmath.org/authors/?q=ai:livan.giacomo "Vivo, Pierpaolo" https://www.zbmath.org/authors/?q=ai:vivo.pierpaolo Determinant of binary circulant matrices. https://www.zbmath.org/1456.15006 2021-04-16T16:22:00+00:00 "Hariprasad, M." https://www.zbmath.org/authors/?q=ai:hariprasad.m Summary: This article gives a closed-form expression for the determinant of binary circulant matrices. A characterization of nonnegativity relative to proper cones. https://www.zbmath.org/1456.15029 2021-04-16T16:22:00+00:00 "Arumugasamy, Chandrashekaran" https://www.zbmath.org/authors/?q=ai:chandrashekaran.a "Jayaraman, Sachindranath" https://www.zbmath.org/authors/?q=ai:jayaraman.sachindranath "Mer, Vatsalkumar N." https://www.zbmath.org/authors/?q=ai:mer.vatsalkumar-n Summary: Let $$A$$ be an $$m \times n$$ matrix with real entries. Given two proper cones $$K_1$$ and $$K_2$$ in $$\mathbb{R}^n$$ and $$\mathbb{R}^m$$, respectively, we say that $$A$$ is nonnegative if $$A(K_1) \subseteq K_2$$. $$A$$ is said to be semipositive if there exists a $$x \in K_1^\circ$$ such that $$Ax \in K_2^\circ$$. We prove that $$A$$ is nonnegative if and only if $$A + B$$ is semipositive for every semipositive matrix $$B$$. Applications of the above result are also brought out. Spectra of large time-lagged correlation matrices from random matrix theory. https://www.zbmath.org/1456.60027 2021-04-16T16:22:00+00:00 "Nowak, Maciej A." https://www.zbmath.org/authors/?q=ai:nowak.maciej-a "Tarnowski, Wojciech" https://www.zbmath.org/authors/?q=ai:tarnowski.wojciech Invertibility via distance for noncentered random matrices with continuous distributions. https://www.zbmath.org/1456.60028 2021-04-16T16:22:00+00:00 "Tikhomirov, Konstantin" https://www.zbmath.org/authors/?q=ai:tikhomirov.konstantin-e Summary: Let $$A$$ be an $$n \times n$$ random matrix with independent rows $$R_1(A),\dots,R_n(A)$$, and assume that for any $$i \leq n$$ and any three-dimensional linear subspace $$F \subset \mathbb R^n$$ the orthogonal projection of $$R_i(A)$$ onto $$F$$ has distribution density $$\rho(x): F\to \mathbb R_+$$ satisfying $$\rho (x) \leq C_1 /\max (1, \| x\|_2^{2000}) (x \in F)$$ for some constant $$C_1>0$$. We show that for any fixed $$n \times n$$ real matrix $$M$$ we have $\mathbb P \{s_{\min} (A+M) \leq tn^{-1/2}\} \leq C' \; t, \;\;\; t>0,$ where $$C' >0$$ is a universal constant. In particular, the above result holds if the rows of $$A$$ are independent centered log-concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for noncentered Gaussian matrices [\textit{A. Sankar} et al., SIAM J. Matrix Anal. Appl. 28, No. 2, 446--476 (2006; Zbl 1179.65033)]. An $$\alpha$$-cut approach for fuzzy product and its use in computing solutions of fully fuzzy linear systems. https://www.zbmath.org/1456.15004 2021-04-16T16:22:00+00:00 "Hassanzadeh, Reza" https://www.zbmath.org/authors/?q=ai:hassanzadeh.reza "Mahdavi, Iraj" https://www.zbmath.org/authors/?q=ai:mahdavi.iraj "Mahdavi-Amiri, Nezam" https://www.zbmath.org/authors/?q=ai:mahdavi-amiri.nezam "Tajdin, Ali" https://www.zbmath.org/authors/?q=ai:tajdin.ali Summary: We propose an approach for computing the product of various fuzzy numbers using $$\alpha$$-cuts. A regression model is used to obtain the membership function of the product. Then, we make use of the approach to compute solutions of fully fuzzy linear systems. We also show how to compute solutions of fully fuzzy linear systems with various fuzzy variables. Examples are worked out to illustrate the approach. A new rank metric for convolutional codes. https://www.zbmath.org/1456.15027 2021-04-16T16:22:00+00:00 "Almeida, P." https://www.zbmath.org/authors/?q=ai:almeida.paulo-j "Napp, D." https://www.zbmath.org/authors/?q=ai:napp.diego Summary: Let $$\mathbb{F}[D]$$ be the polynomial ring with entries in a finite field $$\mathbb{F}$$. Convolutional codes are submodules of $$\mathbb{F} [D]^n$$ that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a new metric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field. Time-inhomogeneous random Markov chains. https://www.zbmath.org/1456.60178 2021-04-16T16:22:00+00:00 "Innocentini, G. C. P." https://www.zbmath.org/authors/?q=ai:innocentini.guilherme-c-p "Novaes, M." https://www.zbmath.org/authors/?q=ai:novaes.marcel|novaes.marcos New upper bounds for the infinity norm of Nekrasov matrices. https://www.zbmath.org/1456.15023 2021-04-16T16:22:00+00:00 "Gao, Lei" https://www.zbmath.org/authors/?q=ai:gao.lei "Liu, Qilong" https://www.zbmath.org/authors/?q=ai:liu.qilong The Nekrasov condition is a form of diagonal dominance. Let $$A=[a_{ij}]$$ be an $$n\times n$$ complex matrix. Then $$A$$ is a Nekrasov matrix if $$\left\vert a_{ii}\right\vert >h_{i}(A)$$ for $$i=1,2,\dots,n$$ where $h_{i}(A):=\sum_{j=1}^{i-1}\frac{|a_{ij}|}{|a_{jj}|}h_{j}(A)+\sum_{j=i+1}^{n}\left\vert a_{ij}\right\vert.