Recent zbMATH articles in MSC 15Ahttps://zbmath.org/atom/cc/15A2023-09-22T14:21:46.120933ZWerkzeugA unified approach to generalized Pascal-like matrices: \(q\)-analysishttps://zbmath.org/1517.050132023-09-22T14:21:46.120933Z"Akkus, Ilker"https://zbmath.org/authors/?q=ai:akkus.ilker"Kizilaslan, Gonca"https://zbmath.org/authors/?q=ai:kizilaslan.gonca"Verde-Star, Luis"https://zbmath.org/authors/?q=ai:verde-star.luisSummary: In this paper, we present a general method to construct \(q\)-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method.The characterization of the minimal weighted acyclic graphshttps://zbmath.org/1517.050682023-09-22T14:21:46.120933Z"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibin"da Fonseca, Carlos M."https://zbmath.org/authors/?q=ai:da-fonseca.carlos-martinsSummary: In this paper, we completely characterize all the trees on \(n\) vertices with diameter \(d\) for which there is a symmetric matrix with nullity \(n-d\) and \(n-d-1\), respectively. These characterizations cover all recent results proved for the standard \(0-1\) adjacency matrices. Here a new technique is developed for the general case, breaking the limitation of star complement technique for the standard \(0-1\) adjacency matrices.The automorphism group and fixing number of the orthogonality graph of the full matrix ringhttps://zbmath.org/1517.050712023-09-22T14:21:46.120933Z"Chen, Zhengxin"https://zbmath.org/authors/?q=ai:chen.zhengxin"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.48Summary: Let \(n \geq 3\), \(M_n(F)\) be the set of all \(n \times n\) matrices over a finite field \(F\), and \(R_n(F)\) the subset of \(M_n(F)\) consisting of all rank one matrices. In this paper, we first determine the automorphism group and the fixing number of the orthogonality graph of \(R_n(F)\), and then characterize the automorphism group and the fixing number of the orthogonality graph of \(M_n(F)\).Saturation of ordered graphshttps://zbmath.org/1517.050822023-09-22T14:21:46.120933Z"Bošković, Vladimir"https://zbmath.org/authors/?q=ai:boskovic.vladimir"Keszegh, Balázs"https://zbmath.org/authors/?q=ai:keszegh.balazsSummary: Recently, the saturation problem of 0-1 matrices has gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy also holds in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinitely many examples exhibiting both possible behaviors, answering a problem of Pálvölgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.On unicyclic non-bipartite graphs with tricyclic inverseshttps://zbmath.org/1517.050882023-09-22T14:21:46.120933Z"Kalita, Debajit"https://zbmath.org/authors/?q=ai:kalita.debajit"Sarma, Kuldeep"https://zbmath.org/authors/?q=ai:sarma.kuldeepSummary: The class of unicyclic non-bipartite graphs with unique perfect matching, denoted by \(\mathcal{U}\), is considered in this article. This article provides a complete characterization of unicyclic graphs in \(\mathcal{U}\) which possess tricyclic inverses.The spectrum of triangle-free graphshttps://zbmath.org/1517.050922023-09-22T14:21:46.120933Z"Balogh, József"https://zbmath.org/authors/?q=ai:balogh.jozsef"Clemen, Felix Christian"https://zbmath.org/authors/?q=ai:clemen.felix-christian"Lidický, Bernard"https://zbmath.org/authors/?q=ai:lidicky.bernard"Norin, Sergey"https://zbmath.org/authors/?q=ai:norine.serguei"Volec, Jan"https://zbmath.org/authors/?q=ai:volec.janSummary: Denote by \(q_n(G)\) the smallest eigenvalue of the signless Laplacian matrix of an \(n\)-vertex graph \(G\). \textit{S. Brandt} [Discrete Math. 183, No. 1--3, 17--25 (1998; Zbl 0895.05042)] conjectured that for regular triangle-free graphs \(q_n(G)\leq\frac{4n}{25}\). We prove a stronger result: If \(G\) is a triangle-free graph, then \(q_n(G)\leq\frac{15n}{94}<\frac{4n}{25}\). Brandt's conjecture is a subproblem of two famous conjectures of \textit{P. Erdős} [ibid. 165--166, 227--231 (1997; Zbl 0872.05020); in: Graph theory and related topics. Proceedings of the conference held in honour of Professor W. T. Tutte on the occasion of his sixtieth birthday, University of Waterloo, July 5--9, 1977. New York - San Francisco - London: Academic Press, A Subsidiary of Harcourt Brace Jovanovich, Publishers. 153--163 (1979; Zbl 0457.05024)]: (1) Sparse-half-conjecture: Every \(n\)-vertex triangle-free graph has a subset of vertices of size \(\lceil\frac{n}{2}\rceil\) spanning at most \(n^2/50\) edges. (2) Every \(n\)-vertex triangle-free graph can be made bipartite by removing at most \(n^2/25\) edges. In our proof we use linear algebraic methods to upper bound \(q_n(G)\) by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.Nonbacktracking spectral clustering of nonuniform hypergraphshttps://zbmath.org/1517.050962023-09-22T14:21:46.120933Z"Chodrow, Philip"https://zbmath.org/authors/?q=ai:chodrow.philip-s"Eikmeier, Nicole"https://zbmath.org/authors/?q=ai:eikmeier.nicole"Haddock, Jamie"https://zbmath.org/authors/?q=ai:haddock.jamieSummary: Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for nonuniform hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigenpairs of our operators. We perform experiments in real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.Comparing the principal eigenvector of a hypergraph and its shadowshttps://zbmath.org/1517.050972023-09-22T14:21:46.120933Z"Clark, Gregory J."https://zbmath.org/authors/?q=ai:clark.gregory-j"Thomaz, Felipe"https://zbmath.org/authors/?q=ai:thomaz.felipe"Stephen, Andrew T."https://zbmath.org/authors/?q=ai:stephen.andrew-tSummary: Graphs (i.e., networks) have become an integral tool for the representation and analysis of relational data. Advances in data gathering have led to multi-relational data sets which exhibit greater depth and scope. In certain cases, this data can be modeled using a hypergraph. However, in practice analysts typically reduce the dimensionality of the data (whether consciously or otherwise) to accommodate a traditional graph model. In recent years spectral hypergraph theory has emerged to study the eigenpairs of the adjacency hypermatrix of a uniform hypergraph. We show how analyzing multi-relational data, via a hypermatrix associated to the aforementioned hypergraph, can lead to conclusions different from those when the data is projected down to its co-occurrence matrix. To this end we consider how the principal eigenvector of a hypergraph and its shadow can vary in terms of their spectral rankings, Pearson/Spearman correlation coefficient, and Chebyshev distance. In particular, we provide an example of a uniform hypergraph where the most central vertex (à la eigencentrality) changes depending on the order of the associated matrix. To the best of our knowledge this is the first known hypergraph to exhibit this property. We further show that the aforementioned eigenvectors have a high Pearson correlation but are uncorrelated under the Spearman correlation coefficient.On the multiplicities of digraph eigenvalueshttps://zbmath.org/1517.051002023-09-22T14:21:46.120933Z"Gavrilyuk, Alexander L."https://zbmath.org/authors/?q=ai:gavrilyuk.alexander-l"Suda, Sho"https://zbmath.org/authors/?q=ai:suda.shoSummary: We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first shown by \textit{P. Delsarte} et al. [Geom. Dedicata 6, 363--388 (1977; Zbl 0376.05015)] and extended by \textit{F. K. Bell} and \textit{P. Rowlinson} [Bull. Lond. Math. Soc. 35, No. 3, 401--408 (2003; Zbl 1023.05097)]. In doing so, we also adapt the \textit{A. Blokhuis}' theory [Few-distance sets. Eindhoven: Technische Hogeschool Eindhoven (1983; Zbl 0516.05017)] of harmonic analysis in real hyperbolic spaces to that in complex hyperbolic spaces.A study on determination of some graphs by Laplacian and signless Laplacian permanental polynomialshttps://zbmath.org/1517.051022023-09-22T14:21:46.120933Z"Khan, Aqib"https://zbmath.org/authors/?q=ai:khan.aqib"Panigrahi, Pratima"https://zbmath.org/authors/?q=ai:panigrahi.pratima"Panda, Swarup Kumar"https://zbmath.org/authors/?q=ai:panda.swarup-kumarSummary: The \textit{permanent} of an \(n \times n\) matrix \(M=(m_{ij})\) is defined as \(\operatorname{per}(M) = \sum_\sigma \prod^n_{i=1} m_{i \sigma (i)}\), where the sum is taken over all permutations \(\sigma\) of \(\{1,2, \dots, n\}\) The \textit{permanental polynomial} of \(M\), denoted by \(\psi (M;x)\) is \(\operatorname{per}(xl_n - M)\) where \(I_n\) is the identity matrix of order \(n\). Let \(G\) be a simple undirected graph on \(n\) vertices and its Laplacian and signless Laplacian matrices be \(L(G)\) and \(Q(G)\) respectively. The permanental polynomials \(\psi (L(G);x)\) and \(\psi (Q(G);x)\) are called the \textit{Laplacian permanental polynomial} and \textit{signless Laplacian permanental polynomial} of \(G\) respectively. A graph \(G\) is said to be \textit{determined by its (signless) Laplacian permanental polynomial} if all the graphs having the same (signless) Laplacian permanental polynomial with \(G\) are isomorphic to \(G\). A graph \(G\) is said to be \textit{combinedly determined by its Laplacian and signless Laplacian permanental polynomials} if all the graphs having \textit{(i)} the same Laplacian permanental polynomial as \(\psi (L(G);x)\) and \textit{(ii)} the same signless Laplacian permanental polynomial as \(\psi(Q(G);x)\), are isomorphic to \(G\). In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph \(S_n^{(r)}\) (whose all \(r\) non-pendant vertices