Recent zbMATH articles in MSC 15https://zbmath.org/atom/cc/152022-11-17T18:59:28.764376ZWerkzeugA fourfold refined enumeration of alternating sign trapezoidshttps://zbmath.org/1496.050052022-11-17T18:59:28.764376Z"Höngesberg, Hans"https://zbmath.org/authors/?q=ai:hongesberg.hansSummary: Alternating sign trapezoids have recently been introduced as a generalisation of alternating sign triangles. \textit{I. Fischer} [J. Comb. Theory, Ser. A 158, 560--604 (2018; Zbl 1391.05040)] established a threefold refined enumeration of alternating sign trapezoids and provided three statistics on column strict shifted plane partitions with the same joint distribution. In this paper, we are able to add a new pair of statistics to these results. More precisely, we consider the number of \(-1s\) on alternating sign trapezoids and introduce a corresponding statistic on column strict shifted plane partitions that has the same distribution. More generally, we show that the joint distributions of the two quadruples of statistics on alternating sign trapezoids and column strict shifted plane partitions, respectively, coincide. In addition, we provide a closed-form expression for the \(2\)-enumeration of alternating sign trapezoids.Spectral symmetry in conference matriceshttps://zbmath.org/1496.050142022-11-17T18:59:28.764376Z"Haemers, Willem H."https://zbmath.org/authors/?q=ai:haemers.willem-h"Majd, Leila Parsaei"https://zbmath.org/authors/?q=ai:majd.leila-parsaeiSummary: A conference matrix of order \(n\) is an \(n\times n\) matrix \(C\) with diagonal entries 0 and off-diagonal entries \(\pm 1\) satisfying \(CC^\top =(n-1)I\). If \(C\) is symmetric, then \(C\) has a symmetric spectrum \(\Sigma\) (that is, \(\Sigma =-\Sigma)\) and eigenvalues \(\pm \sqrt{n-1}\). We show that many principal submatrices of \(C\) also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.Combinatorial invariants for nets of conics in \(\mathrm{PG}(2,q)\)https://zbmath.org/1496.050172022-11-17T18:59:28.764376Z"Lavrauw, Michel"https://zbmath.org/authors/?q=ai:lavrauw.michel"Popiel, Tomasz"https://zbmath.org/authors/?q=ai:popiel.tomasz"Sheekey, John"https://zbmath.org/authors/?q=ai:sheekey.johnSummary: The problem of classifying linear systems of conics in projective planes dates back at least to \textit{C. Jordan}, who classified pencils (one-dimensional systems) of conics over \({\mathbb{C}}\) and \(\mathbb{R}\) in [Journ. de Math. (6) 2, 403--438 (1906; JFM 37.0136.01); ibid. (6) 3, 5--51 (1907; JFM 38.0151.01)]. The analogous problem for finite fields \(\mathbb{F}_q\) with \(q\) odd was solved by \textit{L. E. Dickson} [Quart. J. 39, 316--333 (1907; JFM 39.0148.01)]. \textit{A. H. Wilson} [Am. J. Math. 36, 187--210 (1914; JFM 45.0228.01)] attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson's classification (for \(q\) odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in \(\mathrm{PG}(5,q), q\) odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of \(\mathrm{PGL}(3,q)\) viewed as a subgroup of \(\mathrm{PGL}(6,q)\). Since \(q\) is odd, the orbits of lines in \(\mathrm{PG}(5,q)\) under this action correspond to the aforementioned pencils of conics in \(\mathrm{PG}(2,q)\). The main contribution of this paper is to determine the line-orbit distribution of a plane \(\pi\) corresponding to a net of rank one, namely, the number of lines in \(\pi\) belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of \(\pi\), and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in \(\mathrm{PGL}(3,q)\), and hence the orbit sizes.Ordering the maxima of \(L\)-index and \(Q\)-index: graphs with given size and diameterhttps://zbmath.org/1496.050292022-11-17T18:59:28.764376Z"Jia, Huiming"https://zbmath.org/authors/?q=ai:jia.huiming"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Wang, Shujing"https://zbmath.org/authors/?q=ai:wang.shujingSummary: The \(L\)-index (resp. \(Q\)-index) of a graph \(G\) is the largest eigenvalue of the Laplacian matrix (resp. signless Laplacian matrix) of \(G\). Very recently, \textit{Z. Lou} et al. [Discrete Math. 344, No. 10, Article ID 112533, 9 p. (2021; Zbl 1469.05108)] determined the graph with fixed size and diameter having the maximum \(Q\)-index (resp. \(L\)-index). As a continuance of their result, in this paper we order all the graphs with given size and diameter from the second to the \((\lfloor \frac{d}{2} \rfloor + 1)\)th via their \(Q\)-indices. Consequently, we identify all the graphs of given size and diameter from the second to the \(\lfloor \frac{d}{2} \rfloor\)th via their \(L\)-indices. Furthermore, the graph of given size and diameter with at least one cycle having the largest \(Q\)-index is also characterized.Subsets of Cayley graphs that induce many edgeshttps://zbmath.org/1496.050732022-11-17T18:59:28.764376Z"Gowers, Timothy"https://zbmath.org/authors/?q=ai:gowers.william-timothy"Janzer, Oliver"https://zbmath.org/authors/?q=ai:janzer.oliverSummary: Let \(G\) be a regular graph of degree \(d\) and let \(A\subset V(G)\). Say that \(A\) is \(\eta \)-closed if the average degree of the subgraph induced by \(A\) is at least \(\eta d\). This says that if we choose a random vertex \(x\in A\) and a random neighbour \(y\) of \(x\), then the probability that \(y\in A\) is at least \(\eta \). This paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph \(\Gamma\) whose vertex set is \(\mathbb{F}_2^{n_1}\otimes \dots \otimes \mathbb{F}_2^{n_d}\) with two vertices joined by an edge if their difference is of the form \(u_1\otimes \cdots \otimes u_d\). For the matrix case (that is, when \(d=2)\), such a description was obtained by \textit{K. Subhash} et al. [``Pseudorandom sets in Grassmann graph have near-perfect expansion'', in: Proceedings of the 59th annual IEEE symposium on foundations of computer science, FOCS 2018, Paris, France, October 7--9, 2018. Los Alamitos, CA: IEEE Computer Society. 592--601 (2018; \url{doi:10.1109/FOCS.2018.00062})], a breakthrough that completed the proof of the 2-to-2 conjecture. In this paper, we formulate a conjecture for higher dimensions, and prove it in an important special case. Also, we identify a statement about \(\eta \)-closed sets in Cayley graphs on arbitrary finite abelian groups that implies the conjecture and can be considered as a ``highly asymmetric Balog-Szemerédi-Gowers theorem'' when it holds. We conclude the paper by showing that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph \(\Gamma \).\(A_\alpha\) and \(L_\alpha\)-spectral properties of spider graphshttps://zbmath.org/1496.050952022-11-17T18:59:28.764376Z"Brondani, Andre E."https://zbmath.org/authors/?q=ai:brondani.andre-e"França, Francisca Andrea Macedo"https://zbmath.org/authors/?q=ai:macedo-franca.francisca-andrea"Oliveira, Carla S."https://zbmath.org/authors/?q=ai:oliveira.carla-sSummary: Let \(G\) be a graph with adjacency matrix \(A(G)\) and let \(D(G)\) be the diagonal matrix of the degrees of \(G\). For every real \(\alpha \in [0, 1]\), \textit{V. Nikiforov} [Appl. Anal. Discrete Math. 11, No. 1, 81--107 (2017; Zbl 07481047)] and \textit{S. Wang} et al. [Linear Algebra Appl. 590, 210--223 (2020; Zbl 1437.05151)] defined the matrices \(A_\alpha(G)\) and \(L_\alpha(G)\), respectively, as \(A_\alpha(G) = \alpha D(G)+(1- \alpha) A(G)\) and \(L_\alpha(G) = \alpha D(G)+(\alpha - 1) A(G)\). In this paper, we obtain some relationships between the eigenvalues of these matrices for some families of graphs, a part of the \(A_\alpha\) and \(L_\alpha\)-spectrum of the spider graphs, and we display the \(A_\alpha\) and \(L_\alpha\)-characteristic polynomials when their set of vertices can be partitioned into subsets that induce regular subgraphs. Moreover, we determine some subfamilies of spider graphs that are cospectral with respect to these matrices.Quantum walks do not like bridgeshttps://zbmath.org/1496.050972022-11-17T18:59:28.764376Z"Coutinho, Gabriel"https://zbmath.org/authors/?q=ai:coutinho.gabriel"Godsil, Chris"https://zbmath.org/authors/?q=ai:godsil.christopher-david"Juliano, Emanuel"https://zbmath.org/authors/?q=ai:juliano.emanuel"van Bommel, Christopher M."https://zbmath.org/authors/?q=ai:van-bommel.christopher-mSummary: We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing theorem, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.On eigenvalues and energy of geometric-arithmetic matrix of graphshttps://zbmath.org/1496.050992022-11-17T18:59:28.764376Z"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddin"Rather, Bilal A."https://zbmath.org/authors/?q=ai:rather.bilal-a"Aouchiche, M."https://zbmath.org/authors/?q=ai:aouchiche.mustaphaThis manuscript deals with the calculus of the spectrum and the energy of the geometric-arithmetic matrix associated to a graph \(G\). \par A simple and undirected graph is denoted by \(G(V,E)\), where \(V=\{ v_1, v_2, \ldots, v_n \}\) and \(E\) are the vertex set and the edge set, respectively. If a vertex \(u\) is adjacent to a vertex \(v\), it is denoted \(u \sim v\). The degree of a vertex \(v\), denoted by \(d_G(v)\) or \(d_v\), is the cardinality of the set of vertices adjacent to \(v\), \(N(v)\). A graph is regular if all vertices have the same degree. \par The geometric-arithmetic matrix \(A=\mathcal{GA}(G)\) of \(G\) is a square matrix of size \(n \times n\), indexed by the vertices of \(G\) and defined as
\[
(\mathcal{GA}(G))_{u,v}=\left\{ \begin{array} {ll} \dfrac{2\sqrt{d_v d_u}}{d_v+d_u}, & \mbox{if} \ v \sim u, \\
0, & \mbox{otherwise}. \end{array} \right.
