Search

In this paper, we study the problem of decomposing a given tensor into a tensor train such that the tensors at the vertices are orthogonally decomposable. When the tensor train has length two, and the orthogonally decomposable tensors at the two vertices are symmetric, we recover the decomposition by considering random linear combinations of slices. Furthermore, if the tensors at the vertices are symmetric and low-rank but not orthogonally decomposable, we show that a whitening procedure can transform the problem into the orthogonal case. When the tensor network has length three or more and the tensors at the vertices are symmetric and orthogonally decomposable, we provide an algorithm for recovering them subject to some rank conditions. Finally, in the case of tensor trains of length two in which the tensors at the vertices are orthogonally decomposable but not necessarily symmetric, we show that the decomposition problem reduces to the novel problem of decomposing a matrix into an orthogonal matrix multiplied by diagonal matrices on either side. We provide and compare two solutions, one based on Sinkhorn's theorem and one on Procrustes' algorithm. We conclude with a multitude of open problems in linear and multilinear algebra that arose in our study. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate some properties of an orthogonal complement of a subsemimodule and give some necessary and sufficient conditions for every subsemimodule to be equal to the orthogonal complement of its orthogonal complement. Then we obtain some necessary and sufficient conditions for a set of standard orthogonal vectors to be extended to a standard orthogonal basis of a semimodule. As applications, we present two equivalent descriptions for a semiring to be an exact ring by using terminologies of the orthogonal complements and extending orthogonal subsets of semimodules, respectively. [ABSTRACT FROM AUTHOR]

Let A be a C*-algebra with unit I and let S (A) be the state space of A . Let A ∈ A , an element f ∈ S (A) is said to be maximal for A if f(A*A) = ‖A‖2. Denote the set of all maximal states for A by S max (A) and define the algebraic maximal numerical range of A as follows V 0 (A) := { f (A) : f ∈ S max (A) }. We say that A is orthogonal to I in the Birkhoff–James sense, written as A⊥BJI, whenever ‖A‖ ≤ ‖A − λ‖, for all complex numbers λ. In this paper, we give a characterization of A⊥BJI in terms of the algebraic maximal numerical range V0(A). As applications, we give new numerical radius inequalities, generalize and improve earlier well-known results. [ABSTRACT FROM AUTHOR]

Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors. [ABSTRACT FROM AUTHOR]

In this paper, the super-biderivations on the super Heisenberg-Virasoro algebra are completely determined. We find that there exist non-inner super-skewsymmetric super-biderivations of the super Heisenberg-Virasoro algebra. [ABSTRACT FROM AUTHOR]

In this paper, we propose a feasible and effective iteration method for solving the tensor equation A ∗ N X ∗ M B = C with a tensor inequality constraint D ∗ N X ∗ M E ≥ F . The global convergence is established. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

As one application of the Cauchy–Binet formula, the Ball–Barthe inequality plays a key role in solving reverse isoperimetric inequalities. Recently, a new Grassmannian Ball–Barthe inequality was established. With it, reverse isoperimetric inequalities on Grassmann manifolds were successfully proved. In this paper, we will establish its matrix form with a simpler proof than that of Grassmannian form. The matrix form also recovers the classical Ball–Barthe inequality in a special case. [ABSTRACT FROM AUTHOR]

In this paper, we study 2-local superderivations on the Lie superalgebra of Block type S(q), which is an infinite dimensional Lie superalgebra with an outer derivation. We prove that all 2-local superderivations on S(q) are superderivations. [ABSTRACT FROM AUTHOR]