have the same degree \(n)\) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs \(S_n^{(r)}\) (for \(r=2,3,4,5)\), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials.On the Moore-Penrose pseudo-inversion of block symmetric matrices and its application in the graph theoryhttps://zbmath.org/1517.051082023-09-22T14:21:46.120933Z"Pavlíková, Soňa"https://zbmath.org/authors/?q=ai:pavlikova.sona"Ševčovič, Daniel"https://zbmath.org/authors/?q=ai:sevcovic.danielSummary: The purpose of this paper is to analyze the Moore-Penrose pseudo-inversion of symmetric real matrices with application in the graph theory. We introduce a novel concept of positively and negatively pseudo-inverse matrices and graphs. We also give sufficient conditions on the elements of a block symmetric matrix yielding an explicit form of its Moore-Penrose pseudo-inversion. Using the explicit form of the pseudo-inverse matrix we can construct pseudo-inverse graphs for a class of graphs which are constructed from the original graph by adding pendent vertices or pendant paths.Distance and adjacency spectra and eigenspaces for three (di)graph lifts: a unified approachhttps://zbmath.org/1517.051112023-09-22T14:21:46.120933Z"Wu, Yongjiang"https://zbmath.org/authors/?q=ai:wu.yongjiang"Zhang, Xiaoqian"https://zbmath.org/authors/?q=ai:zhang.xiaoqian"Feng, Lihua"https://zbmath.org/authors/?q=ai:feng.lihua"Wu, Tingzeng"https://zbmath.org/authors/?q=ai:wu.tingzengSummary: By the induced character theory, we first establish the irreducible decomposition of a permutation representation. Then, using it as a unified tool, we derive a decomposition formula for the distance spectrum and eigenspace of a regular lift of a graph, the adjacency spectra and eigenspaces of a relative lift and a ramified uniform lift of a digraph. The latter results are a complement of \textit{C. Dalfó} et al. [J. Algebr. Comb. 54, No. 2, 651--672 (2021; Zbl 1477.05103)] and \textit{A. Deng} et al. [Eur. J. Comb. 28, No. 4, 1099--1114 (2007; Zbl 1114.05059)].A \(q\)-analogue of distance matrix of block graphshttps://zbmath.org/1517.051122023-09-22T14:21:46.120933Z"Xing, Rundan"https://zbmath.org/authors/?q=ai:xing.rundan"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibinSummary: A \(q\)-analogue of the distance matrix (called \(q\)-distance matrix) of graphs, defined by \textit{W. Yan} and \textit{Y.-N. Yeh} [Adv. Appl. Math. 39, No. 3, 311--321 (2007; Zbl 1129.05029)], is revisited, which is formed from the distance matrix by replacing each nonzero entry \(\alpha\) by \(1 + q + \ldots + q^{\alpha - 1}\) (which would be reduced to \(\alpha\) by setting \(q = 1)\). This concept was also proposed by \textit{R. B. Bapat} et al. [Linear Algebra Appl. 416, No. 2--3, 799--814 (2006; Zbl 1092.05041)]. A graph is called a block graph if every block is a clique (not necessarily of the same order). In this paper, the formula for the inverse of \(q\)-distance matrix of block graphs is presented, which generalizes some classical results about the inverse of distance matrix.Balancing permuted copies of multigraphs and integer matriceshttps://zbmath.org/1517.051392023-09-22T14:21:46.120933Z"del Valle, Coen"https://zbmath.org/authors/?q=ai:del-valle.coen"Dukes, Peter J."https://zbmath.org/authors/?q=ai:dukes.peter-jamesSummary: Given a square matrix \(A\) over the integers, we consider the \(\mathbb{Z} \)-module \(M_A\) generated by the set of all matrices that are permutation-similar to \(A\). Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices \(a I + b J\) belonging to \(M_A\). We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over \(\mathbb{Z} \). Several special cases are considered. In particular, the problem for symmetric matrices answers a question of \textit{S. M. Cioabă} and \textit{P. J. Cameron} [Am. Math. Mon. 122, No. 10, 972--981 (2015; Zbl 1338.05175)] on determining the eventual period for integers \(\lambda\) such that the \(\lambda \)-fold complete graph \(\lambda K_n\) has an edge-decomposition into a given (multi)graph.Many nodal domains in random regular graphshttps://zbmath.org/1517.051592023-09-22T14:21:46.120933Z"Ganguly, Shirshendu"https://zbmath.org/authors/?q=ai:ganguly.shirshendu"McKenzie, Theo"https://zbmath.org/authors/?q=ai:mckenzie.theo"Mohanty, Sidhanth"https://zbmath.org/authors/?q=ai:mohanty.sidhanth"Srivastava, Nikhil"https://zbmath.org/authors/?q=ai:srivastava.nikhilSummary: Let \(G\) be a random \(d\)-regular graph on \(n\) vertices. We prove that for every constant \(\alpha > 0\), with high probability every eigenvector of the adjacency matrix of \(G\) with eigenvalue less than \(-2\sqrt{d-2}-\alpha\) has \(\Omega (n/\text{polylog}(n))\) nodal domains.Asymptotic structure of eigenvalues and eigenvectors of certain triangular Hankel matriceshttps://zbmath.org/1517.111092023-09-22T14:21:46.120933Z"Matiyasevich, Yu. V."https://zbmath.org/authors/?q=ai:matiyasevich.yuri-vSummary: The Hankel matrices considered in this article arose in one reformulation of the Riemann hypothesis proposed earlier by the author. Computer calculations showed that, in the case of the Riemann zeta function, the eigenvalues and the eigenvectors of such matrices have an interesting structure. The article studies a model situation when the zeta function is replaced by a function having a single zero. For this case, we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors.Hadamard powers and kernel perceptronshttps://zbmath.org/1517.150022023-09-22T14:21:46.120933Z"Damm, Tobias"https://zbmath.org/authors/?q=ai:damm.tobias"Dietrich, Nicolas"https://zbmath.org/authors/?q=ai:dietrich.nicolasSummary: We study a relation between Hadamard powers and polynomial kernel perceptrons. The rank of Hadamard powers for the special case of a Boolean matrix and for the generic case of a real matrix is computed explicitly. These results are interpreted in terms of the classification capacities of perceptrons.Common properties among various generalized inverses and constrained binary relationshttps://zbmath.org/1517.150032023-09-22T14:21:46.120933Z"Kuang, Ruifei"https://zbmath.org/authors/?q=ai:kuang.ruifei"Deng, Chunyuan"https://zbmath.org/authors/?q=ai:deng.chunyuanSummary: The common characterizations and various individual properties of different generalized inverses are established. Several equivalent conditions for the core-EP, weak group inverse, \(m\)-weak group inverse and weak core inverse are presented. The brand new explicit expressions for the operator binary relations defined by various generalized inverses are obtained. We derive properties and study the relationship between two different elements \(A, B \in \mathcal{B}(\mathcal{H})\) and their \((T, S)\)-outer generalized inverses \(A^{(2)}_{T, S}\) and \(B^{(2)}_{T, S}\) through which the reverse order law results about \((AB)^{\bigcirc \!\!\!\!\!\!\mathrm{W}} = B^{\bigcirc \!\!\!\!\!\!\mathrm{W}} A^{\bigcirc \!\!\!\!\!\!\mathrm{W}}\) and \((AB)^{\bigcirc \!\!\!\!\!\mathrm{d}}=B^{\bigcirc \!\!\!\!\! \mathrm{d}} A^{\bigcirc \!\!\!\!\!\mathrm{d}}\) are obtained.Characterizations of the generalized MPCEP inverse of rectangular matriceshttps://zbmath.org/1517.150042023-09-22T14:21:46.120933Z"Yao, Jiaxuan"https://zbmath.org/authors/?q=ai:yao.jiaxuan"Liu, Xiaoji"https://zbmath.org/authors/?q=ai:liu.xiaoji"Jin, Hongwei"https://zbmath.org/authors/?q=ai:jin.hongweiSummary: In this paper, we introduce a new generalized inverse, called the G-MPCEP inverse of a complex matrix. We investigate some characterizations, representations, and properties of this new inverse. Cramer's rule for the solution of a singular equation \(Ax = B\) is also presented. Moreover, the determinantal representations for the G-MPCEP inverse are studied. Finally, the G-MPCEP inverse being used in solving appropriate systems of linear equations is established.The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a linear treehttps://zbmath.org/1517.150052023-09-22T14:21:46.120933Z"Ding, Wenxuan"https://zbmath.org/authors/?q=ai:ding.wenxuan"Johnson, Charles R."https://zbmath.org/authors/?q=ai:johnson.charles-richard-jun|johnson.charles-royalThis manuscript deals with the minimum number of eigenvalues with multiplicity 1 among matrices whose associated graph is a linear tree.
A High Degree Vertex (HDV) in a tree is a vertex of degree at least 3, and a tree is called linear if all its HDVs lie on a single induced path of the tree.
Let \(T\) be a linear tree and let \(S(T)\) be the set of real symmetric matrices whose graph is \(T\). If \(U(t)\) is the minimum number of eigenvalues with multiplicity 1 among matrices in \(S(T)\), the authors analyze how \(U(T)\) can change when a vertex is added to \(T\), depending upon how the vertex is added: at an HDV, at a degree 2 vertex, at a pendent vertex, or via edge subdivision. The change is proven to never be by more than 1, but not all such changes can occur. The authors determine the exact set of possibilities.
Specifically, if \(T'\) is a linear tree resulting from the addition of one vertex of \(T\), the authors prove that
\[
|U(T')-U(T)| \leq 1.
\]
They also determine the exact set of possible values of \(U(T')-U(T)\). New bounds and refined bounds are given for \(U(T)\), when \(T\) is a \(k\)-linear tree. In particular, \(U(t) \leq d(T)-k\), where \(d(T)\) denotes the diameter of \(T\), the number of vertices in the longest induced path of \(T\).