\]
In this paper, the authors calculate the geometric-arithmetic spectrum of the join of two regular graphs in terms of the adjacency spectrum of the two given graphs. As a consequence, they obtain the spectrum of some well-known families of graphs, including graphs with edge deletion. \par The energy of a graph \(G\), denoted by \(\varepsilon(G)\), is defined as the absolute sum of the adjacency eigenvalues, that is,
\[
\varepsilon(G)=\sum_{i=1}^n |\lambda_i|.
\]
The geometric-arithmetic energy of graph \(G\) is defined in a similar way. The authors obtain several upper and lower bounds for this geometric-arithmetic energy and characterize the graphs attaining such bounds.
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)On the multiplicity of the least signless Laplacian eigenvalue of a graphhttps://zbmath.org/1496.051022022-11-17T18:59:28.764376Z"Tian, Fenglei"https://zbmath.org/authors/?q=ai:tian.fenglei"Guo, Shu-Guang"https://zbmath.org/authors/?q=ai:guo.shuguang"Wong, Dein"https://zbmath.org/authors/?q=ai:wong.deinGiven an \(n\)-vertex connected simple graph \(G,\) let \(A(G),\, D(G)\) be the adjacency matrix and degree diagonal matrix of \(G\), respectively. Then \(Q(G)=D(G)+A(G)\) is the well-known signless Laplacian matrix of \(G\). The eigenvalues of \(Q(G)\) are called the \(Q\)-eigenvalues of \(G\). In this paper, the authors study the multiplicity of the smallest \(Q\)-eigenvalues of \(G\) and identify all the \(n\)-vertex connected graphs with the smallest \(Q\)-eigenvalues of multiplicity \(n-3\). Furthermore, they show that all these graphs are determined by their signless Laplacian spectra. It is a nice paper.
Reviewer: Shuchao Li (Wuhan)Energy of interval-valued fuzzy graphs and its application in ecological systemshttps://zbmath.org/1496.051502022-11-17T18:59:28.764376Z"Patra, Napur"https://zbmath.org/authors/?q=ai:patra.napur"Mondal, Sanjib"https://zbmath.org/authors/?q=ai:mondal.sanjib"Pal, Madhumangal"https://zbmath.org/authors/?q=ai:pal.madhumangal"Mondal, Sukumar"https://zbmath.org/authors/?q=ai:mondal.sukumar(no abstract)Cassini-like formula for generalized hyper-Fibonacci numbershttps://zbmath.org/1496.110182022-11-17T18:59:28.764376Z"Ait-Amrane, Lyes"https://zbmath.org/authors/?q=ai:ait-amrane.lyes"Behloul, Djilali"https://zbmath.org/authors/?q=ai:behloul.djilaliThe authors of the paper under review introduce \textit{generalized hyper-Fibonacci numbers (GHFN) }and present some of these numbers properties. Then they acquire a \textit{Cassini-like} formula for GHFN. Also, they introduce GHFN with negative subscripts and discuss some of their attributes.
Reviewer: Mohammad K. Azarian (Evansville)Relation between matrices and the suborbital graphs by the special number sequenceshttps://zbmath.org/1496.110192022-11-17T18:59:28.764376Z"Akbaba, Ümmügülsün"https://zbmath.org/authors/?q=ai:akbaba.ummugulsun"Değer, Ali Hikmet"https://zbmath.org/authors/?q=ai:deger.ali-hikmetSummary: Continued fractions and their matrix connections have been used in many studies to generate new identities. On the other hand, many examinations have been made in the suborbital graphs under circuit and forest conditions. Special number sequences and special vertex values of minimal length paths in suborbital graphs have been associated in our previous studies. In these associations, matrix connections of the special continued fractions \(\mathcal{K}(-1/-k)\), where \(k\in\mathbb{Z}^+\), \(k\geq 2\) with the values of the special number sequences are used and new identities are obtained. In this study, by producing new matrices, new identities related to Fibonacci, Lucas, Pell, and Pell-Lucas number sequences are found by using both recurrence relations and matrix connections of the continued fractions. In addition, the farthest vertex values of the minimal length path in the suborbital graph \(\mathbf{F}_{u, N}\) and these number sequences are associated.\(k\)-Fibonacci numbers and \(k\)-Lucas numbers and associated bipartite graphshttps://zbmath.org/1496.110262022-11-17T18:59:28.764376Z"Lee, Gwangyeon"https://zbmath.org/authors/?q=ai:lee.gwangyeonSummary: In [\textit{E. Kilic}, J. Comput. Appl. Math. 209, No. 2, 133--145 (2007; Zbl 1162.11013); \textit{G.-Y. Lee} et al., Discrete Appl. Math. 130, No. 3, 527--534 (2003; Zbl 1020.05016); \textit{G.-Y. Lee} and \textit{S.-G. Lee}, Fibonacci Q. 33, No. 3, 273--278 (1995; Zbl 0834.11009)], several authors studied the generalized Fibonacci numbers. Also, in [Linear Algebra Appl. 320, No. 1--3, 51--61 (2000; Zbl 0960.05079)], the author found a class of bipartite graphs whose number of 1-factors is the \(n\)th \(k\)-Lucas numbers. In this paper, we give a new relationship between \(g_n^{(k)}\) and \(l_n^{(k)}\) and the number of 1-factors of a bipartite graph.On the divisibility among power GCD and power LCM matrices on gcd-closed setshttps://zbmath.org/1496.110472022-11-17T18:59:28.764376Z"Zhu, Guangyan"https://zbmath.org/authors/?q=ai:zhu.guangyanGiven \(n,a\in\mathbb{Z}_+\) and \(S=\{x_1,\dots,x_n\}\subset\mathbb{Z}_+\), consider the \(a\)th power GCD matrix \((S^a)\) (with \((i,j)\)th entry \(\gcd(x_i,x_j)^a\)) and \(a\)th power LCM matrix \([S^a]\) (with \((i,j)\)th entry lcm\((x_i,x_j)^a\)). Let \(n\ge 2\) and \(b\in\mathbb{Z}_+\). \textit{S. F. Hong} [Linear Algebra Appl. 428, 1001--1008 (2008; Zbl 1137.11017)] proved that if \(S\) is a divisor chain (i.e., \(x_{\sigma(1)}\mid\dots\mid x_{\sigma(n)}\) for some permutation \(\sigma\)), \(b\in\mathbb{Z}_+\), and \(a\mid b\), then (a) \((S^a)\mid (S^b)\), (b) \((S^a)\mid [S^b]\), (c) \([S^a]\mid [S^b]\). More generally, define that \(d\in S\) is a greatest-type divisor of \(x\in S\) if \(d<x\) and there is no other \(y\in S\) satisfying \(d\mid y\mid x\). Let \(G_S(x)\) be their set. Hong conjectured that if \(S\) is gcd-closed (i.e., \(x,y\in S\Rightarrow\gcd(x,y)\in S\)) and \(\max_{x\in S}|G_S(x)|=1\) (\(|\cdot|\) denotes cardinality) then (a), (b), and (c) hold. The present author proves this for (a) and (b).
Reviewer: Jorma K. Merikoski (Tampere)A note on the maximal rankhttps://zbmath.org/1496.140532022-11-17T18:59:28.764376Z"Bernardi, Alessandra"https://zbmath.org/authors/?q=ai:bernardi.alessandra"Staffolani, Reynaldo"https://zbmath.org/authors/?q=ai:staffolani.reynaldoLet \(X\subset \mathbb P^N\) be an irreducible and non-degenerate projective variety over an algebraically closed field of characteristic zero. This paper investigates upper bounds for the maximal \(X\)-rank.
Given a point \(P\in \mathbb P^N\), the \(X\)-rank of \(P\), denoted by \(r_X(P)\), is the least number of points of \(X\) whose linear span contains \(P\). It is interesting to investigate the maximum possible value for \(r_X(P)\), as the point \(P\) varies in \(\mathbb P^N\). This maximum is called the maximal \(X\)-rank and is here denoted by \(r_{\max}\). This notion has been studied in different contexts depending on \(X\) and, for instance, is connected with the study of the open Waring rank and the usual Waring rank, (see [\textit{J. Jelisiejew}, Arch. Math. 102, No. 4, 329--336 (2014; Zbl 1322.14079); \textit{E. Ballico} and \textit{A. De Paris}, Discrete Comput. Geom. 57, No. 4, 896--914 (2017; Zbl 1369.15004)] for relevant related results).
For an integer \(s\), let \(\sigma_s(X)\) be the \(s\)-secant variety of \(X\), that is the Zariski closure of the set of points \(P\) of \(\mathbb P^N\) such that \(r_X(P)\leq s\). The projective variety \(\sigma_s(X)\) is irreducible and we can consider the maximal \(\sigma_s(X)\)-rank, which is denoted by \(r_{\max,s}\).
The generic \(X\)-rank is the least integer \(g\) such that \(\sigma_g(X)=\mathbb P^N\).
The main result of this paper is the bound \(r_{\max}\leq r_{\max,g-1}+1\), under the hypothesis that \(X\) is smooth and that \(\sigma_{g-1}(X)\) is a hypersurface, with \(g>2\) (see Theorem 2.4). The proof exploits geometric arguments.