$ The authors give variations on different bounds on $$\left\Vert A^{-1}\right\Vert _{\infty}$$ for a Nekrasov matrix. Numerical examples show that these bounds improve bounds given in [\textit{L. Yu. Kolotilina}, J. Math. Sci., New York 199, No. 4, 432--437 (2014; Zbl 1309.15030); translation from Zap. Nauchn. Semin. POMI 419, 111--120 (2013); \textit{Y. Zhu} and \textit{Y. Li}, J. Yunnan Univ., Nat. Sci. 39, No. 1, 13--17 (2017; Zbl 1389.15011)]. Reviewer: John D. Dixon (Ottawa) Faces of the $$5\times 5$$ completely positive cone. https://www.zbmath.org/1456.15032 2021-04-16T16:22:00+00:00 "Zhang, Qinghong" https://www.zbmath.org/authors/?q=ai:zhang.qinghong Summary: The full characterization of extreme rays of the $$5\times 5$$ copositive cone provides an important means to study faces of the $$5\times 5$$ completely positive cone. The maximal faces of the $$5\times 5$$ completely positive cone determined by the Horn matrix or by a Hildebrand matrix can be easily described. In this paper, we study the subfaces of these maximal faces. We prove that most of the subfaces on these maximal faces are exposed. However, a class of subfaces, which are on the maximal face determined by the Horn matrix, are not exposed. Therefore, besides the class of non-exposed faces appearing in a recent paper by the author, another family of non-exposed faces are presented in this paper to show again that the $$n\times n$$ completely positive cone is not facially exposed for $$n\geq 5$$. In this paper, we also prove a necessary and sufficient condition for a strictly positive matrix, which is on the maximal face determined by the Horn matrix, to have a unique cp factorization. On the generalized dimensions of multifractal eigenstates. https://www.zbmath.org/1456.81123 2021-04-16T16:22:00+00:00 "Méndez-Bermúdez, J. A." https://www.zbmath.org/authors/?q=ai:mendez-bermudez.j-a "Alcazar-López, A." https://www.zbmath.org/authors/?q=ai:alcazar-lopez.a "Varga, Imre" https://www.zbmath.org/authors/?q=ai:varga.imre Unit triangular factorization of the matrix symplectic group. https://www.zbmath.org/1456.15013 2021-04-16T16:22:00+00:00 "Jin, Pengzhan" https://www.zbmath.org/authors/?q=ai:jin.pengzhan "Tang, Yifa" https://www.zbmath.org/authors/?q=ai:tang.yifa "Zhu, Aiqing" https://www.zbmath.org/authors/?q=ai:zhu.aiqing New methods based on $$\mathcal{H}$$-tensors for identifying positive definiteness of homogeneous polynomial forms. https://www.zbmath.org/1456.15024 2021-04-16T16:22:00+00:00 "Sun, Deshu" https://www.zbmath.org/authors/?q=ai:sun.deshu Summary: In this paper, some sufficient conditions of identifying the positive definiteness of homogeneous polynomial forms are proposed by identifying $$\mathcal{H}$$-tensors. Numerical examples are given to show the feasibility and effectiveness of the methods. Two-sample hypothesis testing for inhomogeneous random graphs. https://www.zbmath.org/1456.62108 2021-04-16T16:22:00+00:00 "Ghoshdastidar, Debarghya" https://www.zbmath.org/authors/?q=ai:ghoshdastidar.debarghya "Gutzeit, Maurilio" https://www.zbmath.org/authors/?q=ai:gutzeit.maurilio "Carpentier, Alexandra" https://www.zbmath.org/authors/?q=ai:carpentier.alexandra "von Luxburg, Ulrike" https://www.zbmath.org/authors/?q=ai:von-luxburg.ulrike Testing random graphs is challenging problem especially in large dimensions (chemical compounds graphs, brain networks of several patients analysis, and other). This paper focuses on the drawing inference from large sparse networks and consider the graphs on a common vertex set sampled from an inhomogeneous Erdös-Rényi model [\textit{B. Bollobàs} et al., Random Struct. Algorithms 31, No. 1, 3--122 (2007; Zbl 1123.05083)]. The latter model is considered in the case when no structural assumption on the population adjacency matrix is