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)\(l^{p/2,q/2}\)-singular values of a real partially symmetric rectangular tensorhttps://zbmath.org/1517.150062023-09-22T14:21:46.120933Z"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: Let \({\mathscr{A}}\) be a real \((p, q)\)-th order \(m\times n\) dimensional partially symmetric rectangular tensor with \(p\) and \(q\) even. Firstly, in order to judge the positive definiteness of \({\mathscr{A}}\), an \(l^{p/2,q/2}\)-singular value inclusion interval with parameters is constructed. Subsequently, by finding the optimal values of parameters, the optimal parameter inclusion interval of \(l^{p/2,q/2}\)-singular values is derived, which provides a sufficient condition for the positive definiteness of \({\mathscr{A}}\). Secondly, when \({\mathscr{A}}\) is a nonnegative tensor, lower and upper bounds for its \(l^{p/2,q/2}\)-spectral radius are given. Thirdly, a direct method for finding all \(l^{p/2,q/2}\)-singular value/vectors pairs of \({\mathscr{A}}\) with \(p=q=4\) and \(m=n=2\) is presented. Finally, a numerical example is given to verify the theoretical results.On the nonlinear matrix equation \(x^p = A+\sum_{i=1}^m M_i^{\ast} (B+X^{-1})^{-1} M_i\)https://zbmath.org/1517.150072023-09-22T14:21:46.120933Z"Jin, Zhixiang"https://zbmath.org/authors/?q=ai:jin.zhixiang"Zhai, Chengbo"https://zbmath.org/authors/?q=ai:zhai.chengboThe authors derive some necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the equation
\[
x^p = A+\sum_{i=1}^m M_i^{\ast} (B+X^{-1})^{-1} M_i ,
\]
where \(p\) and \(m\) are positive integers, \(M_i\), \ \(i = 1, 2, \ldots, m\), are \(n\times n\) nonsingular complex matrices, and \(A\), \(B\) are HPD matrices. Their approach is based on fixed point theorems. They show that the unique solution exists in a certain partially ordered set, and can be approximated by making an iterative sequence for any given initial point in this set. They analyze three algorithms for obtaining the unique solution, and study the convergence of the iterations obtained with these algorithms. They also show how a real parameter can be added to the above equation, and correspondingly describe the existence of the unique HPD solution. Finally, they provide two numerical examples to illustrate the effectiveness of their results.
Reviewer: Igor Korepanov (Moskva)The consistency and the general common solution to some quaternion matrix equationshttps://zbmath.org/1517.150082023-09-22T14:21:46.120933Z"Xu, Xi-Le"https://zbmath.org/authors/?q=ai:xu.xi-le"Wang, Qing-Wen"https://zbmath.org/authors/?q=ai:wang.qingwenThe authors study the following system of five quaternion linear matrix equations
\[
\begin{cases} AX = B,\\
XC = D,\\
A_1 X B_1 = C_1, \\
A_2 X B_2 = C_2, \\
A_3 X B_3 = C_3, \\
\end{cases}
\]
where \(X\) is the unknown matrix, while all the other matrices are given.
After reviewing some definitions of quaternions and some general results about matrix equations, the authors use the Moore-Penrose inverses and the ranks of the involved matrices to present some new necessary and sufficient conditions for the existence of the general solution to this system. They also present an expression for this solution when the system is solvable. Then a numerical algorithm is given and a numerical example to illustrate the main results is considered in detail.
Additionally, the authors consider a special case of the solution in the case of \(\eta\)-Hermitian matrices. Let us recall that a quaternion matrix \(A\) is said to be \(\eta\)-Hermitian for \(\eta \in \{\mathbf i, \mathbf j, \mathbf k\}\) if \(A=-\eta A^* \eta\). In this case, some solvability conditions are derived and an expression for the solution is given.
Reviewer: Igor Korepanov (Moskva)Null vectors, Schur complements, and Parter verticeshttps://zbmath.org/1517.150092023-09-22T14:21:46.120933Z"Fallat, Shaun"https://zbmath.org/authors/?q=ai:fallat.shaun-m"Parenteau, Johnna"https://zbmath.org/authors/?q=ai:parenteau.johnnaThis manuscript deals with the inverse eigenvalue problem of matrices whose associated graph is a tree. For a given \(n \times n\) real symmetric matrix \(A=(a_{ij})\), the graph of \(A\), \(G(A)\), is the undirected graph whose vertex set is \(\{1,2,\ldots,n\}\) and the edge set is given by \(\{ij \mid i \neq j \ \mbox{and} \ a_{ij} \neq 0\}\). Given an undirected graph \(G\), \(S(G)\) is the set of real symmetric matrices with the given graph \(G\). Let \(T\) be a tree, let \(v\) be a vertex of \(T\). Any component of the induced subgraph \(T \setminus v\) is called a branch of \(T\) at \(v\). If the degree of \(v\) is \(k\), then there are \(k\) branches of \(T\) at \(v\). If \(T \setminus v=B_1 \cup \cdots \cup B_k\), where \(B_i\) are the branches of \(T\) at \(v\), then for any matrix \(A \in S(T)\), we have \(A(v)=A_1 \oplus \cdots \oplus A_k\), where \(A_i=A(B_i)\).
Eigenvalues of graphs have been a central subject bridging matrix theory and graph theory for many years. Among many other known results, the following ones are fundamental and have been established by \textit{S. Parter} [J. Soc. Ind. Appl. Math. 8, 376--388 (1960; Zbl 0115.24804)] and \textit{G. Wiener} [Linear Algebra Appl. 61, 15--29 (1984; Zbl 0549.15004)]. Suppose that \(T\) is a tree and \(A \in S(T)\). If \(\lambda\) is an eigenvalue of \(A\) with \(m_A(\lambda)\geq 2\) (algebraic multiplicity), then there exists a vertex \(v\) in \(T\) such that:
\begin{itemize}
\item[(1)] \(m_{A(v)}(\lambda)=m_A(\lambda)+1\);
\item[(2)] \(\lambda\) is an eigenvalue of at least three distinct branches of \(T\) at \(v\).
\end{itemize}
In this paper, the authors present an alternative elementary proof of the above results, using a basic linear algebra tool known as the Schur complement of a matrix. This allows them to study the nullspace structure of matrices whose graph is a tree.
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Spaces of generators for matrix algebras with involutionhttps://zbmath.org/1517.150102023-09-22T14:21:46.120933Z"Nam, Taeuk"https://zbmath.org/authors/?q=ai:nam.taeuk"Tan, Cindy"https://zbmath.org/authors/?q=ai:tan.cindy"Williams, Ben"https://zbmath.org/authors/?q=ai:williams.benSummary: Let \(k\) be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra \(\mathrm{Mat}_{n \times n} (k)\) can be endowed with a \(k\)-linear involution in one way if \(n\) is odd and in two ways if \(n\) is even. In this paper, we consider \(r\)-tuples \(A_\bullet \in \mathrm{Mat}_{n \times n} (k)^r\) such that the entries of \(A_\bullet\) fail to generate \(\mathrm{Mat}_{n \times n} (k)\) as an algebra with involution. We show that the locus of such \(r\)-tuples forms a closed subvariety \(Z(r;V)\) of \(\mathrm{Mat}_{n \times n} (k)^r\) that is not irreducible. We describe the irreducible components and we calculate the dimension of the largest component of \(Z(r;V)\) in all cases. This gives a numerical answer to the question of how generic it is for an \(r\)-tuple \((a_1, \dots, a_r)\) of elements in \(\mathrm{Mat}_{n \times n} (k)\) to generate it as an algebra with involution.A further improvement of the Ostrowski-Taussky inequality for real matriceshttps://zbmath.org/1517.150112023-09-22T14:21:46.120933Z"Dannan, Fozi Mustafa"https://zbmath.org/authors/?q=ai:dannan.fozi-mustafa"Merikoski, Jorma Kaarlo"https://zbmath.org/authors/?q=ai:merikoski.jorma-kaarloSummary: Let \(A\in \mathbb{C}^{n\times n}\), \(A=H+iK\), \(H=\frac{1}{2}(A+A^\ast)\), \(iK=\frac{1}{2}(A-A^\ast)\). It is well known that if \(H\) is positive definite, then \[ \det H+|\det K| \leq |\det A|.\] We improve this inequality assuming that \(A\in\mathbb{R}^{n\times n}\).A new version of Schur-Horn type theoremhttps://zbmath.org/1517.150122023-09-22T14:21:46.120933Z"Huang, Shaowu"https://zbmath.org/authors/?q=ai:huang.shaowuThe author presents a new symplectic version of the Schur-Horn theorem. The new result gives a relationship in terms of weak supermajorization between symplectic eigenvalues of a real positive definite matrix \(P\) and diagonal entries of a special matrix \(\tilde{P}\) associated with \(P\). More precisely, let
\[
P = \left( \begin{array}{cc} A & C \\
D & B \\
\end{array} \right)
\]
be a \(2n \times 2n\) real positive definite matrix, where \(A\), \(B\), \(C\), \(D\) are \(n\times n\) real matrices, and let \(d_s(P)\) denote the real \(n\)-vector of the symplectic eigenvalues of \(P\). Let \(\tilde{P}\) be a matrix with entries \(\tilde{p}_{ij} = (a_{ij}^2 + b_{ij}^2 + c_{ij}^2 + d_{ij}^2)/2\) and let \(\Delta_h(P)\) be the \(n\)-vector of the diagonal of \((\mathrm{diag}(\tilde{P}))^{1/2}\), where the square root is taken entrywise. Then the theorem claims that
\[
\Delta_h{P} \prec^w d_s(P),
\]
where the weak supermajorization \(x \prec^w y\) means that
\[
\sum_{j=1}^k x_j^{\uparrow} \geq \sum_{j=1}^k y_j^{\uparrow}\text{ for }1\leq k\leq n,
\]
with coordinates of \(x\) and \(y\) rearranged in increasing order, i.e., \(x_1^{\uparrow}\leq\dots\leq x_n^{\uparrow}\) and \(y_1^{\uparrow}\leq\dots\leq y_n^{\uparrow}\).
Conversely, if \(x\), \(y\) are two positive \(n\)-vectors such that \(x \prec^w y\), then there exists a \(2n \times 2n\) positive definite matrix \(P\) such that \(x = \Delta_h{P}\) and \(y = d_s(P)\).
The author also provides a different proof of the second part of Theorem 3 concerning the arithmetic mean stated in [\textit{R. Bhatia} and \textit{T. Jain}, Linear Algebra Appl. 599, 133--139 (2020; Zbl 1451.15012)].
