Abstmct-We reveal an equivalence relation between the construction of a new class of low density MDS array codes, that we call B-Code, and a com-binatorial problem known as perfect one-factorization of complete graphs. We use known perfect one-factors of complete graphs to create constructions and decoding algorithms for both B-Code and its dual code.… (More)
Mason's Conjecture asserts that for an m-element rank r matroid M the sequence I k / m k : 0 ≤ k ≤ r is logarithmically concave, in which I k is the number of independent k-sets of M. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of M satisfies a strong negative correlation property we call… (More)
For a nite multigraph G, the reliability function of G is the probability R G (q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if q 2 C is such that R… (More)
In 1981, Stanley applied the Aleksandrov–Fenchel Inequalities to prove a logarithmic concavity theorem for regular matroids. Using ideas from electrical network theory we prove a generalization of this for the wider class of matroids with the " half–plane property ". Then we explore a nest of inequalities for weighted basis– generating polynomials that are… (More)
A Rayleigh matroid is one which satisfies a set of inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. We show that every matroid of rank three satisfies these inequalities.
Univariate polynomials with only real roots, while special, do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of this… (More)
Let A = a ij be a real n n matrix with non-negative e n tries which are weakly increasing down columns. If B = b ij is the nn matrix where b ij := a ij +z; then we conjecture that all of the roots of the permanent o f B, as a polynomial i n z; are real. Here we establish several special cases of the conjecture.
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial k m k (G)y k has only real nonpositive zeros, in which m k (G) is the number of k-edge matchings of a graph G. There is a stronger multivariate version of this theorem. We provide a general method by which " theorems of Heilmann-Lieb type " can be proved for a wide… (More)