The symmetric Pascal matrix is a square matrix whose entries are given by binomial coefficients modulo 2. In 1997, Christopher and Kennedy defined and studied the binomial graph, which is the graph whose adjacency matrix is the symmetric Pascal matrix. They computed the spectrum of the binomial graph of order a power of 2. In this paper, we study spectral properties of the binomial graph of any order such as eigenvalues and eigenvectors, algebraic connectivities and inertia indices. We also compute the determinant of the symmetric Pascal matrix in modulo 3. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n, let D(G), Tr(G), DL(G) and DQ(G) be, respectively, the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix and the distance signless Laplacian matrix of G. For a real number α ∈ [0, 1], the generalized distance matrix Dα(G) is defined as Dα(G) = αTr(G) + (1 − α)D(G). In this paper, we establish relations between the smallest eigenvalues of D(G), DQ(G) and Dα(G). We obtain sharp upper and lower bounds for the smallest eigenvalue of Dα(G) in terms of various graph parameters like the order n, the diameter d, the Wiener index W(G), the chromatic number χ(G), the transmission degrees and the parameter α. We also identify the extremal graphs attaining the given bounds. [ABSTRACT FROM AUTHOR]

Schur's inequality for the sum of products of the differences of real numbers states that for x, y, z, t ≥ 0, xt(x − y)(x − z) + yt(y − z)(y − x) + zt(z − x)(z − y) ≥ 0. In this paper, we study a generalization of this inequality to more terms, more general functions of the variables and algebraic structures such as vectors and Hermitian matrices. [ABSTRACT FROM AUTHOR]

Let L be a finite-dimensional nilpotent Lie algebra of class two over a field F of characteristic different from two. The aim of the present paper is to give an upper bound for the dimension of the center factor L/Z(L) of a capable Lie algebra L. As a result, we also find a lower bound for the dimension of the derived subalgebra L2 for a capable Lie algebra L. Furthermore, we also give some criteria for determining the capability of L. [ABSTRACT FROM AUTHOR]

The formula for the spectral moment of graphs which is represented by the number of subgraphs is called the RNS spectral moment formula of graphs. In this paper, the research on the RNS spectral moment formula of graphs is generalized to hypergraphs. We give a formula for the spectral moment of a hypertree in terms of the number of some subhypertrees. Using this formula, we obtain the first 3k codegree coefficients of the characteristic polynomial of a k-uniform hypertree in terms of the number of some subhypertrees. [ABSTRACT FROM AUTHOR]

Perfect quantum state transfer plays a crucial role in quantum information processing and quantum computation. There has been extensive study of perfect quantum state transfer on Cayley graphs over abelian groups. In this paper, we consider the existence of perfect quantum state transfer on Cayley graphs over semi-dihedral groups which are non-abelian groups. Using the representations of semi-dihedral groups, we provide some necessary and sufficient conditions for Cayley graphs over semi-dihedral groups admitting perfect quantum state transfer. By those conditions, we present examples of perfect quantum state transfer on Cayley graphs over semi-dihedral groups. In addition, we propose results about whether some new Cayley graphs over non-abelian groups admit perfect quantum state transfer. [ABSTRACT FROM AUTHOR]

Let P and Q be projections on a Hilbert C * -module. This paper deals with the orthogonal complementarity of certain range closures associated with P and Q. Some necessary and sufficient conditions are derived under which (P, Q) is harmonious. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by Gn,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured that among all graphs in Gn,m, n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star Sn with m−n + 1 additional edges all connected to the same vertices for m ≤ n + ⌊ (n − 7) / 2 ⌋ , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy. [ABSTRACT FROM AUTHOR]

In this paper, first we give the cohomology theory of a 3-Lie algebra with a representation (called a 3- L i e R e p pair) and study the linear deformation theory of a 3- L i e R e p pair. We introduce the notion of a Nijenhuis structure on a 3- L i e R e p pair, which generates a trivial linear deformation. Then by adding compatibility conditions between Nijenhuis structures and relative Rota-Baxter operators, we introduce the notion of a relative Rota-Baxter-Nijenhuis structure on a 3- L i e R e p pair. Finally we construct relative Rota-Baxter-Nijenhuis structures on 3- L i e R e p pairs from those on L i e R e p pairs through certain linear functions. [ABSTRACT FROM AUTHOR]

