Recent zbMATH articles in MSC 15https://zbmath.org/atom/cc/152024-05-13T19:39:47.825584ZWerkzeugThe binary rank of circulant block matriceshttps://zbmath.org/1532.050252024-05-13T19:39:47.825584Z"Haviv, Ishay"https://zbmath.org/authors/?q=ai:haviv.ishay"Parnas, Michal"https://zbmath.org/authors/?q=ai:parnas.michalSummary: The binary rank of a \(0, 1\) matrix is the smallest size of a partition of its ones into monochromatic combinatorial rectangles. A matrix \(M\) is called \(( k_1, \ldots, k_m; n_1, \ldots, n_m)\) circulant block diagonal if it is a block matrix with \(m\) diagonal blocks, such that for each \(i \in [m]\), the \(i\)th diagonal block of \(M\) is the circulant matrix whose first row has \(k_i\) ones followed by \(n_i - k_i\) zeros, and all of whose other entries are zeros. In this work, we study the binary rank of these matrices as well as the binary rank of their complement, obtained by replacing the zeros by ones and the ones by zeros. In particular, we compare the binary rank of these matrices to their rank over the reals, which forms a lower bound on the former.
We present a general method for proving upper bounds on the binary rank of block matrices that have diagonal blocks of some specified structure and ones elsewhere. Using this method, we prove that the binary rank of the complement of a \(( k_1, \ldots, k_m; n_1, \ldots, n_m)\) circulant block diagonal matrix for integers satisfying \(n_i > k_i > 0\) for each \(i \in [m]\) exceeds its real rank by no more than the maximum of \(\gcd( n_i, k_i) - 1\) over all \(i \in [m]\). We further present several sufficient conditions for the binary rank of these matrices to strictly exceed their real rank. By combining the upper and lower bounds, we determine the exact binary rank of various families of matrices and, in addition, significantly generalize a result of \textit{D. A. Gregory} [J. Comb. Math. Comb. Comput. 6, 183--187 (1989; Zbl 0691.05017)].
Motivated by a question of \textit{N. J. Pullman} [``Ranks of binary matrices with constant line sums'', Linear Algebra Appl. 104, 193--197 (1988)], we study the binary rank of \(k\)-regular \(0, 1\) matrices, those having precisely \(k\) ones in every row and column, and the binary rank of their complement. As an application of our results on circulant block diagonal matrices, we show that for every \(k \geq 2\), there exist \(k\)-regular \(0, 1\) matrices whose binary rank is strictly larger than that of their complement. Furthermore, we exactly determine for every integer \(r\), the smallest possible binary rank of the complement of a 2-regular \(0, 1\) matrix with binary rank \(r\).The halves of Delannoy matrix and Chung-Feller properties of the \(m\)-Schröder pathshttps://zbmath.org/1532.050262024-05-13T19:39:47.825584Z"Yang, Lin"https://zbmath.org/authors/?q=ai:yang.lin"Zhang, Yu-Yuan"https://zbmath.org/authors/?q=ai:zhang.yujuan"Yang, Sheng-Liang"https://zbmath.org/authors/?q=ai:yang.shengliangSummary: We study the \((m, r, s)\)-halves of the Delannoy matrix and Delannoy matrix of the second kind. We find expression for the \(m\)-central coefficients and the \((m, r, s)\)-halves for these Delannoy matrices in terms of the generating function of the \((m + 1)\)-Schröder numbers. We obtain from these results some new Chung-Feller properties for the \(m\)-Schröder paths, and we also give a bijective proof for one of the results. As an additional application we obtain some identities related to the \(m\)-Schröder numbers.The strong spectral property of graphs: graph operations and barbell partitionshttps://zbmath.org/1532.051022024-05-13T19:39:47.825584Z"Allred, Sarah"https://zbmath.org/authors/?q=ai:allred.sarah-r"Curl, Emelie"https://zbmath.org/authors/?q=ai:curl.emelie"Fallat, Shaun"https://zbmath.org/authors/?q=ai:fallat.shaun-m"Nasserasr, Shahla"https://zbmath.org/authors/?q=ai:nasserasr.shahla"Schuerger, Houston"https://zbmath.org/authors/?q=ai:schuerger.houston"Villagrán, Ralihe R."https://zbmath.org/authors/?q=ai:villagran.ralihe-r"Vishwakarma, Prateek K."https://zbmath.org/authors/?q=ai:vishwakarma.prateek-kumarSummary: The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric matrices possess the Strong Spectral Property (denoted \(\mathcal{G}^{\mathrm{SSP}})\) were studied, and along these lines we aim to study properties of graphs that exhibit a so-called barbell partition. Such a partition is a known impediment to membership in the class \(\mathcal{G}^{\mathrm{SSP}}\). In particular we consider the existence of barbell partitions under various standard and useful graph operations. We do so by considering both the preservation of an already present barbell partition after performing said graph operations as well as barbell partitions which are introduced under certain graph operations. The specific graph operations we consider are the addition and removal of vertices and edges, the duplication of vertices, as well as the Cartesian products, tensor products, strong products, corona products, joins, and vertex sums of two graphs. We also identify a correspondence between barbell partitions and graph substructures called forts, using this correspondence to further connect the study of zero forcing and the Strong Spectral Property.The skew spectral radius and skew Randić spectral radius of general random oriented graphshttps://zbmath.org/1532.051052024-05-13T19:39:47.825584Z"Hu, Dan"https://zbmath.org/authors/?q=ai:hu.dan"Broersma, Hajo"https://zbmath.org/authors/?q=ai:broersma.hajo-j"Hou, Jiangyou"https://zbmath.org/authors/?q=ai:hou.jiangyou"Zhang, Shenggui"https://zbmath.org/authors/?q=ai:zhang.shengguiSummary: Let \(G\) be a simple connected graph on \(n\) vertices, and let \(G^\sigma\) be an orientation of \(G\) with skew adjacency matrix \(S(G^\sigma)\). Let \(d_i\) be the degree of the vertex \(v_i\) in \(G\). The skew Randić matrix of \(G^\sigma\) is the \(n \times n\) real skew symmetric matrix \(\mathcal{R}_S (G^\sigma) = [(\mathcal{R}_S)_{ij}]\), where \((\mathcal{R}_S)_{ij} = -(\mathcal{R}_S)_{ji} = (d_i d_j)^{- \frac{1}{2}}\) if \((v_i, v_j)\) is an arc of \(G^\sigma\), and \((\mathcal{R}_S)_{ij} = (\mathcal{R}_S)_{ji} = 0\) otherwise. The skew spectral radius \(\rho_S (G^\sigma)\) and the skew Randić spectral radius \(\rho_{\mathcal{R}_S}(G^\sigma)\) of \(G^\sigma\) are defined as the spectral radius of \(S(G^\sigma)\) and \(\mathcal{R}_S (G^\sigma)\) respectively. In this paper, we give upper bounds for the skew spectral radius and skew Randić spectral radius of general random oriented graphs.Graph limits and spectral extremal problems for graphshttps://zbmath.org/1532.051072024-05-13T19:39:47.825584Z"Liu, Lele"https://zbmath.org/authors/?q=ai:liu.leleSummary: We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let \(\lambda_1(G)\) be the largest eigenvalue of the adjacency matrix of a graph \(G\) and \(\overline{G}\) be the complement of \(G\). A nice conjecture states that the graph on \(n\) vertices maximizing \(\lambda_1(G)+\lambda_1(\overline{G})\) is the join of a clique and an independent set with \(\lfloor n/3\rfloor\) and \(\lceil 2n/3\rceil\) (also \(\lceil n/3\rceil\) and \(\lfloor 2n/3\rfloor\) if \(n\equiv 2\pmod{3})\) vertices, respectively. We resolve this conjecture for sufficiently large \(n\) using analytic methods. Our second result concerns the \(Q\)-spread of a graph \(G\), which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of \(G\). It was conjectured by \textit{D. Cvetković} et al. [Publ. Inst. Math., Nouv. Sér. 81(95), 11--27 (2007; Zbl 1164.05038)] that the unique \(n\)-vertex connected graph of maximum \(Q\)-spread is the graph formed by adding a pendant edge to \(K_{n-1}\). We confirm this conjecture for sufficiently large \(n\).Extremal problems for the eccentricity matrices of complements of treeshttps://zbmath.org/1532.051082024-05-13T19:39:47.825584Z"Mahato, Iswar"https://zbmath.org/authors/?q=ai:mahato.iswar"Kannan, M. Rajesh"https://zbmath.org/authors/?q=ai:rajesh-kannan.mSummary: The eccentricity matrix of a connected graph \(G\), denoted by \(\mathcal{E}(G)\), is obtained from the distance matrix of \(G\) by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The \(\mathcal{E}\)-eigenvalues of \(G\) are the eigenvalues of \(\mathcal{E}(G)\). The largest modulus of an eigenvalue is the \(\mathcal{E}\)-spectral radius of \(G\). The \(\mathcal{E}\)-energy of \(G\) is the sum of the absolute values of all \(\mathcal{E}\)-eigenvalues of \(G\). In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) \(\mathcal{E}\)-spectral radius among the complements of trees. Then, we prove that the \(\mathcal{E}\)-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least \(\mathcal{E}\)-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest \(\mathcal{E}\)-eigenvalue and the \(\mathcal{E}\)-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum-type lower bounds for the second largest \(\mathcal{E}\)-eigenvalue and \(\mathcal{E}\)-energy of a tree and its complement.On the first two eigenvalues of regular graphshttps://zbmath.org/1532.051112024-05-13T19:39:47.825584Z"Zhang, Shengtong"https://zbmath.org/authors/?q=ai:zhang.shengtongSummary: Let \(G\) be a regular graph with \(m\) edges, and let \(\mu_1, \mu_2\) denote the two largest eigenvalues of \(A_G\), the adjacency matrix of \(G\). We show that, if \(G\) is not complete, then
\[
\mu_1^2 +\mu_2^2 \leq \frac{2(\omega -1)}{\omega} m
\]
where \(\omega\) is the clique number of \(G\). This confirms a conjecture of \textit{B. Bollobás} and \textit{V. Nikiforov} [J. Comb. Theory, Ser. B 97, No. 5, 859--865 (2007; Zbl 1124.05058)] for regular graphs. We also show that equality holds if and only if \(G\) is either a balanced Turán graph or the disjoint union of two balanced Turán graphs of the same size.Generalized Bernoulli numbers and polynomials in the context of the Clifford analysishttps://zbmath.org/1532.110312024-05-13T19:39:47.825584Z"Chandragiri, Sreelatha"https://zbmath.org/authors/?q=ai:chandragiri.sreelatha"Shishkina, Olga A."https://zbmath.org/authors/?q=ai:shishkina.olga-andreevnaSummary: In this paper, we consider the generalization of the Bernoulli numbers and polynomials for the case of the hypercomplex variables. Multidimensional analogs of the main properties of classic polynomials are proved.Avoiding right angles and certain Hamming distanceshttps://zbmath.org/1532.110352024-05-13T19:39:47.825584Z"Bursics, Balázs"https://zbmath.org/authors/?q=ai:bursics.balazs"Matolcsi, Dávid"https://zbmath.org/authors/?q=ai:matolcsi.david"Pach, Péter Pál"https://zbmath.org/authors/?q=ai:pach.peter-pal"Schrettner, Jakab"https://zbmath.org/authors/?q=ai:schrettner.jakabThis paper is a contribution to additive combinatorics (or to geometry over finite fields). More precisely, the first result of the paper shows that the largest size of a subset of \(\mathbb{F}_q^n\) avoiding right angles (namely, 3 distinct vectors \(x, y, z\) such that \(x -z\) and \(y -z\) are perpendicular to each other) is at most \(O(n^{q-2})\).