Following some interesting consequences of this result, the authors formulate the conjecture that \(r_{\max}=\max\{ r_{\max,g-1}+g\}\) under the same hypotheses of the main result (see Conjecture 2.7). They also collect some evidences of this conjecture, while they compare their bound with the existing ones in several known cases. One of the given examples shows that the author's bound is sharp.
Reviewer: Francesca Cioffi (Napoli)Equational theories of upper triangular tropical matrix semigroupshttps://zbmath.org/1496.140622022-11-17T18:59:28.764376Z"Han, Bin Bin"https://zbmath.org/authors/?q=ai:han.bin-bin"Zhang, Wen Ting"https://zbmath.org/authors/?q=ai:zhang.wen-ting.1"Luo, Yan Feng"https://zbmath.org/authors/?q=ai:luo.yan-fengTropical mathematics now interact with many fields and applications, from algebraic geometry to optimization, phylogenetics to economics. The very origin of the word ``tropical'' came from semirings and automata theory. A nice historical overview is given by \textit{J.-E. Pean} [Tropical semirings, idempotency. Bristol. 50--69 (1994)]. This paper falls into this line of work.
A simplistic way to view this line of work is to take the tropical semiring as an interesting algebra, and revisit classical questions for classical objects in semigroup theory over this algebra. For concreteness, one could take the min-plus semiring, where minimum replaces addition and addition replaces multiplication. The set of \(n \times n\) matrices equipped with matrix multiplication over the min-plus semiring becomes a semigroup. Now one could ask, for example, does the semigroup of \(n \times n\) tropical matrices have a finite basis? Does this semigroup have interesting identities? One can also consider other semigroups, such as those of upper triangular \(n \times n\) tropical matrices, denoted \(UT_n\). It turned out that \(UT_n\) is particularly interesting, as their identities have strong relations with the bicyclic and plactic monoid, which are other classical obects in semigroup theory. In particular, various researchers have resolved open questions related to the bicyclic and plactic monoid by first translating the questions in terms of \(UT_n\), and then use combinatorial or geometric techniques to answer them. See the citations on page 3, second paragraph, of this paper.
This paper builds upon previous work and resolved the finite basis problem for the semigroups of upper triangular matrices \(UT_n\) for \(n = 2,3\). It also stated interesting open problems for \(n \geq 4\).
Reviewer: Ngoc Mai Tran (Bonn)Bidiagonal triads and the tetrahedron algebrahttps://zbmath.org/1496.150012022-11-17T18:59:28.764376Z"Funk-Neubauer, Darren"https://zbmath.org/authors/?q=ai:funk-neubauer.darrenA bidiagonal triad on a finite-dimensional vector space \(V\) over a field \(\mathbb{F}\) is a triple of diagonalizable linear endomorphisms \(A_i\) of \(V\), \(i=1,2,3\), such that there exists an ordering \(\{V_i^j\}_{j=0}^{d}\) of the eigenspaces of \(A_i\), with corresponding eigenvalues \(\{\theta_i^j\}_{j=0}^d\), such that the following two conditions hold:
\begin{itemize}
\item \(A_i(V_{i'}^j)\subseteq V_{i'}^j+V_{i'}^{j+1}\) for any \(i\neq i'\) and any \(j=0,\ldots,d\), where \(V_i^{d+1}=0\) for any \(i=1,2,3\);
\item The restriction of \([A_i,A_{i'}]^{d-2j}\) to \(V_{i'}^j\) gives a bijection \(V_{i'}^j\mapsto V_{i'}^{d-j}\) for any \(i\neq i'\) and any \(j\).
\end{itemize}
The first main result of the paper asserts that the eigenvalues of a bidiagonal triad satisfy the linear recurrence relations \[\frac{\theta_i^{j+1}-\theta_i^j}{\theta_i^j-\theta_i^{j-1}}=1, \qquad i=1,2,3.\] \par Then a very natural connection between bidiagonal triads and representations of the tetrahedron Lie algebra \(\boxtimes\) (or the three-point \(\mathfrak{sl}_2\) loop algebra), defined by \textit{B. Hartwig} and \textit{P. Terwilliger} [J. Algebra 308, No. 2, 840--863 (2007; Zbl 1163.17026)] is showcased: any finite-dimensional irreducible module \(V\) for \(\boxtimes\) gives rise to several bidiagonal triads on \(V\), obtained by the action of the three generators of \(\boxtimes\) meeting at a corner of the tetrahedron. Moreover, any `reduced thin' bidiagonal triad on a vector space \(V\) may be extended to a representation of the tetrahedron algebra.
Reviewer: Alberto Elduque (Zaragoza)Methods of Gauss-Jordan elimination to compute core inverse \(A^{\text{\textcircled{\#}}}\) and dual core inverse \(A_{\text{\textcircled{\#}}}\)https://zbmath.org/1496.150022022-11-17T18:59:28.764376Z"Sheng, Xingping"https://zbmath.org/authors/?q=ai:sheng.xingping"Xin, Dawei"https://zbmath.org/authors/?q=ai:xin.daweiLet \(A\) be a complex square matrix such that \(R(A)= R(A^2),\) where \(R(.)\) stands for the range space. A square matrix \(X\) of the same order as \(A\) is called the \textit{core inverse} of \(A\) if \(X\) satisfies the conditions \(AX=AA^{\dagger}\) and \(R(X) \subseteq R(A).\) Here, \(A^{\dagger}\) is the Moore-Penrose inverse of \(A\). From the dual point of view, a matrix \(Y\) of the same order as \(A\) is called the \textit{dual core inverse} of \(A\) if \(YA=A^{\dagger}A\) and \(R(Y) \subseteq R(A^*).\) Two formulae for the core inverse, obtained recently in [\textit{H. Wang} and \textit{X. Liu}, Linear Multilinear Algebra 63, No. 9, 1829--1836 (2015; Zbl 1325.15002); \textit{H. Ma} and \textit{T. Li}, Linear Multilinear Algebra 69, No. 1, 93--103 (2021; Zbl 1460.15006)] are simplified. Employing the Gauss-Jordan elimination, two numerical procedures for computing the core inverse and the dual core inverse are presented and their computational complexities are discussed.
Reviewer: K. C. Sivakumar (Chennai)Permanental representations of negatively subscripted generalized order-\(k\) Fibonacci numbershttps://zbmath.org/1496.150032022-11-17T18:59:28.764376Z"Belbachir, Hacène"https://zbmath.org/authors/?q=ai:belbachir.hacene"Djellas, Ihab-Eddine"https://zbmath.org/authors/?q=ai:djellas.ihab-eddine"Krim, Fariza"https://zbmath.org/authors/?q=ai:krim.farizaThis long article considers a standard generalization of Fibonacci numbers. Here, negative indices are analyzed. The results are trivial and supported with straightforward proofs.
Reviewer: Carlos M. da Fonseca (Safat)Inequations for permanents of matrices over distributive pseudo-latticeshttps://zbmath.org/1496.150042022-11-17T18:59:28.764376Z"Zhang, Guo Yong"https://zbmath.org/authors/?q=ai:zhang.guoyong(no abstract)Extending Putzer's representation to all analytic matrix functions via omega matrix calculushttps://zbmath.org/1496.150052022-11-17T18:59:28.764376Z"Neto, Antônio Francisco"https://zbmath.org/authors/?q=ai:neto.antonio-franciscoSummary: We show that Putzer's method to calculate the matrix exponential in [\textit{E. J. Putzer}, Am. Math. Mon. 73, 2--7 (1966; Zbl 0135.29801)] can be generalized to compute an arbitrary matrix function defined by a convergent power series. The main technical tool for adapting Putzer's formulation to the general setting is the omega matrix calculus; that is, an extension of MacMahon's partition analysis to the realm of matrix calculus and the method in [\textit{R. Ben Taher} and \textit{M. Rachidi}, Linear Algebra Appl. 330, No. 1--3, 15--24 (2001; Zbl 0983.15018)]. Several results in the literature are shown to be special cases of our general formalism, including the computation of the fractional matrix exponentials introduced by \textit{M. R. Rodrigo} [J. Differ. Equations 261, No. 7, 4223--4243 (2016; Zbl 1347.15013)]. Our formulation is a much more general, direct, and conceptually simple method for computing analytic matrix functions. In our approach the recursive system of equations the base for Putzer's method is explicitly solved, and all we need to determine is the analytic matrix functions.Locating eigenvalues of quadratic matrix polynomialshttps://zbmath.org/1496.150062022-11-17T18:59:28.764376Z"Roy, Nandita"https://zbmath.org/authors/?q=ai:roy.nandita"Bora, Shreemayee"https://zbmath.org/authors/?q=ai:bora.shreemayeeIn this very nice paper, the authors present some new methods to localize eigenvalues of quadratic matrix polynomials. The quadratic eigenvalue problem is one of the most important eigenvalue problems in applications (e.g., in mechanics). Locating the eigenvalues with respect to the imaginary axis, the unit circle or the real line can give useful information concerning the stability of the system. The main idea of the paper is to apply some block versions of the Gershgorin circle theorem to some classic linearization of the quadratic matrix polynomial. With this method, the authors obtain some simple sufficient conditions of the coefficient matrices of the polynomial so that its eigenvalues are located in some interesting areas of the complex plane. This allows the authors also to derive new upper and lower bounds on the eigenvalues of quadratic matrix polynomials.
Reviewer: Stef Graillat (Paris)Distinguishability of the descriptor systems with regular pencilhttps://zbmath.org/1496.150072022-11-17T18:59:28.764376Z"Dastgeer, Zoubia"https://zbmath.org/authors/?q=ai:dastgeer.zoubia"Younus, Awais"https://zbmath.org/authors/?q=ai:younus.awais"Tunç, Cemil"https://zbmath.org/authors/?q=ai:tunc.cemilThe authors consider a linear hybrid descriptor system and focus on the property that the distinguishability of this type of systems is imperative.
