A reinforcement-depletion urn model: A contiguity case

Summary An urn contains balls ofs different colors. The problem of the reinforcement of a specified color and random depletion of balls has been considered by Bernard (1977,Bull. Math. Biol.,39, 463–470) and Shenton (1981,Bull. Math. Biol.,43, 327–340), (1983,Bull. Math. Biol.,45, 1–9). Here we consider a special relation between a reinforcement and depletion, leading to a hypergeometric distribution. Research sponsored in part by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract DE-AC05-840R21400 with the Martin Marietta Energy Systems, Inc.     

Some autoregressive moving average processes with generalized Poisson marginal distributions

Some simple models are introduced which may be used for modelling or generating sequences of dependent discrete random variables with generalized Poisson marginal distribution. Our approach for building these models is similar to that of the Poisson ARMA processes considered by Al-Osh and Alzaid (1987,J. Time Ser. Anal.,8, 261–275; 1988,Statist. Hefte,29, 281–300) and McKenzie (1988,Adv. in Appl. Probab.,20, 822–835). The models have the same autocorrelation structure as their counterparts of standard ARMA models. Various properties, such as joint distribution, time reversibility and regression behavior, for each model are investigated.     

Estimation in the Markov-Pólya scheme

The Markov-Pólya urn scheme is considered, in which the balls are sequentially and equiprobably drawn from an urn initially containing a given numberaj of balls of thejth color,j = 1,…,N, and after each draw the ball is returned into the urn together withs new balls of the same color. It is assumed that at the beginning only the total number of balls in the urn is known and one must estimate its structure ā = (a1, …,aN) by observing the frequencies inn trials of the balls of corresponding colors. Various approaches including the Bayes and minimax ones for estimatingā under a quadratic loss function are discussed. The connection of the obtained results with known ones for multinomial and multivariate hypergeometric distributions is also discussed. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 373–382, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00387.     

Stochastic Dynamics of Compact Spins: Ergodicity and Irreducibility

Stochastic dynamics associated with Gibbs measures on M ^{Z d} , where M is a compact Riemannian manifold and Z ^d is an integer lattice, is considered. Equivalence of its L ²-ergodicity and the extremality of the corresponding Gibbs measure is proved.     

Monochromatic Cycle Partitions in Local Edge Colorings

An edge coloring of a graph is said to be an r‐local coloring if the edges incident to any vertex are colored with at most r colors. Generalizing a result of Bessy and Thomassé, we prove that the vertex set of any 2‐locally colored complete graph may be partitioned into two disjoint monochromatic cycles of different colors. Moreover, for any natural number r, we show that the vertex set of any r‐locally colored complete graph may be partitioned into disjoint monochromatic cycles. This generalizes a result of Erd?s, Gyárfás, and Pyber.     

Competition in growth and urns

We study survival among two competing types in two settings: a planar growth model related to two‐neighbor bootstrap percolation, and a system of urns with graph‐based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or , depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one‐for‐one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two‐type growth model on , one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi‐type processes, and list a number of open problems and conjectures.     

Partitioning 2‐Edge‐Colored Ore‐Type Graphs by Monochromatic Cycles

Consider a graph G on n vertices satisfying the following Ore‐type condition: for any two nonadjacent vertices x and y of G, we have . We conjecture that if we color the edges of G with two colors then the vertex set of G can be partitioned to two vertex disjoint monochromatic cycles of distinct colors. In this article, we prove an asymptotic version of this conjecture.     

A probabilistic approach to the first Dirichlet eigenvalue on non-compact Riemannian manifold

By using diffusion process with absorbing boundary, some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+∇h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean's estimate for ∇h=0. Moreover, the first Dirichlet eigenvalue for elliptic operators onR ^d and the first mixed eigenvalue are also studied. Some examples show that our estimates can be sharp even for ∇h≠0. Research supported in part by NFSC and the State Education Commission of China     

Isomorphisms of Galois groups of number fields with restricted ramification

Let K be a number field and S a set of primes of K. We write $K_S/K$ for the maximal extension of K unramified outside S and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: “For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\stackrel{\sim }{\to }{G}_{K_2,S_2}$ an isomorphism. Is σ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?” Here, the main assumption is about the Dirichlet density of $S_i$.     

Monochromatic Kr‐Decompositions of Graphs

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic H‐decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let be the smallest number ? such that any graph G of order n and any coloring of its edges with k colors, admits a monochromatic H‐decomposition with at most ? parts. Here, we study the function for and .