Table of contents

The article studies rings that are direct sums of an infinite family of subrings. The question of the completeness of these rings in maximal ring topologies is investigated for various classes of ring topologies. In particular it is shown that if the continuum hypothesis is assumed, then both complete and non-complete maximal ring topologies exist on a ring that is an infinite direct sum of copies of the same finite field.

We study the subgroups of over a field that comprise a conjugate in of the group of upper-unitriangular matrices of degree 2 over an infinite field such that is an algebraic extension.

The homology groups of the spaces of non-singular polynomial (of degree ≤ 4) embeddings are calculated. General algebraic techniques of such calculations or spaces of polynomial knots of arbitrary degree are described.

In this paper we consider a tree Lie algebra over a field of characteristic zero. This algebra is a module over the full linear group, and the spaces of homogeneous elements are invariant under this action. We study the decomposition of the homogeneous spaces into irreducible components and calculate their multiplicities. One method for calculating these multiplicities involves their connection with the values of the irreducible characters of the symmetric group on conjugacy classes of elements corresponding to a product of independent cycles of the same length. In the second section we give an explicit formula for calculating such character values. This formula is analogous to the hook formula for the dimension of the irreducible modules of the symmetric group. In the second method for calculating multiplicities we make use of Witt's formula for the dimensions of the polyhomogeneous components of a free Lie algebra. The rest of this paper deal with relations between the Hilbert series of a free two-generator Lie algebra and the generating series of the multiplicities of the irreducible modules in this algebra.

A homological characterization is given for groups admitting a presentation by means of defining relations of the form (the  are generators, ). The importance of such groups for geometry is connected with the fact that the finitely presented groups of this class are precisely the groups of knotted compact surfaces in .

Properties of strongly convex sets (that is, of sets that can be represented as intersections of balls of radius fixed for each particular set) are investigated. A connection between strongly convex sets and strongly convex functions is established. The concept of a strongly convex R-hull of a set (the minimal strongly convex set containing the given set) is introduced; an explicit formula for the strongly convex R-hull of a set is obtained. The behaviour of the strongly convex R-hull under the variation of R and of the sets is considered. An analogue of the Carathéodory theorem for strongly convex sets is obtained. The concept of a strongly extreme point is introduced, and a generalization of the Krein-Mil'man theorem for strongly convex sets is proved. Polyhedral approximations of convex and, in particular, of strongly convex compact sets are considered. Sharp error estimates for polyhedral and strongly convex approximations of such sets from inside and outside are established.

The Stone-Cech compactification of a discrete Abelian group is identified with the set of all ultrafilters on . The operation of addition on can be extended naturally to a semigroup operation on . A pair of ultrafilters on is a point of joint continuity for the semigroup if and only if the family of subsets forms an ultrafilter base. The main result of the present paper can be stated as follow: if is countable group with finitely many elements of order 2 and is a point of joint continuity for , then at least one of the ultrafilters of must be principal. Examples demonstrating that the restrictions imposed on are essential are constructed under some further assumptions additional to the standard axioms of ZFC set theory.

For function asymptotic equalities for the Hadamard determinants constructed from its Taylor coefficients are established. Using them, the asymptotics of the deviations from of its Padé approximations and of the corresponding rational functions of best uniform approximation is found for fixed as .