The proof relies on the proof of Lemma 3.6, below. Let z, δ ∈ R, and let δ → 0. Suppose z + δ converges to a point that falls into one of the first three cases given in Section 2. Then, from the statement and proof of Lemma 3.6, an element JPK∗exp (z + δ) of the generalized Jacobian of the projection onto the dual of the exponential cone at z + δ, is just a matrix with fixed entries, since proj...

A polynomial matrix G(z) = Izm −∑m−1 i=0 Ciz with complex coefficients is called discrete-time stable if its characteristic values (i.e. the zeros of detG(z) ) are in the unit disc. A corresponding block companion matrix C is used to study discrete-time stability of G(z) . The main tool is the construction of block diagonal solutions P of a discrete-time Lyapunov inequality P−C∗PC 0 .

The distributions of the smallest and largest eigenvalues for the matrix product Z † Z, where Z is an n × m complex Gaussian matrix with correlations both along rows and down columns, are expressed as m × m determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently fo...

We show that Belavin’s solutions of the quantum Yang–Baxter equation can be obtained by restricting an infinite R-matrix to suitable finite dimensional subspaces. This infinite R-matrix is a modified version of the Shibukawa–Ueno R-matrix acting on functions of two variables. (hep-th/9402011) Shibukawa and Ueno [1] have defined an elliptic R-operator acting on the space of functions of two vari...

Monomers of azobenzene were isolated in argon matrices at 15 K and characterized by infrared spectroscopy and theoretical calculations. When the equilibrium vapors existing over the azobenzene crystals at room temperature were trapped in the matrix, only the thermodynamically most stable E-azobenzene was detected. In an attempt to convert E-azobenzene into the Z isomer, the matrix-isolated E-mo...

Abstract. The matrix valued analog of the Euler’s hypergeometric differential equation was introduced by Tirao in [1]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao’s equation in order to get a generalized matrix valued hypergeometri...

Among them, Y is the n q  random observation matrix; e is the n q  random error matrix, , nm kq X Z is the design matrix that we are known, and   r X m  ,   r Z k  , B is the p k  regression parameter matrix that we are unknown, V and  are positive definite matrices with the size q q  and n n  -order respectively.  expresses product of matrix Kronecker. e expresses a column vector...

In this paper we deal with orthogonal matrix polynomials. First of all, we establish some basic notations and results we need later. A matrix polynomial P is a matrix whose entries are polynomials, or, equivalently, a combination P(t) = A 0 +A 1 t+ +A n t n , where A 0 ; ; A n are numerical matrices (the size of all the matrices which appear in this paper is N N). A positive deenite matrix of m...

Using an invariant of Cuntz, we classify reducible shifts of nite type with two irre-ducible components up to ow equivalence. 0. Introduction Let A be an n n matrix over the non-negative integers Z +. Form a directed graph G A with n vertices and A(i; j) edges from vertex i to vertex j. Let E be the set of all edges of G A endowed with the discrete topology. Then the product space E Z is a comp...

We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking S-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szegö class; on the other hand the transmission coefficient remains the same. This implies the statement given in th...