In this paper, we present results concerning orthogonality in Hilbert C ∗ -modules. Moreover, for a C ∗ -algebra A , we prove theorems concerning the multi- A -linearity and its preservation by A -linear operators. New version of solution of the orthogonality equation on Hilbert C ∗ -modules and mappings preserving orthogonality are also investigated. [ABSTRACT FROM AUTHOR]

Let K n be the second-order cone in R n , where n ≥ ~3. Given a vector q ∈ R n and an n × n matrix G, the second order cone linear complementarity problem SOLCP(G, q) is to find a vector x ∈ R n such that x ∈ K n , y := G x + q ∈ K n and x T y = 0. The matrix G is said to have the Q-property if SOLCP(G, q) has a solution for all q ∈ R n . An n × n matrix G is called a cone automorphism if G K n = K n . In this paper, we obtain a simple characterization for the Q-property of a cone automorphism. This says that G has the Q-property if and only if zero is the only solution to SOLCP(G, 0). [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of tensor equations. Some explicit solutions to these equations are obtained. They are extensions of some results given by Lancaster. At the end of the article, these results are also employed to prove Lyapunov and Stein stability theorems for tensors. [ABSTRACT FROM AUTHOR]

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in Y ∗ and when this association forms a linear operator of norm-1 from Y ∗ to X ∗ . It is proved that, under certain geometric assumptions on X, Y, Z, these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if X ⊗ ε ∨ Y has property-U (SU) in X ⊗ ε ∨ Z. We prove that when X ∗ has the Radon–Nikodým Property for 1 < p < ∞, Lp(μ, Y) has property-U (property-SU) in Lp(μ, X) if and only if Y is so in X. We show that if Z⊆ Y⊆ X and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and Z (⊆ Y) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in c0 which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c0 which have property-U (SU). [ABSTRACT FROM AUTHOR]

In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0, 1) (0 < p ≤ 1). In particular, we prove similar results for the Hardy operator in weighted Lebesgue spaces. Also, it is proved that the grand Lebesgue space Lp)(0, 1) is a quasi-Banach function space. Finally, we establish necessary and sufficient conditions for the boundedness of some integral operator in weighted quasi-Banach Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Perfect state transfer plays a crucial role in quantum information processing and quantum computation. However, perfect state transfer is a rare phenomenon. As a generalization of perfect state transfer, perfect edge state transfer overcomes this disadvantage and there are more graphs admitting perfect edge state transfer than admitting perfect state transfer. In this paper, we reveal the relations between perfect state transfer and perfect edge state transfer on Cayley graphs. Furthermore, we present necessary and sufficient conditions for the existence of perfect edge state transfer on Cayley graphs of dihedral groups. With these conditions, several concrete constructions of Cayley graphs admitting perfect edge state transfer are proposed. Notably, this is the first attempt at perfect edge state transfer on Cayley graphs. [ABSTRACT FROM AUTHOR]

This paper establishes relationships between elliptic functions and Riordan arrays leading to new classes of Riordan arrays which here are called elliptic Riordan arrays. In particular, the case of Riordan arrays derived from Jacobi elliptic functions that are parameterized by the elliptic modulus k will be treated here. Some concrete examples of such Riordan arrays are presented via a recursive formula. [ABSTRACT FROM AUTHOR]

In this paper, a novel method is presented to compute an explicit formula for the inverse of the confluent Vandermonde matrices. The proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences, notably on the situations where the Vandermonde matrices have acquired much usefulness. The method presented here is direct and straightforward, it gives explicit and compact formulas. Several examples are presented to highlight the results. The main tools are some elementary basic linear algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study some parameters which determine the relative position of a pair of quaternionic subspaces and a pair of totally geodesic submanifolds in the real hyperbolic space. We extend some results about principal angles between two real and complex subspaces for quaternionic subspaces. We also prove that the distance and the principal angles of certain tangent spaces determine the congruence class of a non-asymptotic pair of totally geodesic submanifolds of the real hyperbolic space under the action of its isometry group. [ABSTRACT FROM AUTHOR]

