Marc A. Berger

Mixing Markov Chains and Their Images

Recently, orbits of two-dimensional Markov chains have been used to generate computer images. These chains evolve according to products of i.i.d. affine maps. We deal with mixing models, whereby one mixes together several of these Markov chains, so as to create a mixed image. These mixtures involve starting one Markov chain off at the stationary distribution of another, and then running it for a geometrically distributed number of steps. We use this to analyze various mixing scenarios.

Random walks generated by affine mappings

This article is concerned with Markov chains on ℝm constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.

New results for covering systems of residue sets

We announce some new results about systems of residue sets. A residue set R C Z is an arithmetic progression R = {a,a±n,a± 2n,...}. The positive integer n is referred to as the modulus of R. Following Znam [21] we denote this set by a(n). We need several number-theoretic functions. p(ra)-the least prime divisor of a natural number ra, P(m)-the greatest prime divisor of ra, A(m)-the greatest divisor of m which is a power of a single prime: A(ra) = max{d G Z: d\m, d = p s , p prime}, /(ra) = Ylj=i s j(Pj ~ 1) + 1> where ra has the prime factorization ra = Si Si g(m) = rij-iU + X J) ~ Ei=i x 3 ~ !> where Z^k=o Pj Pj 2^k=o Pj and m has the above prime factorization, <p(ra)-Eulers totient function, [x]-the greatest integer in x.a t (nt)), t > 1, which partition Z. The multiplicity of a modulus n = rik is the number of sets in D with that modulus. The multiplicity of D is the maximum multiplicity of its moduli. THEOREM 1. The multiplicity of any modulus n = rik is at least (1) mi = min A (-, r). ni^n \{n,ni)J The multiplicity of D is at least (2) m 2-N + 1,

A non-analytic proof of the Newman:80Zna´m result for disjoint covering systems

A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes.

Disjoint covering systems of rational Beatty sequences

Abstract We prove that for a disjoint covering system with rational moduli, the two largest numerators of the moduli are identical. Furthermore, if the two moduli corresponding to these two identical numerators are distinct, then actually the three largest numerators of the moduli are identical for a system with at least three moduli.

Irreducible disjoint covering systems (with an application to Boolean algebra)

Abstract We develop the lattice geometry which corresponds to irreducible disjoint covering systems of residue sets. For such systems Korec has established a stronger version of Mycielskis inequality, which gives a lower bound on the size of the system in terms of its moduli. We show how the lattice geometry can be used to interpret this and obtain a stronger result. Of particular interest here are the collapsible cell partitions, which are the geometric analogue of the natural disjoint covering systems introduced by Porubský.

Sign patterns of matrices and their inverses

Abstract We study the sign pattern relationship between a matrix and its inverse. To do so we examine the graph whose vertices are the sign pattern matrices, and whose edges connect those which are possible sign patterns of a matrix and its inverse. We have bounds for the degree, connectivity, radius, and diameter of this graph. In addition we have the complete description for the 3 × 3 case.

Lattice parallelotopes and disjoint covering systems

Abstract The purpose of this work is two-fold. First, to establish a connection between certain partitions of lattice point parallelotopes, and disjoint (or exact) covering systems of residue sets. The study of these partitions involves both geometry and combinatorics. Second, as particular consequences of this connection, Theorem 4.1 and Corollary 5.2 are obtained. Theorem 4.1 relates the multiplicity of a disjoint covering system to the prime factors of its moduli. This result can be interpreted either as a lower bound for the multiplicity, given the prime factors, or as an upper bound for the prime factors, given the multiplicity (cf. Burshtein [13]). In particular, it establishes Burshteins conjecture (in fact somewhat more) for general disjoint covering systems. Corollary 5.2 proves a Newman-Znam type lower bound [17, 19] for the multiplicity of maximal moduli in systems of residue sets whose covering functions are identically constant modulo γ, for some number γ. These systems are generalizations of disjoint covering systems, for which the covering function is identically one.

A trotter product formula for random matrices

This note contains a generalization of the Trotter product formula to the setting of multiple linear systems of stochastic differential equations. From the result it follows that the solution of a ...

Product formulas for solutions of initial value partial differential equations, II

Abstract Families of operators which approximate semi-groups or evolution systems generated by partial differential operators are constructed. Product formulas are used to recover these semi-groups or evolution systems through product integrals. Conditions on generators are provided under which its semi-group or evolution system can be approximated in this way by families of specific types of operators.