Let us now move to the second result of the paper. It shown that a subset of \(\mathbb{F}_q^n\) avoiding triangles with all right angles can have size at most \(O(n^{2q-2})\). Furthermore, asymptotically tight bounds are given for the largest possible size of a subset \(A\subseteq \mathbb{F}_q^n\) for which \(x-y\) is not self-orthogonal for any distinct \(x, y\in A\). The exact answer depends on the residue of \(q\) modulo \(3\).
The main technique used in the proofs is the so-called ``slice method''. This method is combined with other ideas, including some extremal set theory.
Reviewer: Juanjo Rué Perna (Barcelona)Legendre symbols related to certain determinantshttps://zbmath.org/1532.110392024-05-13T19:39:47.825584Z"Luo, Xin-Qi"https://zbmath.org/authors/?q=ai:luo.xin-qi"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhi-weiSummary: Let \(p\) be an odd prime. For \(b\), \(c\in{\mathbb{Z}}\), Sun introduced the determinant \(D_p(b,c)=\left| (i^2+bij+cj^2)^{p-2}\right|_{1\le i,j \le p-1}\) and investigated the Legendre symbol \((\frac{D_p(b,c)}{p})\). Recently Wu, She and Ni proved that \((\frac{D_p(1,1)}{p})=(\frac{-2}{p})\) if \(p\equiv 2 \pmod 3\), which confirms a previous conjecture of Sun. In this paper, we determine \((\frac{D_p(1,1)}{p})\) in the case \(p\equiv 1 \pmod 3\). Sun proved that \(D_p(2,2)\equiv 0 \pmod p\) if \(p\equiv 3 \pmod 4\); in contrast, we prove that \((\frac{D_p(2,2)}{p})=1\) if \(p\equiv 1 \pmod 8\), and \((\frac{D_p(2,2)}{p})=0\) if \(p\equiv 5 \pmod 8\). Our tools include generalized trinomial coefficients and Lucas sequences.On the exceptional set of a system of linear equations with prime numbershttps://zbmath.org/1532.111362024-05-13T19:39:47.825584Z"Allakov, Ismail"https://zbmath.org/authors/?q=ai:allakov.ismail-a"Abraev, Bakhrom Kholturaevich"https://zbmath.org/authors/?q=ai:abraev.bakhrom-kholturaevichSummary: Let \(X\) be a sufficiently large real number, \(b_1,b_2\)-integers with \(1\le{{b}_1}\), \({{b}_2}\le X\), \({{a}_{ij}}\), \((i=1,2; j=\overline{1,4})\) positive integers, \( {{p}_{ 1}}, \ldots ,{{p}_4}\)-prime numbers. Let \(B=\max\left\{ 3\left|{{a}_{ij}}\right| \right\}\), \(({{i=1,2;j=\overline{1,4}}})\), \(\bar{b}=(b_1,b_2)\), \(K= 9\sqrt{2}B^3\left|\bar{b}\right|\), \(E_{2,4}(X)= \left\{{{b}_i} \bigm| 1\leq b_i\leq X, {{b}_i}\ne{{a}_{i1}}{{p}_1}+\cdots +{{a}_{i4}}{{p}_4}, i=1,2\right\}.\) The paper studies the solvability of a system of linear equations \({{ b}_i}= {{a}_{i1}}{{p}_1}+\cdots +{{a}_{i4}}{{p}_4}\), \(i=1,2,\) in primes \(p_1,\ldots,p_4\) and for the first time a power estimate for the exceptional set \(E_{2,4}(X)\) and a lower estimate for \(R(\bar b)\) the number of solutions of the system under consideration in prime numbers, are obtained, namely, that if \(X\) is sufficiently large and \(\delta (0<\delta<1)\) is sufficiently small real numbers, then: there exists a sufficiently large number \(A,\) such that for \(X>{{B}^A}\) estimate is fair \({{E}_{2,4}}(X)< {{X}^{2-\delta }};\) and for \(R(\bar b)\) given \(\bar{b}=(b_1,b_2)\),\(1\le b_1,b_2 \le X\) fair estimate \(R(\bar{b})\ge{K}^{2- {\delta }}{{\left( \ln K \right)^{-4}}},\) for all \(\bar b=(b_1,b_2)\) except for at most \({X}^{2-{\delta}}\) pairs of them.Vector-circulant matrices and codes over ringshttps://zbmath.org/1532.130292024-05-13T19:39:47.825584Z"Tapia-Recillas, Horacio"https://zbmath.org/authors/?q=ai:tapia-recillas.horacio"Velazco-Velazco, Juan Armando"https://zbmath.org/authors/?q=ai:velazco-velazco.juan-armandoThe study of vector-circulant matrices over finite fields is extended to commutative ring. Vector circulant based additive codes over a finite commutative ring are explored, and some examples of extremal and near extremal half rate codes over the ring \(\mathbb{F}_2 + u \mathbb{F}_2\) with \(u^2=0\) are provided.
Reviewer: Joël Kabore (Ouagadougou)Jacobian schemes of conic-line arrangements and eigenschemeshttps://zbmath.org/1532.140612024-05-13T19:39:47.825584Z"Beorchia, Valentina"https://zbmath.org/authors/?q=ai:beorchia.valentina"Miró-Roig, Rosa M."https://zbmath.org/authors/?q=ai:miro-roig.rosa-mariaGiven a reduced, singular plane curve \(C = V(f) \subset \mathbb P^2\), its corresponding Jacobian scheme \(\Sigma_f\) is the homogeneous ideal generated by the partials of \(f\). The degree of \(\Sigma_f\) is the \textit{global Tjurina number} \(\tau (C)\). \textit{A. A. Du Plessis} and \textit{C. T. C. Wall} [Math. Proc. Camb. Philos. Soc. 126, No. 2, 259--266 (1999; Zbl 0926.14012)] showed that if \(C\) consists of concurrent lines, the \(\tau (C) = (d-1)^2\), otherwise
\[
(d-1)(d-r-1) \leq \tau (C) \leq (d-1) (d-r-1) + r^2,
\]
where \(r\) is the minimal degree of a syzygy between the three partials. The authors study case \(r=1\) when the curve \(C\) is a conic-line arrangement, so that the global Tjurina number is either equal to \(d^2-3d+2\) or \(d^2-3d+3\), giving a complete characterization as to when each case occurs. If \(\tau (C) = d^2-3d+3\), there are seven possibilities. The first, a line arrangement of \(d-1\) concurrent lines and a general line, was described by \textit{A. Dimca} et al. [Osaka J. Math. 57, No. 4, 847--870 (2020; Zbl 1457.32075)]. The remaining six involve conics, for example the union of conics whose base locus is supported at a point. In these cases the Jacobian schemes \(\Sigma_f\) are eigenschemes of suitable partially symmetric tensors, so that results of \textit{L. Oeding} and \textit{G. Ottaviani} [J. Symb. Comput. 54, 9--35 (2013; Zbl 1277.15019)] and \textit{V. Beorchia} et al. [SIAM J. Appl. Algebra Geom. 5, No. 4, 620--650 (2021; Zbl 1478.13047)] are applicable. If \(\tau (C) = d^2-3d+2\), there are just two possibilities, a conic arrangement given by a union of conics belonging to a bitangent pencil or the union of such a conic arrangement and the line passing through the two bitangency points. The proofs rely on a result of \textit{A. Dimca} et al. relating the syzygy module of a product of polynomials with no common factor [J. Algebra 615, 77--102 (2023; Zbl 1499.14052)] and the characterization of the Hilbert-Burch matrix in the case of particular linear Jacobian syzygies given by \textit{R.-O. Buchweitz} and \textit{A. Conca} [J. Commut. Algebra 5, No. 1, 17--47 (2013; Zbl 1280.32016)].
Reviewer: Scott Nollet (Fort Worth)A course in linear algebrahttps://zbmath.org/1532.150012024-05-13T19:39:47.825584Z"George, Raju K."https://zbmath.org/authors/?q=ai:george.raju-k"Ajayakumar, Abhijith"https://zbmath.org/authors/?q=ai:ajayakumar.abhijithPublisher's description: Designed for senior undergraduate and graduate courses in mathematics and engineering, this self-contained textbook discusses key topics in linear algebra with real-life applications. Split into two parts -- theory in part I and solved problems in part II -- the book makes both theoretical and applied linear algebra easily accessible. Topics such as sets and functions, vector spaces, linear transformations, eigenvalues and eigenvectors, normed spaces, and inner product spaces are discussed in part I; while in part II, over 500 meticulously solved problems show how to use linear algebra in real-life situations. A must-have book for linear algebra courses; it also serves as valuable supplementary material.Deficiency indices of block Jacobi matrices: surveyhttps://zbmath.org/1532.150022024-05-13T19:39:47.825584Z"Budyka, V."https://zbmath.org/authors/?q=ai:budyka.viktoriya-s"Malamud, M."https://zbmath.org/authors/?q=ai:malamud.mark-m"Mirzoev, K."https://zbmath.org/authors/?q=ai:mirzoev.karakhan-agahanSummary: The paper is a survey and concerns with infinite symmetric block Jacobi matrices \textbf{J} with \(m \times m\)-matrix entries. We discuss several results on general block Jacobi matrices to be either self-adjoint or have maximal as well as intermediate deficiency indices. We also discuss several conditions for \textbf{J} to have discrete spectrum.Matrix monotonicity and concavity of the principal pivot transformhttps://zbmath.org/1532.150032024-05-13T19:39:47.825584Z"Beard, Kenneth"https://zbmath.org/authors/?q=ai:beard.kenneth"Welters, Aaron"https://zbmath.org/authors/?q=ai:welters.aaron-tLet \(A\in M_n\) be partitioned into a \(2\times 2\) block matrix,
\begin{gather*}
A=[A_{ij}]_{i,j=1,2}=\begin{bmatrix} A_{11} & A_{12}\\
A_{21} & A_{22} \end{bmatrix},
\end{gather*}
where \(A_{ij}\in M_{n_i\times n_j},\,\, i,j=1,2\). The (generalized) principal pivot transform of \(A\) with respect to the \((2,2)\)-block \(A_{22}\) is defined by
\[
\operatorname{PPT}(A):= \begin{bmatrix} A/A_{22} & A_{12}A_{22}^{+}\\
A_{22}^{+}A_{21} & -A_{22}^{+} \end{bmatrix}\in M_{n},
\]
where \(A/A_{22}=A_{11}-A_{12}A_{22}^{+}A_{21}\) is the (generalized) Schur complement of \(A\) with respect to the \((2,2)\)-block \(A_{22}\) and \(A_{22}^+\) denotes the Moore-Penrose pseudoinverse of \(A_{22}\).
In this paper, the authors prove the minimization variational principle for \(\operatorname{PPT}(\cdot)\) and employ it to prove the matrix concavity of this map, in the sense of the Löwner ordering. They also show that \(\operatorname{PPT}(\cdot)\) is matrix monotone under minimal hypotheses. Their results improve those of \textit{J. E. Pascoe} and \textit{R. Tully-Doyle} [Linear Algebra Appl. 643, 161--165 (2022; Zbl 1490.15009)].
