Let R denote the purely infinite simple unital Leavitt path algebra L(E). We completely determine the pairs of positive integers (c, d) for which there is an isomorphism of matrix rings Mc(R) ∼= Md(R), in terms of the order of [1R] in the Grothendieck group K0(R). For a row-finite directed graph E and field k, the Leavitt path algebra Lk(E) has been defined in [1] and [9], and further investiga...

We consider an autonomous system of partial differential equations for a onedimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Sma...

An old conjecture of Ringel states that every tree with m edges decomposes the complete graph K2m+1. The best lower bound for the order of a complete graph decomposed by a given tree with m edge is O(m). We show that asymptotically almost surely a random tree with m edges and p = 2m + 1 a prime decomposes K2m+1(r) for every r ≥ 2, the graph obtained from the complete graph K2m+1 by replacing ea...

We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. Ideas ...

In this paper we study the number of key exchanges required by Hoare’s FIND algorithm (also called Quickselect) when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed rank. After normalization we give a limit theorem where the limit law is a perpetuity characterized by a recursive distributional equation. To make the limit theorem usable ...

Wolfgang König and Patri k S hmid September 13, 2011 Abstra t We examine the non-exit probability of a multidimensional Brownian motion from a growing trun ated Weyl hamber. More pre isely, we ompute, for xed time t, the probability that the motion does not leave by time t the interse tion of a Weyl hamber and a t-dependent entred box, and we identify its asymptoti s for t → ∞. Di erent regimes...

We develop a weak exact simulation technique for a process X defined by a multidimensional stochastic differential equation (SDE). Namely, for a Lipschitz function g, we propose a simulation based approximation of the expectation E[g(Xt1 , · · · , Xtn)], which by-passes the discretization error. The main idea is to start instead from a wellchosen simulatable SDE whose coefficients are up-dated ...

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabási–Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel & Rácz [7] concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in...