assumed. Reviewer: Denis Sidorov (Irkutsk) A sign pattern with all diagonal entries nonzero whose minimal rank realizations are not diagonalizable over $$\mathbb{C}$$. https://www.zbmath.org/1456.15028 2021-04-16T16:22:00+00:00 "Shitov, Yaroslav" https://www.zbmath.org/authors/?q=ai:shitov.yaroslav-nikolaevich Summary: The rank of the $$9 \times 9$$ matrix $\begin{pmatrix} 1&1&0&0 \text{ }\vline\text{ } 1\text{ }\vline\text{ } 0&0&0&0\\ 1&1&0&0\text{ }\vline\text{ }0\text{ }\vline\text{ }0&0&0&0\\ 0&0&1&1\text{ }\vline\text{ }1\text{ }\vline\text{ }0&0&0&0\\ 0&0&1&1\text{ }\vline\text{ }0\text{ }\vline\text{ }0&0&0&0\\ \hline 0&0&0&0\text{ }\vline\text{ }1\text{ }\vline\text{ }0&1&0&1\\ \hline 0&0&0&0\text{ }\vline\text{ }0\text{ }\vline\text{ }1&1&0&0\\ 0&0&0&0\text{ }\vline\text{ }0\text{ }\vline\text{ }1&1&0&0\\ 0&0&0&0\text{ }\vline\text{ }0\text{ }\vline\text{ }0&0&1&1\\ 0&0&0&0\text{ }\vline\text{ }0\text{ }\vline\text{ }0&0&1&1 \end{pmatrix}$ is 6. If we replace the ones by arbitrary non-zero numbers, we get a matrix $$B$$ with rank $$B \geqslant 6$$, and if rank $$B = 6$$, the $$6 \times 6$$ principal minors of $$B$$ vanish. Geometric mean of partial positive definite matrices with missing entries. https://www.zbmath.org/1456.15030 2021-04-16T16:22:00+00:00 "Choi, Hayoung" https://www.zbmath.org/authors/?q=ai:choi.hayoung "Kim, Sejong" https://www.zbmath.org/authors/?q=ai:kim.sejong "Shi, Yuanming" https://www.zbmath.org/authors/?q=ai:shi.yuanming Summary: In this paper the geometric mean of partial positive definite matrices with missing entries is considered. The weighted geometric mean of two sets of positive matrices is defined, and we show whether such a geometric mean holds certain properties which the weighted geometric mean of two positive definite matrices satisfies. Additionally, counterexamples demonstrate that certain properties do not hold. A Loewner order on partial Hermitian matrices is also defined. The known results for the maximum determinant positive completion are developed with an integral representation, and the results are applied to the weighted geometric mean of two partial positive definite matrices with missing entries. Moreover, a relationship between two positive definite completions is established with respect to their determinants, showing relationship between their entropy for a zero-mean, multivariate Gaussian distribution. Computational results as well as one application are shown. Properties of sparse random matrices over finite fields. https://www.zbmath.org/1456.82482 2021-04-16T16:22:00+00:00 "Alamino, Roberto C." https://www.zbmath.org/authors/?q=ai:alamino.roberto-c "Saad, David" https://www.zbmath.org/authors/?q=ai:saad.david Large deviations of spread measures for Gaussian matrices. https://www.zbmath.org/1456.60017 2021-04-16T16:22:00+00:00 "Cunden, Fabio Deelan" https://www.zbmath.org/authors/?q=ai:cunden.fabio-deelan "Vivo, Pierpaolo" https://www.zbmath.org/authors/?q=ai:vivo.pierpaolo Bethe ansatz solution for one-dimensional directed polymers in random media. https://www.zbmath.org/1456.82496 2021-04-16T16:22:00+00:00 "Dotsenko, Victor" https://www.zbmath.org/authors/?q=ai:dotsenko.victor "Klumov, Boris" https://www.zbmath.org/authors/?q=ai:klumov.boris Universal distribution of random matrix eigenvalues near the birth of a cut'' transition. https://www.zbmath.org/1456.81372 2021-04-16T16:22:00+00:00 "Eynard, B." https://www.zbmath.org/authors/?q=ai:eynard.bertrand Weil representations via abstract data and Heisenberg groups: a comparison. https://www.zbmath.org/1456.20002 2021-04-16T16:22:00+00:00 "Cruickshank, J." https://www.zbmath.org/authors/?q=ai:cruickshank.james "Gutiérrez Frez, L." https://www.zbmath.org/authors/?q=ai:gutierrez-frez.luis "Szechtman, F." https://www.zbmath.org/authors/?q=ai:szechtman.fernando The paper provides Weil representations of unitary groups with even rank over finite rings via Heisenberg groups. The authors use a constructive approach to obtain the explicit matrix form of the Bruhat elements as well as information on generalized Gauss sums. The result is then shown to be identical to the one following from axiomatic considerations. When the ring is local (not necessarily finite) on the other hand, the index of the subgroup generated by the Bruhat elements is computed. Although