It has been proved that every n × n doubly substochastic matrix A = [ a i j ] can be "increased" into a doubly stochastic matrix in the sense that there is a doubly stochastic matrix D = [ d i j ] with a i j ≤ d i j ([Marshall AW, Olkin I, Arnold BC. Inequalities; theory of majorization and its applications. 2nd ed. New York (NY): Springer; 2011], Theorem C.1). In this paper, we show this assertion does not necessarily hold for all infinite doubly substochastic matrices. Then we characterize all such matrices which are called increasable doubly substochastic matrices. We also provide an application of this result to the existence of a non-negative solution for some infinite linear equations. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit structure of ⊗ 3 H and ∧ 3 H where H is a generalized Heisenberg Lie algebra of rank at most 2. Moreover, for a non-abelian nilpotent Lie algebra L, we obtain an upper bound for the dimension of ⊗ 3 L. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, n ≥ 2 a positive integer, V 0 a n-dimensional vector space over F q and T 0 the set of all linear functionals from V 0 to F q . Let V = V 0 ∖ { 0 } and T = T 0 ∖ { 0 }. The linear functional graph of V 0 dented by ϝ (V) , is an undirected bipartite graph, whose vertex set V is partitioned into two sets as V = V ∪ T and two vertices v ∈ V and f ∈ T are adjacent if and only if f sends v to the zero element of F q (i.e. f (v) = 0). In this paper, the structure of all automorphisms of this graph is characterized and formalized. Also the cardinal number of automorphisms group for this graph is determined. [ABSTRACT FROM AUTHOR]

Let Γ be a graph with adjacency matrix A. The transition matrix of Γ corresponding to A is defined by H (t) := exp ⁡ (− ι t A) , where ι = − 1 and t ∈ R . The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence of real numbers { t k } and a complex number γ with unit norm such that lim k → ∞ H (t k) e u = γ e v. In this paper, we explore pretty good state transfer on Cayley graphs over semi-dihedral groups by using the representations of such groups. We show that graphs Cay (S D 8 n , S) have pretty good state transfer for some suitable subsets S if n is a power of 2. Moreover, we present a sufficient and necessary condition for a non-integral graph Cay (S D 8 n , S) to admit pretty good state transfer. Some concrete constructions of Cayley graphs over semi-dihedral groups having PGST are also presented. [ABSTRACT FROM AUTHOR]

This paper mainly deals with applying a general class of preconditioners to accelerate the convergence speed of some iterative schemes for solving multi-linear systems whose coefficient tensors are strong M -tensors. Theoretical results are established to analyse the performance of a general class of preconditioners extracted from the majorization matrix associated with the coefficient tensor. Brief discussions are included in using a Krylov subspace method based on the Hessenberg process to solve the mentioned problem. For the sake of generality, we consider multi-linear systems with multiple right-hand sides. Numerical experiments are reported to illustrate the validity of the presented theoretical results. [ABSTRACT FROM AUTHOR]

