It is known (see [16]) that there are several notions of mixing (i.e. weak mixing, mixing, completely mixing e.c.t.) of measure preserving transformation on a measure space in the ergodic theory. It is important to know how these notions are related with each other. A lot of papers are devoted to this topic. In [1] a rather thorough study of the mentioned concepts of mixing for Markov operators...

Techniques based on minimal graph cuts have become a standard tool for solving combinatorial optimization problems arising in image processing and computer vision applications. These techniques can be used to minimize objective functions written as the sum of a set of unary and pairwise terms, provided that the objective function is submodular. This can be interpreted as minimizing the $l_1$-no...

We develop in finite case sufficient conditions for weak ergodicity of a nonstationary Markov chain r-states having transition matrices (Pn) n≥1 with Pn → P, where P has 2 irreducible and aperiodic closed classes and,possibly, transient states.

This thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and de Verdière (1985) and the quantum unique ergodicity conjecture of Rudnick and Sarnak (1994). The former states that, on any Riemannian manifold with negative curvature or ergodic geodesic flow,...

Weconsider basic ergodicity properties of adaptiveMarkov chainMonteCarlo algorithms under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant.

We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order |ln ~|−1 on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a simila...