Recent zbMATH articles in MSC 15https://www.zbmath.org/atom/cc/152021-04-16T16:22:00+00:00WerkzeugGeneralized Catalan numbers associated with a family of Pascal-like triangles.https://www.zbmath.org/1456.110302021-04-16T16:22:00+00:00"Barry, Paul"https://www.zbmath.org/authors/?q=ai:barry.paulAuthor'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)From interacting particle systems to random matrices.https://www.zbmath.org/1456.826572021-04-16T16:22:00+00:00"Ferrari, Patrik L."https://www.zbmath.org/authors/?q=ai:ferrari.patrik-linoLarge deviations of spread measures for Gaussian matrices.https://www.zbmath.org/1456.600172021-04-16T16:22:00+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.pierpaoloTime-uniform Chernoff bounds via nonnegative supermartingales.https://www.zbmath.org/1456.600542021-04-16T16:22:00+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-sSummary: 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.Commuting maps on rank \(k\) triangular matrices.https://www.zbmath.org/1456.150162021-04-16T16:22:00+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"Tan, Li Yin"https://www.zbmath.org/authors/?q=ai:tan.li-yinSummary: Let \(n\geqslant 2\) be an integer and let \(\mathbb{F}\) be a field with \(\vert \mathbb{F}\vert \geqslant 3\). Let \(T_n(\mathbb{F})\) be the ring of \(n\times n\) upper triangular matrices over \(\mathbb{F}\) with centre \(\mathcal{Z}\). Fixing an integer \(2\leqslant k\leqslant n\), we prove that an additive map \(\psi : T_n(\mathbb{F})\rightarrow T_n(\mathbb{F})\) satisfies \(A\psi(A)=\psi(A)A\) for all rank \(k\) matrices \(A\in T_n(\mathbb{F})\) if and only if there exist an additive map \(\mu : T_n(\mathbb{F})\rightarrow\mathcal{Z}, Z\in\mathcal{Z}\) and \(\alpha\in\mathbb{F}\) in which \(\alpha=0\) when \(\vert \mathbb{F}\vert >3\) or \(k<n\) such that \[\psi(A)=ZA + \mu(A) + \alpha(a_{11}+a_{nn})E_{1n}\] for all \(A=(a_{ij})\in T_n(\mathbb{F})\). Here, \(E_{1n}\in T_n(\mathbb{F})\) is the matrix whose \((1, n)\)th entry is one and zeros elsewhere.An \(\alpha\)-cut approach for fuzzy product and its use in computing solutions of fully fuzzy linear systems.https://www.zbmath.org/1456.150042021-04-16T16:22:00+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.aliSummary: 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.Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without a fixed trace.https://www.zbmath.org/1456.600122021-04-16T16:22:00+00:00"Akemann, Gernot"https://www.zbmath.org/authors/?q=ai:akemann.gernot"Vivo, Pierpaolo"https://www.zbmath.org/authors/?q=ai:vivo.pierpaoloExplicit construction of RIP matrices is Ramsey-hard.https://www.zbmath.org/1456.600202021-04-16T16:22:00+00:00"Gamarnik, David"https://www.zbmath.org/authors/?q=ai:gamarnik.davidSummary: Matrices \(\Phi \in \mathbb{R}^{n \times p}\) satisfying the restricted isometry property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for \(n = \log^{O(1)}p\), the explicit deteministic construction of such matrices defied the repeated efforts, and most of the known approaches hit the so-called \(\sqrt n\) sparsity bottleneck. The notable exception is the work by \textit{J. Bourgain} et al. [Duke Math. J. 159, No. 1, 145--185 (2011; Zbl 1236.94027)] constructing an \(n \times p\) RIP matrix with sparsity \(s = \Theta (n^{1/2 + \epsilon} )\), but in the regime \(n = \Omega (p^{1 - \delta} )\).
In this short note we resolve this open question by showing that an explicit construction of a matrix satisfying the RIP in the regime \(n = O(\log^2 p)\) and \(s = \Theta (n^{1/2})\) implies an explicit construction of a three-colored Ramsey graph on \(p\) nodes with clique sizes bounded by \(O(\log^2 p)\) -- a question in the field of extremal combinatorics that has been open for decades.On a parametrization of non-compact wavelet matrices by Wiener-Hopf factorization.https://www.zbmath.org/1456.150382021-04-16T16:22:00+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-matveySummary: 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.Revisiting Horn's problem.https://www.zbmath.org/1456.150342021-04-16T16:22:00+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-bernardA memory-based method to select the number of relevant components in principal component analysis.https://www.zbmath.org/1456.621162021-04-16T16:22:00+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.tizianaTypical \(l_1\)-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices.https://www.zbmath.org/1456.150362021-04-16T16:22:00+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.1Interior estimates in the sup-norm for a class of generalized functions with integral representations.https://www.zbmath.org/1456.350512021-04-16T16:22:00+00:00"Ariza, Eusebio"https://www.zbmath.org/authors/?q=ai:ariza.eusebio"Di Teodoro, Antonio"https://www.zbmath.org/authors/?q=ai:di-teodoro.antonio-nicola"Vanegas, Judith"https://www.zbmath.org/authors/?q=ai:vanegas.judith-cSummary: In this paper we construct apriori estimates for the first order derivatives in the sup-norm for first order meta-monogenic functions, generalized monogenic functions satisfying a differential equation with an anti-monogenic right hand side and generalized meta-monogenic functions satisfying a differential equation with an anti-meta-monogenic right hand side. We obtain such estimates through integral representations of these classes of functions and give an explicit expression for the corresponding constants appearing in the estimates. Then we show how initial value problems can be solved in case an interior estimate is true in the function spaces under consideration. All related functions are in a Clifford type algebra.The effect on the spectral radius of \(r\)-graphs by grafting or contracting edges.https://www.zbmath.org/1456.051082021-04-16T16:22:00+00:00"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11"Chang, An"https://www.zbmath.org/authors/?q=ai:chang.anSummary: Let \(\mathcal{H}_n^{( r)}\) be the set of all connected \(r\)-graphs with given size \(n\). In this paper, we investigate the effect on the spectral radius of \(r\)-uniform hypergraphs by grafting or contracting an edge and then give the ordering of the \(r\)-graphs with small spectral radius over \(\mathcal{H}_n^{( r)}\), when \(n \geq 20\).Zeon and idem-Clifford formulations of Boolean satisfiability.https://www.zbmath.org/1456.681232021-04-16T16:22:00+00:00"Davis, Amanda"https://www.zbmath.org/authors/?q=ai:davis.amanda"Staples, G. Stacey"https://www.zbmath.org/authors/?q=ai:stacey-staples.g|staples.george-staceySummary: The Boolean satisfiability problem (SAT) is the problem of determining whether the variables of a given Boolean formula can be consistently replaced by true or false in such a way that the formula evaluates to true. In fact, SAT was the first known NP-complete problem. In recent years, SAT has found numerous industrial applications, particularly in model checking tools. In the current work, three approaches to Boolean satisfiability based on Clifford subalgebras are presented. In the first approach, an ``idem-Clifford'' algebraic test for satisfiability is presented. This test is straightforward to implement symbolically (e.g., using \textit{Mathematica}), but does not yield the specific solution sets for a given formula. In the second approach, nilpotent adjacency matrix methods are extended to Boolean formulas in order to determine not only whether or not a Boolean formula is satisfiable but to explicitly obtain all solutions. This approach requires the construction of a graph associated with a given Boolean formula. Finally, a ``new'' algebraic framework is developed that combines the convenience of the first approach with the power of the second, recovering explicit solutions without the need to construct graphs. The algebraic formalism presented here readily lends itself to symbolic computations and provides the theoretical basis of a Clifford-algebraic SAT solver.Geometric matrix midranges.https://www.zbmath.org/1456.150312021-04-16T16:22:00+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-jMaximal acyclic subgraphs and closest stable matrices.https://www.zbmath.org/1456.051002021-04-16T16:22:00+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-yuBilinear expansions of lattices of KP \textbf{\( \tau \)}-functions in BKP \textbf{\( \tau \)}-functions: a fermionic approach.https://www.zbmath.org/1456.813042021-04-16T16:22:00+00:00"Harnad, J."https://www.zbmath.org/authors/?q=ai:harnad.john"Orlov, A. Yu."https://www.zbmath.org/authors/?q=ai:orlov.aleksandr-yuSummary: We derive a bilinear expansion expressing elements of a lattice of Kadomtsev-Petviashvili (KP) \( \tau \)-functions, labeled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP \(\tau \)-functions, labeled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur \(Q\)-functions. It is deduced using the representations of KP and BKP \(\tau \)-functions as vacuum expectation values (VEVs) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by the insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEVs of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial \(\tau \)-functions of KP and BKP type.
