E—matrices and Several Necessary and Sufficient Conditions

AER-x.,Aiscalledanonnegative(nonPositive)matrixifau>O(a,jO(aiirtO),markedA>O(AO(i#j),auO,wecallp(A)=max{IA,l,lA,l'.-.,lAnl},(whereA,,A2,..',A"areeigenva1ueofA)asspectralradiusofA.LemmalIfAe7'.,thenareanonnegativesquarematrxBandapositivenumberksuchthatA=B-kI(whereIistheidentitymatrixofordern).ProofLetbti=…     

Toeplitz and Hankel matrices on ℂ n

It is known that the characterizations of the Toeplitz operatorT onH ² and also the Hankel operatorH onH ² by using the simple unilateral shiftT _z . Recently, some characterization of the normal Toeplitz matrix truncated on ⁿ is given by D.R.Farenick, M. Krupnik, N. Krupnik and W. Y. Lee [1] and, independently, by T.Ito [2]. In this paper, we shall give some characterizations of the Toeplitz matrix and also the Hankel matrix truncated on ⁿ .Dedicated to Professor Masanori Fukamiya on his 88th birthday     

CM–fields and skew–symmetric matrices

Cohen and Odoni prove that every CM–field can be generated by an eigenvalue of some skew–symmetric matrix with rational coefficients. It is natural to ask for the minimal dimension of such a matrix. They show that every CM–field of degree 2n is generated by an eigenvalue of a skew–symmetric matrix over Q of dimension at most 4n+2. The aim of the present paper is to improve this bound.     

Time-band limiting matrices and lamé's equation

The continuity and differentiability for integral operator in Banach spaces are proved with respect to L^p–topology. The weak normal integrand is defined, so that a generalized measurable selection theorem is derived, and the conjugate functional for integral functional is formulated. The duality theorem of optimization problem for integral functional is established by using the conjugate functional.     

A short disproof of Euler’s conjecture based on quasi-difference matrices and difference matrices

In this note, two classes of quasi-difference matrices, $(2n+2,4;1,1;n)$-QDM and $(4n+1,4;1,1;2n?1)$-QDM, are constructed. Combining the known results of quasi-difference matrices and difference matrices, a new short disproof of Euler’s conjecture on mutually orthogonal Latin squares is given.     

Which nonnegative matrices are slack matrices?

In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.     

On eigenvalues of Seidel matrices and Haemers’ conjecture

For a graph G, let S(G) be the Seidel matrix of G and \({\theta }_1(G),\ldots ,{\theta }_n(G)\) be the eigenvalues of S(G). The Seidel energy of G is defined as \(|{\theta }_1(G)|+\cdots +|{\theta }_n(G)|\). Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least \(2n-2\), the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any \(\alpha \) with \(0<\alpha <2,|{\theta }_1(G)|^\alpha +\cdots +|{\theta }_n(G)|^\alpha \geqslant (n-1)^\alpha +n-1\) if and only if \(|\hbox {det}\,S(G)|\geqslant n-1\). This, in particular, implies the Haemers’ conjecture for all graphs G with \(|\hbox {det}\,S(G)|\geqslant n-1\). A computation on the fraction of graphs with \(|\hbox {det}\,S(G)|<n-1\) is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy \(|\hbox {det}\,S(G)|\geqslant n-1\). In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of order \(\Theta (n^{3/2})\). Finally, we prove that self-complementary graphs G of order \(n\equiv 1\pmod 4\) have \(\det S(G)=0\).     

Infinite Töplitz–Lipschitz matrices and operators

We introduce a class of infinite matrices , which are asymptotically (as |s| + |s′| → ∞) close to Hankel–T?plitz matrices. We prove that this class forms an algebra, and that flow-maps of nonautonomous linear equations with coefficients from the class also belong to it.     

Hermitian matrices and cohomology of Kähler varieties

We give some upper bounds on the dimension of the kernel of the cup product map $H^{1}(X,\mathbb{C}) \otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})We give some upper bounds on the dimension of the kernel of the cup product map , where X is a compact K?hler variety without Albanese fibrations.     

Generalized normal matrices

A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.     

Chained permutations and alternating sign matrices—Inspired by three-person chess

We define and enumerate two new two-parameter permutation families, namely, placements of a maximum number of non-attacking rooks on $k$ chained-together $n\times n$ chessboards, in either a circular or linear configuration. The linear case with $k=1$ corresponds to standard permutations of $n$, and the circular case with $n=4$ and $k=6$ corresponds to a three-person chessboard. We give bijections of these rook placements to matrix form, one-line notation, and matchings on certain graphs. Finally, we define chained linear and circular alternating sign matrices, enumerate them for certain values of $n$ and $k$, and give bijections to analogues of monotone triangles, square ice configurations, and fully-packed loop configurations.