the subject of the paper is rather technical, all concepts are explained clearly, results are layed down in great detail and proofs are given in a consistent rigorous manner. The authors also provide several examples at the end as well as a nice selection of references. In view of all this, the article might be interesting not only to specialists in the field, but also to graduate students, due to its pedagogical merits. Reviewer: Danail Brezov (Sofia) Transitions of generalised Bessel kernels related to biorthogonal ensembles. https://www.zbmath.org/1456.15037 2021-04-16T16:22:00+00:00 "Kawamoto, Yosuke" https://www.zbmath.org/authors/?q=ai:kawamoto.yosuke Summary: Biorthogonal ensembles are generalisations of classical orthogonal ensembles such as the Laguerre or the Hermite ensembles. Local fluctuation of these ensembles at the origin has been studied, and determinantal kernels in the limit are described by the Wright generalised Bessel functions. The limit kernels are one parameter deformations of the Bessel kernel and the sine kernel for the Laguerre weight and the Hermite weight, respectively. We study transitions from these generalised Bessel kernels to the sine kernel under appropriate scaling limits in common with classical kernels. Largest Schmidt eigenvalue of random pure states and conductance distribution in chaotic cavities. https://www.zbmath.org/1456.82535 2021-04-16T16:22:00+00:00 "Vivo, Pierpaolo" https://www.zbmath.org/authors/?q=ai:vivo.pierpaolo Several matrix trace inequalities on Hermitian and skew-Hermitian matrices. https://www.zbmath.org/1456.15019 2021-04-16T16:22:00+00:00 "Gao, Xiangyu" https://www.zbmath.org/authors/?q=ai:gao.xiangyu "Wang, Guoqiang" https://www.zbmath.org/authors/?q=ai:wang.guoqiang "Zhang, Xian" https://www.zbmath.org/authors/?q=ai:zhang.xian.1 "Tan, Julong" https://www.zbmath.org/authors/?q=ai:tan.julong Summary: In this paper, we present several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing interior-point methods (IPMs) for semidefinite optimization (SDO). From interacting particle systems to random matrices. https://www.zbmath.org/1456.82657 2021-04-16T16:22:00+00:00 "Ferrari, Patrik L." https://www.zbmath.org/authors/?q=ai:ferrari.patrik-lino On block Gaussian sketching for the Kaczmarz method. https://www.zbmath.org/1456.65023 2021-04-16T16:22:00+00:00 "Rebrova, Elizaveta" https://www.zbmath.org/authors/?q=ai:rebrova.elizaveta "Needell, Deanna" https://www.zbmath.org/authors/?q=ai:needell.deanna Summary: The Kaczmarz algorithm is one of the most popular methods for solving large-scale over-determined linear systems due to its simplicity and computational efficiency. This method can be viewed as a special instance of a more general class of sketch and project methods. Recently, a block Gaussian version was proposed that uses a block Gaussian sketch, enjoying the regularization properties of Gaussian sketching, combined with the acceleration of the block variants. Theoretical analysis was only provided for the non-block version of the Gaussian sketch method. Here, we provide theoretical guarantees for the block Gaussian Kaczmarz method, proving a number of convergence results showing convergence to the solution exponentially fast in expectation. On the flip side, with this theory and extensive experimental support, we observe that the numerical complexity of each iteration typically makes this method inferior to other iterative projection methods. We highlight only one setting in which it may be advantageous, namely when the regularizing effect is used to reduce variance in the iterates under certain noise models and convergence for some particular matrix constructions. Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices. https://www.zbmath.org/1456.15033 2021-04-16T16:22:00+00:00 "Capitaine, Mireille" https://www.zbmath.org/authors/?q=ai:capitaine.mireille "Donati-Martin, Catherine" https://www.zbmath.org/authors/?q=ai:donati-martin.catherine Summary: In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked $$N \times N$$ complex Deformed Wigner matrix $$M_N$$. $$M_N$$ is defined as follows: $$M_N=W_N/\sqrt{N}+A_N$$ where $$W_N$$ is an $$N \times N$$ Hermitian Wigner matrix whose entries have a law $$\mu$$ satisfying a Poincaré inequality and the matrix $$A_N$$ is a block diagonal matrix, with an eigenvalue $$\theta$$ of multiplicity one, generating an outlier in the spectrum of $$M_N$$. We prove that the fluctuations of the norm of the projection of a unit eigenvector corresponding to the outlier of $$M_N$$ onto a unit eigenvector corresponding to $$\theta$$ are not universal. Indeed, we take away a fit approximation of its limit from this norm and prove the convergence to zero as $$N$$ goes to $$\infty$$ of the Lévy-Prohorov distance between this rescaled quantity and the convolution of $$\mu$$ and a centered Gaussian distribution (whose variance may depend depend upon $$N$$ and may not converge). Long and short time asymptotics of the two-time distribution in local random growth. https://www.zbmath.org/1456.82467 2021-04-16T16:22:00+00:00 "Johansson, Kurt" https://www.zbmath.org/authors/?q=ai:johansson.kurt Summary: The two-time distribution gives the limiting joint distribution of the heights at two different times of a local 1D random growth model in the curved geometry. This distribution has been computed in a specific model but is expected to be universal in the KPZ universality class. Its marginals are the GUE Tracy-Widom distribution. In this paper we study two limits of the two-time distribution. The first, is the limit of long time separation when the quotient of the two times goes to infinity, and the second, is the short time limit when the quotient goes to zero. Singular values of large non-central random matrices. https://www.zbmath.org/1456.60016 2021-04-16T16:22:00+00:00 "Bryc, Włodek" https://www.zbmath.org/authors/?q=ai:bryc.wlodzimierz "Silverstein, Jack W." https://www.zbmath.org/authors/?q=ai:silverstein.jack-w Authors' abstract: We study largest singular values of large random matrices, each with mean of a fixed rank K. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest K singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that have block structure. We also deduce asymptotic normality for the largest eigenvalues of a random matrix arising in a model of population genetics. Reviewer: Göran Högnäs (Åbo) An $$L^p$$ multiplicative coboundary theorem for sequences of unitriangular random matrices. https://www.zbmath.org/1456.60024 2021-04-16T16:22:00+00:00 "Morrow, Steven T." https://www.zbmath.org/authors/?q=ai:morrow.steven-t Summary: \textit{R. C. Bradley} [ibid.. 9, No. 3, 659--678 (1996; Zbl 0870.60028)] proved a multiplicative coboundary'' theorem for sequences of unitriangular random matrices with integer entries, requiring tightness of the family of distributions of the entries from the partial matrix products of the sequence. This was an analog of \textit{K. Schmidt}'s result [Cocycles on ergodic transformation groups. Macmillan Lectures in Mathematics 1. Delhi, Bombay, Calcutta, Madras: The Macmillan Company of India Ltd. (1977; Zbl 0421.28017)] for sequences of real-valued random variables with tightness of the family of partial sums. Here is an $$L^p$$ moment analog of Bradley's result which also relaxes the restriction of entries being integers. Large time behavior, bi-Hamiltonian structure, and kinetic formulation for a complex Burgers equation. https://www.zbmath.org/1456.35033 2021-04-16T16:22:00+00:00 "Gao, Yu" https://www.zbmath.org/authors/?q=ai:gao.yu "Gao, Yuan" https://www.zbmath.org/authors/?q=ai:gao.yuan "Liu, Jian-Guo" https://www.zbmath.org/authors/?q=ai:liu.jian-guo Summary: We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner's semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained, and some explicit solutions are given by Wigner's semicircle laws. We also construct a bi-Hamiltonian structure for the system of real and imaginary components of the complex Burgers equation (coupled Burgers system). We establish a kinetic formulation for the coupled Burgers system and prove the existence and uniqueness of entropy solutions. The coupled Burgers system in Lagrangian variable naturally leads to two interacting particle systems, the Fermi-Pasta-Ulam-Tsingou model with nearest-neighbor interactions, and the Calogero-Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit. Extreme value problems in random matrix theory and other disordered systems. https://www.zbmath.org/1456.82488 2021-04-16T16:22:00+00:00 "Biroli, Giulio" https://www.zbmath.org/authors/?q=ai:biroli.giulio "Bouchaud, Jean-Philippe" https://www.zbmath.org/authors/?q=ai:bouchaud.jean-philippe "Potters, Marc" https://www.zbmath.org/authors/?q=ai:potters.marc The maximum number of Parter vertices of acyclic matrices. https://www.zbmath.org/1456.05103 2021-04-16T16:22:00+00:00 "Fonseca, Amélia" https://www.zbmath.org/authors/?q=ai:fonseca.amelia "Mestre, Ângela" https://www.zbmath.org/authors/?q=ai:mestre.angela "Mohammadian, Ali" https://www.zbmath.org/authors/?q=ai:mohammadian.ali "Perdigão, Cecília" https://www.zbmath.org/authors/?q=ai:perdigao.cecilia "Torres, Maria Manuel" https://www.zbmath.org/authors/?q=ai:torres.maria-manuel This manuscript deals with the maximum number of Parter vertices of a singular symmetric matrix whose underlying graph is a tree. In this paper, all graphs are assumed to be finite, undirected and without loops or multiple edges. Let $$G$$ be a graph with vertex set $$V(G)$$ and edge set $$E(G)$$. $$S_F(G)$$ denotes the set of all the symmetric matrices $$A$$ with entries in the field $$F$$, whose rows and columns are indexed by $$V(G)$$, such that for every two distinct vertices $$u, v \in V(G)$$, the $$(u,v)$$-entry of $$A$$ is nonzero if and only if $$(u,v) \in E(G)$$. The adjacency matrix of $$G$$, $$\mathcal{A}(G)$$, is a $$(0,1)$$-matrix in $$S_F(G)$$ all of whose diagonal entries are equal $$0$$. In fact, the matrices in $$S_F(G)$$ can be seen as weighted adjacency matrices of $$G$$. For any tree $$T$$, the elements of $$S_F(T)$$ are referred as acyclic matrices. For every matrix $$A \in S_F(G)$$ and subset $$X$$ of $$V(G)$$, the principal submatrix of $$A$$ obtained by deleting the rows and columns indexed by $$X$$ is denoted by $$A(X)$$. Let $$G$$ be a graph with $$n=|V(G)|$$ and let $$A \in S_F(G)$$. A vertex $$v \in V(G)$$ is a Parter vertex of $$A$$ if $$\eta(A(v))=\eta(A)+1$$, where $$\eta(A)$$ denotes the dimension of $$\ker{A}$$. In this paper, the authors are interested in the maximum number of Parter vertices of singular acyclic matrices. It is known that this number, for a singular matrix with rank $$r$$ whose underlying graph has no isolated vertices, is at most $$r-1$$. In addition, the maximum number of Parter vertices of $$n \times n$$ singular acyclic matrices is $$2\lfloor\frac{n-1}{2}\rfloor-1$$. As a generalization, the authors prove that the number of Parter vertices of singular acyclic matrices with rank $$r$$ is at most $$2\lfloor\frac{r}{2}\rfloor-1$$. They also characterize the structure of trees which achieve this upper bound. Reviewer: Juan Ramón Torregrosa Sánchez (Valencia) Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. https://www.zbmath.org/1456.15005 2021-04-16T16:22:00+00:00 "Dai, Hui" https://www.zbmath.org/authors/?q=ai:dai.hui "Geary, Zachary" https://www.zbmath.org/authors/?q=ai:geary.zachary "Kadanoff, Leo P." https://www.zbmath.org/authors/?q=ai:kadanoff.leo-p Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques. https://www.zbmath.org/1456.82838 2021-04-16T16:22:00+00:00 "Sasamoto, T." https://www.zbmath.org/authors/?q=ai:sasamoto.tomohiro Generalized Catalan numbers associated with a family of Pascal-like triangles. https://www.zbmath.org/1456.11030 2021-04-16T16:22:00+00:00 "Barry, Paul" https://www.zbmath.org/authors/?q=ai:barry.paul Author's abstract: We find closed-form expressions and continued fraction generating functions for a family of generalized Catalan numbers associated with a set of Pascal-like number triangles that are defined by Riordan arrays. We express these generalized Catalan numbers as the moments of appropriately defined orthogonal polynomials. We also describe them as the row sums of related Riordan arrays. Links are drawn to the Narayana numbers and to lattice paths. We further generalize this one-parameter family to a three-parameter family. We use the generalized Catalan numbers to define generalized Catalan triangles. We define various generalized Motzkin numbers defined by these general Catalan numbers. Finally we indicate that the generalized Catalan numbers can be associated with certain generalized Eulerian numbers by means of a special transform. Reviewer: Thomas Ernst (Uppsala) On the Toeplitz and polar decompositions of an involutive matrix. https://www.zbmath.org/1456.15012 2021-04-16T16:22:00+00:00 "Ikramov, Kh. D." https://www.zbmath.org/authors/?q=ai:ikramov.khakim-d The Toeplitz decomposition of a square complex matrix $$A$$ is its representation in the form $$A=B+iC, B=B^\ast , C=C^\ast.