Reviewer: Ctirad Matonoha (Praha)Strict positivity and \(D\)-majorizationhttps://zbmath.org/1517.150132023-09-22T14:21:46.120933Z"vom Ende, Frederik"https://zbmath.org/authors/?q=ai:vom-ende.frederikSummary: Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector \(d\) to majorization on square matrices with respect to a positive definite matrix \(D\). For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of \(D\) has multiplicity one.When does \(\Vert f(A) \vert = f(\Vert A \Vert)\) hold true?https://zbmath.org/1517.150142023-09-22T14:21:46.120933Z"Bünger, Florian"https://zbmath.org/authors/?q=ai:bunger.florian"Rump, Siegfried M."https://zbmath.org/authors/?q=ai:rump.siegfried-michaelThe authors obtain conditions on norms on the set \(M_n\) of complex \(n\)-by-\(n\) matrices.
They obtain the following result:
Theorem. Let \(\|\cdot\|\) be a norm on the set \(M_n\), and let \(f(x):=\sum_{k=0}^{\infty}c_k x^k\) with \(c_k>0\), radius of convergence \(R\in (0,\infty]\) and \(f(0)(\|I\|-1)=0\). Suppose that one of the following cases holds true: the norm is
\begin{itemize}
\item[(1)] induced by a uniformly convex vector norm and \(c_k c_{k+1}\neq 0\) for some \(k\geq 0\);
\item[(2)] unitarily invariant and \(c_k c_{k+1}\neq 0\) for some \(k\geq 0\);
\item[(3)] the numerical radius and \(c_k c_{k+1}\neq 0\) for some \(k\geq 1\);
\item[(4)] the \(\ell^{1}\)- or \(\ell^{\infty}\)-norm and \(c_k\neq 0\) for all \(k\geq k_0\) and some \(k_0\geq 0\).
\end{itemize}
Then, for \(A\in M_n\) with \(\|A\|\leq R\), we have \(\|f(A)\|=f(\|A\|)\) if and only if \(\|A\|\) is an eigenvalue of \(A\).
This is a generalization of the result in [\textit{Yu. A. Abramovich} et al., J. Funct. Anal. 97, No. 1, 215--230 (1991; Zbl 0770.47005)] concerning the so-called Daugavet equation \(\|I+A\|=1+\|A\|\).
Reviewer: Takeaki Yamazaki (Kawagoe)An infinity norm bound for the inverse of strong \(\mathrm{SDD}_1\) matrices with applicationshttps://zbmath.org/1517.150152023-09-22T14:21:46.120933Z"Wang, Yinghua"https://zbmath.org/authors/?q=ai:wang.yinghua"Song, Xinnian"https://zbmath.org/authors/?q=ai:song.xinnian"Gao, Lei"https://zbmath.org/authors/?q=ai:gao.leiBy imposing an additional condition on the main diagonal entries, the authors introduce a new class of matrices called strong SDD1 matrices (SDD stands for ``strictly diagonally dominant'', while SDD1 matrices were proposed by \textit{J. M. Peña} [Adv. Comput. Math. 35, No. 2--4, 357--373 (2011; Zbl 1254.65057)]) and present an infinity norm bound for the inverse of strong SDD1 matrices. Based on the proposed infinity norm bound, a new pseudospectra localization for matrices is obtained. Moreover, a new lower bound for the distance to instability is derived.
Reviewer: Minghua Lin (Xi'an)Minimality of tensors of fixed multilinear rankhttps://zbmath.org/1517.150162023-09-22T14:21:46.120933Z"Heaton, Alexander"https://zbmath.org/authors/?q=ai:heaton.alexander"Kozhasov, Khazhgali"https://zbmath.org/authors/?q=ai:kozhasov.khazhgali"Venturello, Lorenzo"https://zbmath.org/authors/?q=ai:venturello.lorenzoSummary: We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, it is shown that real tensors of the fixed multilinear rank form a minimal submanifold of the Euclidean space of tensors endowed with the Frobenius inner product. We also establish the absence of local extrema for linear functionals restricted to the submanifold of rank-one tensors, finding application in statistics.On centres and direct sum decompositions of higher degree formshttps://zbmath.org/1517.150172023-09-22T14:21:46.120933Z"Huang, Hua-Lin"https://zbmath.org/authors/?q=ai:huang.hua-lin"Lu, Huajun"https://zbmath.org/authors/?q=ai:lu.huajun"Ye, Yu"https://zbmath.org/authors/?q=ai:ye.yu"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4The authors show that almost all direct sum decompositions of higher degree forms have trivial centre, i.e., isomorphic to the ground field. This means that they are a priori absolutely indecomposable. They also prove that the centre of the algebra of a higher-degree form is semisimple iff the form is not a limit of direct sums forms, as discussed in [\textit{W. Buczyńska} et al., Mich. Math. J. 64, No. 4, 675--719 (2015; Zbl 1339.13012)]. Furthermore, for forms with semisimple centre the authors develop an elementary criterion for the direct sum decomposability, which is equivalent to computing the rank of a finite set of vectors.
In Theorem 3.2, it is shown that a generic higher degree form is central; with the help of Theorems 3.6 and 3.7, a central form may be expressed as an indecomposable non-LDS (here LDS means limit of direct sums). In Theorem 3.11, an algorithm is discussed, purely in terms of linear algebra, for direct sum decompositions of any higher degree form. Such an algorithm can be obtained by slightly extending the algorithm for forms with semisimple centres again using the Jordan decomposition theorem.
The authors discuss several related results and examples.
Reviewer: M. P. Chaudhary (New Delhi)Centers of multilinear forms and applicationshttps://zbmath.org/1517.150182023-09-22T14:21:46.120933Z"Huang, Hua-Lin"https://zbmath.org/authors/?q=ai:huang.hua-lin"Lu, Huajun"https://zbmath.org/authors/?q=ai:lu.huajun"Ye, Yu"https://zbmath.org/authors/?q=ai:ye.yu"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4Summary: The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.An alternating shifted inverse power method for the extremal eigenvalues of fourth-order partially symmetric tensorshttps://zbmath.org/1517.150192023-09-22T14:21:46.120933Z"Wang, Chunyan"https://zbmath.org/authors/?q=ai:wang.chunyan"Chen, Haibin"https://zbmath.org/authors/?q=ai:chen.haibin"Wang, Yiju"https://zbmath.org/authors/?q=ai:wang.yiju"Yan, Hong"https://zbmath.org/authors/?q=ai:yan.hongSummary: Extremal \(M\)-eigenvalues of fourth-order partially symmetric tensors play an important role in the nonlinear elastic material analysis and the entanglement problem in quantum physics. In this paper, we propose an alternating shifted inverse power method for computing the extremal \(M\)-eigenvalues of fourth-order partially symmetric tensors. The proposed algorithm is simple to operate and easy to understand for convergence analysis. Numerical experiments show the effectiveness of the proposed method.Z-eigenvalue localization sets for tensors and the applications in rank-one approximation and quantum entanglementhttps://zbmath.org/1517.150202023-09-22T14:21:46.120933Z"Zhang, Juan"https://zbmath.org/authors/?q=ai:zhang.juan.1|zhang.juan.2|zhang.juan.7"Chen, Xuechan"https://zbmath.org/authors/?q=ai:chen.xuechanSummary: In this paper, we propose two Z-eigenvalue inclusive sets of tensors, and prove that our new inclusion sets are more precise than some existing results. Using the derived inclusion sets, we present new upper and lower bounds of the spectral radius of nonnegative weakly symmetric tensors. Further, we offer two applications of the obtained upper and lower bounds. One application is the best rank-one approximation rate. The other application is the geometric measure of quantum pure state entanglement with nonnegative amplitudes. Finally, numerical examples are given to illustrate the validity of the derived results.T-product factorization based method for matrix and tensor completion problemshttps://zbmath.org/1517.150212023-09-22T14:21:46.120933Z"Yu, Quan"https://zbmath.org/authors/?q=ai:yu.quan"Zhang, Xinzhen"https://zbmath.org/authors/?q=ai:zhang.xinzhenSummary: Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data of low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. For a matrix completion problem, we establish a relationship between matrix rank and tensor tubal rank, and reformulate matrix completion problem as a third order tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank of two tensors, original tensor and the reshaped tensor of the mode-3 unfolding matrix of original tensor. Based on this, we propose a reweighted tensor factorization algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.Linear transformations preserving algebraic elements of degree 2https://zbmath.org/1517.150222023-09-22T14:21:46.120933Z"Franca, Willian"https://zbmath.org/authors/?q=ai:franca.willian"Alves, Magno"https://zbmath.org/authors/?q=ai:alves.magno-brancoLet us consider the ring \(M_n(\mathbb{F})\) of all \(n\times n\) matrices over an algebraically closed field \(\mathbb{F}\) of characteristic zero. If \(X \in M_{n}(\mathbb{F})\), \(p_{m,X}\) will denote the monic polynomial of lowest degree such that \(p_{m,X}(X)=0\) and we say that \(X\) is an algebraic element of degree \(2\) if \(p_{m,X}(z)=z^2+a_1 z+a_0\).
The main result of this paper is the characterization of the linear maps \(f\) of \(M_n(\mathbb{F})\), with \(n \geq 3\), such that \(f(S)=S\), where \(S\) denotes the set of algebraic elements of degree \(2\). The case \(n=2\) is left open but there are some interesting comments at the end of the paper concerning this problem.