Reviewer: Mohammad Sal Moslehian (Mashhad)Index-MP and MP-index matriceshttps://zbmath.org/1532.150042024-05-13T19:39:47.825584Z"Mosić, Dijana"https://zbmath.org/authors/?q=ai:mosic.dijana"Stanimirović, Predrag S."https://zbmath.org/authors/?q=ai:stanimirovic.predrag-s"Kyrchei, Ivan I."https://zbmath.org/authors/?q=ai:kyrchei.ivanSummary: To extend the concepts of the DMP, MPD and CMP inverses and solve several types of systems of equations, we define three new kinds of square matrices, called the index-MP, MP-index and MP-index-MP matrices. We prove several characterizations for new classes of matrices and their relations with significant generalized inverses. Several types of representations for these new matrices are established. We define new binary relations applying index-MP, MP-index and MP-index-MP matrices and we show that these relations are partial orders on the set including square matrices of index one.Permanents as formulas of summation over an algebra with a unique \(n\)-ary operationhttps://zbmath.org/1532.150052024-05-13T19:39:47.825584Z"Egorychev, Georgy P."https://zbmath.org/authors/?q=ai:egorychev.georgii-petrovichSummary: We give a new general definition for permanents over an algebra with a unique \(n\)-ary operation and study their properties. In particular, it is shown that properties of these permanents coincide with the basic properties of the classical Binet-Cauchy permanent (1812).A projection method based on discrete normalized dynamical system for computing C-eigenpairshttps://zbmath.org/1532.150062024-05-13T19:39:47.825584Z"Cui, Lu-Bin"https://zbmath.org/authors/?q=ai:cui.lubin"Yao, Jia-Le"https://zbmath.org/authors/?q=ai:yao.jia-le"Yuan, Jin-Yun"https://zbmath.org/authors/?q=ai:yuan.jin-yun|yuan.jinyunSummary: The largest C-eigenvalue of piezoelectric tensors determines the highest piezoelectric coupling constant, which reflects the coupling between the elastic and dielectric properties of crystal. Here, a projection method based on discrete normalized dynamical system (PDND) is established for computing the largest C-eigenvalue. Theoretical analysis of the convergence for PDND algorithm is given. In numerical experiments, the longitudinal piezoelectric modulus and the unit uniaxial direction that the extreme piezoelectric effect along took place of different piezoelectric materials are given to display the physical meaning of the C-eigenvalues and eigenvectors. Furthermore, the largest C-eigenvalue and all the corresponding eigenvectors can be obtained, which is the advantage of the proposed method.On the Smith normal form of the least common multiple of matrices from some class of matriceshttps://zbmath.org/1532.150072024-05-13T19:39:47.825584Z"Romaniv, A. M."https://zbmath.org/authors/?q=ai:romaniv.a-mSummary: For nonsingular matrices of arbitrary order over a commutative principal ideal domain under certain restrictions imposed on their canonical diagonal forms, we construct the Smith normal form of their least common right multiple.Generic eigenstructures of Hermitian pencilshttps://zbmath.org/1532.150082024-05-13T19:39:47.825584Z"De Terán, Fernando"https://zbmath.org/authors/?q=ai:de-teran.fernando"Dmytryshyn, Andrii"https://zbmath.org/authors/?q=ai:dmytryshyn.andrii-r"Dopico, Froilán M."https://zbmath.org/authors/?q=ai:dopico.froilan-mThe authors consider complex Hermitian \(n\times n\) matrix pencils with rank at most \(r\) (\(r\leqslant n\)). They study their generic eigenstructures. In order to do this, they prove that the set of those pencils is the union of (a finite number of) closures of bundles of certain pencils. The latter ones are in the Hermitian Kronecker canonical form from which one can derive the desired eigenstructures. Moreover, the authors determine the number of aforementioned bundles, as well as their codimension.
Reviewer: Roksana Słowik (Gliwice)Harmony of asymmetric variants of the Filbert and Lilbert matrices in \(q\)-formhttps://zbmath.org/1532.150092024-05-13T19:39:47.825584Z"Kiliç, Emrah"https://zbmath.org/authors/?q=ai:kilic.emrah"Ersanli, Didem"https://zbmath.org/authors/?q=ai:ersanli.didemSummary: We define a new matrix whose entries are defined by the idea of combining two different asymmetric rules. We will derive explicit formulae for the matrices \(L\) and \(U\) come from \(LU\)-decomposition, \(L^{-1}\), \(U^{-1}\), inverse of the main matrix as well as its determinant. To prove the claimed results, we use backward induction method.Nonnegative matrix factorization and log-determinant divergenceshttps://zbmath.org/1532.150102024-05-13T19:39:47.825584Z"Ndour, Macoumba"https://zbmath.org/authors/?q=ai:ndour.macoumba"Ndaw, Mactar"https://zbmath.org/authors/?q=ai:ndaw.mactar"Ngom, Papa"https://zbmath.org/authors/?q=ai:ngom.papaSummary: Nonnegative matrix factorization (NMF) decomposes any nonnegative matrix into the product of two low dimensional nonnegative matrices. Traditional NMF has the risk learning rank-deficient based on high-dimensional dataset. In this work, we propose a new class of multiplicative algorithms for Nonnegative Matrix Factorization (NMF) based on the family of Alpha-Beta logarithm determinant (AB log-det) divergences, which are parametrized by two parameters \(\alpha\) and \(\beta )\) and the logarithm determinant function. We have developed a multiplicative updating rule to optimize AB log-det in the frame of block coordinate descent, which theoretically proves its convergence. Experimental results on popular datasts show the AB-log-det NMF that have been introduced, present more advanced than traditional NMF methods. The proposed family of AB log-det-NMF multiplicative updating rules algorithms is shown to improve robustness. The AB log-det-divergences are very promising for the rank reduction problem.
For the entire collection see [Zbl 1515.35013].Explicit solutions of conjugate, periodic, time-varying Sylvester equationshttps://zbmath.org/1532.150112024-05-13T19:39:47.825584Z"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li.4|ma.li.3|ma.li.5|ma.li.8|ma.li|ma.li.6"Chang, Rui"https://zbmath.org/authors/?q=ai:chang.rui"Han, Mengqi"https://zbmath.org/authors/?q=ai:han.mengqi"Jiao, Yongmei"https://zbmath.org/authors/?q=ai:jiao.yongmei(no abstract)On the bounds of the eigenvalues of matrix polynomialshttps://zbmath.org/1532.150122024-05-13T19:39:47.825584Z"Shah, Wali Mohammad"https://zbmath.org/authors/?q=ai:shah.wali-mohammad"Monga, Zahid Bashir"https://zbmath.org/authors/?q=ai:monga.zahid-bashirSummary: Let \(P(z):=\displaystyle{\sum\limits_{j=0}^n} A_jz^j,~A_j\in \mathbb{C}^{m\times m}\), \(0\leq j\leq n\) be a matrix polynomial of degree \(n,\) such that
\[A_n\geq A_{n-1}\geq \ldots \geq A_0\geq 0,~A_n>0.\]
Then the eigenvalues of \(P(z)\) lie in the closed unit disk.
This theorem proved by \textit{G. Dirr} and \textit{H. K. Wimmer} [IEEE Trans. Autom. Control 52, No. 11, 2151--2153 (2007; Zbl 1366.15033)]
is in fact a matrix extension of a famous and elegant result on the distribution of zeros of polynomials known as Eneström-Kakeya theorem. In this paper, we prove a more general result which inter alia includes the above result as a special case. We also prove an improvement of a result due to \textit{C.-T. Lê} et al. [Oper. Matrices 13, No. 4, 937--954 (2019; Zbl 1432.15008)]
besides a matrix extension of a result proved by \textit{Q. G. Mohammad} [Am. Math. Mon. 74, 290--292 (1967; Zbl 0152.06101)].Inequalities for partial determinants of accretive block matriceshttps://zbmath.org/1532.150132024-05-13T19:39:47.825584Z"Fu, Xiaohui"https://zbmath.org/authors/?q=ai:fu.xiaohui"Hu, Lihong"https://zbmath.org/authors/?q=ai:hu.lihong"Salarzay, Abdul Haseeb"https://zbmath.org/authors/?q=ai:salarzay.abdul-haseeb(no abstract)On sign-real spectral radii and sign-real expansive matriceshttps://zbmath.org/1532.150142024-05-13T19:39:47.825584Z"Bünger, Florian"https://zbmath.org/authors/?q=ai:bunger.florian"Seeger, Alberto"https://zbmath.org/authors/?q=ai:seeger.albertoLet \({\mathbb M}_n\) be the space of \(n\times n\) real matrices. The sign-real spectral radius of \(A\in {\mathbb M}_n\) is
\[
\xi(A):=\max\{\lambda\in {\mathbb R}_+ : |Ax|\ge \lambda |x| \mbox{ for some } x\in {\mathbb R}^n\setminus\{0\}\},
\]
where \(|x|:=(|x_1|, \dots, |x_n|)^\top\) if \(x = (x_1, \dots, x_n)^\top\in {\mathbb R}^n\) and the vector inequality \(x\le y\) is defined component-wise. It has been studied by researchers since its introduction by \textit{S. M. Rump} [Linear Algebra Appl. 266, 1--42 (1997; Zbl 0901.15002)]
in 1997. The notion is related to estimating the component-wise distance to singularity. The function \(\xi\) bears some resemblance of or is related to the unitarily invariant matrix norms in some ways. A matrix \(A\in {\mathbb M}_n\) is called sign-real expansive if \(\xi (A) \ge 1\). Let
\[
\Omega_n =\{ A\in {\mathbb M}_n: \xi(A) \ge 1\}
\]
be the set of sign-real expansive matrices.
The authors obtain new properties of \(\xi\) and study the structure of \(\Omega_n\), such as checking membership based on computing principal minors. Another method presented is new volumetric technique for testing sign-real expansiveness. Some topological and invariance properties are given, for example, they prove that \(\Omega_n\) is a regular closed, unbounded, path-connected, and semi-algebraic set. They come up with a conjecture which is stronger than the conjecture in [\textit{S. M. Rump}, ``100 Euro Problem'', \url{https://www.tuhh.de/ti3/rump/100EuroProblem.pdf}] and further provide evidence supporting their conjecture.