For a probability measure of compact support μ on the set P n of all positive definite matrices and t ∈ (0 , 1 ] , let P t (μ) be the unique positive solution of X = ∫ P n X ♯ t Z d μ (Z). In this paper, we show that if μ and ν are probability measures on P n and P m , respectively, and Φ : M n × M m → M k is a unital positive bilinear map, then Φ (P t (μ) , P t (ν)) ≤ P t (λ) for all t ∈ [ − 1 , 1 ] ∖ { 0 } , where λ is the push-forward of the product measure μ × ν by the mapping Φ | P n × P m . In particular, P t (μ) ⊗ P t (ν) ≤ P t (λ). Moreover, we show that P t (μ) ≤ I ⟹ P t p (ν) ≤ P t (μ) , for every p ≥ 1 , where ν (Z) = μ (Z 1 / p) and I denotes the identity matrix. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field of characteristic zero and G a finite cyclic group. Let V G be a variety of associative G-graded PI-algebras over F of finite basic rank. In this paper, we prove that if V G is minimal with respect to a given G-exponent, then there exist finite-dimensional G-simple F-algebras A 1 , ... , A m such that V G is generated by a suitable G-graded upper block triangular matrix algebra U T G (A 1 , ... , A m) endowed with an elementary grading and where the diagonal blocks are given by the A i 's. Moreover, for a fixed m-tuple (A 1 , ... , A m) of finite-dimensional G-simple F-algebras, we prove the converse of the above result for some important classes of G-graded algebras A = U T G (A 1 , ... , A m) endowed with elementary gradings. In particular, we conclude that the variety generated by A is minimal when A has one or two G-simple blocks as well whenever all (except for at most one) the G-simple components of A are G-regular. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the extensions of the Bessel matrix function and the modified Bessel matrix function. We first introduce the extended Bessel matrix function and the extended modified Bessel matrix function of the first kind by using the extended Beta matrix function. Then we establish the integral representations, differentiation formula, and hypergeometric representation of such functions. Finally, as an application, we study a kind of second-order matrix differential equations. We prove that the extended modified Bessel matrix function is a particular solution to this kind of differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we determine the energies of complements of unitary one-matching bi-Cayley graphs over finite commutative rings. We also determine the energies of some one-matching bi-gcd-graphs as well as the energies of complements of some one-matching bi-gcd-graphs, which are generalizations of unitary one-matching bi-Cayley graphs. [ABSTRACT FROM AUTHOR]

In this paper, we firstly establish new numerical radius inequalities which refine a result of Kittaneh in [Studia Math. 168, 73–80 (2005)], then present some numerical radius inequalities involving non-negative increasing convex functions for n × n operator matrices, which generalize the related results of Shebrawi in [Linear Algebra Appl. 523(15), 1–12 (2017)]. [ABSTRACT FROM AUTHOR]

In this paper we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution. We also present examples of all the different cases that our analysis exhibits for the nilpotency indices of the inner superderivations. [ABSTRACT FROM AUTHOR]

Recently, duality principles in g-frame theory were studied. This paper addresses a relaxation of g-R-duals. We introduce the notion of weak g-R-dual in g-frame theory setting, and present the link between frame properties of a bounded operator sequence and its weak g-R-duals. Using weak g-R-duals, we characterize g-frames and (unitary) equivalence between g-frames. And using pseudo-inverse operators, we represent the canonical duals of weak g-R-duals. Some examples are also provided. [ABSTRACT FROM AUTHOR]

The proper power graph P ∗ (G) of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups G 1 and G 2 , where at least one of G 1 and G 2 is an EPO group. Also we provide the full spectrum and hence the energy of P ∗ (G 1 × G 2) whenever both G 1 and G 2 are EPO groups. [ABSTRACT FROM AUTHOR]

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings. [ABSTRACT FROM AUTHOR]

A generalized tournament matrix M is a nonnegative matrix that satisfies M + M t = J − I , where J is the all ones matrix and I is the identity matrix. In this paper, a characterization of generalized tournament matrices with the same principal minors of orders 2, 3, and 4 is given. In particular, it is proven that the principal minors of orders 2, 3, and 4 determine the rest of the principal minors. [ABSTRACT FROM AUTHOR]

Brualdi and Hoffman [On the spectral radius of (0, 1)-matrices. Linear Algebra Appl. 1985;65:133–146] proposed the problem of determining the maximal spectral radius of graphs with a given size. In this paper, we consider the Brualdi–Hoffman-type problem for graphs with a given matching number. The maximal Q-spectral radius of graphs with a given size and matching number is obtained, and the corresponding extremal graphs are completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we prove the fundamental theorem of Hopf modules associated with a Hopf brace. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of banded matrices to matrices where two bands are allowed, called double banded matrices. Our main aim is to establish the periodicity of the determinants for (0 , 1) double banded matrices. As a corollary, we answer to two recent conjectures and other extensions. Several illustrative examples are provided as well. [ABSTRACT FROM AUTHOR]