The paper contains two results. The first result is related to the distinguishability of the descriptor system under study, for which some criteria and equivalent conditions related to polynomial input distinguishability, analytic and smooth input distinguishability are developed.
The second result concerns equivalent criteria for input distinguishability of descriptor systems with a regular pencil by using the Laplace transform and the Cayley-Hamilton theorem.
Reviewer: Ioannis Dassios (Dublin)Matrix pencils with coefficients that have positive semidefinite Hermitian partshttps://zbmath.org/1496.150082022-11-17T18:59:28.764376Z"Mehl, C."https://zbmath.org/authors/?q=ai:mehl.christian"Mehrmann, V."https://zbmath.org/authors/?q=ai:mehrmann.volker-ludwig"Wojtylak, M."https://zbmath.org/authors/?q=ai:wojtylak.michalLet \(A^*\) denote the conjugate transpose of a matrix \(A\) and let \(\mathbb F\) be either \(\mathbb R\) or \(\mathbb C\). \textit{Dissipative Hamiltonian} matrix pencils are pencils of the form
\[
P(\lambda)=\lambda E-A=\lambda E-(J-R)Q,\tag{1}
\]
where \(E,J,R\in\mathbb F^{n,n}\) satisfy the conditions \(J^*=-J\) and the matrices \(Q^*E=E^*Q\), \(R^*=R\) are positive semidefinite. The authors prove that an arbitrary matrix pencil \(L(\lambda)\in\mathbb F^{n,n}[\lambda]\) is strictly equivalent to a pencil of form (1) if and only of the following conditions hold:
\begin{itemize}
\item[1.] the spectrum of \(L(\lambda)\) is contained in the closed left half plane;
\item[2.] the finite nonzero eigenvalues on the imaginary axis are semisimple, and the partial multiplicities of the eigenvalue zero are at most two;
\item[3.] the index of \(L(\lambda)\) is at most two;
\item[4.] the left minimal indices are all zero, and the right minimal indices are at most one.
\end{itemize}
They introduce the concept of a posH pencils, i.e., pencils of the form
\[
\lambda(J_1+R_1)+(J_2+R_2),
\]
where \(J_1=-J_1^*\), \(J_2=-J_2^*\), and \(R_1\), \(R_2\) are positive semidefinite. Such pencils arise as linearizations of matrix polynomials with positive definite or semidefinite Hermitian coefficients. The authors characterize the possible Kronecker structures for posH matrix pencils. The obtained criterion implies some necessary or sufficient conditions for the regularity of such pencils. The numerical range of posH pencils is also studied.
The last part of the paper is devoted to the matrix polynomials \(P(\lambda)=\sum_{i=0}^d\lambda^iA_i\) whose skew-Hermitian parts of the coefficients are equal to zero. The authors prove that if for even \(d\) the matrix \(A_0\) is invertible, then the index of \(P(\lambda)\) does not exceed \(d\). They also localize the spectrum of \(P(\lambda)\) for the case when \(d=3\), \(A_1\) is positive semidefinite, and \(A_0\), \(A_2\), \(A_3\), \(A_2+A_1\) are positive definite.
Reviewer: Roksana Słowik (Gliwice)Idempotent factorization of matrices over a Prüfer domain of rational functionshttps://zbmath.org/1496.150092022-11-17T18:59:28.764376Z"Cossu, L."https://zbmath.org/authors/?q=ai:cossu.lauraSummary: We consider the smallest subring \(D\) of \(\mathbb{R}(X)\) containing every element of the form \(1/(1+x^2)\), with \(x \in \mathbb{R}(X)\). \(D\) is a Prüfer domain called the \textit{minimal Dress ring of} \(\mathbb{R}(X)\). In this paper, addressing a general open problem for Prüfer non Bézout domains, we investigate whether \(2\times 2\) singular matrices over \(D\) can be decomposed as products of idempotent matrices. We show some conditions that guarantee the idempotent factorization in \(M_2(D)\).On tensor products of matrix factorizationshttps://zbmath.org/1496.150102022-11-17T18:59:28.764376Z"Fomatati, Yves Baudelaire"https://zbmath.org/authors/?q=ai:fomatati.yves-baudelaireSummary: Let \(K\) be a field. Let \(f \in K [[x_1, \dots, x_r]]\) and \(g \in K [[ y_1,\dots, y_s]]\) be nonzero elements. If \(X\) (resp. \(Y)\) is a matrix factorization of \(f\) (resp. \(g)\), Yoshino had constructed a tensor product (of matrix factorizations) \(\widehat{\otimes}\) such that \(X \widehat{\otimes} Y\) is a matrix factorization of \(f + g \in K [[x_1, \dots, x_r, y_1, \dots, y_s]]\). In this paper, we propose a bifunctorial operation \(\widetilde{\otimes}\) and its variant \(\widetilde{\otimes}^\prime\) such that \(X \widetilde{\otimes} Y\) and \(X \widetilde{\otimes}^\prime Y\) are two different matrix factorizations of \(f g \in K [[x_1, \dots, x_r, y_1, \dots, y_s]]\). We call \(\widetilde{\otimes}\)\textit{the multiplicative tensor product} of \(X\) and \(Y\). Several properties of \(\widetilde{\otimes}\) are proved. Moreover, we find three functorial variants of Yoshino's tensor product \(\widehat{\otimes}\). Then, \(\widetilde{\otimes}\) (or its variant) is used in conjunction with \(\widehat{\otimes}\) (or any of its variants) to give an improved version of the standard algorithm for factoring polynomials using matrices on the class of \textit{summand-reducible polynomials} defined in this paper. Our algorithm produces matrix factors whose size is at most one half the size one obtains using the standard method.Manifold expressions of all solutions of the Yang-Baxter-like matrix equation for rank-one matriceshttps://zbmath.org/1496.150112022-11-17T18:59:28.764376Z"Lu, Linzhang"https://zbmath.org/authors/?q=ai:lu.linzhangSummary: Let \(A\) be a complex rank-one matrix, we derive simple sufficient and necessary conditions for a complex matrix \(X\) being a (commuting, non-commuting) solution of the quadratic matrix equation \(A X A = X A X\). On the basis, we construct the manifolds, each of which is a product of one free parameter matrix and another one to two matrices fixed by \(A\), to express concisely all the (commuting) solutions of the matrix equation.Can one hear a matrix? Recovering a real symmetric matrix from its spectral datahttps://zbmath.org/1496.150122022-11-17T18:59:28.764376Z"Maciążek, Tomasz"https://zbmath.org/authors/?q=ai:maciazek.tomasz"Smilansky, Uzy"https://zbmath.org/authors/?q=ai:smilansky.uzyIt is known that the spectrum of a symmetric real \(N \times N\) matrix determines the matrix up to unitary equivalence. So, more spectral data is needed together with some sign indicators in order to remove this unitary ambiguities.
In this paper, the spectral information about the matrix comes from the spectra of its \(N\) nested main minors. Such a spectral information determines the matrix up to a finite number of possibilities. The reconstruction can be then made unique by providing additional discrete data encoded in the signs of certain expressions involving entries of the desired matrix.
The first part of this manuscript deals with the inverse problem of full, real symmetric matrices of size \(N \times N\), where the number of unknown entries is \(\dfrac{1}{2}N(N+1)\). The spectral data to be used is the union of the spectra of the first \(N\) nested main minors of the matrix of size \(1,2,\ldots,N\), and \(\dfrac{1}{2}N(N-1)\) sign indicators needed for the complete reconstruction. The technique, called telescopic construction, is inductive: given a matrix \(A\) and a minor \(A^{(n)}\) of dimension \(n\), its next neighbor \(A^{(n+1)}\) is obtained by computing the \((n+1)\)-th column from the given spectra and sign indicators. The uniqueness of the resulting solution is proved. \par The values \(s^{(n)}_j\), that we are going to define, play the role of the sign indicators in this paper. Let \(A\) be a real symmetric matrix of size \(N \times N\). \(A^{(n)}\), \(1 \leq n \leq N\), denotes its \(n \times n\) upper main diagonal minor, so \(A^{(1)}=A_{11}\) and \(A^{(N)}=A\). \(|x^{(n)}\rangle\) denotes the column vector in dimension \(n\) with entries \(x^{(n)}_j\), \(j=1,2,\ldots,n\), and the scalar product is denoted by \(\langle x^{(n)} | y^{(n)}\rangle\). The authors also introduce the unit vectors \(|e^{(n)}(j)\rangle\), \(1 \leq j \leq n\), whose entries all vanish but for the \(j\)'th which is unity.
For every minor \(A^{(n)}\), \(1 \leq n \leq (N-1)\), define the upper off-diagonal part of the next column
\[
|a^{(n)}\rangle:=\left( \begin{array}{c} A_{1,n+1}^{(n+1)} \\
\vdots \\
A_{n,n+1}^{(n+1)} \end{array} \right) = \sum_{j=1}^n A_{j,n+1}^{(n+1)} | e^{(n+1)}(j)\rangle.