$$ The Hermitian matrices $$B$$ and $$C$$ are determined uniquely by the formulas $$B={\frac{1}{2}}(A+A^\ast), C={\frac{1}{2i}}(A-A^\ast).$$ The polar decompositions of $$A$$ are its representations of the form $$A = PU = UQ.$$ Here, $$U$$ is a unitary matrix, and the Hermitian matrices $$P$$ and $$Q$$ are positive semi-definite. Moreover, $$P=(AA^\ast)^{1/2} , \,\, Q=(A^\ast A)^{1/2}.$$ The main result that the author obtains is the following: Theorem. Assume that an involution $$A$$ is not a Hermitian matrix. Then, by a unitary similarity transformation, $$A$$ can be brought to a block diagonal matrix in which the first order blocks are either $$1$$ or $$-1$$. The other blocks have even orders. Every block of order $$2k$$ is associated with a pair of eigenvalues $$\pm\cosh t$$ of the matrix $$B$$ and the corresponding pair of eigenvalues $$\pm\sinh t$$ of $$C$$. Each of these eigenvalues is of multiplicity $$k$$. Reviewer: Erich W. Ellers (Toronto) Typical $$l_1$$-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices. https://www.zbmath.org/1456.15036 2021-04-16T16:22:00+00:00 "Kabashima, Yoshiyuki" https://www.zbmath.org/authors/?q=ai:kabashima.yoshiyuki "Vehkaperä, Mikko" https://www.zbmath.org/authors/?q=ai:vehkapera.mikko "Chatterjee, Saikat" https://www.zbmath.org/authors/?q=ai:chatterjee.saikat.1 Geometric matrix midranges. https://www.zbmath.org/1456.15031 2021-04-16T16:22:00+00:00 "Mostajeran, Cyrus" https://www.zbmath.org/authors/?q=ai:mostajeran.cyrus "Grussler, Christian" https://www.zbmath.org/authors/?q=ai:grussler.christian "Sepulchre, Rodolphe" https://www.zbmath.org/authors/?q=ai:sepulchre.rodolphe-j Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős-Rényi graphs. https://www.zbmath.org/1456.60063 2021-04-16T16:22:00+00:00 "Augeri, Fanny" https://www.zbmath.org/authors/?q=ai:augeri.fanny Summary: We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds, except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation of the traces of powers of Wigner matrices with sub-Gaussian entries and the upper tail of cycles counts in sparse Erdős-Rényi graphs down to the sparsity threshold $$n^{-1/2}$$. Time-uniform Chernoff bounds via nonnegative supermartingales. https://www.zbmath.org/1456.60054 2021-04-16T16:22:00+00:00 "Howard, Steven R." https://www.zbmath.org/authors/?q=ai:howard.steven-r "Ramdas, Aaditya" https://www.zbmath.org/authors/?q=ai:ramdas.aaditya-k "McAuliffe, Jon" https://www.zbmath.org/authors/?q=ai:mcauliffe.jon-d "Sekhon, Jasjeet" https://www.zbmath.org/authors/?q=ai:sekhon.jasjeet-s Summary: We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960--80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980--2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Peña; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cramér-Chernoff method, self-normalized processes, and other parts of the literature. Maximal acyclic subgraphs and closest stable matrices. https://www.zbmath.org/1456.05100 2021-04-16T16:22:00+00:00 "Cvetković, Aleksandar" https://www.zbmath.org/authors/?q=ai:cvetkovic.aleksandar-s "Protasov, Vladimir Yu." https://www.zbmath.org/authors/?q=ai:protasov.vladimir-yu Rank functions. https://www.zbmath.org/1456.15001 2021-04-16T16:22:00+00:00 "Beasley, LeRoy B." https://www.zbmath.org/authors/?q=ai:beasley.leroy-b A rank function on an additive monoid $$\mathcal{Q}$$ is a function $$f:\mathcal{Q}\rightarrow\mathbb{N}$$ such that (i) $$f(A)=0$$ if and only if $$A=0$$ and (ii) $$f(A+B)\leq f(A)+f(B)$$ for all $$A$$ and $$B$$. This paper is a short catalogue of examples of rank functions for graphs and for matrices over semirings. For example: for matrices the usual definitions of matrix rank involve rank functions which are all equivalent for matrices over a field but can differ for matrices over a semiring; for any vector space norm $$\left\Vert ~\right\Vert$$ over a field $$v\longmapsto\left\lceil \left\Vert v\right\Vert \right\rceil$$ is a rank function; and various covering and partition numbers in graphs are rank functions. For the entire collection see [Zbl 1433.05003]. Reviewer: John D. Dixon (Ottawa) Large deviations of the shifted index number in the Gaussian