{\copyright 2021 American Institute of Physics}Towards a geometric approach to Strassen's asymptotic rank conjecture.https://www.zbmath.org/1456.140672021-04-16T16:22:00+00:00"Conner, Austin"https://www.zbmath.org/authors/?q=ai:conner.austin"Gesmundo, Fulvio"https://www.zbmath.org/authors/?q=ai:gesmundo.fulvio"Landsberg, Joseph M."https://www.zbmath.org/authors/?q=ai:landsberg.joseph-m"Ventura, Emanuele"https://www.zbmath.org/authors/?q=ai:ventura.emanuele"Wang, Yao"https://www.zbmath.org/authors/?q=ai:wang.yaoIn this manuscript, first the authors give a short review of the classes of tensors they will work with, namely \textit{tight}, \textit{oblique} and \textit{free} tensors. They also recall briefly the kind of tensor ranks that they will use such as border rank and asymptotic rank.
After this short introduction they state a series of conjectures both previously stated and new, and explain how this conjectures relate to each other. The authors also compute the dimension of the set of tight, oblique and free tensors. All this results are connected with Strassen's asymptotic rank conjecture which in simple terms says that the exponent of matrix multiplication is two.
It is also proved some results regarding compressibility and slice rank of tensors, both in a general setting and in particular cases. These results give evidence to favor some of the conjectures.
Reviewer: Rick Rischter (Itajubá)Large time behavior, bi-Hamiltonian structure, and kinetic formulation for a complex Burgers equation.https://www.zbmath.org/1456.350332021-04-16T16:22:00+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-guoSummary: 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.Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices.https://www.zbmath.org/1456.811072021-04-16T16:22:00+00:00"Haah, Jeongwan"https://www.zbmath.org/authors/?q=ai:haah.jeongwanSummary: We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with the code distance being the linear system size is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code stabilizer group (abelian discrete gauge theory). This means that under local Clifford circuits, the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit codes or the Calderbank-Shor-Steane structure for prime qudit codes; we do not assume any of these.
{\copyright 2021 American Institute of Physics}Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques.https://www.zbmath.org/1456.828382021-04-16T16:22:00+00:00"Sasamoto, T."https://www.zbmath.org/authors/?q=ai:sasamoto.tomohiroThe maximum number of Parter vertices of acyclic matrices.https://www.zbmath.org/1456.051032021-04-16T16:22:00+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-manuelThis 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)A data-driven McMillan degree lower bound.https://www.zbmath.org/1456.370982021-04-16T16:22:00+00:00"Hokanson, Jeffrey M."https://www.zbmath.org/authors/?q=ai:hokanson.jeffrey-mTan's epsilon-determinant and ranks of matrices over semirings.https://www.zbmath.org/1456.150082021-04-16T16:22:00+00:00"Mohindru, Preeti"https://www.zbmath.org/authors/?q=ai:mohindru.preeti"Pereira, Rajesh"https://www.zbmath.org/authors/?q=ai:pereira.rajeshSummary: 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.Solvability of new constrained quaternion matrix approximation problems based on core-EP inverses.https://www.zbmath.org/1456.150142021-04-16T16:22:00+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-sSummary: 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.Existence of semiclassical solutions for some critical Dirac equation.https://www.zbmath.org/1456.811662021-04-16T16:22:00+00:00"Ding, Yanheng"https://www.zbmath.org/authors/?q=ai:ding.yanheng"Guo, Qi"https://www.zbmath.org/authors/?q=ai:guo.qi"Yu, Yuanyang"https://www.zbmath.org/authors/?q=ai:yu.yuanyangSummary: In this paper, we study the following critical Dirac equation \(- i \varepsilon \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x) u = P(x) f(| u |) u + Q(x) | u | u, x \in \mathbb{R}^3\), where \(\varepsilon > 0\) is a small parameter; \(a > 0\) is a constant; \( \alpha_1, \alpha_2, \alpha_3\), and \(\beta\) are \(4 \times 4\) Pauli-Dirac matrices; and \(V, P, Q\), and \(f\) are continuous but are not necessarily of class \(\mathcal{C}^1\). We prove the existence and concentration of semiclassical solutions under suitable assumptions on the potentials \(V(x), P(x)\), and \(Q(x)\) by using variational methods. We also show the semiclassical solutions \(\omega_\varepsilon\) with maximum points \(x_\varepsilon\) of |\( \omega_\varepsilon\)| concentrating at a special set \(\mathcal{H}_P\) characterized by \(V(x), P(x)\), and \(Q(x)\) and for any sequence \(x_\varepsilon \to x_0 \in \mathcal{H}_P, v_\varepsilon(x) := \omega_\varepsilon(\varepsilon x + x_\varepsilon)\) converges in \(W^{1, q}(\mathbb{R}^3, \mathbb{C}^4)\) for \(q \geq 2\) to a ground state solution \(u\) of \(- i \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x_0) u = P(x_0) f(| u |) u + Q(x_0) | u | u, \text{in} \mathbb{R}^3\). Finally, we estimate the exponential decay properties of solutions.
{\copyright 2021 American Institute of Physics}A new rank metric for convolutional codes.https://www.zbmath.org/1456.150272021-04-16T16:22:00+00:00"Almeida, P."https://www.zbmath.org/authors/?q=ai:almeida.paulo-j"Napp, D."https://www.zbmath.org/authors/?q=ai:napp.diegoSummary: 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.Cullis-Radić determinant of a rectangular matrix which has a number of identical columns.https://www.zbmath.org/1456.150072021-04-16T16:22:00+00:00"Makarewicz, Anna"https://www.zbmath.org/authors/?q=ai:makarewicz.anna"Pikuta, Piotr"https://www.zbmath.org/authors/?q=ai:pikuta.piotr\textit{C. E. Cullis} [Matrices and determinoids, Vol. 1. Cambridge: University Press XII (1913; JFM 44.0171.12)] and \textit{M. Radic} [Glas. Mat., III. Ser. 1(21), 17--22 (1966; Zbl 0168.02703)], independently, proposed a definition of the determinant of an \(m\times n\) matrix, with \(m\leq n\). Notice that there are other definitions of determinant of rectangular matrices which are not equivalent to the Cullis-Radić one. Interestingly, the proposed definition shares several properties with the classic determinant, such as the evaluation by Laplace expansion relatively to a row. In the present paper, it is investigated how the existence of identical columns affects the Cullis-Radić determinant of a rectangular matrix.
