Vol 71, No 1 (2016)

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in R^m that isometrically contains all finite metric spaces which can be embedded into R^m, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on R^m and, in particular, for the Euclidean metric.

Interrelation between full moduli of smoothness of natural orders in the metrics L₁ and L_∞ is considered in the paper.

Natural mechanical systems describing the motion of a particle on a two-dimensional Riemannian manifold of revolution in the field of a central smooth potential are studied. A complete classification of such Riemannian manifolds and potentials on them satisfying the strengthened Bertrand property, i.e., any orbit not contained in any meridian is closed, is obtained.

The paper is focused on some problems related to existence of periodic structures in words from formal languages. Squares, i.e., fragments of the form xx, where x is some word, and squares with one error, i.e. fragments of the form xy, where the word x is different from the word y in only one letter, are considered. We study the existence of arbitrarily long words not containing squares with the length exceeding l₀ and squares with one error and the length more than l₁ depending on the natural numbers l₀, l₁ For all possible pairs l₁ > l₀ we find the minimal alphabet such that there exists an arbitrarily long word with these properties over this alphabet.

A convergence criterion of a Noor-type iteration scheme with errors is proved for approximation of common fixed points of three sequences of uniformly quasi-Lipschitzian self-mappings of a closed convex subset in a complete convex cone metric space.