Reviewer: Tin Yau Tam (Reno)Connections between matrix products for 3-vectors and geometric algebrahttps://zbmath.org/1532.150152024-05-13T19:39:47.825584Z"Bongardt, Bertold"https://zbmath.org/authors/?q=ai:bongardt.bertold"Löwe, Harald"https://zbmath.org/authors/?q=ai:lowe.haraldSummary: The \textit{geometric product} represents a core concept for establishing geometric algebras and in case of vectors matches the formal sum of their \textit{inner product} and their \textit{wedge product}. The geometric product is reconsidered for the case of 3-vectors by means of usual matrix algebra in this article. Therefore, a symmetric matrix product and an antisymmetric matrix product are introduced, whose matrix sum yields a third product that renders the information of a vector pair's geometric product in terms of a matrix associated to the vectors. The three matrices -- that correspond to inner product, wedge product, and geometric product, respectively -- are named \textit{wheel product}, \textit{curl product}, and \textit{full product}. The observation about the structural correspondence of the geometric product with matrix theory may be used for future practical computations and unveils connections of geometric algebra with related disciplines.Geometry of linear algebrashttps://zbmath.org/1532.150162024-05-13T19:39:47.825584Z"Burlakov, I. M."https://zbmath.org/authors/?q=ai:burlakov.igor-mSummary: In this paper, we consider spaces whose geometry is generated by a homogeneous function of degree \(m \geq 2\), which is invariant under the action of some subgroup of the linear group of the given space. A general method is proposed and examples of realization of such spaces on linear algebras are given.Composition algebras of arbitrary degreeshttps://zbmath.org/1532.150172024-05-13T19:39:47.825584Z"Guseva, N. I."https://zbmath.org/authors/?q=ai:guseva.nadezhda-ivanovna"Lukyanova, E. V."https://zbmath.org/authors/?q=ai:lukyanova.e-vSummary: In this paper, a generalization of composition algebras is given, in which the quadratic form is replaced by a form of arbitrary degree. Two series of generalized composition algebras are found: cyclic algebras of arbitrary order and algebras that are obtained by the generalized doubling procedure applied to cyclic algebras.Kernels of algebraic curvature tensors of symmetric and skew-symmetric buildshttps://zbmath.org/1532.150182024-05-13T19:39:47.825584Z"Klos, B."https://zbmath.org/authors/?q=ai:klos.birgitSummary: The kernel of an algebraic curvature tensor is a fundamental subspace that can be used to distinguish between different algebraic curvature tensors. Kernels of algebraic curvature tensors built only of canonical algebraic curvature tensors of a single build have been studied in detail. We consider the kernel of an algebraic curvature tensor \(R\) that is a sum of canonical algebraic curvature tensors of symmetric and skew-symmetric build. An obvious way to ensure that the kernel of \(R\) is nontrivial is to choose the involved bilinear forms such that the intersection of their kernels is nontrivial. We present a construction wherein this intersection is trivial but the kernel of \(R\) is nontrivial. We also show how many bilinear forms satisfying certain conditions are needed in order for \(R\) to have a kernel of any allowable dimension.Efficient trace-free decomposition of symmetric tensors of arbitrary rankhttps://zbmath.org/1532.150192024-05-13T19:39:47.825584Z"Toth, Viktor T."https://zbmath.org/authors/?q=ai:toth.viktor-t"Turyshev, Slava G."https://zbmath.org/authors/?q=ai:turyshev.slava-gSummary: Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well as in the study of gravitational radiation and gravitational lensing. Their use allows integration of the relevant wave propagation equations to arbitrary order. We present an improved iterative method for the trace-free decomposition of symmetric tensors of arbitrary rank. The method can be used both in coordinate-free symbolic derivations using a computer algebra system and in numerical modeling. We obtain a closed-form representation of the trace-free decomposition in arbitrary dimensions. To demonstrate the results, we compute the coordinate combinations representing the symmetric trace-free (STF) mass multipole moments for rank 5 through 8, not readily available in the literature.Minimal integrity bases of invariants of two second order antisymmetric or symmetric tensors in the Minkowski spacehttps://zbmath.org/1532.150202024-05-13T19:39:47.825584Z"Chen, Yannan"https://zbmath.org/authors/?q=ai:chen.yannan"Huang, Zheng-Hai"https://zbmath.org/authors/?q=ai:huang.zheng-hai"Qi, Liqun"https://zbmath.org/authors/?q=ai:qi.liqunSummary: We study invariants of two second order antisymmetric or symmetric tensors in the Minkowski space. First, we discuss some properties of tensor polynomials of two second order tensors, which provide a theoretical basis for us to investigate integrity bases of invariants of two second order antisymmetric or symmetric tensors. Second, we present a minimal integrity basis of invariants of two second order antisymmetric tensors in the Minkowski space; and give two examples to illustrate the obtained results. Finally, we apply similar approach to the case of a second order symmetric tensor and a second order antisymmetric tensor and the case of two second order symmetric tensors, and establish their minimal integrity bases of invariants.Solving linear equations over maxmin-\(\omega\) systemshttps://zbmath.org/1532.150212024-05-13T19:39:47.825584Z"Syifa'ul Mufid, Muhammad"https://zbmath.org/authors/?q=ai:mufid.muhammad-syifaul"Patel, Ebrahim"https://zbmath.org/authors/?q=ai:patel.ebrahim-l"Sergeev, Sergeĭ"https://zbmath.org/authors/?q=ai:sergeev.sergei-m|sergeev.sergei-n|sergeev.sergei-alekseevichSummary: Maxmin-\(\omega\) dynamical systems were previously introduced as an ``all-in-one package'' that can yield a solely min-plus, a solely max-plus, or a max-min-plus dynamical system by varying a parameter \(\omega \in (0, 1]\). With such systems in mind, it is natural to introduce and consider maxmin-\(\omega\) linear systems of equations of the type \(A \otimes_{\omega} x=b\). However, to our knowledge, such maxmin-\(\omega\) linear systems have not been studied before and in this paper we present an approach to solve them. We show that the problem can be simplified by performing normalization and then generating a ``canonical'' matrix which we call the principal order matrix. Instead of directly trying to find the solutions, we search the possible solution indices which can be identified using the principal order matrix and the parameter \(\omega\). The fully active solutions are then immediately obtained from these solution indices. With the fully active solutions at hand, we then present the method to find other solutions by applying a relaxation, i.e., increasing or decreasing some components of fully active solutions. This approach can be seen as a generalization of an approach that could be applied to solve max-plus or min-plus linear systems. Our results also shed more light on an unusual feature of maxmin-\(\omega\) linear systems, which, unlike in the usual linear algebra, can have a finite number of solutions in the case where their solution is non-unique.Contractive symmetric matrix completion problems related to graphshttps://zbmath.org/1532.150222024-05-13T19:39:47.825584Z"Chun, Sangmin"https://zbmath.org/authors/?q=ai:chun.sangmin"Kim, In Hyoun"https://zbmath.org/authors/?q=ai:kim.in-hyoun"Kim, Jaewoong"https://zbmath.org/authors/?q=ai:kim.jaewoong"Yoon, Jasang"https://zbmath.org/authors/?q=ai:yoon.jasangSummary: In this paper, we consider the contractive real symmetric matrix completion problems motivated in part by studies on sparse (or dense) matrices for weighted sparse recovery problems and rating matrices with rating density in recommender systems. We completely characterize symmetric patterns \(P\) with the property (C) that every partially contractive real symmetric matrix with pattern \(P\) has a contractive real symmetric completion using graphs.Pairs of linear maps sending matrices having a fixed product into matrices with a fixed product at some vectorhttps://zbmath.org/1532.150232024-05-13T19:39:47.825584Z"Constantin, Costara"https://zbmath.org/authors/?q=ai:constantin.costaraLet \(n\geq 2\) be a natural number, and denote by \(\mathcal{M}_n\) the algebra of all \(n\times n\) matrices over the complex field \(\mathbb{C}\). Let \(C \in\mathcal{M}_n\) be a fixed matrix, and let \(x_0, y_0\in \mathbb{C}^n\) be fixed vectors with \(x_0\) different from the zero vector. The author characterizes linear maps \(\phi , \psi :\mathcal{M}_n \rightarrow\mathcal{M}_n\), with \(\phi\) bijective, having the property that \(\phi (A)\psi (B) x_0 = y_0\) whenever \(AB = C\).
The main result is Theorem 1. Its proof is long and is divided it into 6 sub-results. The proofs of the results are easy to read and the reader has examples at hand that show why some statements in the results are important.
Reviewer: Rosário Fernandes (Lisboa)Linear maps preserving matrices annihilated by a fixed polynomialhttps://zbmath.org/1532.150242024-05-13T19:39:47.825584Z"Li, Chi-Kwong"https://zbmath.org/authors/?q=ai:li.chi-kwong"Tsai, Ming-Cheng"https://zbmath.org/authors/?q=ai:tsai.ming-cheng"Wang, Ya-Shu"https://zbmath.org/authors/?q=ai:wang.ya-shu"Wong, Ngai-Ching"https://zbmath.org/authors/?q=ai:wong.ngai-chingSummary: Let \(\mathbf{M}_n(\mathbb{F})\) be the algebra of \(n\times n\) matrices over an arbitrary field \(\mathbb{F}\). We consider linear maps \(\Phi:\mathbf{M}_n(\mathbb{F})\to \mathbf{M}_r(\mathbb{F})\) preserving matrices annihilated by a fixed polynomial \(f(x)=(x-a_1)\cdots(x-a_m)\) with \(m\geq 2\) distinct zeroes \(a_1,a_2,\dots,a_m\in\mathbb{F}\); namely,
\[
f(\Phi(A))=0\text{ whenever }f(A)=0.
\]
Suppose that \(f(0)=0\), and the zero set \(Z(f)=\{a_1,\dots,a_m\}\) is not an additive group. Then \(\Phi\) assumes the form
\[
A\mapsto S\begin{pmatrix} A\otimes D_1 & & \\ & & A^{\mathrm{t}}\otimes D_2 & \\ & & & 0_s\end{pmatrix} S^{-1}\tag{\(\dagger\)},
\]
for some invertible matrix \(S\in\mathbf{M}_r(\mathbb{F})\), invertible diagonal matrices \(D_1\in \mathbf{M}_p(\mathbb{F})\) and \(D_2\in \mathbf{M}_q(\mathbb{F})\), where \(s=r-np-nq\geq 0\). The diagonal entries \(\lambda\) in \(D_1\) and \(D_2\), as well as 0 in the zero matrix \(0_s\), are zero multipliers of \(f(x)\) in the sense that \(\lambda Z(f)\subseteq Z(f)\).
In general, assume that \(Z(f)-a_1\) is not an additive group. If \(\Phi( I_n)\) commutes with \(\Phi(A)\) for all \(A\in\mathbf{M}_n(\mathbb{F})\), or if \(f(x)\) has a unique zero multiplier \(\lambda=1\), then \(\Phi\) assumes the form \((\dagger)\). The above assertions follow from the special case when \(f(x)=x(x-1)=x^2-x\), for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map \(\Phi:\mathbf{M}_n(\mathbb{F})\to \mathbf{M}_r(\mathbb{F})\) sending disjoint rank one idempotents to disjoint idempotents always assume the above form \((\dagger)\) with \(D_1=I_p\) and \(D_2=I_q\), unless \(\mathbf{M}_n(\mathbb{F})=\mathbf{M}_2(\mathbb{Z}_2)\).A note on semi-orthogonal (G-matrix) and semi-involutory MDS matriceshttps://zbmath.org/1532.150252024-05-13T19:39:47.825584Z"Chatterjee, Tapas"https://zbmath.org/authors/?q=ai:chatterjee.tapas"Laha, Ayantika"https://zbmath.org/authors/?q=ai:laha.ayantikaSummary: Maximum Distance Separable (MDS) matrices are widely used in various cryptographic constructions since they provide perfect diffusion. Further, MDS matrices with easy-to-implement inverses are useful in designing diffusion layers in block ciphers. It is known that the inverse of an MDS matrix is computationally inexpensive if the matrix is either orthogonal or involutory. Generalizing the notion of orthogonal matrices, \textit{M. Fiedler} and \textit{T. L. Markham} [Linear Algebra Appl. 438, No. 1, 231--241 (2013; Zbl 1255.15035)] introduced semi-orthogonal property in 2012. Following this, \textit{G.-S. Cheon} et al. [Linear Algebra Appl. 622, 294--315 (2021; Zbl 1465.05022)] introduced semi-involutory property to generalize the involutory property in 2021. In both these cases, the aim of the authors was to construct matrices having computationally simple inverses.