\]
The spectra of \(A^{(n)}\), \(1 \leq n \leq N\), is denoted by \(\sigma^{(n)}= \{ \lambda^{(n)}_j\}_{j=1}^n\), with \(\lambda^{(n)}_k \geq \lambda^{(n)}_j\), \(\forall k \geq j\), and the corresponding normalized eigenvectors are \(\{ |v^{(n)}(j)\rangle \}_{j=1}^n\). The overlaps of \(|a^{(n)}\rangle\) with the eigenvectors of \(A^{(n)}\) is denoted by \(s^{(n)}_j \xi^{(n)}_j\) where
\[
\left \{ s^{(n)}_j:= \mathrm{Sign} \left(\langle a^{(n)} | v^{(n)}(j)\rangle \right) \right\}_{j=1}^n,
\]
with
\[
\mathrm{Sign}(x)=\left\{ \begin{array} {ccc} +1,& & \mbox{for} \ x\geq 0, \\
-1,& & \mbox{for} \ x < 0. \end{array} \right.
\]
\par In the second part of the paper, the authors use the telescopic method restricted to banded matrices with band width \(D=2d+1\) much smaller than \(N\). The only sign information required consists of the signs of the off-diagonal matrix elements in the diagonal which is \(d\) steps away from the main diagonal. The proposed method is proved to provide a unique solution only for generic \(D\)-diagonal matrices. \par Finally, in the last part, it is shown that the large redundancy which exists in the telescopic approach can be reduced by studying a different construction: one considers only the \(d+1\) dimensional principal minors \(M^{(n)}\) which are distinguished by the position of their upper corner of \(M^{(n)}\) along the diagonal.
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Realizable list by circulant and skew-circulant matriceshttps://zbmath.org/1496.150132022-11-17T18:59:28.764376Z"Nazari, A. M."https://zbmath.org/authors/?q=ai:nazari.alimohammad|nazari.ali-mohamad"Mohammadi, R."https://zbmath.org/authors/?q=ai:mohammadi.rezaSummary: In this paper for two given sets of eigenvalues, which one of them is the eigenvalues of circulant matrix and the other is the eigenvalues of skew-circulant matrix, we find a nonnegative matrix, such that the union of two sets be the spectrum of nonnegative matrices.Solution of the linearly structured partial polynomial inverse eigenvalue problemhttps://zbmath.org/1496.150142022-11-17T18:59:28.764376Z"Rakshit, Suman"https://zbmath.org/authors/?q=ai:rakshit.suman"Khare, S. R."https://zbmath.org/authors/?q=ai:khare.swanand-rThis paper deals with the linearly structured partial polynomial inverse eigenvalue problem (LPPIEP) for matrices \(A_i \in \mathbb{R}^{n \times n}\), \(i=0,1,\ldots,k-1\) of such that the matrix polynomial
\[
P(\lambda)=\lambda^k I_n + \sum_{i=0}^{k-1} \lambda^{i} A_i
\]
has \(m\) (\(1 \leq m \leq kn\)) prescribed eigenvalues and eigenvectors. Note that the coefficients of \(P(\lambda)\) may have different linear structures. For instance, some of the matrices can be diagonal matrices, other symmetric tridiagonal, etc. In order to address these situations, the authors study another problem, namely:
Problem (LPPIEP-I). Given two positive integers \(k\), \(n\) and a set of partial eigenpairs \((\lambda_j, \phi_j)_{j=1}^m\), \(1 \leq m \leq kn\), construct a monic matrix polynomial \(P(\lambda)\) of degree \(k\) in such a way that matrices \(A_i \in\mathcal L_i\), \(i=0,1,\ldots,k-1\) and \(P(\lambda)\) has the specified set \((\lambda_j, \phi_j)_{j=1}^m\), \(1 \leq m \leq kn\) as its eigenpairs.
The authors propose an approach for the original LPPIEP. Their results solve some open problems in the theory of polynomial inverse eigenvalue problems. The proposed solution gives an alternative answer to the inverse eigenvalue problems considered in the literature for a specific structure such as symmetric, skew-symmetric, etc. However, in this paper the framework is generalized to an arbitrary linear structure. In particular, for any given specific linear structure, this paper presents a way that systematically generates a system of linear equations and obtain the solution of the corresponding inverse eigenvalue problem.
The contributions made in this paper can be summarized as follows:
\begin{itemize}
\item[(i)] the proposed methods are capable to solve LPPIEP and LPPIEP-I using a set of \(m\), \(1 \leq m \leq kn\), eigenpairs without imposing any restrictions on it;
\item[(ii)] it is possible to study the partial inverse eigenvalue problem for different linear structures at once;
\item[(iii)] the authors derive some necessary and sufficient conditions on the eigendata for the existence of solution to these problems;
\item[(iv)] explicit analytical expressions for the family of matrix polynomials are provided;
\item[(v)] sensitivity of the solutions to the perturbations of the eigendata is also discussed;
\item[(vi)] the minimum norm solutions of LPPIEP and LPPIEP-I are obtained.
\end{itemize}
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Logarithmic mean of multiple accretive matriceshttps://zbmath.org/1496.150152022-11-17T18:59:28.764376Z"Luo, Wenhui"https://zbmath.org/authors/?q=ai:luo.wenhuiSummary: Using the integral representation of logarithmic mean, we define the logarithmic mean of multiple accretive matrices. When the number of matrices is two, it coincides with the recent studies carried out by \textit{F. Tan} and \textit{A. Xie} [Filomat 33, No. 15, 4747--4752 (2019; Zbl 07537437)]. Several inequalites are presented along with the studies.On strongly non-singular polynomial matriceshttps://zbmath.org/1496.150162022-11-17T18:59:28.764376Z"Abramov, Sergei A."https://zbmath.org/authors/?q=ai:abramov.sergei-a"Barkatou, Moulay A."https://zbmath.org/authors/?q=ai:barkatou.moulay-aSummary: We consider matrices with infinite power series as entries and suppose that those matrices are represented in an ``approximate'' form, namely, in a truncated form. Thus, it is supposed that a polynomial matrix \(P\) which is the \(l\)-truncation (\(l\) is a non-negative integer, \(\deg P=l\)) of a power series matrix \(M\) is given, and \(P\) is non-singular, i.e., \(\det P\neq 0\). We prove that the strong non-singularity testing, i.e., the testing whether \(P\) is not a truncation of a singular matrix having power series entries, is algorithmically decidable. Supposing that a non-singular power series matrix \(M\) (which is not known to us) is represented by a strongly non-singular polynomial matrix \(P\), we give a tight lower bound for the number of initial terms of \(M^{-1}\) which can be determined from \(P^{-1}\). In addition, we report on possibility of applying the proposed approach to ``approximate'' linear differential systems.
For the entire collection see [Zbl 1391.33001].Sharp estimate on the resolvent of a finite-dimensional contractionhttps://zbmath.org/1496.150172022-11-17T18:59:28.764376Z"Fouchet, Karine"https://zbmath.org/authors/?q=ai:fouchet.karineSummary: We compute an asymptotic formula for the supremum of the resolvent norm \(\| (\zeta -T)^{-1}\|\) over \(|\zeta |\geq 1\) and contractions \(T\) acting on an \(n\)-dimensional Hilbert space, whose spectral radius does not exceed a given \(r \in (0,1)\). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.3-parameter generalized quaternionshttps://zbmath.org/1496.150182022-11-17T18:59:28.764376Z"Şentürk, Tuncay Deniz"https://zbmath.org/authors/?q=ai:senturk.tuncay-deniz"Ünal, Zafer"https://zbmath.org/authors/?q=ai:unal.zaferSummary: In this article, we give a general form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study various properties and applications. Firstly we present the definiton, the multiplication table and algebraic properties of 3PGQs. We give matrix representation and Hamilton operators for 3PGQs. We derive the polar represenation, De Moivre's and Euler's formulas with the matrix representations for 3PGQs. Additionally, we derive relations between the powers of the matrices associated with 3PGQs. Finally, Lie groups and Lie algebras are studied and their matrix representations are given. Also the Lie multiplication and the Killing bilinear form are given.General tail bounds for random tensors summation: majorization approachhttps://zbmath.org/1496.150192022-11-17T18:59:28.764376Z"Chang, Shih Yu"https://zbmath.org/authors/?q=ai:chang.shih-yu"Wei, Yimin"https://zbmath.org/authors/?q=ai:wei.yimin.1|wei.yiminSummary: In recent years, tensors have been applied to different applications in science and engineering fields. In order to establish theory about tail bounds of the tensors summation behavior, this research extends previous work by considering the tensors summation tail behavior of the top \(k\)-largest singular values of a function of the tensors summation, instead of the largest/smallest singular value of the tensors summation directly (identity function) explored in [\textit{S. Y. Chang}, ``Convenient tail bounds for sums of random tensors'', Preprint, \url{arXiv:2012.15428}]. Majorization and antisymmetric tensor product tools are main techniques utilized to establish inequalities for unitarily invariant norms of multivariate tensors. The Laplace transform method is integrated with these inequalities for unitarily invariant norms of multivariate tensors to give us tail bounds estimation for the Ky Fan \(k\)-norm for a function of the tensors summation. By restricting different random tensor conditions, we obtain generalized tensor Chernoff and Bernstein inequalities.Tensor polynomial identitieshttps://zbmath.org/1496.150202022-11-17T18:59:28.764376Z"Huber, Felix"https://zbmath.org/authors/?q=ai:huber.felix-m|huber.felix-michael"Procesi, Claudio"https://zbmath.org/authors/?q=ai:procesi.claudioIn [J. Math. Phys. 62, No. 2, 022203, 27 p. (2021; Zbl 1459.81023)], the first author proved that
\[
\sum_{\sigma \in S_4} \epsilon_\sigma X_{\sigma(1)}X_{\sigma(2)} \otimes X_{\sigma(3)}X_{\sigma(4)}=0
\]
for all \(2 \times 2 \) complex matrices \(X_1,X_2,X_3,X_4,\) where \( \epsilon_\sigma\) is the sign of the permutation \(\sigma \in S_n.\) The main results of the present paper generalize this equality for tensor polynomial identities. More precisely, the authors describe a method to characterize a certain class of tensor polynomial identities in terms of their associated Young diagrams.