Reviewer: Natalia Bebiano (Coimbra)Singular values of large non-central random matrices.https://www.zbmath.org/1456.600162021-04-16T16:22:00+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-wAuthors' 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)Unit triangular factorization of the matrix symplectic group.https://www.zbmath.org/1456.150132021-04-16T16:22:00+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.aiqingSolving the problem of simultaneous diagonalization of complex symmetric matrices via congruence.https://www.zbmath.org/1456.150102021-04-16T16:22:00+00:00"Bustamante, Miguel D."https://www.zbmath.org/authors/?q=ai:bustamante.miguel-d"Mellon, Pauline"https://www.zbmath.org/authors/?q=ai:mellon.pauline"Velasco, M. Victoria"https://www.zbmath.org/authors/?q=ai:velasco.maria-victoriaSpectra of large time-lagged correlation matrices from random matrix theory.https://www.zbmath.org/1456.600272021-04-16T16:22:00+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.wojciechLower bounds for matrices and its applications in frame theory.https://www.zbmath.org/1456.150222021-04-16T16:22:00+00:00"Talebi, Gholamreza"https://www.zbmath.org/authors/?q=ai:talebi.gholamrezaSummary: Lower bounds of non-negative triangular matrices and generalized Hausdorff matrices on the Taylor sequence space are considered. Some estimates are found for their lower bounds that depend on the \(\ell_1\)-norm of the columns of the Taylor matrix. Further, we show that similar estimates are obtained if we consider such matrices on the domain space of an arbitrary summability matrix \(\varLambda\) in \(\ell_p\), in which again the \(\ell_1\)-norm of the columns of the matrix \(\varLambda\) appears. As an application of such estimates to frame theory, we present the concept of \(\varLambda\)-frames for a separable Hilbert space \(\mathcal{H}\), as a special case of \(\mathrm{E}\)-frames which were recently introduced in [\textit{G. Talebi} and \textit{M. A. Dehghan}, Banach J. Math. Anal. 9, No. 3, 43--74 (2015; Zbl 1311.42096)]. We study some properties of \(\varLambda\)-frames and specially Taylor frames. We characterize all Taylor orthonormal bases, Taylor Riesz bases and Taylor frames starting with an arbitrary orthonormal basis for \(\mathcal{H}\). Finally, we characterize all dual Taylor frames for a given Taylor frame.A max-plus algebra approach for generating non-delay schedule.https://www.zbmath.org/1456.900832021-04-16T16:22:00+00:00"Žužek, Tena"https://www.zbmath.org/authors/?q=ai:zuzek.tena"Peperko, Aljoša"https://www.zbmath.org/authors/?q=ai:peperko.aljosa"Kušar, Janez"https://www.zbmath.org/authors/?q=ai:kusar.janezSummary: Max-plus algebra is one of the promising mathematical approaches, that can be used for scheduling operations. It was already applied for Johnson's algorithm and cyclic job shop problem. In this article, max-plus algebra is used for generating non-delay schedule. When using non-delay schedule approach, in each stage, task with earliest possible start is scheduled. If there is more than one task eligible for scheduling, we apply the priority (dispatching) rule, and in case of another tie, the tie-breaking rule is applied. We present simple step-by-step procedure for generating matrices of starting and finishing times of operations, using Max-plus algebra. We apply LRPT (Longest Remaining Processing Time) as priority rule and SPT (Shortest Processing Time) as tie-breaking rule.Recovering the normal form and symmetry class of an elasticity tensor.https://www.zbmath.org/1456.740122021-04-16T16:22:00+00:00"Abramian, S."https://www.zbmath.org/authors/?q=ai:abramian.s"Desmorat, B."https://www.zbmath.org/authors/?q=ai:desmorat.boris"Desmorat, R."https://www.zbmath.org/authors/?q=ai:desmorat.rodrigue"Kolev, B."https://www.zbmath.org/authors/?q=ai:kolev.boris"Olive, M."https://www.zbmath.org/authors/?q=ai:olive.marcSummary: We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor. We produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants. An algorithm to detect the symmetry class of an Elasticity tensor is finally formulated.Directed nonabelian sandpile models on trees.https://www.zbmath.org/1456.826072021-04-16T16:22:00+00:00"Ayyer, Arvind"https://www.zbmath.org/authors/?q=ai:ayyer.arvind"Schilling, Anne"https://www.zbmath.org/authors/?q=ai:schilling.anne"Steinberg, Benjamin"https://www.zbmath.org/authors/?q=ai:steinberg.benjamin"Thiéry, Nicolas M."https://www.zbmath.org/authors/?q=ai:thiery.nicolas-marcSummary: We define two general classes of nonabelian sandpile models on directed trees (or arborescences), as models of nonequilibrium statistical physics. Unlike usual applications of the well-known abelian sandpile model, these models have the property that sand grains can enter only through specified reservoirs.
In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the rate of convergence to stationarity. The proofs use wreath products and the representation theory of monoids.Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős-Rényi graphs.https://www.zbmath.org/1456.600632021-04-16T16:22:00+00:00"Augeri, Fanny"https://www.zbmath.org/authors/?q=ai:augeri.fannySummary: 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 \(n^{-1/2}\).Parameterized structure-preserving transformations of matrix polynomials.https://www.zbmath.org/1456.150112021-04-16T16:22:00+00:00"Kawano, Daniel T."https://www.zbmath.org/authors/?q=ai:kawano.daniel-tAn \(n\times n\) matrix polynomial of degree \(\ell\) is a polynomial of the form
\(P(\lambda)=\sum_{i=0}^{\ell}\lambda^{i}A_{i}\), where each \(A_{i}\) is a complex
\(n\times n\) matrix, \(A_{\ell}\neq0\) and \(f(\lambda):=\det P(\lambda)\) is not
identically \(0\) (regularity). The roots of \(f(\lambda)\) are called the finite
eigenvalues of \(P(\lambda)\). Clearly \(f(\lambda)\) has degree at most \(\ell n\).
If \(f(\lambda)\) has degree \(m<\ell n\) then the determinant \(f_{\mathrm{rev}}(\lambda)\)
of the reverse polynomial \(P_{\mathrm{rev}}(\lambda):=\sum_{i=0}^{\ell}\lambda^{\ell
-i}A_{i}\) has \(0\) as a root with multiplicity \(\ell n-m\). In this case
\(P(\lambda)\) is said to have an infinite eigenvalue of multiplicity \(\ell
n-m\). Each eigenvalue corresponds to a Jordan block, and we write \(J_{F}\) as
the sum of the blocks for the finite eigenvalues of \(P(\lambda)\) and
\(J_{\infty}\) as the sum of the blocks for the zero eigenvalue of
\(P_{\mathrm{rev}}(\lambda)\).
Generalizing the idea of a companion matrix of a scalar
polynomial, there exists a construction of an \(\ell n\times\ell n\) matrix
\(A\lambda-B\) (where entries of \(A\) and \(B\) are polynomials in \(\lambda\))
called a companion form of \(P(\lambda)\) such that its Weierstrass canonical
form has the form \(\mathfrak{W}(\lambda):=(I\lambda-J_{F})\oplus(J_{\infty
}\lambda-I)\). If \(\tilde{A}\lambda-\tilde{B}\) is the companion form for a
second matrix polynomial \(\tilde{P}(\lambda)\) which is isospectral to
\(P(\lambda)\) then the definition of the Weierstrass form implies that
\(\tilde{A}\lambda-\tilde{B}=U(A\lambda-B)V\) for some nonsingular matrices
\(U,V\in\mathbb{C}^{\ell n\times\ell n}\).
The aim of the paper is to show how
each such pair \((U,V)\) can be generated by a matrix \(S\) which commutes with
\(\mathfrak{W}(\lambda)\). This work extends the results of \textit{P. Lancaster} and \textit{I. Zaballa} [Oper. Theory: Adv. Appl. 218, 403--424 (2012; Zbl 1252.15010)]. In the earlier
paper [the author et al., Z. Angew. Math. Phys. 69, No. 6, Paper No. 137, 19 p. (2018; Zbl 1403.34012)], the authors proved the result only for the case where the leading
coefficient \(A_{\ell}\) of \(P(\lambda)\) is nonsingular. The present author shows
that this extension is of interest, even in the case \(\ell=2\).