In this work, we show that some existing Cauchy and Vandermonde based constructions of MDS matrices satisfy semi-orthogonal properties. We give some characterization of \(3 \times 3\) and \(4 \times 4\) semi-involutory and semi-orthogonal matrices in light of the MDS property. We also provide some results on circulant matrices with semi-involutory and semi-orthogonal properties. Finally we give a characterization of \(4 \times 4\) semi-involutory matrices which is a generalization of the \(3 \times 3\) case of {G.-S. Cheon} et al. [loc. cit.].Stable quaternion principal component pursuithttps://zbmath.org/1532.150262024-05-13T19:39:47.825584Z"Li, Wenxin"https://zbmath.org/authors/?q=ai:li.wenxin"Zhang, Ying"https://zbmath.org/authors/?q=ai:zhang.ying.4|zhang.ying.7|zhang.ying|zhang.ying.37|zhang.ying.34|zhang.ying.12|zhang.ying.36|zhang.ying.49|zhang.ying.2|zhang.ying.17|zhang.ying.6|zhang.ying.8|zhang.ying.25|zhang.ying.9|zhang.ying.39Summary: The relaxed quaternion principal component pursuit is studied to recover low-rank quaternion matrix and sparse quaternion matrix with small entry-wise noise. Stable estimates of the low-rank quaternion matrix and the sparse quaternion matrix are provided by solving a convex minimization problem. The result in this paper generalizes the relaxed principal component pursuit from the case of real matrices to the case of quaternion matrices.Linearity of Cartan and Wasserstein meanshttps://zbmath.org/1532.150272024-05-13T19:39:47.825584Z"Choi, Hayoung"https://zbmath.org/authors/?q=ai:choi.hayoung"Kim, Sejong"https://zbmath.org/authors/?q=ai:kim.sejong"Lim, Yongdo"https://zbmath.org/authors/?q=ai:lim.yongdoThe authors consider the cone of positive definite Hermitian matrices as a Cartan-Hadamard-Riemannian manifold endowed with the metrics:
\[
\langle X,Y\rangle_A=\mathrm{tr} ( A^{-1}XA^{-1}Y) ,
\]
\[
d(X,Y)=\left[\mathrm{tr}(X+Y)-2\mathrm{tr}(X^{\frac 12}YX^{\frac 12})^{\frac 12}\right]^{\frac 12}.
\]
For these metrics the means of matrices \(A\), \(B\) are, respectively, equal to
\[
A\sharp_tB=A^{\frac 12}\left(A^{-\frac 12}BA^{-\frac 12}\right)^tA^{\frac 12}, \quad t\in[0,1],
\]
and
\[
A\diamond_tB=(1-t)^2A+t^2B+t(1-t)\left[(AB)\frac 12+(BA)^{\frac 12}\right], \quad t\in[0,1].
\]
In most cases the geodesic segment \(t\mapsto A\sharp_tB\) (resp., \(t\mapsto A\diamond_tB\)), is not linear, i.e., except of \(t=0,1\) it lies outside the linear span of \(A\), \(B\). It is a natural problem then to find out when the equations
\[
A\sharp_tB=xA+yB, \qquad A\diamond_tB=xA+yB,
\]
have solutions \((x,y)\in\mathbb R^2\).
The first main result claims that if equation \(A\diamond_tB=xA+yB\) has a solution, then
\[
A\diamond_tB=(1-t)(1-at)A+t(1-b+bt)B
\]
for \(a\), \(b\) satisfying conditions \(a<1\), \(b<1\), \((1-a)(1-b)<1\).
The second main result states that equation \(A\sharp_tB=xA+yB\) has a solution if and only if the matrix \(A^{-1}B\) has exactly two eigenvalues \(a\), \(b\), and in this case there holds
\[
A\sharp_tB=\frac{ab^t-ba^t}{a-b}A+\frac{a^t-b^t}{a-b}B.
\]
One of the consequences of the above result is that
\[
A\sharp B=xA+yB \quad\textrm{iff}\quad (xA+yB)^{-1}=yA^{-1}+xB^{-1},
\]
with \(x,y>0,\; xy<1/4\).
Moreover, the authors discuss the linearity problem for the case when matrices \(A\), \(B\) commute. It is solvable if and only if \(A^{-1}B\) has exactly two eigenvalues.
Reviewer: Roksana Słowik (Gliwice)On the fourth moment of a random determinanthttps://zbmath.org/1532.150282024-05-13T19:39:47.825584Z"Beck, Dominik"https://zbmath.org/authors/?q=ai:beck.dominikSummary: In this paper, we generalize the formula for the fourth moment of a random determinant to account for entries with asymmetric distribution. We also derive the second moment of a random Gram determinant.Rank 1 perturbations in random matrix theory -- a review of exact resultshttps://zbmath.org/1532.150292024-05-13T19:39:47.825584Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-jSummary: A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank \(1\) perturbation. Considered in this review are the additive rank \(1\) perturbation of the Hermitian Gaussian ensembles, the multiplicative rank \(1\) perturbation of the Wishart ensembles, and rank \(1\) perturbations of Hermitian and unitary matrices giving rise to a two-dimensional support for the eigenvalues. The focus throughout is on exact formulas, which are typically the result of various integrable structures. The simplest is that of a determinantal point process, with others relating to partial differential equations implied by a formulation in terms of certain random tridiagonal matrices. Attention is also given to eigenvector overlaps in the setting of a rank \(1\) perturbation.Jacobi beta ensemble and \(b\)-Hurwitz numbershttps://zbmath.org/1532.150302024-05-13T19:39:47.825584Z"Ruzza, Giulio"https://zbmath.org/authors/?q=ai:ruzza.giulioThe Jacobi \(\beta\)-ensemble is the random point process with joint probability density function
\[
w_\beta(x_1,\dots,x_n;c,d)= \frac{1}{Z_\beta} \prod_{i=1}^n x_i^{c\beta/2-1} (1-x_i)^{d\beta/2-1} \cdot \prod_{1 \le i<j \le n} |x_i-x_j|^\beta,
\]
where \(x_1,\dots,x_n \in (0,1)\), \(\beta,c,d>0\), and the normalization constant \(Z_\beta\) can be explicitly given. For \(\beta=1\), \(2\), or \(4\), this gives the eigenvalue distribution of Jacobi random matrix models. The first result of this paper is to find an expression of the following quantity (called correlator):
\[
\mathcal{C}_{(\lambda_1,\dots,\lambda_\ell)}(n,\beta,c,d)= \int_{(0,1)^n} \left(\prod_{k=1}^\ell (x_1^{\lambda_k}+\cdots+x_n^{\lambda_k} )\right) w_\beta(x_1,\dots,x_n;c,d) \, \mathrm{d}x_1 \cdots \mathrm{d}x_n.
\]
In terms of matrix models, this represents the expected value of the product of traces of powers. The author proves that this quantity is a generating function for \(b\)-Hurwitz numbers by using Selberg-Jack-type integral formula due to \textit{K. W. J. Kadell} [Adv. Math. 130, No. 1, 33--102 (1997; Zbl 0885.33009)].
The second important result of this paper is a combinatorial interpretation of general \(b\)-Hurwitz numbers. A class of \(b\)-Hurwitz numbers was recently introduced by \textit{G. Chapuy} and \textit{M. Dołęga} [Adv. Math. 409, Part A, Article ID 108645, 72 p. (2022; Zbl 1498.05278)] for the study of Jack symmetric functions. Those \(b\)-Hurwitz numbers count a special kind of combinatorial maps related to graphs on surfaces. The present study of Jacobi \(\beta\)-ensembles suggests the possibility of a combinatorial interpretation for general \(\beta>0\).
Reviewer: Sho Matsumoto (Kagoshima)Tracy-Widom law for the extreme eigenvalues of large signal-plus-noise matriceshttps://zbmath.org/1532.150312024-05-13T19:39:47.825584Z"Zhang, Zhixiang"https://zbmath.org/authors/?q=ai:zhang.zhixiang"Liu, Yiming"https://zbmath.org/authors/?q=ai:liu.yiming"Pan, Guangming"https://zbmath.org/authors/?q=ai:pan.guangmingLet \(\mathbf{R}\) be a deterministic, rectangular matrix, called ``signal matrix'', and \(\textbf{X}\) be a real random matrix with independent entries, called ``noise matrix''. The so called ``signal-plus-noise matrix model'' \(\mathbf{S} = \mathbf{R} + \mathbf{X}\) has been studied in the past, with applications in machine learning and signal processing.
The authors prove that under mild assumptions on \(\mathbf{R}\) and \(\mathbf{S}\), most notably a tail assumption on the entries of \(\mathbf{X}\), the largest eigenvalue of \(\mathbf{S}\mathbf{S}^{*}\) follows the Tracy-Widom distribution of type 1 in the large dimension limit, as do matrices of the Gaussian orthogonal ensemble, see [\textit{C. A. Tracy} and \textit{H. Widom}, Commun. Math. Phys. 177, No. 3, 727--754 (1996; Zbl 0851.60101)].
Furthermore, it is shown that the tail condition is necessary to obtain the convergence to the Tracy-Widom distribution.
The proof is done by analyzing the Stieltjes transform of the model. The authors first prove the result in the case of Gaussian noise, using in particular an interpolation procedure relating the signal-plus-noise matrix to a Wishart matrix. In this part, their approach is similar to the one of [\textit{J. O. Lee} and \textit{K. Schnelli}, Rev. Math. Phys. 27, No. 8, Article ID 1550018, 94 p. (2015; Zbl 1328.15051)]. A technical part of the paper is devoted to deriving the so-called optical theorems, which give control over the Green kernel of the random matrix.