As a consequence they prove that
\[
\sum_{\sigma \in S_n} \epsilon_\sigma X_{\sigma(1)}\cdots X_{\sigma(m)} \otimes \cdots \otimes X_{\sigma((n-1)m+1)}\cdots X_{\sigma(mn)} =0
\]
for all \(d \times d \) complex matrices \(X_1,\dots,X_{mn},\) if and only if \(n>[d;m],\) where \(d,m\leq 2d\) are two integers and
\[
[d;m]=\lfloor (2d-1)/m\rfloor + \lfloor (2d-3)/m\rfloor + \cdots + \lfloor 3/m\rfloor + \lfloor 1/m\rfloor.
\]
Reviewer: Rick Rischter (Itajubá)Polynomial matrices, splitting subspaces and Krylov subspaces over finite fieldshttps://zbmath.org/1496.150212022-11-17T18:59:28.764376Z"Aggarwal, Divya"https://zbmath.org/authors/?q=ai:aggarwal.divya"Ram, Samrith"https://zbmath.org/authors/?q=ai:ram.samrithSummary: Let \(T\) be a linear operator on a vector space \(V\) of dimension \(n\) over \(\mathbb{F}_q\). For any divisor \(m\) of \(n\), an \(m\)-dimensional subspace \(W\) of \(V\) is \(T\)-splitting if
\[
V = W \oplus T W \oplus \cdots \oplus T^{d - 1} W,
\] where \(d = n / m\). Let \(\sigma(m, d; T)\) denote the number of \(m\)-dimensional \(T\)-splitting subspaces. Determining \(\sigma(m, d; T)\) for an arbitrary operator \(T\) is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for \(\sigma(m, d; T)\) in the case where the invariant factors of \(T\) satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed.On characterization of tripotent matrices in triangular matrix ringshttps://zbmath.org/1496.150222022-11-17T18:59:28.764376Z"Petik, Tuğba"https://zbmath.org/authors/?q=ai:petik.tugbaSummary: Let \(\mathfrak{R}\) be a ring with identity 1 whose tripotents are only \(-1\), 0, and 1. It is characterized the structure of tripotents in \(\mathcal{T}(\mathfrak{R})\) which is the ring of triangular matrices over \(\mathfrak{R}\). In addition, when \(\mathfrak{R}\) is finite, it is given number of the tripotents in \(\mathcal{T}_n(\mathfrak{R})\) which is the ring of \(n\times n\) dimensional triangular matrices over \(\mathfrak{R}\) with \(n\) being a positive integer.Properties of the index lattice of a Boolean matrixhttps://zbmath.org/1496.150232022-11-17T18:59:28.764376Z"Zeng, Zhi Xuan"https://zbmath.org/authors/?q=ai:zeng.zhixuan"Luo, Cong Wen"https://zbmath.org/authors/?q=ai:luo.congwen(no abstract)On degree of nonlinearity of the coordinate polynomials for a product of transformations of a binary vector spacehttps://zbmath.org/1496.150242022-11-17T18:59:28.764376Z"Fomichëv, Vladimir Mikhaĭlovich"https://zbmath.org/authors/?q=ai:fomichev.vladimir-mikhailovichSummary: We construct a nonnegative integer matrix to evaluate the matrix of nonlinearity characteristics for the coordinate polynomials of a product of transformations of a binary vector space. The matrix of the characteristics of the transformation is defined by the degrees of nonlinearity of the derivatives of all coordinate functions with respect to each input variable. The entries of the evaluation matrix are expressed in terms of the characteristics of the coordinate polynomials of the multiplied transformations. Calculation of the evaluation matrix is easier than calculating the exact values of the characteristics. The estimation method is extended to an arbitrary number of multiplied transformations. Computational examples are given that in particular show the accuracy of the obtained estimates and the domain of their nontriviality.On the condition number of the shifted real Ginibre ensemblehttps://zbmath.org/1496.150252022-11-17T18:59:28.764376Z"Cipolloni, Giorgio"https://zbmath.org/authors/?q=ai:cipolloni.giorgio"Erdös, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schröder, Dominik"https://zbmath.org/authors/?q=ai:schroder.dominikHamming spaces and locally matrix algebrashttps://zbmath.org/1496.160292022-11-17T18:59:28.764376Z"Bezushchak, Oksana"https://zbmath.org/authors/?q=ai:bezushchak.oksana-o"Oliynyk, Bogdana"https://zbmath.org/authors/?q=ai:oliynyk.bogdanaPolynomial representations of \(\mathrm{GL}(m|n)\)https://zbmath.org/1496.170172022-11-17T18:59:28.764376Z"Flicker, Yuval Z."https://zbmath.org/authors/?q=ai:flicker.yuval-zThe paper under review develops a theory of polynomial representations of the super general linear group \(\mathrm{GL}(m|n,A)\), defined over an arbitrary commutative superalgebra \(A\). The methods used adapt and parallel Green's approach to the usual Schur algebra via comodules, as presented in \S 2 of [\textit{J. A. Green}, Polynomial representations of \(\mathrm{GL}_n\). With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker. 2nd corrected and augmented edition. Berlin: Springer (2007; Zbl 1108.20044)]. Thus, rather than work directly with super modules for \(\mathrm{GL}(m|n,A)\), the author first defines, for each suitable sub-super coalgebra \(D\) of the super algebra of finitary functions \(\mathrm{GL}(m|n,A)\), a module category \(\mathrm{mod}_D(A\mathrm{GL}(m|n,A))\) of representations of the super group algebra \(A\mathrm{GL}(m|n,A)\) such that the coefficient functions of the representing matrices lie in \(D\). There is an equivalence of categories
\[
\mathrm{mod}_D(A\mathrm{GL}(m|n,A)) \simeq \mathrm{com}(D)
\]
between this category and the category of super \(D\)-comodules. Taking \(D\) to be the super coalgebra of polynomial functions \(\mathrm{GL}(m|n,A)\) of degree \(r\) and dualizing, the author obtains a superalgebra \(S_A(m|n,r)\) that is the super analogue of the usual Schur algebra. A key result on the Schur algebra is that if \(F\) is an infinite field then category of polynomial representations of \(\mathrm{GL}(n,F)\) of polynomial degree \(r\) is equivalent to the category of representations of the Schur algebra \(S_F(n,r)\), defined over the field \(F\). The author proves the analogous result for representations of \(S_A(m|n,r)\) in Proposition 4.2.
The main result of this paper, described as `super Schur duality' is Theorem 5.1. In it \(A\) is a commutative superalgebra over an infinite field \(F\) and \(E_A = A^{m|n}\) is a free rank \(m|n\) \(A\)-module, generated by \(m\) even elements and \(n\) odd elements. The free \(A\)-module \(E_A^{\otimes r}\) is acted on by \(\mathrm{GL}(m|n,A)\) with coefficient functions of polynomial degree \(r\). It may therefore be regarded as a representation of \(S_A(m|n,r)\). The symmetric group \(S_r\) acts on \(E_A^{\otimes r}\) by permuting the factors (with signs coming from the super structure). Theorem 5.1 states that the natural action map
\[
S_A(m|n,r) \rightarrow \mathrm{End}_A(E_A^{\otimes r}) \tag{\(\star\)}
\]
is injective, and its image is precisely those \(A\)-endomorphisms of \(E_A^{\otimes r}\) which commute with the action of the symmetric group \(S_r\). This is the super version of Schur's theorem (see [loc. cit., Theorem 2.6c]) that \(S_F(n,r) \cong \mathrm{End}_{S_r}(V^{\otimes r})\), where \(V\) is the natural \(\mathrm{GL}_n(F)\)-module. As is the case with Schur's theorem, the main force of the result is that \emph{every} \(S_r\) endomorphism of the tensor algebra comes from an element of the super Schur algebra, and so is induced by the action of a suitable linear combination of elements in the super group algebra \(A\mathrm{GL}(m|n,A)\). An important corollary is that if \(F\) has infinite characteristic or prime characteristic \(p > r\) then \(S_A(m|n,r)\) is semisimple.
The author begins with a brief but useful survey of other approaches to Schur-Weyl duality, emphasising that the main novel feature in his paper is to work with the supergroup \(\mathrm{GL}(m|n)\) rather than its super Lie algebra \(\mathfrak{gl}(m|n)\). In this connection we mention that modules for \(S_A(m|n,r)\) are direct sums of the special class of covariant \(\mathfrak{gl}(m|n)\)-modules: see Chapter 3 of Moens' Ph.D.~thesis [Supersymmetric Schur functions and Lie superalgebra representations. Universiteit Gent (2007)] for an excellent introduction. In general, and in contrast to the case for \(S_A(m|n,r)\), modules for \(\mathfrak{gl}(m|n)\) are not completely reducible. Another reference one might add to the author's list is [\textit{D. Benson} and \textit{S. Doty}, Arch. Math. 93, No. 5, 425--435 (2009; Zbl 1210.20039)], which shows (amongst other results) that the Schur algebra analogue of (\(\star\)) holds over any field \(F\) such that \(|F| > r\).
The paper under review includes all the results needed to perform \(p\)-modular reduction on the category of representations of the Super Schur algebra \(S(m|n,r)\). The author remarks that `It seems to us such a modular theory is needed for a geometric theory'.