Reviewer: John D. Dixon (Ottawa)Several matrix trace inequalities on Hermitian and skew-Hermitian matrices.https://www.zbmath.org/1456.150192021-04-16T16:22:00+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.julongSummary: 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).Matrix Richard inequality via the geometric mean.https://www.zbmath.org/1456.150182021-04-16T16:22:00+00:00"Fujimoto, Masayuki"https://www.zbmath.org/authors/?q=ai:fujimoto.masayuki"Seo, Yuki"https://www.zbmath.org/authors/?q=ai:seo.yukiSummary: In this paper, we show the matrix version of Richard inequality by virture of Cauchy-Schwartz type inequalities via the matrix geometric mean. As an application, we show a matrix Buzano inequality.On block Gaussian sketching for the Kaczmarz method.https://www.zbmath.org/1456.650232021-04-16T16:22:00+00:00"Rebrova, Elizaveta"https://www.zbmath.org/authors/?q=ai:rebrova.elizaveta"Needell, Deanna"https://www.zbmath.org/authors/?q=ai:needell.deannaSummary: 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.Invertibility via distance for noncentered random matrices with continuous distributions.https://www.zbmath.org/1456.600282021-04-16T16:22:00+00:00"Tikhomirov, Konstantin"https://www.zbmath.org/authors/?q=ai:tikhomirov.konstantin-eSummary: 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)].Parallel reduction of four matrices to condensed form for a generalized matrix eigenvalue algorithm.https://www.zbmath.org/1456.651892021-04-16T16:22:00+00:00"Bosner, Nela"https://www.zbmath.org/authors/?q=ai:bosner.nelaSummary: The VZ algorithm proposed by \textit{C. F. Van Loan} [SIAM J. Numer. Anal. 12, 819--834 (1975; Zbl 0321.65023)] attempts to solve the generalized type of matrix eigenvalue problem \(ACx = \lambda BDx\), where \(A, B \in R^{n \times m}, C, D \in R^{m \times n}\), and \(m \geq n\), without forming products and inverses. Especially, this algorithm is suitable for solving the generalized singular value problem. Van Loan's approach first reduces the matrices \(A, B, C\), and \(D\) to a condensed form by the finite step initial reduction. The reduction finds orthogonal matrices \(Q, U, V\), and \(Z\), such that \textit{QAZ} is upper Hessenberg, and \(QBV, \ Z^TCU\), and \(V^TDU\) are upper triangular. In this initial reduction, \(A\) is reduced to upper Hessenberg form, while simultaneously preserving triangularity of other three matrices. This is done by Givens rotations, annihilating one by one element of \(A\), and by generating three more rotations applied to other matrices per each annihilation. Such an algorithm is quite inefficient. In our work, we propose a blocked algorithm for the initial reduction, based on aggregated Givens rotations and matrix-matrix multiplications, which are applied in the outer loop updates. This algorithm has another level of blocking, exploited in the inner loop. Further, we also consider a variant of the algorithm in a hybrid CPU-GPU framework, where compute-intensive outer loop updates are performed on GPU, and can be overlapped with the reduction in the next step performed on CPU. On the other hand, application of a sequence of rotations in the inner loop is parallelized on CPU, with balanced operation count per thread. Since a large number of aggregated rotations are produced in every outer loop step, they are simultaneously accumulated before outer loop updates. These adjustments speed up original initial reduction considerably which is confirmed by numerical experiments, and the efficiency of the whole VZ algorithm is increased.An \(L^p\) multiplicative coboundary theorem for sequences of unitriangular random matrices.https://www.zbmath.org/1456.600242021-04-16T16:22:00+00:00"Morrow, Steven T."https://www.zbmath.org/authors/?q=ai:morrow.steven-tSummary: \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.Adaptive total variation and second-order total variation-based model for low-rank tensor completion.https://www.zbmath.org/1456.650292021-04-16T16:22:00+00:00"Li, Xin"https://www.zbmath.org/authors/?q=ai:li.xin.11|li.xin.6|li.xin.14|li.xin.13|li.xin|li.xin.12|li.xin.1|li.xin.10|li.xin.7|li.xin.5|li.xin.9|li.xin.2|li.xin.4|li.xin.3|li.xin.15"Huang, Ting-Zhu"https://www.zbmath.org/authors/?q=ai:huang.ting-zhu"Zhao, Xi-Le"https://www.zbmath.org/authors/?q=ai:zhao.xile"Ji, Teng-Yu"https://www.zbmath.org/authors/?q=ai:ji.teng-yu"Zheng, Yu-Bang"https://www.zbmath.org/authors/?q=ai:zheng.yubang"Deng, Liang-Jian"https://www.zbmath.org/authors/?q=ai:deng.liangjianSummary: Recently, low-rank regularization has achieved great success in tensor completion. However, only considering the global low-rankness is not sufficient, especially for a low sampling rate (SR). Total variation (TV) is introduced into low-rank tensor completion (LRTC) problem to promote the local smoothness by incorporating the first-order derivatives information. However, TV usually leads to undesirable staircase effects. To alleviate these staircase effects, we suggest a first- and second-order TV-based parallel matrix factorization model for LRTC problem, which integrates the local smoothness and global low-rankness by simultaneously exploiting the first- and second-order derivatives information. To solve the proposed model, an efficient proximal alternating optimization (PAO)-based algorithm is developed with theoretical guarantee. Moreover, we suggest a regularization parameter selection strategy to automatically update two regularization parameters, which is able to take advantage of the best properties of each of the two regularization terms. Extensive experiments on different tensor data show the superiority of the proposed method over other methods, particularly for extremely low SRs.Retraction note to: ``A determinantal expression for the Fibonacci polynomials in terms of a tridiagonal determinant''.https://www.zbmath.org/1456.110222021-04-16T16:22:00+00:00"Qi, Feng"https://www.zbmath.org/authors/?q=ai:qi.feng"Wang, Jing-Lin"https://www.zbmath.org/authors/?q=ai:wang.jing-lin"Guo, Bai-Ni"https://www.zbmath.org/authors/?q=ai:guo.bai-niThe Editor-in-Chief has retracted the original article [the authors, ibid. 45, No. 6, 1821--1829 (2019; Zbl 1423.11035)] because it significantly overlaps with a number of previously published articles [the first and third author, Matematiche 72, No. 1, 167--175 (2017; Zbl 1432.11014); the first author, \textit{V. Čerňanová} and \textit{Y. S. Semenov}, ``Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials'', Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 81, No. 1, 123--136 (2019); the first author and \textit{A.-Q. Liu}, Acta Univ. Sapientiae, Math. 10, No. 2, 287--297 (2018; Zbl 07042987)]. This article is therefore redundant.On the Toeplitz and polar decompositions of an involutive matrix.https://www.zbmath.org/1456.150122021-04-16T16:22:00+00:00"Ikramov, Kh. D."https://www.zbmath.org/authors/?q=ai:ikramov.khakim-dThe 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)Analytical nonlinear shrinkage of large-dimensional covariance matrices.https://www.zbmath.org/1456.621052021-04-16T16:22:00+00:00"Ledoit, Olivier"https://www.zbmath.org/authors/?q=ai:ledoit.olivier"Wolf, Michael"https://www.zbmath.org/authors/?q=ai:wolf.michael.2This note provides a closed form expression for the nonlinear shrinkage estimation of large covariance matrices. The work can be viewed as the analytic estimation analogue of \textit{O. Ledoit} and \textit{M. Wolf} [J. Multivariate Anal. 88, No. 2, 365--411 (2004; Zbl 1032.62050)] in which estimation is accomplished by numerical techniques. As opposed to Ledoit and Wolf [loc. cit] where linear shrinkage estimation is employed, the present approach relies on the connection between nonlinear shrinkage and nonparametric estimation of the Hilber transform of the sample spectral density. The advantage is the high accuracy and increased speed of the approach as well as the fact that it covers the case where the dimension exceeds the sample size.
