Table of contents

In the present paper an area theorem is established for certain regular functions associated with multivalent mappings of a finitely connected domain onto a surface with parallel slits. Several consequences of this theorem generalize well-known results from the theory of univalent conformal mappings. The notion of the generalized span of a domain is introduced. It is then shown that it possesses certain properties completely analogous to the basic extremal properties of the span of a domain. Grötzsch's theorem concerning the range of the first coefficient of the regular part of the normalized Laurent expansion of a univalent function about a pole is extended to multivalent functions. Bibliography: 7 titles.

The author considers a stochastic differential equation in a half-space with boundary conditions of Ventcel'. Under mild regularity conditions imposed on the coefficients the existence of a solution is proved, and the Markov property thereof is studied. From the results the existence of a Markov process in a half-space, with corresponding generating operator and boundary conditions, is deduced. An ancillary estimate of the distribution density of a stochastic integral, which is of independent interest, is also proved. Bibliography: 20 titles.

A general theorem on estimating an integral of I. M. Vinogradov is proved, from which a number of consequences of independent interest are obtained. Bibliography: 16 titles.

Let be an entire function of order and an entire function of order with simple zeros . A series is assigned (according to a specific rule) to an arbitrary entire function of order . Necessary and sufficient conditions on are found under which this series always converges to in some topology. Bibliography: 5 titles.

A method of reducing the computation of -widths of compact sets of functions to the analogous problem for finite-dimensional compact sets is presented. Using this method the author obtains best possible (in the "power scale") estimates for Kolmogorov, Aleksandrov and entropy -widths of the class of functions , , that are -periodic in each variable, satisfy the inequality and have the property that any Fourier coefficients with at least one zero index must be equal to zero. Bibliography: 21 titles.

In this paper obstructions to the commutativity and associativity of multiplications in cobordism theories with singularities are determined. Obstructions to the existence and commutativity of multiplications are expressed in terms of Steenrod-tom Dieck operations in cobordism.

General theorems are applied to the cobordism theories SO*, U*, SU*, Sp* and Sc* with singularities. Associativity of multiplication is proved in those cases where it exists, as well as the existence of a commutative and associative multiplication if the operation of division by 2 can be carried out in the ring of scalars of a theory with singularities.

As an application of the main theorems, a uniqueness theorem for "generalized K-theories" is proved. Bibliography: 15 titles.

In this paper the author establishes the existence of a series

in some rearranged trigonometric system, which converges everywhere to a Lebesgue integrable function and whose coefficients and , , are not the Fourier-Lebesgue coefficients of this function. Similar results are established for the Haar system and some other systems. Bibliography: 17 titles.

Since I. N. Sanov proved the local finiteness of groups of exponent 4 in 1940, interest in such groups revived in the 1950's in connection with a question posed by G. Higman and M. Hall: Is the variety of groups of exponent 4 solvable? In recent years several attempts have been made to solve this problem in one aspect or another. In this paper a negative answer is given to the question of Higman and Hall. Bibliography: 8 titles.

In this paper the unirationality of supersingular K3 surfaces over a field of characteristic 2 is proved and a classification of such surfaces is given. Bibliography: 14 titles.

Uniform convergence of the Bieberbach polynomials is proved in the case of a simply connected region bounded by a Lipschitz Jordan curve. Bibliography: 11 titles.

In this paper a Hermitian analog of reduced K-theory is constructed. The author studies the reduced unitary Whitehead groups SUK₁(A) of simple finite-dimensional central algebras A over a field K, which arise both in unitary K-theory and in the theory of algebraic groups. In the case of discretely valued Hensel fields K, with this end in mind groups of unitary projective conorms are introduced, with the aid of which the groups SUK₁(A) are included in exact sequences whose terms are computable in many important cases. For a number of special fields K of significant interest the triviality of the groups SUK₁(A) is deduced from this. In addition, for an important class of simple algebras a formula is proved that reduces the computation of SUK₁(A) to the calculation of so-called relative involutory Brauer groups, which are easily computable in many cases. Furthermore, for an arbitrary field K the behavior of SUK₁(A) is described when K undergoes a purely transcendental extension, which in the case of division rings of odd index is a stability theorem important for many applications. Bibliography: 31 titles.