Let a finite partition F of the real interval (0, 1) be given. We show that if every member of F is measurable or if every member of F is a Baire set, then one member of F must contain a sequence with all of its finite sums and products (and, in the measurable case, all of its infinite sums as well). These results are obtained by using the algebraic structure of the Stone-Čech compactification ...

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a poi...

A pro-C∗-algebra is a (projective) limit of C∗-algebras in the category of topological ∗algebras. From the perspective of non-commutative geometry, pro-C∗-algebras can be seen as non-commutative k-spaces. An element of a pro-C∗-algebra is bounded if there is a uniform bound for the norm of its images under any continuous ∗-homomorphism into a C∗-algebra. The ∗-subalgebra consisting of the bound...

In this paper we study a notion of preorder that arises in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. In particular, we prove that there exist ultrafilters maximal for...

The aim of this work is to systematize the knowledge resulting from research on the impact of the feature selection on the quality of diagnostic procedures in the diagnosis of nonlinear systems. Particular attention was devoted to the selection of appropriate comparative criteria and optimization algorithms necessary for the selection of defects in the studied nonlinear systems, so that the inc...

The paper is concerned with the recognition of faces at application of different methods of global feature generation. We check the selected choice of transformations of images, leading to the numerical representation of the face image. The investigated approaches include the linear and nonlinear methods of transformation: principal component analysis (PCA), Kernel PCA, Fisher linear discrimina...

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)), Homeo(X)), where Exp(X) stands for the hype...

Introduction The Cech-Stone compactiication of the real line R seems to have been much less investigated than the space !. One reason for this is that ! can be attacked with two kinds of weapons: Boolean algebraic and topological. The space R is a continuum, so it is not susceptible to Boolean algebraic treatment. Be that as it may, R deserves more of our attention than it has received so far. ...