ensemble. https://www.zbmath.org/1456.82978 2021-04-16T16:22:00+00:00 "Pérez Castillo, Isaac" https://www.zbmath.org/authors/?q=ai:perez-castillo.isaac On reverses of the Golden-Thompson type inequalities. https://www.zbmath.org/1456.15020 2021-04-16T16:22:00+00:00 "Ghaemi, Mohammad Bagher" https://www.zbmath.org/authors/?q=ai:ghaemi.mohammad-bagher "Kaleibary, Venus" https://www.zbmath.org/authors/?q=ai:kaleibary.venus "Furuichi, Shigeru" https://www.zbmath.org/authors/?q=ai:furuichi.shigeru Summary: In this paper we present some reverses of the Golden-Thompson type inequalities: Let $H$ and $K$ be Hermitian matrices such that $e^s e^H \preceq_{ols} e^K \preceq_{ols} e^t e^H$ for some scalars $s \leq t$, and $\alpha \in [0 , 1]$. Then for all $p>0$ and $k =1,2,\ldots, n$ $\lambda_k (e^{(1-\alpha)H + \alpha K} ) \leq (\max \lbrace S(e^{sp}), S(e^{tp})\rbrace)^{\frac{1}{p}} \lambda_k (e^{pH} \sharp_\alpha e^{pK})^{\frac{1}{p}},$ where $A\sharp_\alpha B = A^\frac{1}{2} \big ( A^{-\frac{1}{2}} B^\frac{1}{2} A^{-\frac{1}{2}} \big) ^\alpha A^\frac{1}{2}$ is $\alpha$-geometric mean, $S(t)$ is the so called Specht's ratio and $\preceq_{ols}$ is the so called Olson order. The same inequalities are also provided with other constants. The obtained inequalities improve some known results. A matrix model for plane partitions. https://www.zbmath.org/1456.82437 2021-04-16T16:22:00+00:00 "Eynard, B." https://www.zbmath.org/authors/?q=ai:eynard.bertrand Condition numbers for real eigenvalues in the real elliptic Gaussian ensemble. https://www.zbmath.org/1456.60019 2021-04-16T16:22:00+00:00 "Fyodorov, Yan V." https://www.zbmath.org/authors/?q=ai:fyodorov.yan-v "Tarnowski, Wojciech" https://www.zbmath.org/authors/?q=ai:tarnowski.wojciech Summary: We study the distribution of the eigenvalue condition numbers $$\kappa_i = \sqrt{(\mathbf{l}_i^* \mathbf{l}_i) (\mathbf{r}_i^* \mathbf{r}_i)}$$ associated with real eigenvalues $$\lambda_i$$ of partially asymmetric $$N \times N$$ random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa_i$$ signal the non-orthogonality of the (bi-orthogonal) set of left $$\mathbf{l}_i$$ and right $$\mathbf{r}_i$$ eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite $$N$$ expression for the joint density function (JDF) $$\mathcal{P}_N (z, t)$$ of $$t = \kappa_i^2 - 1$$ and $$\lambda_i$$ taking value $$z$$, and investigate its several scaling regimes in the limit $$N \rightarrow \infty$$. When the degree of asymmetry is fixed as $$N \rightarrow \infty$$, the number of real eigenvalues is $$\mathcal{O} (\sqrt{N})$$, and in the bulk of the real spectrum $$t_i = \mathcal{O}(N)$$, while on approaching the spectral edges the non-orthogonality is weaker: $$t_i = \mathcal{O} (\sqrt{N})$$. In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of $$N$$ eigenvalues remain real as $$N \rightarrow \infty$$. In such a regime eigenvectors are weakly non-orthogonal, $$t = \mathcal{O}(1)$$, and we derive the associated JDF, finding that the characteristic tail $$\mathcal{P} (z, t) \sim t^{-2}$$ survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices. The probability that all eigenvalues are real for products of truncated real orthogonal random matrices. https://www.zbmath.org/1456.15035 2021-04-16T16:22:00+00:00 "Forrester, Peter J." https://www.zbmath.org/authors/?q=ai:forrester.peter-j "Kumar, Santosh" https://www.zbmath.org/authors/?q=ai:kumar.santosh.3|kumar.santosh.4|kumar.santosh.2|kumar.santosh.1 Summary: The probability that all eigenvalues of a product of $$m$$ independent $$N \times N$$ subblocks of a Haar distributed random real orthogonal matrix of size $$(L_i+N) \times (L_i+N)$$, $$(i=1,\dots ,m)$$ are real is calculated as a multidimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any $$m$$ and with each $$L_i$$ even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases. Tan's epsilon-determinant and ranks of matrices over semirings. https://www.zbmath.org/1456.15008 2021-04-16T16:22:00+00:00 "Mohindru, Preeti" https://www.zbmath.org/authors/?q=ai:mohindru.preeti "Pereira, Rajesh" https://www.zbmath.org/authors/?q=ai:pereira.rajesh Summary: We use the $$\varepsilon$$-determinant introduced by \textit{Ya-Jia Tan} [Linear Multilinear Algebra 62, No. 4, 498--517 (2014; Zbl 1298.15014)] to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.