Reviewer: Jerónimo Alaminos Prats (Granada)Linear maps on nonnegative symmetric matrices preserving the independence numberhttps://zbmath.org/1517.150232023-09-22T14:21:46.120933Z"Hu, Yanan"https://zbmath.org/authors/?q=ai:hu.yanan"Huang, Zejun"https://zbmath.org/authors/?q=ai:huang.zejunSummary: The independence number of a square matrix \(A\), denoted by \(\alpha (A)\), is the maximum order of its principal zero submatrices. Given any integer \(n\), let \(S_n^+\) be the set of \(n \times n\) nonnegative symmetric matrices with zero diagonal. We characterize the linear maps on \(S_n^+\) preserving the independence number of all matrices in \(S_n^+\).Hadamard product on quaternion Hermitian matriceshttps://zbmath.org/1517.150242023-09-22T14:21:46.120933Z"Chang, Yu-Lin"https://zbmath.org/authors/?q=ai:chang.yulin"Hu, Chu-Chin"https://zbmath.org/authors/?q=ai:hu.chu-chin"Yang, Ching-Yu"https://zbmath.org/authors/?q=ai:yang.chingyu"Chen, Jein-Shan"https://zbmath.org/authors/?q=ai:chen.jein-shanSummary: Recently, \textit{S. Kum} et al. [Linear and Nonlinear Algebra 71, No. 12, 2049--2065 (2023; \url{https://doi.org/10.1080/03081087.2022.2093319})]
define an Hadamard product in the setting of Jordan spin algebra, under the scheme of the Peirce decomposition. It is shown that this novel product satisfies an analog of the Schur product theorem as well as the inequalities of Hadamard, Oppenheim, Fiedler, etc. In fact, they also try to extend their results to Euclidean Jordan algebras, but they encounter troubles on Hermitian quaternion matrices. In this paper, we propose an Hadamard product on Hermitian quaternion matrices, and we show this new product satisfies the same inequalities as well.Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard producthttps://zbmath.org/1517.150272023-09-22T14:21:46.120933Z"Silverstein, Jack W."https://zbmath.org/authors/?q=ai:silverstein.jack-wSummary: This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices \(\frac{1}{N}(D_n \circ X_n) (D_n \circ X_n )^{\ast}\), studied in [\textit{V. L. Girko}, Theory of stochastic canonical equations. Vol. 1. Dordrecht: Kluwer Academic Publishers (2001; Zbl 0996.60002)]. Here, \(X_n =(x_{ij})\) is an \(n\times N\) random matrix consisting of independent complex standardized random variables, \(D_n =(d_{ij}), n\times N\), has nonnegative entries, and \(\circ\) denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of \(X_n\) and \(D_n\) which are different from those in [loc. cit.], which include a Lindeberg condition on the entries of \(D_n \circ X_n\), as well as a bound on the average of the rows and columns of \(D_n \circ D_n\). The present paper separates the assumptions needed on \(X_n\) and \(D_n\). It assumes a Lindeberg condition on the entries of \(X_n\), along with a tightness-like condition on the entries of \(D_n\).A variant of Rosset's approach to the Amitsur-Levitzki theorem and some \(\mathbb{Z}_2\)-graded identities of \(\mathrm{M}_n(E)\)https://zbmath.org/1517.160202023-09-22T14:21:46.120933Z"Homolya, Szilvia"https://zbmath.org/authors/?q=ai:homolya.szilvia"Szigeti, Jenő"https://zbmath.org/authors/?q=ai:szigeti.jenoThe celebrated theorem of \textit{A. S. Amitsur} and \textit{J. Levitzki} [Proc. Am. Math. Soc. 1, 449--463 (1950; Zbl 0040.01101)] states that the matrix algebra \(M_n(F)\) over a field \(F\) (or more generally over a commutative unital ring) satisfies the standard identity \(s_{2n}\), the latter being the alternating sum of all multilinear monomials of degree \(2n\) in the variables \(x_1, \dots, x_{2n}\). Moreover this is the identity of the least degree for \(M_n(F)\). There are several different proofs of this theorem, the shortest of them probably that of \textit{S. Rosset} [Isr. J. Math. 23, 187--188 (1976; Zbl 0322.15020)].
The full matrix algebras \(M_n(F)\) together with \(M_n(E)\) where \(E\) is the infinite dimensional Grassmann algebra, are of extreme importance in PI theory. In characteristic 0 they, together with the algebras \(M_{a,b}(E)\) generate the T-prime varieties. Recall that \(E=E_0\oplus E_1\) is naturally a superalgebra (a 2-graded algebra), and \(M_{a,b}(E)\) is the subalgebra of all matrices of order \(a+b\) over \(E\) divided into four blocks. The two blocks on the main diagonal have entries coming from \(E_0\) while the off-diagonal blocks have their entries from \(E_1\).
By following ideas from the above mentioned paper by Rosset, the authors of the paper under review deduce a new 2-graded polynomial identity for the algebra \(M_n(E)\).
Reviewer: Plamen Koshlukov (Campinas)A notion of rank for noncommutative quadratic forms on four generatorshttps://zbmath.org/1517.160252023-09-22T14:21:46.120933Z"Cain, Jessica G."https://zbmath.org/authors/?q=ai:cain.jessica-g"Frauendienst, Leah R."https://zbmath.org/authors/?q=ai:frauendienst.leah-r"Veerapen, Padmini"https://zbmath.org/authors/?q=ai:veerapen.padmini-pSummary: In this paper, we extend work from [\textit{M. Vancliff} and \textit{P. P. Veerapen}, Contemp. Math. 592, 241--250 (2013; Zbl 1326.16019)], where a notion of rank, called \(\mu\)-rank, was proposed for noncommutative quadratic forms on two and three generators. In particular, we provide a definition of \(\mu\)-rank one and two for noncommutative quadratic forms on four generators. We apply this definition to determine the number of point modules over certain quadratic AS-regular algebras of global dimension four.Lie triple derivations on trivial extension algebrashttps://zbmath.org/1517.160362023-09-22T14:21:46.120933Z"Ansari, Mohammad Afajal"https://zbmath.org/authors/?q=ai:ansari.mohammad-afajal"Ashraf, Mohammad"https://zbmath.org/authors/?q=ai:ashraf.mohammad"Akhtar, Mohd Shuaib"https://zbmath.org/authors/?q=ai:akhtar.mohd-shuaibSummary: Let \(\mathcal{U}\) be a unital algebra over a unital commutative ring \(\mathcal{R}\) and \(\mathcal{M}\) be a \(\mathcal{U}\)-bimodule. A trivial extension algebra \(\mathcal{U} \ltimes \mathcal{M}\) is defined as an \(\mathcal{R}\)-algebra with the usual operations of the \(\mathcal{R}\)-linear space \(\mathcal{U} \oplus \mathcal{M}\) and the multiplication defined by \((u_1,m_1)(u_2,m_2)=(u_1u_2,u_1m_2+m_1u_2)\) for all \(u_1,u_2\in \mathcal{U},m_1,m_2\in \mathcal{M}\). In this article, the structure of Lie triple derivations on \(\mathcal{U} \ltimes \mathcal{M}\) are given, and it is shown that under some mild restrictions every Lie triple derivation on \(\mathcal{U} \ltimes \mathcal{M}\) is of the form \(\delta +\tau\), where \(\delta : \mathcal{U} \ltimes \mathcal{M} \rightarrow \mathcal{U} \ltimes \mathcal{M}\) is a derivation and \(\tau : \mathcal{U} \ltimes \mathcal{M} \rightarrow \mathcal{Z}(\mathcal{U} \ltimes \mathcal{M})\) is an \(\mathcal{R}\)-linear mapping.Asymptotic of the largest Floquet multiplier for cooperative matriceshttps://zbmath.org/1517.340172023-09-22T14:21:46.120933Z"Carmona, Philippe"https://zbmath.org/authors/?q=ai:carmona.philippeLet \(A(t):\mathbb{R} \rightarrow \mathcal{M}_{d\times d}\) be a Lipschitz continuous 1-periodic matrix function and
\[
\dot{x}(t)=A(t)x(t)
\]
the associated linear differential equation with fundamental matrix \(\phi(t)\). Let \(\phi^{(T)}(t)\) be the fundamental matrix of the \(T\) periodic matrix \(t\mapsto A(t/T)\). The author proves the following result.
Assuming that for every \(t\geq 0\) the matrix \(A(t)=[a_{ij}(t)]\) is cooperative i.e. \(a_{ij}(t)\geq 0\) for \(i\neq j\), and irreductible then
\[
\lim_{T\rightarrow +\infty}\ln \rho(\phi^{(T)}(T)) = \int_{0}^{1}s(A(u)) du
\]
where \(\rho(\phi^{(T)}(T))\) is the \textit{spectral radius} of \(\phi^{(T)}(T)\) and \(\int_{0}^{1}s(A(u))du\) is the mean \textit{spectral abscissa} (introduced in this paper).
The motivation for the result comes from the study of the spread of infectious diseases for periodic systems in populations whose individuals can be divided into a finite number of distinct groups.
Reviewer: Predrag Punosevac (Pittsburgh)Construction of the Fuchs differential equation with \(3\times 3\) residue-matrices and three singular points using logarithmization methodhttps://zbmath.org/1517.341172023-09-22T14:21:46.120933Z"Khvoshchinskaya, L. A."https://zbmath.org/authors/?q=ai:khvoshchinskaya.l-aThe famous Riemann problem is the problem to find analytic functions or a Fuchsian differential equation they satisfy with a given monodromy group around a finite number of singular points. In some previous papers (e.g., [\textit{L. A. Khvoshchinskaya}, ``Representation of the logarithm of the product of nonsingular matrices of the 2nd order'', in: Proceedings of the 16th international scientific conference devoted to Acad. M. Krawtchuk. Kyiv: Kyiv Polytechnic Institute. 192--195 (2015); \textit{L. Khvostchinskaya} and \textit{S. Rogosin}, ``On a solution method for the Riemann problem with two pairs of unknown functions'', in: Proceedings of the 9th international conference on analytic methods of analysis and differential equations, AMADE 2018. Cottenham: Cambridge Scientific Publishers. 79-111 (2020); \textit{S. Rogosin} and \textit{L. Khvoshchinskaya}, Lobachevskii J. Math. 42, No. 4, 830--849 (2021; Zbl 1515.34086)]) the author of the paper under review obtained the logarithmization method, which gives a closed formula for a logarithm of the product of two \(2\times 2\) matrices, and applied it to the Riemann problem for a given monodromy group of order two. In this paper, extending the logarithmization method to a logarithm of the product of two \(3\times 3\) matrices, the author discusses the Riemann problem for a given monodromy group of order three around three singular points.