Reviewer: Mark Wildon (Egham)On generalized Heisenberg groups: the symmetric casehttps://zbmath.org/1496.200502022-11-17T18:59:28.764376Z"Sangkhanan, Kritsada"https://zbmath.org/authors/?q=ai:sangkhanan.kritsada"Suksumran, Teerapong"https://zbmath.org/authors/?q=ai:suksumran.teerapong(no abstract)Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted rangehttps://zbmath.org/1496.201162022-11-17T18:59:28.764376Z"Sommanee, W."https://zbmath.org/authors/?q=ai:sommanee.woracheadSummary: Let \(V\) be a vector space over a field and let \(T(V)\) denote the semigroup of all linear transformations from \(V\) into \(V\). For a fixed subspace \(W\) of \(V\), let \(F(V,W)\) be the subsemigroup of \(T(V\) ) formed by all linear transformations \(\alpha\) from \(V\) into \(W\) such that \(V \alpha \subseteq W \alpha \). We prove that any regular semigroup \(S\) can be embedded in \(F(V,W)\) with \(\dim (V) = |S^1|\) and \(\dim (W) = |S|,\) and determine all maximal subsemigroups of \(F(V,W)\) in the case where \(W\) is a finite-dimensional subspace of \(V\) over a finite field.A few results on permittivity variations in electromagnetic cavitieshttps://zbmath.org/1496.353772022-11-17T18:59:28.764376Z"Luzzini, Paolo"https://zbmath.org/authors/?q=ai:luzzini.paolo"Zaccaron, Michele"https://zbmath.org/authors/?q=ai:zaccaron.micheleSummary: We study the eigenvalues of time-harmonic Maxwell's equations in a cavity upon changes in the electric permittivity \(\varepsilon\) of the medium. We prove that all the eigenvalues, both simple and multiple, are locally Lipschitz continuous with respect to \(\varepsilon \). Next, we show that simple eigenvalues and the symmetric functions of multiple eigenvalues depend real analytically upon \(\varepsilon\) and we provide an explicit formula for their derivative in \(\varepsilon \). As an application of these results, we show that for a generic permittivity all the Maxwell eigenvalues are simple.Further results on eigenvalues of symmetric decomposable tensors from multilinear dynamical systemshttps://zbmath.org/1496.370252022-11-17T18:59:28.764376Z"Chen, Haibin"https://zbmath.org/authors/?q=ai:chen.haibin"Li, Mengzhen"https://zbmath.org/authors/?q=ai:li.mengzhen"Yan, Hong"https://zbmath.org/authors/?q=ai:yan.hong"Zhou, Guanglu"https://zbmath.org/authors/?q=ai:zhou.guangluThe authors consider a discrete-time version of multilinear dynamical systems associated with symmetric decomposable tensors, including orthogonal and non-orthogonal decomposable tensors as subclasses. Special emphasis is given to the study of the asymptotic stability of such systems. The results are supported by numerical analysis.
Reviewer: P. Shakila Banu (Erode)Positive definiteness and infinite divisibility of certain functions of hyperbolic cosine functionhttps://zbmath.org/1496.420102022-11-17T18:59:28.764376Z"Kosaki, Hideki"https://zbmath.org/authors/?q=ai:kosaki.hidekiLet \(\alpha \geq 0\), \(t>-1\) and \(f_{\alpha },\) \(g_{\alpha }\) two real functions defined by
\[
f_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (3x)+t\cosh x}
\]
and
\[
g_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (2x)+t\cosh x}.
\]
The author investigates infinite divisibility and positive definiteness of the functions \(f_{\alpha }\) and of \(g_{\alpha }\). Furthermore, he uses the positive definiteness criterion to study certain norm comparison results for operator means.
Reviewer: Elhadj Dahia (Bou Saâda)On the algebraic dimension of Riesz spaceshttps://zbmath.org/1496.460012022-11-17T18:59:28.764376Z"Baziv, N. M."https://zbmath.org/authors/?q=ai:baziv.n-m"Hrybel, O. B."https://zbmath.org/authors/?q=ai:hrybel.o-bSummary: We prove that the algebraic dimension of an infinite dimensional \(C\)-\(\sigma \)-complete Riesz space (in particular, of a Dedekind \(\sigma \)-complete and a laterally \(\sigma \)-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.Some refinements of numerical radius inequalities for Hilbert space operatorshttps://zbmath.org/1496.470142022-11-17T18:59:28.764376Z"Alizadeh, Ebrahim"https://zbmath.org/authors/?q=ai:alizadeh.ebrahim"Farokhinia, Ali"https://zbmath.org/authors/?q=ai:farokhinia.aliSummary: The main goal of this paper is to obtain some refinements of numerical radius inequalities for Hilbert space operators.Drazin-star and star-Drazin inverses of bounded finite potent operators on Hilbert spaceshttps://zbmath.org/1496.470342022-11-17T18:59:28.764376Z"Pablos Romo, Fernando"https://zbmath.org/authors/?q=ai:pablos-romo.fernandoThe notions of the Drazin-star and the star-Drazin of matrices were introduced by \textit{D. Mosić} [Result. Math. 75, No. 2, Paper No. 61, 21 p. (2020; Zbl 1437.15004)]. In the paper under review, the author extends the above notions to bounded finite potent endomorphisms on arbitrary Hilbert spaces. He demonstrates the existence, structure and main properties of these operators. Moreover, for bounded finite potent endomorphisms with index less or equal to~1, he studies the group-star and the star-group inverses. He also applies the properties of the Drazin-star inverse of a bounded finite potent endomorphism for studying the consistence and the general solutions of linear systems on Hilbert spaces.
Reviewer: Mohammad Sal Moslehian (Mashhad)Tensor trigonometry. Theory and applicationshttps://zbmath.org/1496.510012022-11-17T18:59:28.764376Z"Ninul, Anatoliĭ Sergeevich"https://zbmath.org/authors/?q=ai:ninul.anatolii-sergeevichFor the English edition see [Zbl 1482.51002].Projective points over matrices and their separability propertieshttps://zbmath.org/1496.540072022-11-17T18:59:28.764376Z"Agnew, Alfonso F."https://zbmath.org/authors/?q=ai:agnew.alfonso-f"Rathbun, Matt"https://zbmath.org/authors/?q=ai:rathbun.matt"Terry, William"https://zbmath.org/authors/?q=ai:terry.williamThe authors focus their work on topological quotients of real and complex matrices, by various subgroups, and they study their separation properties, as this finds an immediate application to twistor spaces in mathematical physics. As a main conclusion, the authors find that as the group one quotients by gets smaller, the separability properties of the quotient improve. The authors then pose a number of topological questions. For example, the quotient by the general linear group is compact, but the others are not; is there a ground of a further topological study, apart from the separation properties? What about the homotopy or homology properties?
Reviewer: Kyriakos Papadopoulos (Madīnat al-Kuwait)Riemannian conjugate gradient methods for computing the extreme eigenvalues of symmetric tensorshttps://zbmath.org/1496.650552022-11-17T18:59:28.764376Z"Wen, Ya-qiong"https://zbmath.org/authors/?q=ai:wen.yaqiong"Li, Wen"https://zbmath.org/authors/?q=ai:li.wen.1Summary: In this paper, we propose two kinds of Riemannian conjugate gradient methods for computing the extreme eigenvalues of even order symmetric tensors. One is a sufficient descent Dai-Yuan type conjugate gradient method, and another is Zhu's Riemannian nonmonotone conjugate gradient method. The global convergence of two proposed algorithms can be guaranteed, respectively. Numerical results are reported to demonstrate the feasibility and efficiency of the proposed methods.An inverse eigenvalue problem for Jacobi matrices with a missing eigenvaluehttps://zbmath.org/1496.650562022-11-17T18:59:28.764376Z"He, Bin"https://zbmath.org/authors/?q=ai:he.bin"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min|wang.min.2|wang.min.1"Wei, Guangsheng"https://zbmath.org/authors/?q=ai:wei.guangshengSummary: We consider an inverse eigenvalue problem for constructing an \(n \times n\) Jacobi matrix \(J_n\) under the circumstance that its all eigenvalues, except for one and a part of the matrix \(J_n\) are given. To be precise, the known partial data of \(J_n\) means either its leading principal submatrix \(J_{[(n+1)/2]}\) when \(n\) is odd, or the submatrix \(J_{[(n+1)/2]}\) together with the \([(n+1)/2]\times(n/2+1)\) codiagonal element when \(n\) is even. The necessary and sufficient conditions for the solvability of the problem is derived, also the numerical algorithm and a numerical example are provided.Algorithms for approximating means of semi-infinite quasi-Toeplitz matriceshttps://zbmath.org/1496.650592022-11-17T18:59:28.764376Z"Bini, Dario A."https://zbmath.org/authors/?q=ai:bini.dario-andrea"Iannazzo, Bruno"https://zbmath.org/authors/?q=ai:iannazzo.bruno"Meng, Jie"https://zbmath.org/authors/?q=ai:meng.jie.1Summary: We provide algorithms for computing the Karcher mean of positive definite semi-infinite quasi-Toeplitz matrices. After showing that the power mean of quasi-Toeplitz matrices is a quasi-Toeplitz matrix, we obtain a first algorithm based on the fact that the Karcher mean is the limit of a family of power means. A second algorithm, that is shown to be more effective, is based on a generalization to the infinite-dimensional case of a reliable algorithm for computing the Karcher mean in the finite-dimensional case. Numerical tests show that the Karcher mean of infinite-dimensional quasi-Toeplitz matrices can be effectively approximated with a finite number of parameters.
