A real or complex n × n matrix is generalized doubly stochastic if all of its row sums and column sums equal one. Denote by V the linear space spanned by such matrices. We study the reducibility of V under the group Γ of linear operators of the form A 7→ PAQ, where P and Q are n×n permutation matrices. Using this result, we show that every linear operator φ : V → V mapping the set of generalize...

We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium if the row vectors of the random matrices, defining the stochastic constraints of each player, are elliptically symmetric distributed random vectors. We furth...

The polytope Q, of the convex combinations of the permutation matrices of order n is well known (Birkhoff’s theorem) to be the polytope of doubly stochastic matrices of order n. In particular it is easy to decide whether a matrix of order n belongs to Q,. . check to see that the entries are nonnegative and that all row and columns sums equal 1. Now the permutations z of { 1, 2, . . . . n} are i...

(1) 2) vertex models and analytic Bethe ansätze We introduce fusion U q (G (1) 2) vertex models related to fundamental representations. The eigenvalues of their row to row transfer matrices are derived through analytic Bethe ansätze. By combining these results with our previous studies on functional relations among transfer matrices(the T-system), we conjecture explicit eigenvalues for a wide c...

It is well known that the inverse C = c i;j ] of an irreducible nonsingular symmetric tridiagonal matrix is a Green matrix, i.e. it satisses c i;j = u i v j for i j and two sequences of real numbers fu i g and fv i g. A similar result holds for nonsymmetric matrices A. There the inverse is given by four sequences fu i g; fv i g; fy i g; and fy i g: Here we characterize certain properties of A i...

We present a multivariate generating function for all n × n nonnegative integral matrices with all row and column sums equal to a positive integer t , the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for t...

In this paper, we consider the relationships between the second order linear recurrences, and the generalized doubly stochastic permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Lucas Sequence, fLng ; is de…ned by the recurrence relation, for n 1 Ln+1 = Ln + Ln 1 (1.2) where ...

We consider the convex polytope x(x) that consists of those n X n (row) stochastic matrices having a common nonnegative (left) fixed vector rt. We examine the l-skeleton of d(x) and show how to construct all extreme points adjacent to a given one (as vertices of the l-skeleton). Connections with transportation polytopes are discussed. Further, we give a formula for the degree of an extreme poin...

A cluster algorithm for graphs called the Markov Cluster algorithm (MCL algorithm) is introduced. The algorithm provides basically an interface to an algebraic process de ned on stochastic matrices, called the MCL process. The graphs may be both weighted (with nonnegative weight) and directed. Let G be such a graph. The MCL algorithm simulates ow in G by rst identifying G in a canonical way wit...