Reviewer: Yoshitsugu Takei (Kyoto)Functional equations characterizing certain determinants and permanentshttps://zbmath.org/1517.390132023-09-22T14:21:46.120933Z"Suriyacharoen, Wuttichai"https://zbmath.org/authors/?q=ai:suriyacharoen.wuttichai"Laohakosol, Vichian"https://zbmath.org/authors/?q=ai:laohakosol.vichianSummary: General solution functions \(g, h, f:\mathbb{R}^2\to \mathbb{R}\) of the following three functional equations \[ \begin{aligned} g(ux-vy, uy+v(x+y)) &=g(x, y)g(u, v) \\ h(ux-vy, uy+v(xy)) &=h(x, y)h(u, v), \\ f(vx+uy, vy-ux) &=f(x, y)f(u, v)\end{aligned}\] are determined without any regularity assumptions on the unknown functions. These equations arise from the problem of characterizing certain determinants and permanents.Geometric harmonic analysis I. A sharp divergence theorem with nontangential pointwise traceshttps://zbmath.org/1517.420012023-09-22T14:21:46.120933Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusThe present book is the first in a series of five volumes, at the confluence of Harmonic Analysis,
Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically
branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles:
Volume I: A Sharp Divergence Theorem with Nontangential Pointwise Traces;
Volume II: Function Spaces Measuring Size and Smoothness on Rough Sets;
Volume III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering;
Volume IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis;
Volume V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems.
The present review is concerned with the first volume. In the first chapter, starting from the classical work
of De Giorgi-Federer, the authors develop a new generation of divergence theorems both in the Euclidean space
as well as in the setting of Riemannian manifolds. The most striking feature is that the vector field in question is
strictly defined in the underlying open set and its boundary trace is considered in a pointwise nontangential fashion.
In the second chapter, a wealth of examples and counter-examples are presented, indicating that the main results are
optimal from a variety of perspectives.
The third chapter gathers foundational material from measure theory and topology.
Chapter four contains a variety of selected topics from (or inspired by) distribution theory.
For example, the authors develop a brand of distribution theory on arbitrary subsets of the Euclidean space, taking
Lipschitz functions with bounded support as test functions. Here they also coin the notion of
``bullet product'' which, in essence, is a weak version (modeled upon integration by parts) of the inner product of
the normal vector to a domain with a given vector field satisfying only some very mild integrability properties in that domain.
Among other things, a proof of Leibniz's product rule for weak derivatives is provided, and the chapter ends with what the
authors call the contribution at infinity of a vector field.
In the fifth chapter, the author discusses basic results from Geometric Measure Theory, including thick sets,
the corkscrew condition, the geometric measure theoretic boundary, area and coarea formulas, countable rectifiability,
approximate tangent planes, functions of bounded variation, sets of locally finite perimeter, Ahlfors regularity,
uniformly rectifiable sets, the local John condition, and nontangentially accessible domains.
The sixth chapter is focused on tools from Harmonic Analysis, such as the regularized distance function,
Whitney's Extension Theorem, and the fractional Hardy-Littlewood maximal operator in non-metric settings.
This chapter also contains an informative review of Clifford algebras (which are higher-dimensional versions
of the field of complex numbers, that happen to be highly non-commutative, in which a brand of complex analysis may be developed),
and a discussion of reverse Hölder inequalities and interior estimates. The authors close this chapter by introducing
the solid maximal function and defining maximal Lebesgue spaces.
The seventh chapter, entitled Quasi-Metric Spaces and Spaces of Homogeneous Type, consists of the following sections:
Quasi-Metric Spaces and a Sharp Metrization Result; Estimating Integrals Involving the Quasi-Distance;
Hölder Spaces on Quasi-Metric Spaces; Functions of Bounded Mean Oscillations on Spaces of Homogeneous Type;
Whitney Decompositions on Geometrically Doubling Quasi-Metric Spaces; The Hardy-Littlewood Maximal Operator
on Spaces of Homogeneous Type; Muckenhoupt Weights on Spaces of Homogeneous Type; The Fractional Integration Theorem.
The eighth chapter is entitled Open Sets with Locally Finite Surface Measures and Boundary Behavior.
The first section focuses on nontangential approach regions in arbitrary open sets. The second and third sections
deal with the basic properties of the nontangential maximal operator. The fourth section contains size estimates
for the nontangential maximal operator involving a doubling measure, while the fifth one is reserved for a
comparison between the nontangential and tangential maximal operators. In the sixth section, the authors establish
off-diagonal Carleson measure estimates of reverse Hölder type, which are crucial ingredients in the proofs of the main results.
The seventh section elaborates on estimates for Marcinkiewicz type integrals and applications. The eighth and the ninth sections
are on what the authors call the nontangentially accessible boundary and, respectively, the nontangential boundary trace operator.
The tenth section treats the averaged nontangential maximal operator.
The last chapter contains the proofs of the main results pertaining to the family of divergence theorems stated in chapter one.
Reviewer: Mohammed El Aïdi (Bogotá)Alternation points, weights and orthogonal polynomialshttps://zbmath.org/1517.420242023-09-22T14:21:46.120933Z"Harris, Lawrence A."https://zbmath.org/authors/?q=ai:harris.lawrence-aSummary: In this note we characterize the orthogonal polynomials having alternation points in terms of algebraic properties of the coefficients of their recurrence relations. As an example, we deduce that the four types of Chebyshev polynomials are the only Jacobi polynomials of degree greater than 3 that have alternation points.
We also show that if there are alternation points for polynomials orthogonal with respect to a measure \(\mu \), then the inner product defined by \(\mu\) is a multiple of the discrete inner product generated by the alternation points. Finally, we produce orthonormal polynomials having alternation points that are the roots of a given trigonometric equation.Cycles cross ratio: an invitationhttps://zbmath.org/1517.510022023-09-22T14:21:46.120933Z"Kisil, Vladimir V."https://zbmath.org/authors/?q=ai:kisil.vladimir-vThis paper offers an engaging presentation of various properties of a notion of cross ratio introduced in Lie sphere geometry.
A \(2\times 2\) complex matrix \(C=\begin{pmatrix} \overline{L} & -m\\
k & -L \end{pmatrix}\) is associated, via the Fillmore-Springer-Cnops construction, to the cycle \(k(x^2+y^2)-2lx-2ny+m=0\) with coefficients \((k, l, n, m)\), where \(L\) stands for \(l+in\). A cycle product \(\langle C_1, C_2\rangle\) for two matrices \(C_1\) and \(C_2\) if defined by \(\langle C_1, C_2\rangle=-\operatorname{tr}(C_1\overline{C_2})\), where \(\operatorname{tr}\) denotes the trace, allowing the definition, one which turns out to be well-defined and a fractional linear transformation invariant of quadruples of cycles, of the cross ratio of four cycles \(C_1, C_2, C_3, C_4\) by (assuming \({\langle C_1, C_4\rangle}{\langle C_2, C_3\rangle}\neq 0\)):
\[
\langle C_1, C_2; C_3, C_4\rangle=\frac{\langle C_1, C_3\rangle}{\langle C_1, C_4\rangle}:\frac{\langle C_2, C_3\rangle}{\langle C_2, C_4\rangle}.
\]
Among others, a distance between two cycles is defined, which is shown to be Möbius invariant, and which coincides with the hyperbolic metric on the upper-half plane in the case of zero radius cycles.
Reviewer: Victor V. Pambuccian (Glendale)Affine invariants of an immersion of a topological space in the \(n\)-dimensional real vector spacehttps://zbmath.org/1517.530142023-09-22T14:21:46.120933Z"Khadjiev, Djavvat"https://zbmath.org/authors/?q=ai:khadzhiev.dzhavvat"Ayupov, Shavkat"https://zbmath.org/authors/?q=ai:ayupov.sh-a"Ören, İdris"https://zbmath.org/authors/?q=ai:oren.idris(no abstract)Generalized elliptical quaternions with some applicationshttps://zbmath.org/1517.530162023-09-22T14:21:46.120933Z"Çolakoğlu, Harun Bariş"https://zbmath.org/authors/?q=ai:colakoglu.harun-baris"Özdemir, Mustafa"https://zbmath.org/authors/?q=ai:ozdemir.mustafa-kemalGiven an ellipsoid \(E\) in \(\mathbb{R}^3\), the authors say that an affine transformation \(T:\mathbb{R}^3\to \mathbb{R}^3\) is an elliptical rotation if \(T(E)=E\). A \((3\times 3)\)-matrix \(M\) is called the elliptical rotation matrix of \(T\) if there is a vector \(c\) such that \(Tx=Mx+c\) holds true for all \(x\). They propose an algorithm for the computation of the rotation matrix of an elliptical rotation that maps a point on the ellipsoid \(a_1x^2+a_2y^2+a_3z^2=1\) to another point on the same ellipsoid.
Reviewer: Victor Alexandrov (Novosibirsk)Maximality of Laplacian algebras, with applications to invariant theoryhttps://zbmath.org/1517.530182023-09-22T14:21:46.120933Z"Mendes, Ricardo A. E."https://zbmath.org/authors/?q=ai:mendes.ricardo-a-e"Radeschi, Marco"https://zbmath.org/authors/?q=ai:radeschi.marcoThe authors extend their previous work [Transform. Groups 25, No. 1, 251--277 (2020; Zbl 1445.53018)] on Laplacian algebras and prove their conjecture stating that being Laplacian implies being maximal. It should be noted that this work is done for finite-dimensional real inner product spaces on which a compact group \(G\) acts by orthogonal transformations.
Laplacian algebras are defined as the subalgebras \(A\subset \mathbb{R}[V]\) which are preserved under the Laplacian operator \(\Delta = \sum_i \partial/\partial x_i\) and also contain the distance squared polynomial \(r^2=\sum_i x_i^2,\) where \(\mathbb{R}[V] \equiv\mathbb{R}[x_1,\dotsc,x_n],\) for \(n=\dim V.\)
Many implications of the obtained results are given further as applications. These applications include inverse invariant theory, separating sets, polarizations (for vector invariants) and first fundamental theorems with some examples.
The authors also introduce \textit{generalized polarizations} and \(k\) normal spaces, also providing sufficient criteria to obtain the invariant ring for vector invariants. In this part of the work, they use infinitesimal manifold submetry technique developed in their previous work [loc. cit.].
