Table of contents

This paper considers a boundary value problem for the equation

in some conical domains , where , , is a homogeneous polynomial of degree with real coefficients, and . An essential restriction on the domain is the following condition: the boundary contains no rays parallel to the -axis. The first part of the paper studies, for a wide class of domains , the asymptotics of a fundamental solution and the solution of a boundary value problem subject to the condition that the right-hand side and the boundary data tend rapidly to zero at infinity. In §3, for a specific domain and , a more involved case is examined, in which the right-hand side and the boundary data are unbounded. Bibliography: 13 titles.

It is shown that the "relaxed" Lie algebra of conformal transformations of the space generates quasiconformal flows. Analogous results are established for the Lie algebra of the group of isometries of Euclidean space. Bibliography: 19 titles.

In this paper it is proved that a topological foliation of codimension one on a three-dimensional sphere must have a compact leaf. Figures: 3. Bibliography: 12 titles.

This paper deals with inversion of the well-known theorems of Sylow and Hall on conjugacy of Sylow and Hall subgroups of finite and solvable finite groups. Bibliography: 17 titles.

In this paper the problem of describing blocking sets of terms is considered. A description of the singleton blocking sets, i.e. the blocking terms, is obtained. It is also shown that a blocking set whose terms collectively depend on a finite number of variables contains a blocking term. Bibliography. 5 titles.

The paper proves that the groups are trivial for all natural numbers , where is an arbitrary -algebra. Bibliography: 2 titles.

A homomorphism of a group into the multiplicative group of the ring of integers is called, in algebraic topology, an orientation homomorphism of the group . If is an element of the integral group ring , we will let denote the element . An element of the multiplicative group is called -unitary if the inverse coincides with or . The collection of all -unitary elements of the group form a subgroup . If , the group is said to be -unitary. Our study of the group is motivated by its appearance in algebraic topology, and was suggested by S. P. Novikov. The main result of this article consists of necessary conditions, given in terms of the kernel and an element such that , for the group to be -unitary. We also consider to what extent these conditions are sufficient. Bibliography: 3 titles.

Formulas are derived for solving an optimization problem with piecewise linear convex functions. Bibliography: 1 title.

Let a finite group have a -subgroup of order whose normalizer differs from and , and let the order of be odd and each coset of , for , contain an involution. Earlier the author (Math. USSR Sb. 36 (1980), 577-601) posed the question of the existence of simple groups other than with the indicated properties. In this paper it is proved that . The result includes theorems of Feit and Ito on Zassenhaus groups. Bibliography: 11 titles.

This paper establishes a general trace formula for Hecke operators acting on the space of vector-valued forms of real weight for an arbitrary Fuchsian group of the first kind in . Bibliography: 7 titles.

An existence theorem is proved in the control problem , where is a bounded functional of the sample functions of a counting process with intensity . It is assumed that satisfies a certain condition of weak dependence on the "tail" of the sample function. The proof is based on compactness considerations and makes essential use of a description of the extreme points of the set of admissible local densities. The Appendix gives a description of the set of extreme points for the family of distribution densities of diffusion-type processes relative to Wiener measure. Bibliography: 17 titles.

A number of properties of the Green function of the second mixed problem for a parabolic equation of second order on ( an arbitrary domain in ) are established. By means of these results a criterion is proved for the uniform stabilization of a solution: the existence of a uniform limit of the spherical mean of the initial function (extended by zero outside ) is necessary and sufficient for the uniform stabilization of a solution of the problem considered, with a bounded initial function under a certain condition on the unbounded domain . The basic properties of the Green function are obtained on the basis of an estimate of the solution of the problem with a compactly supported initial function in terms of the norm of the initial function in . Bibliography: 53 titles.

A scheme of finitely dependent (and, generally speaking, nonstationary) distribution of particles in a countable collection of cells is considered. Sufficient conditions are given for asymptotic normality of the random variables (the number of cells containing exactly particles each), (the number of occupied cells), and (the number of -fold repetitions). For these conditions correspond to the "left intermediate domain of variation of the parameters", while for they include also the "central domain". The method of moments is used in the proof. Bibliography: 6 titles.

It is shown that every irreducible unitary representation of a locally compact separable group with nontrivial first cohomology is unseparable from the identity representation in the space of irreducible unitary representations. For compactly generated groups the statement can be sharpened: a nontrivial cocycle in such a representation is in a certain sense the limit of trivial cocycles in irreducible representations. A survey is given of the connections between the notions indicated in the title. Bibliography: 17 titles.

The problem of averaging a system of ordinary differential equations is considered. When a number of conditions are imposed on the functions contained in the equation, the averaged equation is described, and it is shown that in classes of smooth functions the sequence of solutions of the original problems converges to the solution of the averaged problem with G-convergence or strong G-convergence depending on the conditions imposed. Bibliography: 15 titles.

This paper develops a procedure which enables one to construct microlocal asymptotic solutions of

(pseudo-)

differential equations with weighted symbols. Some theorems concerning local unsolvability and nonhypoellipticity are proved. Bibliography: 7 titles.

This work gives an estimate for the quasinorm of the th derivative of a function in the Hardy space , , using a modulus of continuity of order , specially introduced for . Also considered are applications of the results to the problem of imbedding of Hardy classes and the theory of approximation. Bibliography: 26 titles.