Reviewer: Thomas Buc-d'Alché (Lyon)Beyond the 10-fold way: 13 associative \(\mathbb{Z}_2\times\mathbb{Z}_2\)-graded superdivision algebrashttps://zbmath.org/1532.160142024-05-13T19:39:47.825584Z"Kuznetsova, Zhanna"https://zbmath.org/authors/?q=ai:kuznetsova.zhanna-g"Toppan, Francesco"https://zbmath.org/authors/?q=ai:toppan.francescoSummary: The ``10-fold way'' refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, \(\mathbb{Z}_2\)-graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in \(\mathbb{Z}_2\times\mathbb{Z}_2\)-graded physics (classical and quantum invariant models, parastatistics) we classify the associative \(\mathbb{Z}_2\times\mathbb{Z}_2\)-graded superdivision algebras and show that 13 inequivalent cases have to be added to the 10-fold way. Our scheme is based on the ``alphabetic presentation of Clifford algebras'', here extended to graded superdivision algebras. The generators are expressed as equal-length words in a 4-letter alphabet (the letters encode a basis of invertible \(2\times 2\) real matrices and in each word the symbol of tensor product is skipped). The 13 inequivalent \(\mathbb{Z}_2\times\mathbb{Z}_2\)-graded superdivision algebras are split into real series (4 subcases with 4 generators each), complex series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators). As an application, the connection of \(\mathbb{Z}_2\times\mathbb{Z}_2\)-graded superdivision algebras with a parafermionic Hamiltonian possessing time-reversal and particle-hole symmetries is presented.Generalized Jacobson's Lemma in a Banach algebrahttps://zbmath.org/1532.160312024-05-13T19:39:47.825584Z"Chen, Huanyin"https://zbmath.org/authors/?q=ai:chen.huanyin"Abdolyousefi, Marjan Sheibani"https://zbmath.org/authors/?q=ai:sheibani-abdolyousefi.marjanSummary: We present an extension, for Banach algebras, of the extension, due to \textit{G. Corach} et al. [Commun. Algebra 41, No. 2, 520--531 (2013; Zbl 1269.47002)], of Jacobson's lemma on the g-Drazin inverse.Superspace realizations of the Bannai-Ito algebrahttps://zbmath.org/1532.170032024-05-13T19:39:47.825584Z"Crampé, Nicolas"https://zbmath.org/authors/?q=ai:crampe.nicolas"De Bie, Hendrik"https://zbmath.org/authors/?q=ai:de-bie.hendrik"Iliev, Plamen"https://zbmath.org/authors/?q=ai:iliev.plamen"Vinet, Luc"https://zbmath.org/authors/?q=ai:vinet.lucSummary: A model of the Bannai-Ito algebra in a superspace is introduced. It is obtained from the threefold tensor product of the basic realization of the Lie superalgebra \(\mathfrak{osp}(1|2)\) in terms of operators in one continuous and one Grassmanian variable. The basis vectors of the resulting Bannai-Ito algebra module involve Jacobi polynomials.One-dimensional topological theories with defects: the linear casehttps://zbmath.org/1532.180092024-05-13T19:39:47.825584Z"Im, Mee Seong"https://zbmath.org/authors/?q=ai:im.mee-seong"Khovanov, Mikhail"https://zbmath.org/authors/?q=ai:khovanov.mikhail-gUniversal construction [\textit{C. Blanchet} et al., Topology 34, No. 4, 883--927 (1995; Zbl 0887.57009); \textit{M. Khovanov}, ``Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series'', Preprint, \url{arXiv:2010.05730}] starts with an evaluation function for closed \(n\)-manifolds to produce state space for closed \(\left( n-1\right) \)-manifolds and maps between these spacces associated to \(n\)-cobordisms, which results in a functor from the category of \(n\)-dimensional cobordisms to the category of vector spaces usually failing to be a TQFT with the tensor product of states for two \(\left( n-1\right) \)-manifolds \(N_{1},N_{2}\)\ properly embedded into the state space for their union
\[
A\left( N_{1}\right) \otimes A\left( N_{2}\right) \hookrightarrow A\left( N_{1}\sqcup N_{2}\right)
\]
The universal construction turns out to be riveting already in low-dimensions, including in dimensions two [\textit{M. Khovanov}, ``Universal construction of topological theories in two dimensions'', Preprint, \url{arXiv:2007.03361}; \textit{M. Khovanov} et al., Sel. Math., New Ser. 28, No. 4, Paper No. 71, 68 p. (2022; Zbl 1496.18018); \textit{M. Khovanov} et al., Commun. Math. Phys. 385, No. 3, 1835--1870 (2021; Zbl 1490.57039); \textit{M. Khovanov} and \textit{R. Sazdanovic}, J. Pure Appl. Algebra 225, No. 6, Article ID 106592, 24 p. (2021; Zbl 1480.16037)] and one [\textit{P. Gustafson} et al., Lett. Math. Phys. 113, No. 5, Paper No. 93, 38 p. (2023; Zbl 07743387); \textit{M. S. Im} and \textit{M. Khovanov}, ``Topological theories and automata'', Preprint, \url{arXiv:2202.13398}; \textit{M. S. Im} et al., ``Universal construction in monoidal and non-monoidal settings, the Brauer envelope, and pseudocharacters'', Preprint, \url{arXiv:2303.02696}; \textit{M. S. Im} and \textit{P. Zimmer}, Involve 15, No. 2, 319--331 (2022; Zbl 1499.18039)]. In the latter case, one needs to add zero-dimensional defects with labels in a set \(\Sigma\). An oriented interval with a collection of \(\Sigma\)-labelled defects encodes a word \(\omega\), that is an element of the free monoid \(\Sigma^{\ast}\)\ on the set \(\Sigma\). An oriented circle with labels in \(\Sigma\)\ encodes a word up to cyclic eigenvalues. Given an evaluation of each word and a separate evaluation of words up to cyclic equivalence, there is an associated rigid linear monoidal category [\textit{M. Khovanov}, ``Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series'', Preprint, \url{arXiv:2010.05730}].
This paper studies this category for a rational evaluation \(\alpha\). The Karoubi closure of the resulting category can be reduced to the Karoubi closure of a category built from a symmetric Frobenius algebra \(\mathcal{K}\) that can be extracted from \(\alpha\) (\S 2.4). \S \S 2.1--2.3 deal with the setup, basic theory and various examples. \S 3 reviews thin flat surface 2D TQFTs associated to symmetric Frobenius algebras, explaining how to enhance these TQFTs\ by 0-dimensional defects floating along the boundary that carry elements of the algebra. Comparisons between one-dimensional theories with defects and two-dimensional theories without defects are addressed throughout the paper. The Boolean analogues of these categories and their relation to automata and regular languages were investigated in [\textit{M. S. Im} and \textit{M. Khovanov}, ``Topological theories and automata'', Preprint, \url{arXiv:2202.13398}].
For the entire collection see [Zbl 1531.17003].
Reviewer: Hirokazu Nishimura (Tsukuba)Clifford-valued fractal interpolationhttps://zbmath.org/1532.280112024-05-13T19:39:47.825584Z"Massopust, Peter R."https://zbmath.org/authors/?q=ai:massopust.peter-rSummary: In this short note, we merge the areas of hypercomplex algebras with that of fractal interpolation and approximation. The outcome is a new holistic methodology that allows the modeling of phenomena exhibiting a complex self-referential geometry and which require for their description an underlying algebraic structure.
For the entire collection see [Zbl 1531.00060].Matrix Jacobi biorthogonal polynomials via Riemann-Hilbert problemhttps://zbmath.org/1532.330152024-05-13T19:39:47.825584Z"Branquinho, Amílcar"https://zbmath.org/authors/?q=ai:branquinho.amilcar"Foulquié-Moreno, Ana"https://zbmath.org/authors/?q=ai:foulquie-moreno.ana-pilar"Fradi, Assil"https://zbmath.org/authors/?q=ai:fradi.assil"Mañas, Manuel"https://zbmath.org/authors/?q=ai:manas.manuelSummary: We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential-difference relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-abelian extensions of a family of discrete Painlevé d-P\(_{IV}\) equations are obtained for the three term recurrence relation coefficients.Stability analysis for a coupled Schrödinger system with one boundary dampinghttps://zbmath.org/1532.354012024-05-13T19:39:47.825584Z"Zhang, Hua-Lei"https://zbmath.org/authors/?q=ai:zhang.hualei(no abstract)Numerical integration of Schrödinger maps via the Hasimoto transformhttps://zbmath.org/1532.354152024-05-13T19:39:47.825584Z"Banica, Valeria"https://zbmath.org/authors/?q=ai:banica.valeria"Maierhofer, Georg"https://zbmath.org/authors/?q=ai:maierhofer.georg"Schratz, Katharina"https://zbmath.org/authors/?q=ai:schratz.katharinaSummary: We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realization based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e., under lower-regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.Invariant Gibbs dynamics for the two-dimensional Zakharov-Yukawa systemhttps://zbmath.org/1532.354312024-05-13T19:39:47.825584Z"Seong, Kihoon"https://zbmath.org/authors/?q=ai:seong.kihoonSummary: We study the Gibbs dynamics for the Zakharov-Yukawa system on the two-dimensional torus \(\mathbb{T}^2\), namely a Schrödinger-wave system with a Zakharov-type coupling \((-\Delta)^\gamma\). We first construct the Gibbs measure in the weakly nonlinear coupling case (\(0 \leq \gamma <\)). Combined with the non-construction of the Gibbs measure in the strongly nonlinear coupling case (\(\gamma = 1\)) by \textit{T. Oh} et al. [``A remark on Gibbs measures with log-correlated Gaussian fields'', Preprint, \url{arXiv:2012.06729}], this exhibits a phase transition at \(\gamma = 1\). We also study the dynamical problem and prove almost sure global well-posedness of the Zakharov-Yukawa system and invariance of the Gibbs measure under the resulting dynamics for the range \(0 \leq \gamma < \frac{1}{3}\). In this dynamical part, the main step is to prove local well-posedness. Our argument is based on the first order expansion and the operator norm approach via the random matrix/tensor estimate from a recent work [\textit{Y. Deng} et al., Invent. Math. 228, No. 2, 539--686 (2022; Zbl 1506.35208)]. In the appendix, we briefly discuss the Hilbert-Schmidt norm approach and compare it with the operator norm approach.Description of reversibility of 9-cyclic 1D finite linear cellular automata with periodic boundary conditionshttps://zbmath.org/1532.370132024-05-13T19:39:47.825584Z"Akin, Hasan"https://zbmath.org/authors/?q=ai:akin.hasanSummary: We consider a family of one-dimensional (1D) finite linear cellular automata (FLCA) with radius 4 on the periodic boundary conditions (PBC, shortly) over a finite field \(\mathbb{F} \mathbf{p}\) where \(\mathbf{p}\geq 2\) is prime number. We show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata. We solve the reversibility problem for 9-Cyclic 1D FLCA with radius 4 and periodic boundary conditions. Computing the determinant of relevant rule matrices for some finite number of cells and any given values of the coefficients of the local rules, we test the reversibility of the cellular automaton.Pearson equations for discrete orthogonal polynomials. I: Generalized hypergeometric functions and Toda equationshttps://zbmath.org/1532.370632024-05-13T19:39:47.825584Z"Mañas, Manuel"https://zbmath.org/authors/?q=ai:manas.manuel"Fernández-Irisarri, Itsaso"https://zbmath.org/authors/?q=ai:fernandez-irisarri.itsaso"González-Hernández, Omar F."https://zbmath.org/authors/?q=ai:gonzalez-hernandez.omar-fSummary: The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semiinfinite matrix that models the shifts by \(\pm 1\) in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well-known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous 1D Toda equation for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It is also shown that the Kadomtesev-Petviashvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case, the deformation does not satisfy a Pearson equation.
{{\copyright} 2021 The Authors. \textit{Studies in Applied Mathematics} published by Wiley Periodicals LLC}Polynomial approximation on disjoint segments and amplification of approximationhttps://zbmath.org/1532.410052024-05-13T19:39:47.825584Z"Malykhin, Yu."https://zbmath.org/authors/?q=ai:malykhin.yu-v"Ryutin, K."https://zbmath.org/authors/?q=ai:ryutin.konstantin-sSummary: We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical analysis, complexity theory, quantum algorithms, etc. The one, most relevant for us, is the amplification of approximation method: it allows to construct approximations of higher degree \(M\) and better accuracy from the approximations of degree \(m\).The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersectionhttps://zbmath.org/1532.410292024-05-13T19:39:47.825584Z"Bauschke, Heinz H."https://zbmath.org/authors/?q=ai:bauschke.heinz-h"Singh, Shambhavi"https://zbmath.org/authors/?q=ai:singh.shambhavi"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfuThe paper investigates recent splitting methods proposed by Ryu, Malitsky and Tam, and Campoy within the framework of normal cone operators for subspaces. The authors have established that all three algorithms not only find some solution but specifically the projection of the initial point onto the intersection of the subspaces. Furthermore, they have demonstrated strong convergence of the iterates even in infinite-dimensional settings. Numerical experiments suggest that Ryu's method tends to converge faster, although both Malitsky-Tam splitting and Campoy splitting are not restricted to three subspaces. Finally, the paper discusses two potential directions for future research.