For the entire collection see [Zbl 1482.94007].WARPd: a linearly convergent first-order primal-dual algorithm for inverse problems with approximate sharpness conditionshttps://zbmath.org/1496.650742022-11-17T18:59:28.764376Z"Colbrook, Matthew J."https://zbmath.org/authors/?q=ai:colbrook.matthew-jDNT preconditioner for one-sided space fractional diffusion equationshttps://zbmath.org/1496.651212022-11-17T18:59:28.764376Z"Lin, Fu-Rong"https://zbmath.org/authors/?q=ai:lin.fu-rong"Qu, Hai-Dong"https://zbmath.org/authors/?q=ai:qu.haidong"She, Zi-Hang"https://zbmath.org/authors/?q=ai:she.zihangSummary: In this paper, we consider diagonal-times-nonsymmetric-Toeplitz (DNT) preconditioner for Toeplitz-like linear systems arising from one-sided space fractional diffusion equations (OSFDE) with variable coefficients. We first correct the result of Lemma 3.3 in [\textit{X.-L. Lin} et al., BIT 58, No. 3, 729--748 (2018; Zbl 1404.65136)] and prove the corrected result. We then extend the DNT preconditioner to Toeplitz-like linear systems arising from OSFDEs, where high order difference operators are applied to discretize the fractional derivative. Theoretically, we prove that the condition number of the preconditioned matrix is uniformly bounded by a constant under mild assumptions and verify that several discretization schemes from the literature satisfy the required assumptions. Numerical results are reported to show the efficiency of the DNT preconditioner.Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensionshttps://zbmath.org/1496.652202022-11-17T18:59:28.764376Z"Kazeev, Vladimir"https://zbmath.org/authors/?q=ai:kazeev.vladimir-a"Oseledets, Ivan"https://zbmath.org/authors/?q=ai:oseledets.ivan-v"Rakhuba, Maxim V."https://zbmath.org/authors/?q=ai:rakhuba.maxim-v"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophA sparse deep learning model for privacy attack on remote sensing imageshttps://zbmath.org/1496.682892022-11-17T18:59:28.764376Z"Wang, Eric Ke"https://zbmath.org/authors/?q=ai:kewang.eric"Zhe, Nie"https://zbmath.org/authors/?q=ai:zhe.nie"Li, Yueping"https://zbmath.org/authors/?q=ai:li.yueping"Liang, Zuodong"https://zbmath.org/authors/?q=ai:liang.zuodong"Zhang, Xun"https://zbmath.org/authors/?q=ai:zhang.xun"Yu, Juntao"https://zbmath.org/authors/?q=ai:yu.juntao"Ye, Yunming"https://zbmath.org/authors/?q=ai:ye.yunming(no abstract)Spin Hurwitz theory and Miwa transform for the Schur Q-functionshttps://zbmath.org/1496.810772022-11-17T18:59:28.764376Z"Mironov, A."https://zbmath.org/authors/?q=ai:mironov.andrei-d"Morozov, A."https://zbmath.org/authors/?q=ai:morozov.alexei-yurievich"Zhabin, A."https://zbmath.org/authors/?q=ai:zhabin.aSummary: Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary \(N \times N\) matrices \(X\), known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras \(\mathfrak{q}(N)\). The corresponding spin \(\hat{\mathcal{W}}\)-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix \(\Psi\) in addition to \(X\), but its realization using supermatrices is \textit{not} quite naive.Differential recurrences for the distribution of the trace of the \(\beta\)-Jacobi ensemblehttps://zbmath.org/1496.810912022-11-17T18:59:28.764376Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh.2|kumar.santosh.1|kumar.santosh.4|kumar.santosh|kumar.santosh.3Summary: Examples of the \(\beta\)-Jacobi ensemble in random matrix theory specify the joint distribution of the transmission eigenvalues in scattering problems. For this application, the trace is of relevance as determining the conductance. Earlier, in the case \(\beta = 1\), the trace statistic was isolated in studies of covariance matrices in multivariate statistics. There, Davis showed that for \(\beta = 1\) the trace statistic could be characterised by \((N + 1) \times (N + 1)\) matrix differential equations, now understood for general \(\beta > 0\) as part of the theory of Selberg correlation integrals. However the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameters \(b\) and Dyson index \(\beta\) non-negative integers. The distribution then has the functional form of a series of piecewise power functions times a polynomial, and our characterisation gives a recurrence for the computation of the polynomials. For all \(\beta > 0\) we express the Fourier-Laplace transform of the trace statistic in terms of a generalised hypergeometric function based on Jack polynomials.Sketched Newton-Raphsonhttps://zbmath.org/1496.901122022-11-17T18:59:28.764376Z"Yuan, Rui"https://zbmath.org/authors/?q=ai:yuan.rui"Lazaric, Alessandro"https://zbmath.org/authors/?q=ai:lazaric.alessandro"Gower, Robert M."https://zbmath.org/authors/?q=ai:gower.robert-manselGeometric structure guided model and algorithms for complete deconvolution of gene expression datahttps://zbmath.org/1496.920642022-11-17T18:59:28.764376Z"Chen, Duan"https://zbmath.org/authors/?q=ai:chen.duan"Li, Shaoyu"https://zbmath.org/authors/?q=ai:li.shaoyu"Wang, Xue"https://zbmath.org/authors/?q=ai:wang.xueSummary: Complete deconvolution analysis for bulk RNA-seq data is important and helpful to distinguish whether the differences of disease-associated GEPs (gene expression profiles) in tissues of patients and normal controls are due to changes in cellular composition of tissue samples, or due to GEPs changes in specific cells. One of the major techniques to perform complete deconvolution is nonnegative matrix factorization (NMF), which also has a wide-range of applications in the machine learning community. However, the NMF is a well-known strongly ill-posed problem, so a direct application of NMF to RNA-seq data will suffer severe difficulties in the interpretability of solutions. In this paper, we develop an NMF-based mathematical model and corresponding computational algorithms to improve the solution identifiability of deconvoluting bulk RNA-seq data. In our approach, we combine the biological concept of marker genes with the solvability conditions of the NMF theories, and develop a geometric structures guided optimization model. In this strategy, the geometric structure of bulk tissue data is first explored by the spectral clustering technique. Then, the identified information of marker genes is integrated as solvability constraints, while the overall correlation graph is used as manifold regularization. Both synthetic and biological data are used to validate the proposed model and algorithms, from which solution interpretability and accuracy are significantly improved.Random matrices and controllability of dynamical systemshttps://zbmath.org/1496.930232022-11-17T18:59:28.764376Z"Leventides, John"https://zbmath.org/authors/?q=ai:leventides.john"Poulios, Nick"https://zbmath.org/authors/?q=ai:poulios.nick"Poulios, Costas"https://zbmath.org/authors/?q=ai:poulios.costasSummary: We introduce the concept of \(\epsilon\)-uncontrollability for random linear systems, i.e. linear systems in which the usual matrices have been replaced by random matrices. We also estimate the \(\varepsilon\)-uncontrollability in the case where the matrices come from the Gaussian orthogonal ensemble. Our proof utilizes tools from systems theory, probability theory and convex geometry.The notion of almost zeros and randomnesshttps://zbmath.org/1496.930512022-11-17T18:59:28.764376Z"Leventides, J."https://zbmath.org/authors/?q=ai:leventides.john"Karcanias, N."https://zbmath.org/authors/?q=ai:karcanias.nicos|karcanias.n-k"Poulios, N. C."https://zbmath.org/authors/?q=ai:poulios.n-cSummary: We investigate the problem of almost zeros of polynomial matrices as used in system theory. It is related to the controllability and observability notion of systems as well as the determination of Macmillan degree and complexity of systems. We also present some new results on this important invariant in the light of randomness and we prove an uncertainty type relation appearing in such ensembles of operators.The study of the projective transformation under the bilinear strict equivalencehttps://zbmath.org/1496.930562022-11-17T18:59:28.764376Z"Kalogeropoulos, Grigoris I."https://zbmath.org/authors/?q=ai:kalogeropoulos.grigoris-i"Karageorgos, Athanasios D."https://zbmath.org/authors/?q=ai:karageorgos.athanasios-d"Pantelous, Athanasios A."https://zbmath.org/authors/?q=ai:pantelous.athanasios-aSummary: The study of linear time invariant descriptor systems has intimately been related to the study of matrix pencils. It is true that a large number of systems can be reduced to the study of differential (or difference) systems, \(S(F,G)\),
\[
S(F,G):F\dot{x}(t)=Gx(t)(\text{or the dual, }Fx(t)=G\dot{x}(t)),
\]
and
\[
S(F,G):Fx_{k+1}=Gx_k(\text{or the dual, } Fx_k=Gx_{k+1}),F,G\in\mathbb{C}^{m\times n},
\]
and their properties can be characterized by homogeneous matrix pencils, \(sF-\hat{s}G\). Based on the fact that the study of the invariants for the projective equivalence class can be reduced to the study of the invariants of the matrices of set \(\mathbb{C}^{k\times 2}\) (for \(k\geqslant 3\) with all \(2\times 2\)-minors non-zero) under the \textit{extended Hermite equivalence}, in the context of the bilinear strict equivalence relation, a novel projective transformation is analytically derived.Color image inpainting via robust pure quaternion matrix completion: error bound and weighted losshttps://zbmath.org/1496.940072022-11-17T18:59:28.764376Z"Chen, Junren"https://zbmath.org/authors/?q=ai:chen.junren"Ng, Michael K."https://zbmath.org/authors/?q=ai:ng.michael-k|ng.michael-ka-shingEncryption schemes using anti-orthogonal of type I matriceshttps://zbmath.org/1496.940542022-11-17T18:59:28.764376Z"Kamyun, Nilobol"https://zbmath.org/authors/?q=ai:kamyun.nilobol"Pingyot, Krittiya"https://zbmath.org/authors/?q=ai:pingyot.krittiya"Sompong, Supunnee"https://zbmath.org/authors/?q=ai:sompong.supunneeSummary: Applications of the orthogonal matrix have been applied to many circumstance such as signal processing, image processing, coding theory, cryptology. In this paper, we are not only introduce the new type of matrices, anti-orthogonal and \(H\)-anti-orthogonal of type I matrices but also apply these matrices in cryptology.