Let G be the group of n×n upper-triangular matrices with elements in a finite field and ones on the diagonal. This paper applies the character theory of Andre, Carter and Yan to analyze a natural random walk based on adding or subtracting a random row from the row above.

Let R = (r1, . . . , rm) and C = (c1, . . . , cn) be positive integer vectors such that r1 + . . .+ rm = c1 + . . .+ cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (“t...

Let Mn,m be the set of all n × m matrices with entries in F, where F is the field of real or complex numbers. A matrix R ∈ Mn with the property Re=e, is said to be a g-row stochastic (generalized row stochastic) matrix. Let A,B∈ Mn,m, so B is said to be gw-majorized by A if there exists an n×n g-row stochastic matrix R such that B=RA. In this paper we characterize all linear operators that stro...

We show that a cubic graph G with girth g(G) ≥ 5 has a Hamiltonian Circuit if and only if the matrix A+ I can be row permuted such that each column has at most 2 blocks of consecutive 1’s, where A is the adjacency matrix of G, I is the unit matrix, and a block can be consecutive in circular sense, i.e., the first row and the last row are viewed as adjacent rows. Then, based on this necessary an...

Let ∆n denote the set of n×n matrices of non-negative integers which have each row and column sum equal to k. Let Λn denote the subset of all binary matrices (matrices of zeroes and ones) in ∆n. If G is a bipartite multigraph let B(G) denote the usual ‘biadjacency’ matrix of G. That is, B(G) is the matrix with rows and columns respectively corresponding to the vertices in the two parts of G, an...

Let %I( R, S) denote the class of all m X n matrices of O’s and l’s having row sum vector R and column sum vector S. The interchange graph G( R, S) is the graph where the vertices are the matrices in %(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We characterize those 81 (R, S) for which the graph C( R, S) has diameter at most 2 and those YI( R, S) ...

We present results regarding row and column spaces of matrices whose entries are elements of residuated lattices. In particular, we define the notions of a row and column space for matrices over residuated lattices, provide connections to concept lattices and other structures associated to such matrices, and show several properties of the row and column spaces, including properties that relate ...

Let V be a vector space of dimension n over a fieldK of characteristic equal to 0 or ≥ n/2. Let g = gln(V ) and n be the nilcone of g, i.e., the cone of nilpotent matrices of g. We write elements of V and V ∗ as column and row vectors, respectively. In this paper we study the variety N := {(X,Y, i, j) ∈ n× n × V × V ∗ | [X,Y ] + ij = 0} and prove that it has n irreducible components: 2 of dimen...

Three decompositions of a substochastic transition function are shown to yield substochastic parts. These are the Lebesgue decomposition with respect to a finite measure, the decomposition into completely atomic and continuous parts, and on Rn, a decomposition giving a part with continuous distribution function and a part with discontinuous distribution function. Introduction. The present paper...