Reviewer: Ugur Madran (Eqaila)Twisted Blanchfield pairings and twisted signatures. I: Algebraic backgroundhttps://zbmath.org/1517.570042023-09-22T14:21:46.120933Z"Borodzik, Maciej"https://zbmath.org/authors/?q=ai:borodzik.maciej"Conway, Anthony"https://zbmath.org/authors/?q=ai:conway.anthony"Politarczyk, Wojciech"https://zbmath.org/authors/?q=ai:politarczyk.wojciechIn this article, the authors lay the algebraic foundations for forthcoming papers devoted to the study of twisted linking forms of knots and three-manifolds. Their goal is to describe how to define and calculate signature invariants associated to a linking form \(M \times M \rightarrow \mathbb{F} (t)/ \mathbb{F} [t^{\pm1}]\) for \(\mathbb{F} = \mathbb{R}, \mathbb{C}\), where \(M\) is a torsion \(\mathbb{F} [t^{\pm1}]\)-module.
They begin by stating and proving a classification result for \(\mathbb{F} [t ^{\pm1}]\)-linking forms. Then they apply these methods to linking forms over local rings and study their representability and their classification up to isometry and Witt equivalence. Finally they define signature jumps and the signature function of a \(\mathbb{F} [t ^{\pm1}]\)-linking form.
Reviewer: Leila Ben Abdelghani (Monastir)Universal scaling limits of the symplectic elliptic Ginibre ensemblehttps://zbmath.org/1517.600082023-09-22T14:21:46.120933Z"Byun, Sung-Soo"https://zbmath.org/authors/?q=ai:byun.sung-soo"Ebke, Markus"https://zbmath.org/authors/?q=ai:ebke.markusSummary: We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behavior of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel-Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.Asymptotic behavior of the prediction error for stationary sequenceshttps://zbmath.org/1517.600402023-09-22T14:21:46.120933Z"Babayan, Nikolay M."https://zbmath.org/authors/?q=ai:babayan.nikolay-m"Ginovyan, Mamikon S."https://zbmath.org/authors/?q=ai:ginovyan.mamikon-sSummary: One of the main problem in prediction theory of discrete-time second-order stationary processes \(X(t)\) is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting \(X(0)\) given \(X(t), - n \leq t \leq - 1\), as \(n\) goes to infinity. This behavior depends on the regularity (deterministic or nondeterministic) and on the dependence structure of the underlying observed process \(X(t)\). In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case -- deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process \(X(t)\) and the differential properties of its spectral density \(f\), while for deterministic processes it is determined by the geometric properties of the spectrum of \(X(t)\) and singularities of its spectral density \(f\).An urn model for the Jacobi-Piñeiro polynomialshttps://zbmath.org/1517.600842023-09-22T14:21:46.120933Z"Grünbaum, F. Alberto"https://zbmath.org/authors/?q=ai:grunbaum.francisco-alberto"de la Iglesia, Manuel D."https://zbmath.org/authors/?q=ai:dominguez-de-la-iglesia.manuelSummary: The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix.A tensor-EM method for large-scale latent class analysis with binary responseshttps://zbmath.org/1517.620982023-09-22T14:21:46.120933Z"Zeng, Zhenghao"https://zbmath.org/authors/?q=ai:zeng.zhenghao"Gu, Yuqi"https://zbmath.org/authors/?q=ai:gu.yuqi"Xu, Gongjun"https://zbmath.org/authors/?q=ai:xu.gongjunSummary: Latent class models are powerful statistical modeling tools widely used in psychological, behavioral, and social sciences. In the modern era of data science, researchers often have access to response data collected from large-scale surveys or assessments, featuring many items (large \(J)\) and many subjects (large \(N)\). This is in contrary to the traditional regime with fixed \(J\) and large \(N\). To analyze such large-scale data, it is important to develop methods that are both computationally efficient and theoretically valid. In terms of computation, the conventional EM algorithm for latent class models tends to have a slow algorithmic convergence rate for large-scale data and may converge to some local optima instead of the maximum likelihood estimator (MLE). Motivated by this, we introduce the tensor decomposition perspective into latent class analysis with binary responses. Methodologically, we propose to use a moment-based tensor power method in the first step and then use the obtained estimates as initialization for the EM algorithm in the second step. Theoretically, we establish the clustering consistency of the MLE in assigning subjects into latent classes when \(N\) and \(J\) both go to infinity. Simulation studies suggest that the proposed tensor-EM pipeline enjoys both good accuracy and computational efficiency for large-scale data with binary responses. We also apply the proposed method to an educational assessment dataset as an illustration.Two-parameter double-step scale splitting real-valued iterative method for solving complex symmetric linear systemshttps://zbmath.org/1517.650242023-09-22T14:21:46.120933Z"Xie, Xiaofeng"https://zbmath.org/authors/?q=ai:xie.xiaofeng"Huang, Zhengge"https://zbmath.org/authors/?q=ai:huang.zhengge"Cui, Jingjing"https://zbmath.org/authors/?q=ai:cui.jingjing"Li, Beibei"https://zbmath.org/authors/?q=ai:li.beibeiSummary: In this work, we first apply the parameter accelerating strategy to the double-step scale splitting (DSS) real-valued algorithm one derived by \textit{J. Zhang} et al. [Appl. Math. Comput. 353, 338--346 (2019; Zbl 1429.65073)] and establish a two-parameter DSS real-valued (TDSS) iterative method for solving the large sparse complex symmetric linear systems. Furthermore, we explore the convergence conditions of this method. Moreover, we derive the optimal parameters which minimize an upper bound of the spectral radius of the iteration matrix for the TDSS iterative method. In addition, by utilizing the latest information of the TDSS iterative scheme, we deduce an improved TDSS (ITDSS) iterative method and analyze its convergence properties. At last, the correctness of the theories and the advantages of the ITDSS iterative method over some existing ones are verified by three numerical experiments.An extended Ulm-like method for inverse singular value problems with multiple and/or zero singular valueshttps://zbmath.org/1517.650272023-09-22T14:21:46.120933Z"Wang, Jinhua"https://zbmath.org/authors/?q=ai:wang.jinhua.1"Shen, Weiping"https://zbmath.org/authors/?q=ai:shen.weiping"Li, Chong"https://zbmath.org/authors/?q=ai:li.chong.2"Jin, Xiaoqing"https://zbmath.org/authors/?q=ai:jin.xiaoqingSummary: In this paper, we propose an extended Ulm-like method for solving the inverse singular value problem (ISVP for short) with multiple and/or zero singular values. Compared with the Ulm-like method, the proposed method reduces the amount of calculations and is well-defined even when multiple and/or zero singular values appear. Under the nonsingularity assumption for the relative generalized Jacobian at a solution and by using a new technique, a convergence analysis is provided and the quadratic convergence is proved. Our results in the present paper improve and extend significantly the corresponding ones of \textit{S.-W. Vong} et al. [SIAM J. Matrix Anal. Appl. 32, No. 2, 412--429 (2011; Zbl 1232.65063)] and \textit{W. Shen} et al. [Numer. Algorithms 79, No. 2, 375--398 (2018; Zbl 1397.65059)] for the ISVP with distinct and positive singular values and/or with square matrices. In particular, we solve the interesting problem raised in [Vong et al., loc. cit.]: whether the Ulm-like method and its convergence result can be extended to the cases of multiple singular values and of zero singular values. Numerical results reveal that the extended Ulm-like method is an efficient algorithm for the ISVPs even when multiple and/or zero singular values appear.The eigenspace spectral regularization method for solving discrete ill-posed systemshttps://zbmath.org/1517.650292023-09-22T14:21:46.120933Z"Wireko, Fredrick Asenso"https://zbmath.org/authors/?q=ai:wireko.fredrick-asenso"Barnes, Benedict"https://zbmath.org/authors/?q=ai:barnes.benedict"Sebil, Charles"https://zbmath.org/authors/?q=ai:sebil.charles"Ackora-Prah, Joseph"https://zbmath.org/authors/?q=ai:ackora-prah.josephSummary: This paper shows that discrete linear equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, TST matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. Gauss least square method (GLSM), QR factorization method (QRFM), Cholesky decomposition method (CDM), and singular value decomposition (SVDM) failed to regularize these ill-posed problems. This paper introduces the eigenspace spectral regularization method (ESRM), which solves ill-posed discrete equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, and banded and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularizes such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator \(\kappa(K) = \|K^{-1}K\| = 1\). Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.Higher-order QR with tournament pivoting for tensor compressionhttps://zbmath.org/1517.650302023-09-22T14:21:46.120933Z"Beaupère, Matthias"https://zbmath.org/authors/?q=ai:beaupere.matthias"Frenkiel, David"https://zbmath.org/authors/?q=ai:frenkiel.david"Grigori, Laura"https://zbmath.org/authors/?q=ai:grigori.lauraSummary: We present in this paper a parallel algorithm that generates a low-rank approximation of a distributed tensor using QR decomposition with tournament pivoting (QRTP). The algorithm, which is a parallel variant of the higher-order singular value decomposition, generates factor matrices for a Tucker decomposition by applying QRTP to the unfolding matrices of a tensor distributed blockwise (by subtensor) on a set of processors. For each unfolding mode the algorithm logically reorganizes (unfolds) the processors so that the associated unfolding matrix has a suitable logical distribution. We also establish error bounds between a tensor and the compressed version of the tensor generated by the algorithm.A convergence analysis for an algorithm computing a symmetric low rank orthogonal approximation of a symmetric tensorhttps://zbmath.org/1517.650312023-09-22T14:21:46.120933Z"Du, Wenxin"https://zbmath.org/authors/?q=ai:du.wenxin"Hu, Shenglong"https://zbmath.org/authors/?q=ai:hu.shenglongSummary: We present an algorithm for solving the problem of the symmetric low rank orthogonal tensor approximation for a given symmetric tensor. Proximality technique and shifted power technique are tailored into this algorithm. Interestingly, we can show that this algorithm converges globally without any assumption once the parameters are chosen appropriately, and moreover the convergence rate is sublinear with an explicitly