Table of contents

In this paper we construct an algorithm recognizing the solvability of arbitrary equations in a free semigroup. Bibliography: 4 titles.

Conditions are obtained for the existence of "exterior" asymptotics for orthogonal polynomials. In particular, it is shown that if almost everywhere on the interval ( is a nondecreasing function on ), then for the corresponding orthonormal polynomials the relation holds on compact subsets of . The branch of the square root is chosen so that in the region described. Bibliography: 6 titles.

In this paper a necessary and sufficient condition is obtained for the existence of a proper limit distribution, not concentrated at one point, of the number of particles in a critical Markov branching process with several types of particles on the set of nondegenerate trajectories under the classical normalization. In the case where the limit distribution in question exists, limit distributions for the distance to the nearest common ancestor are obtained. Bibliography: 10 titles.

We show that the least uniform rational deviations from the function , continuous and convex on the interval , satisfy the condition as , and that uniformly for the continuous convex functions whose absolute values are bounded by unity. These estimates are precise with respect to the rate of decrease of the right-hand sides. Bibliography: 16 titles.

In the paper we find the structure of groups given by systems of generators and power relations of a special kind, and we prove that the word problem and the conjugacy problem are solvable. The groups of birational automorphisms of minimal cubic surfaces belong to this class, and this enables us to solve a problem of Manin on the enumeration of automorphisms of finite order. Bibliography: 4 titles.

It is proved that a periodic loop imbeddable in an alternative PI-ring is locally finite, and also that a periodic loop of bounded index which is imbeddable in a Cayley-Dickson algebra over a field of characteristic zero is finite. Bibliography: 11 titles.