We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, and

We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the Hammersley–Clifford theorem; however, we show that for nondecomposable graphical models they are not. We also show that nondecomposable models can have nonrational maximum likelihood estimates. These results are used to give several novel characterizations of decomposable graphical models.

The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987, see [S-Ch(1987)] and an improved version in [K-S-Sz(1990)]. It provides sucient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently ([B-Ch-SzT (2000)]) that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are de ned by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models { among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Rehacek stadia.

It is well known that an interleaver with random properties, quite often generated by pseudo-random algorithms, is one of the essential building blocks of turbo codes. However, randomly generated interleavers have two major drawbacks: lack of an adequate analysis that guarantees their performance and lack of a compact representation that leads to a simple implementation. In this paper we present several new classes of deterministic interleavers of length , with construction complexity ( ), that permute a sequence of bits with nearly the same statistical distribution as a random interleaver and perform as well as or better than the average of a set of random interleavers. The new classes of deterministic interleavers have a very simple representation based on quadratic congruences and hence have a structure that allows the possibility of analysis as well as a straightforward implementation. Using the new interleavers, a turbo code of length 16384 that is only 0.7 dB away from capacy at a bit-error rate (BER) of 10 5 is constructed. We also generalize the theory of previously known deterministic interleavers that are based on block interleavers, and we apply this theory to the construction of a nonrandom turbo code of length 16384 with a very regular structure whose performance is only 1.1 dB away from capacity at a BER of 10 5 .

We consider the system of N (# 2) hard balls with masses m 1 , . . . , mN and radius r in the flat torus T of size L, # 3.

This paper addresses the widely observed breakdown in multigrid performance for turbulent

1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8

Abstract. We show that the simultaneous stabilizability of three linear systems, that is the question of knowing whether three linear systems are simultaneously stabilizable, is an undecidable question. It is undecidable in the sense that it is not possible to find necessary and sufficient conditions for simultaneous stabilization of the three systems that involve only a combination of arithmetical operations (additions, substractions, multiplications and divisions), logical operations (`and' and `or') and sign tests operations (equal to, greater than, greater than or equal to,...) on the coefficients of the three systems.

Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial Branch-and-Bound (sBB) algorithms.

... (k 2 N), where x 2 [a; b] and a! 0! b; the derived relaxation is continuous and differentiable everywhere in [a; b]. We then make a comparison with an existing convex relaxation for odd power monomials where we show that the novel convex relaxation gains better results in a Branch-and-Bound implementation.