Table of contents

We prove that a transitive topological Markov chain has almost-periodic points of all -periods. Moreover, every -period is realized by continuously many distinct minimal sets. We give a simple constructive proof of the result which asserts that any transitive topological Markov chain has periodic points of almost all periods, and study the structure of the finite set of positive integers that are not periods.

We give a classification of all compact homogeneous manifolds of dimension at most 7 up to a finite covering. Earlier classifications of this kind up to dimension 6 are obtained by a unified method. The main focus of the paper is on the case of dimension 7.

We use the Delaunay triangulation to construct a series of examples showing that in the multidimensional case, in contrast to the two-dimensional case, the approximation property for derivatives fails to hold for piecewise-linear approximations of a smooth function. These examples are closely related to the classical Schwarz example concerning piecewise-linear approximations of the lateral surface of a right circular cylinder.

We consider orthorecursive expansions (a generalization of orthogonal series) over families of non-orthogonal wavelets formed by the dyadic dilations and integer shifts of a given function . We estimate the rate of convergence of such expansions under some fairly relaxed restrictions on  and give examples of these estimates in some concrete cases.

We prove the equation for every nuclear Fréchet-Arens-Michael algebra  of finite weak bidimension, where is the weak global dimension and the weak bidimension of . Assuming that has a projective bimodule resolution of finite type, we establish the estimate , where is the global dimension and the bidimension of . We also prove that for all nuclear Fréchet-Arens-Michael algebras satisfying the Van den Bergh conditions . As an application, we calculate the homological dimensions of smooth and complex-analytic quantum tori.

We get an asymptotic formula for the mean value of Frobenius numbers with three arguments when averaged with respect to three parameters.

We find all homogeneous spaces of algebraic groups with one-dimensional stabilizer for which the action is co-isotropic (that is, the tangent space of a generic -orbit is co-isotropic).

We briefly review the results related to the notion of stability of convex sets and consider some of their applications. We prove a corollary of the stability property which enables us to develop an approximation technique for concave functions on a wide class of convex sets. This technique yields necessary and sufficient conditions for the local continuity of concave functions. We describe some examples of their applications.