given rate and it is better than the usual \(O(\frac1p)\) of first order methods in optimization.Parallel algorithms for computing the tensor-train decompositionhttps://zbmath.org/1517.650322023-09-22T14:21:46.120933Z"Shi, Tianyi"https://zbmath.org/authors/?q=ai:shi.tianyi"Ruth, Maximilian"https://zbmath.org/authors/?q=ai:ruth.maximilian"Townsend, Alex"https://zbmath.org/authors/?q=ai:townsend.alexSummary: The tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we propose four parallelizable algorithms that compute the TT format from various tensor inputs: (1) Parallel-TTSVD for traditional format, (2) PSTT and its variants for streaming data, (3) Tucker2TT for Tucker format, and (4) TT-fADI for solutions of Sylvester tensor equations. We provide theoretical guarantees of accuracy, parallelization methods, scaling analysis, and numerical results. For example, for a \(d\)-dimension tensor in \(\mathbb{R}^{n\times\dots\times n}\), a two-sided sketching algorithm PSTT2 is shown to have a memory complexity of \(\mathcal{O}(n^{\lfloor d/2\rfloor})\), improving upon \(\mathcal{O}(n^{d-1})\) from previous algorithms.Tensor regression networkshttps://zbmath.org/1517.683352023-09-22T14:21:46.120933Z"Kossaifi, Jean"https://zbmath.org/authors/?q=ai:kossaifi.jean"Lipton, Zachary C."https://zbmath.org/authors/?q=ai:lipton.zachary-c"Kolbeinsson, Arinbjorn"https://zbmath.org/authors/?q=ai:kolbeinsson.arinbjorn"Khanna, Aran"https://zbmath.org/authors/?q=ai:khanna.aran"Furlanello, Tommaso"https://zbmath.org/authors/?q=ai:furlanello.tommaso"Anandkumar, Anima"https://zbmath.org/authors/?q=ai:anandkumar.animaSummary: Convolutional neural networks typically consist of many convolutional layers followed by one or more fully connected layers. While convolutional layers map between high-order activation tensors, the fully connected layers operate on flattened activation vectors. Despite empirical success, this approach has notable drawbacks. Flattening followed by fully connected layers discards multilinear structure in the activations and requires many parameters. We address these problems by incorporating tensor algebraic operations that preserve multilinear structure at every layer. First, we introduce Tensor Contraction Layers (TCLs) that reduce the dimensionality of their input while preserving their multilinear structure using tensor contraction. Next, we introduce Tensor Regression Layers (TRLs), which express outputs through a low-rank multilinear mapping from a high-order activation tensor to an output tensor of arbitrary order. We learn the contraction and regression factors end-to-end, and produce accurate nets with fewer parameters. Additionally, our layers regularize networks by imposing low-rank constraints on the activations (TCL) and regression weights (TRL). Experiments on ImageNet show that, applied to VGG and ResNet architectures, TCLs and TRLs reduce the number of parameters compared to fully connected layers by more than 65\% while maintaining or increasing accuracy. In addition to the space savings, our approach's ability to leverage topological structure can be crucial for structured data such as MRI. In particular, we demonstrate significant performance improvements over comparable architectures on three tasks associated with the UK Biobank dataset.Spinor representations of positional adapted frame in the Euclidean 3-spacehttps://zbmath.org/1517.700022023-09-22T14:21:46.120933Z"İşbilir, Zehra"https://zbmath.org/authors/?q=ai:isbilir.zehra"Özen, Kahraman Esen"https://zbmath.org/authors/?q=ai:ozen.kahraman-esen"Güner, Mehmet"https://zbmath.org/authors/?q=ai:guner.mehmet(no abstract)Structured derivation of moment equations and stable boundary conditions with an introduction to symmetric, trace-free tensorshttps://zbmath.org/1517.760572023-09-22T14:21:46.120933Z"Bünger, Jonas"https://zbmath.org/authors/?q=ai:bunger.jonas"Christhuraj, Edilbert"https://zbmath.org/authors/?q=ai:christhuraj.edilbert"Hanke, Andrea"https://zbmath.org/authors/?q=ai:hanke.andrea"Torrilhon, Manuel"https://zbmath.org/authors/?q=ai:torrilhon.manuelSummary: Moment equations are used to approximate kinetic equations in a physically meaningful and numerically efficient way. This paper discusses the derivation of general moment equations including various caveats that must be taken care of when moment systems get implemented in simulation tools. We suggest to use a constant Maxwellian with arbitrary constant reference values for density, velocity and temperature for the underlying expansion of the distribution. This seems restrictive in comparison to using a local Maxwellian, but full nonlinear fluid dynamics remains possible as long as the reference values do not differ too much from the local values and a sufficiently large moment vector is considered. Additionally, this approach allows to derive globally hyperbolic equations in a simple way and the transport operator becomes linear. The resulting equations are discussed in detail and compared to various existing approaches.
The paper also presents boundary conditions that render the moment systems stable in an \(L^2\)-sense. This is done both by general considerations and for the concrete case of the Maxwell accommodation model for kinetic equations, generalizing earlier stable boundary conditions for special cases of moment equations. The extensive appendix of the paper will help readers unfamiliar with tensor variables and symmetric, trace-free decompositions to understand the general concept needed for moment approximations without the need of group-theoretical expertise.A Landau-Ginzburg mirror theorem via matrix factorizationshttps://zbmath.org/1517.810642023-09-22T14:21:46.120933Z"He, Weiqiang"https://zbmath.org/authors/?q=ai:he.weiqiang"Polishchuk, Alexander"https://zbmath.org/authors/?q=ai:polishchuk.alexander-e"Shen, Yefeng"https://zbmath.org/authors/?q=ai:shen.yefeng"Vaintrob, Arkady"https://zbmath.org/authors/?q=ai:vaintrob.arkadySummary: For an invertible quasihomogeneous polynomial \(\boldsymbol{w}\) we prove an all-genus mirror theorem relating two cohomological field theories of Landau-Ginzburg type. On the \(B\)-side it is the Saito-Givental theory for a specific choice of a primitive form. On the \(A\)-side, it is the matrix factorization CohFT for the dual singularity \(\boldsymbol{w}^T\) with the maximal diagonal symmetry group.All identically conserved gravitational tensors are metric variations of invariant actionshttps://zbmath.org/1517.830162023-09-22T14:21:46.120933Z"Deser, S."https://zbmath.org/authors/?q=ai:deser.stanleySummary: I prove an old unsolved conjecture, the hard -- necessary -- part of its obvious sufficiency, namely that all identically conserved symmetric 2-ortensors are necessarily metric variations of invariant actions, thus sparing the wld more alternative gravity theories. The proof is reasonably simple, if perhaps a ``physicist's''.The spin-one DKP oscillator in the plane with an external magnetic fieldhttps://zbmath.org/1517.830742023-09-22T14:21:46.120933Z"Chargui, Yassine"https://zbmath.org/authors/?q=ai:chargui.yassine"Dhahbi, Anis"https://zbmath.org/authors/?q=ai:dhahbi.anis-benSummary: We successfully obtained the complete exact solution of the Duffin-Kemmer-Petiau oscillator (DKPO) for spin-1 bosons in \((1 + 2)\)-dimensional space time using a \(10 \times 10\) representation of the beta-matrices. We show that the problem can be decoupled into three independent subproblems, each of which is assigned with one of the spin-projection numbers \(m_s = 0\) and \(m_s = \pm 1\). Moreover, the solutions such that \(m_s = 0\) and those for which \(m_s = \pm 1\) are found to be respectively associated with \((-1)^J\) and \((-1)^{J + 1}\) parities. In addition, we investigate the effect of the presence of an external transverse homogeneous magnetic field on the dynamics of the oscillator: different settings, as well as interesting particular cases, are considered and the non-relativistic limit of the model is also discussed.The iterative solution of a class of tensor equations via Einstein product with a tensor inequality constrainthttps://zbmath.org/1517.901442023-09-22T14:21:46.120933Z"Huang, Baohua"https://zbmath.org/authors/?q=ai:huang.baohua"Ma, Changfeng"https://zbmath.org/authors/?q=ai:ma.changfengSummary: In this paper, we propose a feasible and effective iteration method for solving the tensor equation \(\mathcal{A} *_N \mathcal{X} *_M \mathcal{B} = \mathcal{C}\) with a tensor inequality constraint \(\mathcal{D} *_N \mathcal{X} *_M \mathcal{E} \geq \mathcal{F}\). The global convergence is established. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed algorithm.A geometric algebra approach to invariance control in sliding regimes for switched systemshttps://zbmath.org/1517.930152023-09-22T14:21:46.120933Z"Sira-Ramírez, H."https://zbmath.org/authors/?q=ai:sira-ramirez.hebertt-j"Gómez-León, B. C."https://zbmath.org/authors/?q=ai:gomez-leon.brian-c"Aguilar-Orduña, M. A."https://zbmath.org/authors/?q=ai:aguilar-orduna.mario-andresSliding mode control systems, affine in the binary-valued control law, are considered. Known results are presented by using in a systematic way a geometric algebra framework, as follows: necessary and sufficient conditions for the existence of a sliding regime, a characterization of the equivalent control, and the invariance control approach. Within this framework it is shown that the twisting and super twisting algorithms may be obtained as a limiting case of first order regimes. Examples are discussed in detail.
Reviewer: Tullio Zolezzi (Genova)A new image restoration model associated with special elliptic quaternionic least-squares solutions based on LabVIEWhttps://zbmath.org/1517.940172023-09-22T14:21:46.120933Z"Atali, Gokhan"https://zbmath.org/authors/?q=ai:atali.gokhan"Kosal, Hidayet Huda"https://zbmath.org/authors/?q=ai:kosal.hidayet-huda"Pekyaman, Muge"https://zbmath.org/authors/?q=ai:pekyaman.mugeSummary: In this paper, we take advantage of the elliptic complex matrix representation of elliptic quaternion matrices. Then we obtain the methods of the elliptic quaternionic least-squares solution, the pure elliptic quaternionic least-squares solution, and the real least-squares solution with the least norm of the elliptic quaternion matrix equation \(A X = B\). We also apply the newly obtained method of the pure elliptic quaternionic least-squares solution with the least norm to the color image restoration based on the LabVIEW program. In this context, we propose a new image restoration model called ``ELSI image restoration model'' associated with elliptic quaternionic least-squares solutions.