Reviewer: Dimitrios Bagkavos (Ioannina)Corrigendum to: ``Completely bounded norms of right module maps''.https://www.zbmath.org/1456.460472021-04-16T16:22:00+00:00"Levene, Rupert H."https://www.zbmath.org/authors/?q=ai:levene.rupert-h"Timoney, Richard M."https://www.zbmath.org/authors/?q=ai:timoney.richard-mThis correction refers to [the authors, ibid. 436, No. 5, 1406--1424 (2012; Zbl 1244.46026)].Dimensional reduction and scattering formulation for even topological invariants.https://www.zbmath.org/1456.814942021-04-16T16:22:00+00:00"Schulz-Baldes, Hermann"https://www.zbmath.org/authors/?q=ai:schulz-baldes.hermann"Toniolo, Daniele"https://www.zbmath.org/authors/?q=ai:toniolo.danieleSummary: Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the \(K\)-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.On spectra and real energy of complex weighted digraphs.https://www.zbmath.org/1456.050982021-04-16T16:22:00+00:00"Bhat, Mushtaq A."https://www.zbmath.org/authors/?q=ai:bhat.mushtaq-a"Pirzada, S."https://www.zbmath.org/authors/?q=ai:pirzada.shariefuddin"Rada, J."https://www.zbmath.org/authors/?q=ai:rada.juanThe authors studied spectral properties of complex weighted digraphs. Also, they showed that a complex weighted digraph \(D\) is balanced if and only if \(D\) and \(|D|\) have the same spectrum, where \(|D|\) is the absolute value weighted digraph of \(D\), that is, the digraph obtained by replacing the weight of each arc by its absolute value. They extended the concept of real energy to complex weighted digraphs and obtained extremal energy unicyclic complex weighted digraphs with cycle weight in the punctured disk \(\{z\in C : |z|\leq 1\}\{0\}\).
The authors considered a family of complex weighted digraphs \(D_{n,h}\), in which each digraph has order \(n\) and cycles of length \(h\geq 2\) only with constant complex weight \(c=a+ib\). For each \(D\in D_{n,h}\), the real energy of \(D\) is related to the real energy of the unweighted cycle of length \(h\) and in some special cases real energy can be compared using quasiorder relations on coefficients of the characteristic polynomial. Finally, they obtained upper bounds on the real energy which generalize those known for unweighted digraphs and signed digraphs.
The article has many avenues and interesting results. It is useful to researchers working on graphs and matrices.
Reviewer: V. Lokesha (Bangalore)Geometrical inverse matrix approximation for least-squares problems and acceleration strategies.https://www.zbmath.org/1456.650272021-04-16T16:22:00+00:00"Chehab, Jean-Paul"https://www.zbmath.org/authors/?q=ai:chehab.jean-paul"Raydan, Marcos"https://www.zbmath.org/authors/?q=ai:raydan.marcosSummary: We extend the geometrical inverse approximation approach to the linear least-squares scenario. For that, we focus on the minimization of \(1 - \cos (X(A^T A), I)\), where \(A\) is a full-rank matrix of size \(m \times n\), with \(m \geq n\), and \(X\) is an approximation of the inverse of \(A^T A\). In particular, we adapt the recently published simplified gradient-type iterative scheme MinCos to the least-squares problem. In addition, we combine the generated convergent sequence of matrices with well-known acceleration strategies based on recently developed matrix extrapolation methods, and also with some line search acceleration schemes which are based on selecting an appropriate steplength at each iteration. A set of numerical experiments, including large-scale problems, are presented to illustrate the performance of the different accelerations strategies.Numerical solution of separable nonlinear equations with a singular matrix at the solution.https://www.zbmath.org/1456.650332021-04-16T16:22:00+00:00"Shen, Yunqiu"https://www.zbmath.org/authors/?q=ai:shen.yunqiu"Ypma, Tjalling J."https://www.zbmath.org/authors/?q=ai:ypma.tjalling-jSummary: We present a numerical method for solving the separable nonlinear equation \(A(y)z + b(y) = 0\), where \(A(y)\) is an \(m \times N\) matrix and \(b(y)\) is a vector, with \(y \in \mathbf{R}^n\) and \(z \in \mathbf{R}^N \). We assume that the equation has an exact solution \((y^\ast, z^\ast)\). We permit the matrix \(A(y)\) to be singular at the solution \(y^\ast\) and also possibly in a neighborhood of \(y^\ast \), while the rank of the matrix \(A(y)\) near \(y^\ast\) may differ from the rank of \(A(y^\ast )\) itself. We previously developed a method for this problem for the case \(m = n + N\), that is, when the number of equations equals the number of variables. That method, based on bordering the matrix \(A(y)\) and finding a solution of the corresponding extended system of equations, could produce a solution of the extended system that does not correspond to a solution of the original problem. Here, we develop a new quadratically convergent method that applies to the more general case \(m \geq n + N\) and produces all of the solutions of the original system without introducing any extraneous solutions.New upper bounds for the infinity norm of Nekrasov matrices.https://www.zbmath.org/1456.150232021-04-16T16:22:00+00:00"Gao, Lei"https://www.zbmath.org/authors/?q=ai:gao.lei"Liu, Qilong"https://www.zbmath.org/authors/?q=ai:liu.qilongThe 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)Largest Schmidt eigenvalue of random pure states and conductance distribution in chaotic cavities.https://www.zbmath.org/1456.825352021-04-16T16:22:00+00:00"Vivo, Pierpaolo"https://www.zbmath.org/authors/?q=ai:vivo.pierpaoloA geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications.https://www.zbmath.org/1456.150262021-04-16T16:22:00+00:00"Zimmermann, Ralf"https://www.zbmath.org/authors/?q=ai:zimmermann.ralfSummary: 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.Equivalence classes of e-matrices and associated eigenvalue localization regions.https://www.zbmath.org/1456.150092021-04-16T16:22:00+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-jThe 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)Perturbation analysis of rational Riccati equations.https://www.zbmath.org/1456.150152021-04-16T16:22:00+00:00"Weng, Peter Chang-Yi"https://www.zbmath.org/authors/?q=ai:weng.peter-chang-yiSummary: In this paper, we consider the perturbation analyses of the continuous-time rational Riccati equations using the norm-wise, mixed and component-wise analyses, which arises from the stochastic \(H_\infty\) problems and the indefinite stochastic linear quadratic control problems. We derive sufficient conditions for the existence of stabilizing solutions of the perturbed rational Riccati equations. Moreover, we obtain the perturbation bounds for the relative errors with respect to the stabilizing solutions of the rational Riccati equations under three kinds of perturbation analyses. Numerical results are presented to illustrate sharper perturbation bounds under the normwise, mixed and componentwise perturbation analyses.An optimal preconditioner for tensor equations involving Einstein product.https://www.zbmath.org/1456.150252021-04-16T16:22:00+00:00"Xie, Ze-Jia"https://www.zbmath.org/authors/?q=ai:xie.ze-jia"Jin, Xiaoqing"https://www.zbmath.org/authors/?q=ai:jin.xiaoqing"Sin, Vaikuong"https://www.zbmath.org/authors/?q=ai:sin.vaikuongSummary: An optimal preconditioner for tensor equations involving the Einstein product is considered. This preconditioner actually is an approximation to any given tensor in some tensor subspaces such as the subspace of all circulant tensors. Numerical examples show that the optimal preconditioner is efficient for some Toeplitz tensor equations.Two-sample hypothesis testing for inhomogeneous random graphs.https://www.zbmath.org/1456.621082021-04-16T16:22:00+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.ulrikeTesting 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)Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem.https://www.zbmath.org/1456.150032021-04-16T16:22:00+00:00"Diao, Huai-An"https://www.zbmath.org/authors/?q=ai:diao.huaian"Sun, Yang"https://www.zbmath.org/authors/?q=ai:sun.yangSummary: In this paper, we consider the mixed and componentwise condition numbers for a linear function \(Lx\) of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques under both unstructured and structured componentwise perturbations. The sharp upper bounds for condition numbers are obtained. An efficient condition estimation algorithm is proposed, which can be integrated into the iterative method for solving large scale TLS problems. Moreover, the new derived condition number expressions can recover the previous results on the condition analysis for the TLS problem when \(L = I_n\). Numerical experiments show the effectiveness of the introduced condition numbers and condition estimation algorithm.A characterization of nonnegativity relative to proper cones.https://www.zbmath.org/1456.150292021-04-16T16:22:00+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-nSummary: 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.On the generalized dimensions of multifractal eigenstates.https://www.zbmath.org/1456.811232021-04-16T16:22:00+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.imreOn pole-swapping algorithms for the eigenvalue problem.https://www.zbmath.org/1456.650252021-04-16T16:22:00+00:00"Camps, Daan"https://www.zbmath.org/authors/?q=ai:camps.daan"Mach, Thomas"https://www.zbmath.org/authors/?q=ai:mach.thomas"Vandebril, Raf"https://www.zbmath.org/authors/?q=ai:vandebril.raf"Watkins, David S."https://www.zbmath.org/authors/?q=ai:watkins.david-sSummary: Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms.Transitions of generalised Bessel kernels related to biorthogonal ensembles.https://www.zbmath.org/1456.150372021-04-16T16:22:00+00:00"Kawamoto, Yosuke"https://www.zbmath.org/authors/?q=ai:kawamoto.yosukeSummary: 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.Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices.https://www.zbmath.org/1456.150332021-04-16T16:22:00+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.catherineSummary: 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).A matrix model for plane partitions.https://www.zbmath.org/1456.824372021-04-16T16:22:00+00:00"Eynard, B."https://www.zbmath.org/authors/?q=ai:eynard.bertrandCondition numbers for real eigenvalues in the real elliptic Gaussian ensemble.https://www.zbmath.org/1456.600192021-04-16T16:22:00+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.wojciechSummary: 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.Complexifications of real Banach spaces and their isometries.https://www.zbmath.org/1456.460122021-04-16T16:22:00+00:00"Ilišević, Dijana"https://www.zbmath.org/authors/?q=ai:ilisevic.dijana"Kuzma, Bojan"https://www.zbmath.org/authors/?q=ai:kuzma.bojan"Li, Chi-Kwong"https://www.zbmath.org/authors/?q=ai:li.chi-kwong"Poon, Edward"https://www.zbmath.org/authors/?q=ai:poon.edwardLet \((\mathcal{X}, \|\cdot\|)\) be a real normed space and \(\mathbb{C}\mathcal{X}:=\mathcal{X}+i\mathcal{X}\) be its complexification endowed with the so-called Taylor complexification norm given by \[\|x+iy\|_{\mathbb{C}}:=\sup_{0\le\theta\le 2\pi}\|x\cos\theta+y\sin\theta\|.\]
When \(\mathcal{X}\) is finite-dimensional, the authors determine the group of isometries on the space \((\mathbb{C} \mathcal{X},\|\cdot\|_{\mathbb{C}})\) in terms of those of \((\mathcal{X}, \|\cdot\|)\). The proof is very technical and uses a series of auxiliary lemmas and results that are of independent interest. Among them, the authors give a description of the Taylor complexification norm in terms of extreme points of the unit ball for the dual of the original norm \(\|\cdot\|\). As an application, they compute the group of isometries for the numerical radius and related norms. Other results, including a partial extension to infinite-dimensional Banach spaces, are also discussed. Moreover, many examples and remarks are given which nicely illustrate the obtained results.
Reviewer: Abdellatif Bourhim (Syracuse)Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)Vertex operators, solvable lattice models and metaplectic Whittaker functions.https://www.zbmath.org/1456.820972021-04-16T16:22:00+00:00"Brubaker, Ben"https://www.zbmath.org/authors/?q=ai:brubaker.ben"Buciumas, Valentin"https://www.zbmath.org/authors/?q=ai:buciumas.valentin"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Gustafsson, Henrik P. A."https://www.zbmath.org/authors/?q=ai:gustafsson.henrik-p-aThis paper discusses two mechanisms by which the quantum groups \(U_q (\hat{\mathfrak{g}})\), for a simple Lie algebra or superalgebra \(\mathfrak{g}\), produce families of special functions with a number of interesting properties related to functional equations, branching rules and unexpected algebraic relations. The first mechanism uses solvable lattice models associated to finite-dimensional modules of \(U_q (\hat{\mathfrak{g}})\). The second mechanism uses actions of Heisenberg and Clifford algebras on a fermionic Fock space, exploiting the boson-fermion correspondence arising in connection with soliton theory, dating back to [\textit{M. Jimbo} and \textit{T. Miwa}, Publ. Res. Inst. Math. Sci. 19, 943--1001 (1983; Zbl 0557.35091)] and pushed forward by \textit{T. Lam} [Math. Res. Lett. 13, No. 2--3, 377--392 (2006; Zbl 1160.05056)] and especially by [\textit{M. Kashiwara} et al., Sel. Math., New Ser. 1, No. 4, 787--805 (1995; Zbl 0857.17013)]. These two points of view provide new insight into the theory of metaplectic Whittaker functions for the general linear group and relate them to LLT polynomials (known also as ribbon symmetric functions). The main theorem of the paper considers two solvable lattice models, named Gamma ice and Delta, and details in Section 4 their row transfer matrices. In this study, metaplectic ice models are exploited, whose partition functions are metaplectic Whittaker functions. In the process, the authors introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions. It is explained that half vertex operators agree with Lam's construction, and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials [\textit{A. Lascoux} et al., J. Math. Phys. 38, No. 2, 1041--1068 (1997; Zbl 0869.05068)] can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models. A number of links with the existing literature is identified as well.