Reviewer: Ravindra Kishor Bisht (Pune)Schoenberg correspondence for \(k\)-(super)positive maps on matrix algebrashttps://zbmath.org/1532.460472024-05-13T19:39:47.825584Z"Bhat, B. V. Rajarama"https://zbmath.org/authors/?q=ai:bhat.b-v-rajarama"Chakraborty, Purbayan"https://zbmath.org/authors/?q=ai:chakraborty.purbayan"Franz, Uwe"https://zbmath.org/authors/?q=ai:franz.uweSummary: We prove a Schoenberg-type correspondence for non-unital semigroups which generalizes an analogous result for unital semigroup proved by \textit{M.~Schürmann} [Quantum probability and applications II, Proc. 2nd Workshop, Heidelberg/Ger. 1984, Lect. Notes Math. 1136, 475--492 (1985; Zbl 0581.16007)].
It characterizes the generators of semigroups of linear maps on \(M_n (\mathbb{C})\) which are \(k\)-positive, \(k\)-superpositive, or \(k\)-entanglement breaking. As a corollary we reprove Lindblad, Gorini, Kossakowski, Sudarshan's theorem
[\textit{G.~Lindblad}, Commun. Math. Phys. 48, 119--130 (1976; Zbl 0343.47031), \textit{V.~Gorini} et al., J. Math. Phys. 17, 821--825 (1976; Zbl 1446.47009)].
We present some concrete examples of semigroups of operators and study how their positivity properties can improve with time.A Grassmann manifold handbook: basic geometry and computational aspectshttps://zbmath.org/1532.530012024-05-13T19:39:47.825584Z"Bendokat, Thomas"https://zbmath.org/authors/?q=ai:bendokat.thomas"Zimmermann, Ralf"https://zbmath.org/authors/?q=ai:zimmermann.ralf"Absil, P.-A."https://zbmath.org/authors/?q=ai:absil.pierre-antoineSummary: The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.Clifford structures, bilegendrian surfaces, and extrinsic curvaturehttps://zbmath.org/1532.530112024-05-13T19:39:47.825584Z"Smith, Graham"https://zbmath.org/authors/?q=ai:smith.graham-andrew|smith.graham-mThe author uses Clifford algebras to give a unified treatment for the study of Constant Extrinsic Curvature (CEC) surfaces in pseudo-Riemannian 3-manifolds through the notion of bilegendrian surfaces. It also gives many examples where this general framework is applied to surfaces of constant positive curvature in hyperbolic space and constant negative curvature in Euclidean space. Moreover, the author gives an elementary proof of Hilbert's theorem on the nonexistence of complete CEC surfaces with curvature equal to \((-k)\) in a 3-dimensional Riemannian space form of curvature \(c\) if \((-k)<\min(0,-c)\).
Reviewer: Andrea Tamburelli (Houston)A note on invariant description of \(\mathrm{SU}(2)\)-structures in dimension 5https://zbmath.org/1532.530752024-05-13T19:39:47.825584Z"Niedziałomski, Kamil"https://zbmath.org/authors/?q=ai:niedzialomski.kamilThe aim of this paper is to study \(\mathrm{SU}(2)\)-geometry from the perspective of spinors, focusing on the invariant approach, i.e., independent of the choice of the defining spinor. The author focuses on the action of vectors on spinors. He develops an invariant approach to \(\mathrm{SU}(2)\)-structures on spin 5-manifolds. It is shown that there are different choices of spinors in \(\Delta=\mathbb{C}^4\) defining the same subgroup isomorphic to \(\mathrm{SU}(2)\). In fact, the author shows that the subspace \(V\subset\Delta\) of spinors defining the same group isomorphic to \(\mathrm{SU}(2)\) is of real dimension four. He characterizes via the spinor approach the subspaces in the spinor bundle which induce the same group isomorphic to \(\mathrm{SU}(2)\). The characterization of these spaces is done from two angles: complex and quaternionic. The quaternionic approach seems to be well known to the experts in the field, whereas the complex approach is most likely new. The other case is concerned with a characterization of the intrinsic torsion modules. The author shows how to induce quaternionic structure on the contact distribution of the considered \(\mathrm{SU}(2)\)-structure. He proves the invariance of certain components of the covariant derivative \(\nabla\varphi\), where \(\varphi\) is any spinor field defining the \(\mathrm{SU}(2)\)-structure. This implies, as expected, that (at least some of) the intrinsic torsion modules can be derived invariantly with the spinorial approach. The author concludes with the explicit description of the intrinsic torsion and the characteristic connection.
The paper is organized as follows.
Section 1 deals with an algebraic approach to spinors defining \(\mathrm{SU}(2)\), or equivalently, its Lie algebra \(su(2)\). The author gives a characterization of the subspaces of spinors defining a given Lie algebra \(su(2)\) in terms of complex and quaternionic structures. Moreover, he studies a correspondence between the complex structures on the spinor space \(\Delta\) and an associated four-dimensional space \(D\) of vectors acting on spinors. He shows how to obtain, in a canonical way, a quaternionic structure on \(D\) by the invariant spinorial approach. The map which assigns a complex structure from the quaternionic structure to a unit spinor is, in fact, the Hopf fibration. Moreover, the author shows nonexistence of a complex structure on \(D\) induced from the complex structure on \(V\).
Section 2 is devoted to an invariant description of \(\mathrm{SU}(2)\)-structures. Here the author shows how the algebraic approach developed in the first section induces a \(\mathrm{SU}(2)\)-structure on a 5-dimensional spin manifold, and shows relations with the approaches from [\textit{G. Bazzoni} et al., ``Spin-harmonic structures and nilmanifolds'', Preprint, \url{arXiv:1904.01462}; \textit{D. Conti} and \textit{S. Salamon}, Trans. Am. Math. Soc. 359, No. 11, 5319--5343 (2007; Zbl 1130.53033)].
Section 3 deals with a characterization of the intrinsic torsion and its modules. In this section the author wants to derive a decomposition of the module of all possible intrinsic torsions via spinorial approach. He shows that with a slight modification the spinorial approach developed in [loc. cit.] is invariant, i.e., independent of the choice of a defining spinor. Moreover, he derives explicit formula for the intrinsic torsion as well as for the characteristic connection.
Reviewer: Ahmed Lesfari (El Jadida)On the CLT for linear eigenvalue statistics of a tensor model of sample covariance matriceshttps://zbmath.org/1532.600072024-05-13T19:39:47.825584Z"Dembczak-Kołodziejczyk, Alicja"https://zbmath.org/authors/?q=ai:dembczak-kolodziejczyk.alicja"Lytova, Anna"https://zbmath.org/authors/?q=ai:lytova.annaSummary: In [\textit{A. Lytova}, J. Theor. Probab. 31, No. 2, 1024--1057 (2018; Zbl 1394.15025)], there was proved the CLT for linear eigenvalue statistics \({Tr} \varphi(M_n)\) of sample covariance matrices of the form \(M_n=\sum_{\alpha=1}^m \mathbf{y}_{\alpha}^{(1)} \otimes \mathbf{y}_{\alpha}^{(2)}(\mathbf{y}_{\alpha}^{(1)} \otimes \mathbf{y}_{\alpha}^{(2)})^T\), where \((\mathbf{y}_{\alpha}^{(1)}\), \(\mathbf{y}_{\alpha}^{(2)})_{\alpha}\) are iid copies of \(\mathbf{y}\in \mathbb{R}^n\) satisfying \(\mathbf{Eyy}^T=n^{-1} I_n\), \(\mathbf{Ey}^2_i\mathbf{y}^2_j=(1+\delta_{ij}d)n^{-2}+a(1+\delta_{ij}d_1)n^{-3}+O(n^{-4})\) for some \(a, d,d_1\in \mathbb{R} \). It was shown that given a smooth enough test function \(\varphi\), \(\mathbf{Var} {Tr} \varphi(M_n)=O(n)\) as \(m, n\to\infty\), \(m/n^2\to c>0\), and \(({Tr} \varphi(M_n)-\mathbf{E} {Tr} \varphi(M_n))/\sqrt{n}\) converges in distribution to a Gaussian mean zero random variable with variance \(V[\varphi]\) proportional to \(a+d\). It was noticed that if \(\mathbf{y}\) is uniformly distributed on the unit sphere then \(a+d=0\) and \(V[\varphi]\) vanishes. In this note we show that in this case \(\mathbf{Var} {Tr}(M_n-zI_n)^{-1}=O(1)\), so that the CLT should be valid for linear eigenvalue statistics themselves without a normalising factor in front (in contrast to the Gaussian case.)Gaussian diagrammatics from circular ensembles of random matriceshttps://zbmath.org/1532.600092024-05-13T19:39:47.825584Z"Novaes, Marcel"https://zbmath.org/authors/?q=ai:novaes.marcelSummary: We uncover a hidden Gaussian ensemble inside each of the three circular ensembles of random matrices, providing novel diagrammatic rules for the calculation of moments. The matrices involved are generic complex for \(\beta = 2\), complex symmetric for \(\beta = 1\) and complex self-dual for \(\beta = 4\), and at the last step their dimension must be set to \(1 - 2/\beta\). As an application, we compute moments of traces of submatrices.
{{\copyright} 2024 IOP Publishing Ltd}Limiting spectral distribution of stochastic block modelhttps://zbmath.org/1532.600112024-05-13T19:39:47.825584Z"Su, Giap Van"https://zbmath.org/authors/?q=ai:su.giap-van"Chen, May-Ru"https://zbmath.org/authors/?q=ai:chen.mayru"Guo, Mei-Hui"https://zbmath.org/authors/?q=ai:guo.meihui"Huang, Hao-Wei"https://zbmath.org/authors/?q=ai:huang.haoweiSummary: The stochastic block model (SBM) is an extension of the Erdős-Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with \(K\) communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the size of the matrix tends to infinity. We also obtain an upper bound for the operator norm of such random matrices by means of the Stieltjes transform and random matrix theory.Extrema of a multinomial assignment processhttps://zbmath.org/1532.600142024-05-13T19:39:47.825584Z"Lifshits, Mikhail"https://zbmath.org/authors/?q=ai:lifshits.mikhail-a"Mordant, Gilles"https://zbmath.org/authors/?q=ai:mordant.gillesSummary: We study the asymptotic behaviour of the expectation of the maxima and minima of a random assignment process generated by a large matrix with multinomial entries. A variety of results is obtained for different sparsity regimes.Strong limit theorem for largest entry of large-dimensional random tensorhttps://zbmath.org/1532.600602024-05-13T19:39:47.825584Z"Ding, Xue"https://zbmath.org/authors/?q=ai:ding.xueSummary: Suppose that \(\mathbf{y}_{\alpha}^{(i)}, i=1, \ldots, k, \alpha =1,\ldots, m\) are i.i.d. copies of random vector \(\mathbf{y}= (y_1, y_2, \ldots, y_p )^T\). Let
\[
\mathbf{Y}_{\alpha}= \mathbf{y}_{\alpha}^{(1)} \otimes\cdots\otimes \mathbf{y}_{\alpha}^{(k)},
\]
then the random tensor product constructed by \(\mathbf{Y}_{\alpha}\) is defined by
\[
\mathcal{S}= \sum_{\alpha =1}^m \mathbf{Y}_{\alpha}.
