Let X(t) , t >1 O, be a Polish space-valued r a n d o m process defined on a probabi l i ty space (~, ~ , P), where ~ is the P-comple t ion of a countab ly generated a-field ~ o . Let ~(t + ) (the ' l ook-ahead ' a-field at t) and r ) (the ' l ook-back ' a-field at t) denote the P-comple t ions of c~>,a(X(y), t <~ y <~ s) and c~s<ta(X(y), s ~< y ~< t) respectively. The aim of this note is to p ...

The study relates to the problem of cluster management in the conditions of sustainable development. Against the background of the assumptions and conditions for sustainable development, the specificity of the cluster activity in the conditions of the Polish economy has been presented. The objective of the paper is to characterize the cluster management standards in Poland. Such standards have ...

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K > 1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K > 1, K-bi-Lipschitz embed into the real line.

We present a simple proof of the fact that every countable group Γ is weak Rohlin, that is, there is in the Polish space AΓ of measure preserving Γ-actions an action T whose orbit in AΓ under conjugations is dense. In conjunction with earlier results this in turn yields a new characterization of non-Kazhdan groups as those groups which admit such an action T which is also ergodic.

A simple language is defined, a probabilistic semantics and a partial correctness logic is proposed in terms of probabilistic relations. It is shown how a derandomized semantics can be constructed through the Eilenberg-Moore algebras. We investigate the category of EilenbergMoore algebras for the Giry monad associated with stochastic relations over Polish spaces with continuous maps as morphism...

We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that Iso (U) has a dense locally fi...

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space X, showing that these groups do not admit a compatible Polish group topology and, in the case of Z-actions, are coanalytic nonBorel inside Homeo(X). We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a...

We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alon...