Reviewer: Piotr Garbaczewski (Opole)Alternating sign hypermatrix decompositions of Latin-like squares.https://www.zbmath.org/1456.050262021-04-16T16:22:00+00:00"O'Brien, Cian"https://www.zbmath.org/authors/?q=ai:obrien.cianSummary: 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 \(2n\).On the sum of \(k\) largest Laplacian eigenvalues of a graph and clique number.https://www.zbmath.org/1456.051042021-04-16T16:22:00+00:00"Ganie, Hilal A."https://www.zbmath.org/authors/?q=ai:ganie.hilal-ahmad"Pirzada, S."https://www.zbmath.org/authors/?q=ai:pirzada.shariefuddin"Trevisan, Vilmar"https://www.zbmath.org/authors/?q=ai:trevisan.vilmarSummary: For a simple graph \(G\) with order \(n\) and size \(m\) having Laplacian eigenvalues \(\mu_1, \mu_2, \dots, \mu_{n-1},\mu_n=0\), let \(S_k(G)=\sum_{i=1}^k\mu_i\), be the sum of \(k\) largest Laplacian eigenvalues of \(G\). We obtain upper bounds for the sum of \(k\) largest Laplacian eigenvalues of two large families of graphs. As a consequence, we prove Brouwer's Conjecture for large number of graphs which belong to these families of graphs.A norm inequality for positive block matrices.https://www.zbmath.org/1456.150212021-04-16T16:22:00+00:00"Lin, Minghua"https://www.zbmath.org/authors/?q=ai:lin.minghuaSummary: Any positive matrix \(M = (M_{i, j})_{i, j = 1}^m\) with each block \(M_{i, j}\) square satisfies the symmetric norm inequality \(\| M \| \leq \| \sum_{i = 1}^m M_{i, i} + \sum_{i = 1}^{m - 1} \omega_i I \|\), where \(\omega_i\) (\(i = 1, \ldots, m - 1\)) are quantities involving the width of numerical ranges. This extends the main theorem of \textit{J.-C. Bourin} and \textit{A. Mhanna} [C. R., Math., Acad. Sci. Paris 355, No. 10, 1077--1081 (2017; Zbl 06806461)]
to a higher number of blocks.Geometric mean of partial positive definite matrices with missing entries.https://www.zbmath.org/1456.150302021-04-16T16:22:00+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.yuanmingSummary: 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.A sign pattern with all diagonal entries nonzero whose minimal rank realizations are not diagonalizable over \(\mathbb{C}\).https://www.zbmath.org/1456.150282021-04-16T16:22:00+00:00"Shitov, Yaroslav"https://www.zbmath.org/authors/?q=ai:shitov.yaroslav-nikolaevichSummary: 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.Classical adjoint commuting and determinant preserving linear maps on Kronecker products of Hermitian matrices.https://www.zbmath.org/1456.150022021-04-16T16:22:00+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-heongLet \(H_n\) 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)The probability that all eigenvalues are real for products of truncated real orthogonal random matrices.https://www.zbmath.org/1456.150352021-04-16T16:22:00+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.1Summary: 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.Bethe ansatz solution for one-dimensional directed polymers in random media.https://www.zbmath.org/1456.824962021-04-16T16:22:00+00:00"Dotsenko, Victor"https://www.zbmath.org/authors/?q=ai:dotsenko.victor"Klumov, Boris"https://www.zbmath.org/authors/?q=ai:klumov.borisInclusion regions and bounds for the eigenvalues of matrices with a known eigenpair.https://www.zbmath.org/1456.150172021-04-16T16:22:00+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-jLet \(A=[a_{ij}]\) be an \(n\times n\) real matrix with a real eigenpair \((\lambda,v)\).
The authors use a particular similarity transform, defined by the matrix \(S\) which depends on the eigenpair, to get the matrix \(S^{-1}AS\) which is a real constant row sum matrix. For such matrices the results of \textit{F. J. Hall} and \textit{R. Marsli} [Bull. Korean. Math. Soc. 55, 1691--1701 (2018; Zbl 1406.15013); Linear Multi. Alg. 67, 672--684 (2019; Zbl 1412.15020)] can be applied to obtain inclusion sets for the rest eigenvalues of \(A\).
Many numerical examples and comparisons with known inclusion sets are discussed.
Reviewer: Antoine Mhanna (Kfardebian)Image encryption algorithm for synchronously updating Boolean networks based on matrix semi-tensor product theory.https://www.zbmath.org/1456.680342021-04-16T16:22:00+00:00"Wang, Xingyuan"https://www.zbmath.org/authors/?q=ai:wang.xingyuan"Gao, Suo"https://www.zbmath.org/authors/?q=ai:gao.suoSummary: 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.Invariant sums of random matrices and the onset of level repulsion.https://www.zbmath.org/1456.820102021-04-16T16:22:00+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.pierpaoloFaces of the \(5\times 5\) completely positive cone.https://www.zbmath.org/1456.150322021-04-16T16:22:00+00:00"Zhang, Qinghong"https://www.zbmath.org/authors/?q=ai:zhang.qinghongSummary: 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.New methods based on \(\mathcal{H}\)-tensors for identifying positive definiteness of homogeneous polynomial forms.https://www.zbmath.org/1456.150242021-04-16T16:22:00+00:00"Sun, Deshu"https://www.zbmath.org/authors/?q=ai:sun.deshuSummary: 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.Parameter-related projection-based iterative algorithm for a kind of generalized positive semidefinite least squares problem.https://www.zbmath.org/1456.650422021-04-16T16:22:00+00:00"Li, Chengjin"https://www.zbmath.org/authors/?q=ai:li.chengjinSummary: A projection-based iterative algorithm, which is related to a single parameter (or the multiple parameters), is proposed to solve the generalized positive semidefinite least squares problem introduced in this paper. The single parameter (or the multiple parameters) projection-based iterative algorithms converges to the optimal solution under certain condition, and the corresponding numerical results are shown too.Universal distribution of random matrix eigenvalues near the ``birth of a cut'' transition.https://www.zbmath.org/1456.813722021-04-16T16:22:00+00:00"Eynard, B."https://www.zbmath.org/authors/?q=ai:eynard.bertrandA trust region algorithm for computing extreme eigenvalues of tensors.https://www.zbmath.org/1456.650302021-04-16T16:22:00+00:00"Chen, Yannan"https://www.zbmath.org/authors/?q=ai:chen.yannan"Chang, Jingya"https://www.zbmath.org/authors/?q=ai:chang.jingyaSummary: Eigenvalues and eigenvectors of high order tensors have crucial applications in sciences and engineering. For computing H-eigenvalues and Z-eigenvalues of even order tensors, we transform the tensor eigenvalue problem to a nonlinear optimization with a spherical constraint. Then, a trust region algorithm for the spherically constrained optimization is proposed in this paper. At each iteration, an unconstrained quadratic model function is solved inexactly to produce a trial step. The Cayley transform maps the trial step onto the unit sphere. If the trial step generates a satisfactory actual decrease of the objective function, we accept the trial step as a new iterate. Otherwise, a second order line search process is performed to exploit valuable information contained in the trial step. Global convergence of the proposed trust region algorithm is analyzed. Preliminary numerical experiments illustrate that the novel trust region algorithm is efficient and promising.Convergence of a randomized Douglas-Rachford method for linear system.https://www.zbmath.org/1456.650212021-04-16T16:22:00+00:00"Hu, Leyu"https://www.zbmath.org/authors/?q=ai:hu.leyu"Cai, Xingju"https://www.zbmath.org/authors/?q=ai:cai.xingjuSummary: In this article, we propose a randomized Douglas-Rachford (DR) method for linear system. This algorithm is based on the cyclic DR method. We consider a linear system as a feasible problem of finding intersection of hyperplanes. In each iteration, the next iteration point is determined by a random DR operator. We prove the convergence of the iteration points based on expectation. And the variance of the iteration points declines to zero. The numerical experiment shows that the proposed algorithm performs better than the cyclic DR method.An improved algorithm for generalized least squares estimation.https://www.zbmath.org/1456.621392021-04-16T16:22:00+00:00"Chang, Xiao-Wen"https://www.zbmath.org/authors/?q=ai:chang.xiaowen"Titley-Peloquin, David"https://www.zbmath.org/authors/?q=ai:titley-peloquin.davidSummary: The textbook direct method for generalized least squares estimation was developed by \textit{C. C. Paige} [Math. Comput. 