\]
In this paper, we obtain the strong limit theorems of the largest entry of large-dimensional random tensor product \(\mathcal{S}\) under two high-dimensional settings: the polynomial rate and the exponential rate. The conclusions are established under weaker moment condition than the existing papers and the relationship between \(m\) and \(p\) is more flexible.The second class particle process at shockshttps://zbmath.org/1532.602142024-05-13T19:39:47.825584Z"Ferrari, Patrik L."https://zbmath.org/authors/?q=ai:ferrari.patrik-lino"Nejjar, Peter"https://zbmath.org/authors/?q=ai:nejjar.peterSummary: We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities \(\lambda\) to the left of the origin and \(\rho\) to the right of it and \(\lambda < \rho \). We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.Numerically stable iterative methods for computing matrix sign functionhttps://zbmath.org/1532.650212024-05-13T19:39:47.825584Z"Rani, Litika"https://zbmath.org/authors/?q=ai:rani.litika"Kansal, Munish"https://zbmath.org/authors/?q=ai:kansal.munishSummary: The main objective of this study is to develop two novel iterative schemes for computing the sign of a matrix with no pure imaginary eigenvalues. Detailed convergence analysis and asymptotical stability of the proposed methods are discussed. It is shown that the new schemes converge globally by drawing attraction basins. In addition, we extend the obtained results to compute nontrivial solutions of the Yang-Baxter-like equation, provided the given matrix has no eigenvalues on the imaginary axis. To illustrate the effectiveness of theoretical developments, a class of numerical examples for matrices of various dimensions, including eigenvalue clustering, are worked out.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Computing the Tracy-Widom distribution for arbitrary \(\beta>0\)https://zbmath.org/1532.650572024-05-13T19:39:47.825584Z"Trogdon, Thomas"https://zbmath.org/authors/?q=ai:trogdon.thomas"Zhang, Yiting"https://zbmath.org/authors/?q=ai:zhang.yitingSummary: We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by \textit{A. Bloemendal} in his Ph.D. Thesis [Finite rank perturbations of random matrices and their continuum limits (Ph.D. thesis). Toronto, Canada: University of Toronto (2011)]. The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since \(\beta\) is simply a parameter in the boundary-value problem, any \(\beta> 0\) can be used, in principle. The limiting distribution of the \(n\)th largest eigenvalue can also be computed. Our methods are available in the \textsc{Julia} package \texttt{TracyWidomBeta.jl}.Lu decomposition and Toeplitz decomposition of a neural networkhttps://zbmath.org/1532.680822024-05-13T19:39:47.825584Z"Liu, Yucong"https://zbmath.org/authors/?q=ai:liu.yucong"Jiao, Simiao"https://zbmath.org/authors/?q=ai:jiao.simiao"Lim, Lek-Heng"https://zbmath.org/authors/?q=ai:lim.lek-hengIn providing a justification for the importance of feed-forward neural networks, it is now known that several universal approximation theorems exist and are proved using the following types of networks: (i) shallow wide networks: neural networks of fixed depth and arbitrary width; (ii) deep narrow networks: neural networks with fixed width and arbitrary depth.
The theorems themselves use neural networks to show that under different settings and in various well-defined frameworks, these networks can approximate various classes of functions to arbitrary accuracy under various measures of accuracy. This is an extremely important idea in many areas for example in approximation theory. As examples of such results, Theorems 1.1 and 1.2 in the paper under review are density results and are due to Pinkus and Kidger and Lyons.
Typically, in all these results, the neural networks are fully connected. In this interesting paper, the authors show that for weight matrices with special structure, for example upper and lower triangular, Toeplitz or Hankel, the same type of universal approximation results exist for both shallow wide and deep narrow networks. Numerically, the authors show that when kept at the same depth and width, a neural network with these structured weight matrices suffers almost no loss in expressive power, but requires only a fraction of the parameters cost of \(O(n\log n)\) operations as opposed to the usual \(O(n^2)\).
To give a small amount of insight, we note that any matrix \(A\) has an LU decomposition up to a permutation. This means a decomposition up to a permutation of an upper triangular and lower triangular matrix. Similarly, any matrix \(A\) has a Toeplitz decomposition as well. The authors essentially prove that any continuous function \(f:\mathbb R^n\to \mathbb R^m\) can be approximated to arbitrary accuracy by a neural network that maps an \(x\in \mathbb R^n\) to \(L_1\sigma_1U_1\sigma_2L_2\sigma_3U_2...L_r\sigma_{2r-1}U_r,\, x\) \(\in \mathbb R^m\) where the weight matrices alternate between upper and lower triangular matrices, \(\sigma_i(x):=\sigma(x-b_i)\), with some bias vector \(b_i\) and \(\sigma\) any nonpolynomial function, uniformly continuous. The authors establish the same kind of results for Toeplitz matrices and Hankel ones. A consequence of the Toeplitz result is a fixed-width universal approximation theorem for convolutional neural networks, which so far have only arbitrary width versions. Since the results apply in particular to the case when \(f\) is a general neural network, one may regard these results as LU and Toeplitz decompositions of a neural network. Thus one may reduce the number of weight parameters in a neural network without sacrificing its power of universal approximation as is shown above.
Reviewer: Steven B. Damelin (Ann Arbor)Dual semi-supervised convex nonnegative matrix factorization for data representationhttps://zbmath.org/1532.680832024-05-13T19:39:47.825584Z"Peng, Siyuan"https://zbmath.org/authors/?q=ai:peng.siyuan"Yang, Zhijing"https://zbmath.org/authors/?q=ai:yang.zhijing"Ling, Bingo Wing-Kuen"https://zbmath.org/authors/?q=ai:ling.bingo-wing-kuen"Chen, Badong"https://zbmath.org/authors/?q=ai:chen.badong"Lin, Zhiping"https://zbmath.org/authors/?q=ai:lin.zhipingSummary: Semi-supervised nonnegative matrix factorization (NMF) has received considerable attention in machine learning and data mining. A new semi-supervised NMF method, called dual semi-supervised convex nonnegative matrix factorization (DCNMF), is proposed in this paper for fully using the limited label information. Specifically, DCNMF simultaneously incorporates the pointwise and pairwise constraints of labeled samples as dual supervisory information into convex NMF, which results in a better low-dimensional data representation. Moreover, DCNMF imposes the nonnegative constraint only on the coefficient matrix but not on the base matrix. Consequently, DCNMF can process mixed-sign data, and hence enlarge the range of applications. We derive an efficient alternating iterative algorithm for DCNMF to solve the optimization, and analyze the proposed DCNMF method in terms of the convergence and computational complexity. We also discuss the relationships between DCNMF and several typical NMF based methods. Experimental results illustrate that DCNMF outperforms the related state-of-the-art NMF methods on nonnegative and mixed-sign datasets for clustering applications.Graph clustering by hierarchical singular value decomposition with selectable range for number of clusters membershttps://zbmath.org/1532.680852024-05-13T19:39:47.825584Z"Sadeghian, Azam"https://zbmath.org/authors/?q=ai:sadeghian.azam"Shahzadeh Fazeli, Seyed Abolfazl"https://zbmath.org/authors/?q=ai:fazeli.seyed-abolfazl-shahzadeh"Karbassi, Seyed Mehdi"https://zbmath.org/authors/?q=ai:karbassi.seyed-mehdiSummary: Graphs have so many applications in real world problems.
When we deal with huge volume of data, analyzing data is difficult or
sometimes impossible and clustering data is a useful tool for these data
analysis. Singular value decomposition(SVD) is one of the best algorithms
for clustering graph but we do not have any choice to select the number
of clusters and the number of members in each cluster. In this paper,
we use hierarchical SVD to cluster graphs to desirable number of clusters
and the number of members in each cluster. In this algorithm, users
can select a range for the number of members in each cluster and the
algorithm hierarchically cluster each clusters to achieve desirable range.
The results show in hierarchical SVD algorithm, clustering measurement
parameters are more desirable and clusters are as dense as possible. In
this paper, simple and bipartite graphs are studied.Essential tensor learning for multi-view spectral clusteringhttps://zbmath.org/1532.680872024-05-13T19:39:47.825584Z"Wu, Jianlong"https://zbmath.org/authors/?q=ai:wu.jianlong"Lin, Zhouchen"https://zbmath.org/authors/?q=ai:lin.zhouchen"Zha, Hongbin"https://zbmath.org/authors/?q=ai:zha.hongbinEditorial remark: No review copy delivered.Control of the rotation of a solid (spacecraft) with a combined optimality criterion based on quaternionshttps://zbmath.org/1532.700302024-05-13T19:39:47.825584Z"Levskii, M. V."https://zbmath.org/authors/?q=ai:levskii.m-v(no abstract)Stochastic response determination of structural systems modeled via dependent coordinates: a frequency domain treatment based on generalized modal analysishttps://zbmath.org/1532.740412024-05-13T19:39:47.825584Z"Pirrotta, Antonina"https://zbmath.org/authors/?q=ai:pirrotta.antonina"Kougioumtzoglou, Ioannis A."https://zbmath.org/authors/?q=ai:kougioumtzoglou.ioannis-a"Pantelous, Athanasios A."https://zbmath.org/authors/?q=ai:pantelous.athanasios-aSummary: Generalized independent coordinates are typically utilized within an analytical dynamics framework to model the motion of structural and mechanical engineering systems. Nevertheless, for complex systems, such as multi-body structures, an explicit formulation of the equations of motion by utilizing generalized, independent, coordinates can be a daunting task. In this regard, employing a set of redundant coordinates can facilitate the formulation of the governing dynamics equations. In this setting, however, standard response analysis techniques cannot be applied in a straightforward manner. For instance, defining and determining a transfer function within a frequency domain response analysis framework is challenging due to the presence of singular matrices, and thus, the machinery of generalized matrix inverses needs to be employed. An efficient frequency domain response analysis methodology for structural dynamical systems modeled via dependent coordinates is developed herein. This is done by resorting to the Moore-Penrose generalized matrix inverse in conjunction with a recently proposed extended modal analysis treatment. It is shown that not only the formulation is efficient in drastically reducing the computational cost when compared to a straightforward numerical evaluation of the involved generalized inverses, but also facilitates the derivation and implementation of the celebrated random vibration input-output frequency domain relationship between the excitation and the response power spectrum matrices. The validity of the methodology is demonstrated by considering a multi-degree-of-freedom shear type structure and a multi-body structural system as numerical examples.Quantum curves from refined topological recursion: the genus 0 casehttps://zbmath.org/1532.810492024-05-13T19:39:47.825584Z"Kidwai, Omar"https://zbmath.org/authors/?q=ai:kidwai.omar"Osuga, Kento"https://zbmath.org/authors/?q=ai:osuga.kentoSummary: We formulate geometrically (without reference to physical models) a refined topological recursion applicable to genus zero curves of degree two, inspired by Chekhov-Eynard and Marchal, introducing new degrees of freedom in the process. For such curves, we prove the fundamental properties of the recursion analogous to the unrefined case. We show the quantization of spectral curves due to Iwaki-Koike-Takei can be generalized to this setting and give the explicit formula, which turns out to be related to the unrefined case by a simple transformation. For an important collection of examples, we write down the quantum curves and find that in the Nekrasov-Shatashvili limit, they take an especially simple form.Block-sparse recovery and rank minimization using a weighted \(l_p-l_q\) modelhttps://zbmath.org/1532.900862024-05-13T19:39:47.825584Z"Nigam, H. K."https://zbmath.org/authors/?q=ai:nigam.hare-krishna"Yadav, Saroj"https://zbmath.org/authors/?q=ai:yadav.saroj-r(no abstract)