The Smith Normal Form Distribution of A Random Integer Matrix

We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μ p of SNF over ℤ/p s Z with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and compute the density μ for several interesting types of sets. As an application, we determine the probability that the cokernel of a random integer square matrix has at most ℓ generators for a positive integer ℓ, and establish its asymptotics as ℓ → ∞, which extends a result of Ekedahl (1991) on the case ℓ = 1

The Smith normal form distribution of a random integer matrix

International audienceWe show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets

Natural Density Distribution of Hermite Normal Forms of Integer Matrices

The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present an explicit computation of this distribution together with some applications. More precisely, for integer matrices whose entries are upper bounded in absolute value by a large bound, we compute the asymptotic number of such matrices whose HNF has a prescribed diagonal structure. We apply these results to the analysis of some procedures and algorithms whose dynamics depend on the HNF of randomly looking integer matrices

Joint Mixability of Elliptical Distributions and Related Families

In this paper, we further develop the theory of complete mixability and joint mixability for some distribution families. We generalize a result of R\"uschendorf and Uckelmann (2002) related to complete mixability of continuous distribution function having a symmetric and unimodal density. Two different proofs to a result of Wang and Wang (2016) which related to the joint mixability of elliptical distributions with the same characteristic generator are present. We solve the Open Problem 7 in Wang (2015) by constructing a bimodal-symmetric distribution. The joint mixability of slash-elliptical distributions and skew-elliptical distributions is studied and the extension to multivariate distributions is also investigated.Comment: 15page