We characterize extreme contractions defined between finite-dimensional polyhedral Banach spaces using k-smoothness of operators. As application of results obtained, we explicitly compute the number of extreme contractions in some special Banach spaces. Our approach, in this paper, in studying extreme contractions lead to the improvement and generalization of previously known results. [ABSTRACT FROM AUTHOR]

Let G be a graph with order n and size m. For each real number α ∈ [ 0 , 1 ] , the A α -matrix of a graph G is defined as A α (G) := α D (G) + (1 − α) A (G) , where D (G) = d i a g (d 1 , ... , d n) is the degree matrix of G, and A (G) is its adjacency matrix. The functions s (G) = ∑ 1 ≤ i ≤ n d i − 2 m n a n d v a r (G) = 1 n ∑ 1 ≤ i ≤ n d i − 2 m n 2 have been already used elsewhere as measures of graph irregularity. In this paper, we find several bounds for the spectral radius λ (G) of the matrix A α (G). Especially, when G is non-empty we prove for α ∈ [ 0 , 1) that s 2 (G) 2 n 2 n (2 α − α 2) Δ 2 + 2 (1 − α) 2 m ≤ λ (G) − 2 m n ≤ 1 + (2 α − α 2) n Ω (G) 2 + (1 − α) 2 s (G) , where Ω (G) = Δ (G) − ⌈ 2 m / n ⌉ + 1 and Δ (G) is the maximum vertex degree of G. These inequalities extend to the A α -matrix a Nikiforov's result involving the spectral radius of the adjacency matrix. In addition, for fixed order and size, we determine the graphs minimizing the degree variance v a r (G) and compute the A α -spectrum of some blown-up graphs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some inequalities for the Berezin number of operators on reproducing kernel Hilbert spaces by using the Krein–Lin inequality and refinements of the Young inequality. Also, we give upper bounds for b e r 2 (A) − b e r (A 2) on reproducing kernel Hilbert spaces. [ABSTRACT FROM AUTHOR]

Let G n , β be the set of simple graphs of order n with given matching number β. In this paper, we characterize the extremal graphs with maximal A α -spectral radius in G n , β for 0 ≤ α < 1 , which generalizes the results for adjacency matrix in [Feng L, Yu G, Zhang X-D. Spectral radius of graphs with given matching number. Linear Algebra Appl. 2007;422:133–138, Theorem 1.1] and signless Laplacian matrix in [Yu G. On the maximal signless Laplacian spectral radius of graphs. Proc Japan Acad Ser A. 2018, Theorem 2.5]. [ABSTRACT FROM AUTHOR]

In Markov chain theory, stochastic matrices are used to describe inter-state transitions. Powers of such transition matrices are computed to determine the behaviour within a Markov system. For this, diagonalizable matrices are preferred because of their useful properties. The non-diagonalizable matrices are therefore undesirable. The aim is to determine a nearby diagonalizable matrix A ~ , starting from a non-diagonalizable matrix A. Previous studies tackled this problem, limited to 3 × 3 stochastic matrices. In this paper, these results are generalized for n × n stochastic matrices. Spectral properties of A are preserved in this process, such that A and A ~ have coinciding semisimple eigenvalues and coinciding corresponding eigenvectors. This problem is examined and solved in this study and an algorithm is presented to find such a diagonalizable matrix A ~ . [ABSTRACT FROM AUTHOR]

In this paper, we propose generalized inverse power methods with variable shifts for finding the smallest H-/Z-eigenvalue and associated H-/Z-eigenvector of symmetric tensors. The methods are guaranteed to always converge to a H-/Z-eigenpair. Furthermore, for an even order nonsingular symmetric M -tensor, the proposed method with any positive initial point always converges to the smallest H-eigenvalue. Numerical results are reported to illustrate that the proposed methods often can find the smallest H-/Z-eigenvalue instead of other H-/Z-eigenvalues of symmetric tensors. Moreover, we can always get the smallest H-eigenvalue for an even order nonsingular symmetric M -tensor. [ABSTRACT FROM AUTHOR]