33, 171--183 (1979; Zbl 0405.65018); SIAM J. Numer. Anal. 16, 165--171 (1979; Zbl 0402.65006)] about 40 years ago. He proposed two algorithms. Suppose that the noise covariance matrix, rather than its factor, is available. Both of the Paige's algorithms involve three matrix factorizations. The first does not exploit the matrix structure of the problem, but it can be implemented by blocking techniques to reduce data communication time on modern computer processors. The second takes advantage of the matrix structure, but its main part cannot be implemented by blocking techniques. In this paper, we propose an improved algorithm. The new algorithm involves only two matrix factorizations, instead of three, and can be implemented by blocking techniques. We show that, in terms of flop counts, the improved algorithm costs less than Paige's first algorithm in any case and less than his second algorithm in some cases. Numerical tests show that in terms of CPU running time, our improved algorithm is faster than both of the existing algorithms when blocking techniques are used.Large deviations of the shifted index number in the Gaussian ensemble.https://www.zbmath.org/1456.829782021-04-16T16:22:00+00:00"Pérez Castillo, Isaac"https://www.zbmath.org/authors/?q=ai:perez-castillo.isaacPartial smoothness of the numerical radius at matrices whose fields of values are disks.https://www.zbmath.org/1456.490172021-04-16T16:22:00+00:00"Lewis, A. S."https://www.zbmath.org/authors/?q=ai:lewis.adrian-s"Overton, M. L."https://www.zbmath.org/authors/?q=ai:overton.michael-lLimiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices.https://www.zbmath.org/1456.621132021-04-16T16:22:00+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.guangmingLet \(\mathbf{Y}=\mathbf{\Gamma X}\) 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)Long and short time asymptotics of the two-time distribution in local random growth.https://www.zbmath.org/1456.824672021-04-16T16:22:00+00:00"Johansson, Kurt"https://www.zbmath.org/authors/?q=ai:johansson.kurtSummary: 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.Rank functions.https://www.zbmath.org/1456.150012021-04-16T16:22:00+00:00"Beasley, LeRoy B."https://www.zbmath.org/authors/?q=ai:beasley.leroy-bA 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)Dimensional and scaling analysis.https://www.zbmath.org/1456.001052021-04-16T16:22:00+00:00"Meinsma, Gjerrit"https://www.zbmath.org/authors/?q=ai:meinsma.gjerritWhen do cross-diffusion systems have an entropy structure?https://www.zbmath.org/1456.351042021-04-16T16:22:00+00:00"Chen, Xiuqing"https://www.zbmath.org/authors/?q=ai:chen.xiuqing"Jüngel, Ansgar"https://www.zbmath.org/authors/?q=ai:jungel.ansgarSummary: In this note, necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix \(A(u)\) are given, based on results from matrix factorization. The entropy structure is important in the analysis for such equations since \(A(u)\) is typically neither symmetric nor positive definite. In particular, the normal ellipticity of \(A(u)\) for all \(u\) and the symmetry of the Onsager matrix implies its positive definiteness and hence an entropy structure. If \(A\) is constant or constant up to nonlinear perturbations, the existence of an entropy structure is equivalent to the normal ellipticity of \(A\). The results are applied to various examples from physics and biology. Finally, the normal ellipticity of the \(n\)-species population model of Shigesada, Kawasaki, and Teramoto is proved.On the eigenpoints of cubic surfaces.https://www.zbmath.org/1456.140432021-04-16T16:22:00+00:00"Celik, Turku Ozlum"https://www.zbmath.org/authors/?q=ai:celik.turku-ozlum"Galuppi, Francesco"https://www.zbmath.org/authors/?q=ai:galuppi.francesco"Kulkarni, Avinash"https://www.zbmath.org/authors/?q=ai:kulkarni.avinash"Sorea, Miruna-Ştefana"https://www.zbmath.org/authors/?q=ai:sorea.miruna-stefanaThe aim of the paper is to study the eigenscheme of order three partially symmetric and symmetric tensors. They also show that a subvariety of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenscheme of \(4 \times 4 \times 4\) symmetric tensors.
The spectral theory of tensors is a multi-linear generalization of the study of eigenvalues and eigenvectors in the case of matrices. The eigenscheme \(E(\mathcal{T})\) of a tensor \(\mathcal{T}\) can be roughly thought as the set of eigenpoints of the tensor, i.e. eigenvectors of \(\mathbb{C}^{n+1}\) of a particular contraction of the tensor. In the case of partial symmetric or symmetric tensors of order \(3\) a contraction may be the following. A partial symmetric tensor \(\mathcal{T} \in \operatorname{Sym}^2 \mathbb{C}^{n+1} \otimes \mathbb{C}^{n+1}\) can be seen as an \((n+1)\)-tuple of quadratic forms \((q_0,\dots,q_n)\) in the variables \(x_i\). Analogously, given a a symmetric tensor \(f \in \operatorname{Sym}^3 \mathbb{C}^{n+1}\), i.e. a homogeneous cubic polynomial, one can associate to it an \((n+1)\)-tuple of quadratic forms given by its derivatives \(\frac{\partial f}{\partial x_i}\). The authors investigates the eigenscheme and some its particular subschemes of the aforementioned tensors with those contractions.
At first they recall some basic notions regarding the theory. In particular they introduce the irregular eigenscheme \(\operatorname{Irr}(\mathcal{T})\) and the regular eigenscheme \(\operatorname{Reg}(\mathcal{T})\). The first can be thought as the subscheme of \(E(\mathcal{T})\) given by points with zero eigenvalue, while the second is the residue of \(E(\mathcal{T})\) with respect to \(\operatorname{Irr}(\mathcal{T})\). After that they focus on the case of order \(3\) symmetric tensors providing bounds on the dimensions and geometric properties of the irregular and regular eigenschemes. Numerous examples of symmetric tensors satisfying all the described properties are provided. As they observe, if the regular eigenscheme of a cubic polynomial is \(0\) dimensional, then it consists of at most \(2^{n+1}-1\) points. Therefore they investigate in the ternary and quaternary case whether there exists a cubic polynomial with a prescribed number of regular eigenpoints. Eventually they show that a open subvariety of a linear subspace of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenschemes of order \(3\) quaternary symmetric tensors.
Reviewer: Reynaldo Staffolani (Trento)Averages of products and ratios of characteristic polynomials in polynomial ensembles.https://www.zbmath.org/1456.600112021-04-16T16:22:00+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-rSummary: 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.Determinant of binary circulant matrices.https://www.zbmath.org/1456.150062021-04-16T16:22:00+00:00"Hariprasad, M."https://www.zbmath.org/authors/?q=ai:hariprasad.mSummary: This article gives a closed-form expression for the determinant of binary circulant matrices.Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices.https://www.zbmath.org/1456.150052021-04-16T16:22:00+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-pTime-inhomogeneous random Markov chains.https://www.zbmath.org/1456.601782021-04-16T16:22:00+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.marcosWeil representations via abstract data and Heisenberg groups: a comparison.https://www.zbmath.org/1456.200022021-04-16T16:22:00+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.fernandoThe 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)Properties of sparse random matrices over finite fields.https://www.zbmath.org/1456.824822021-04-16T16:22:00+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.davidOn reverses of the Golden-Thompson type inequalities.https://www.zbmath.org/1456.150202021-04-16T16:22:00+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.shigeruSummary: 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.Extreme value problems in random matrix theory and other disordered systems.https://www.zbmath.org/1456.824882021-04-16T16:22:00+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.marcAnalysis of the multiplicative Schwarz method for matrices with a special block structure.https://www.zbmath.org/1456.650202021-04-16T16:22:00+00:00"Echeverría, Carlos"https://www.zbmath.org/authors/?q=ai:echeverria.carlos"Liesen, Jörg"https://www.zbmath.org/authors/?q=ai:liesen.jorg"Tichý, Petr"https://www.zbmath.org/authors/?q=ai:tichy.petrSummary: Abstract We analyze the convergence of the (algebraic) multiplicative Schwarz method applied to linear algebraic systems with matrices having a special block structure that arises, for example, when a (partial) differential equation is posed and discretized on a two-dimensional domain that consists of two subdomains with an overlap. This is a basic situation in the context of domain decomposition methods. Our analysis is based on the algebraic structure of the Schwarz iteration matrices, and we derive error bounds that are based on the block diagonal dominance of the given system matrix. Our analysis does not assume that the system matrix is symmetric (positive definite), or has the \(M\)- or \(H\)-matrix property. Our approach is motivated by, and significantly generalizes, an analysis for a special one-dimensional model problem of Echeverría et al. given in [\textit{C. Echeverría} et al., ETNA, Electron. Trans. Numer. Anal. 48, 40--62 (2018; Zbl 1390.15072)].