Table of contents

The title of the paper by E. I. Khukhro (pp. 51-63) should read: Groups and Lie rings admitting an almost regular automorphism of prime order.

The authors solve the problem of recovering the matrix-valued potential , , from the given reaction operator , . They show the connections between this problem and the theory of boundary control, which allows them to obtain analogues of the classical Gel'fand-Levitan-Krein equations. They establish the basis property for a family of vector-valued exponentials; this property is connected with the spectral characteristics of the boundary value problem. They prove the controllability of the corresponding system under a boundary control .

It is proved that there are continuous positive weights such that the orthogonal polynomials constructed with respect to them are not uniformly bounded at a given point, both for the circle and for a closed interval. Furthermore, in the case of the circle the orthogonal polynomials have logarithmic growth. Also determined is a minimal (in a certain sense) class of positive continuous functions in which there exists a weight function having the property indicated.

The author studies the structure of the curvature tensor of -manifolds, i.e., almost Hermitian manifolds whose metric is, at least locally, conformally related to a Kählerian metric (with the same structure operator).

For second-order partial differential equations the question of whether they can have solutions decaying superexponentially at infinity is studied. An example is constructed of an equation Δu = q(x)u on the plane with bounded coefficients q having a nonzero solution decaying superexponentially. This example provides a negative answer to a familiar question of E. M. Landis. These questions are also studied for hyperbolic and parabolic equations on manifolds. An example is constructed of a parabolic equation having a nonzero solution u(x, t) decaying superexponentially as t→∞.

An orthogonal trigonometric basis in the space is constructed whose degrees have growth rate .

The time-dependent system of equations describing plane-parallel motion of Ostwald-de Waele media is considered in the approximation of magnetohydrodynamics. The system differs from the classical equations of magnetohydrodynamics by the presence in the leading term of power nonlinearities. The initial boundary value problem is solved with the conditions of diffraction of the physical characteristics on the boundary separating the media. Existence of a generalized solution is proved on the basis of the Faedo-Galerkin method and the method of monotone operators.

The asymptotic behavior is investigated for the classical Chebyshev-Hahn orthogonal polynomials , which form an orthogonal system on the set with the weight

and are such that . A weighted estimate is established as a corollary.

Conditions are found that are necessary and sufficient for a curve to bound the domain of convergence of a series in homogeneous polynomials. The conditions have a nonlocal character and can be expressed in terms of the Fourier coefficients of the Fourier coefficients of .

The inverse problem of recovering differential operators

from the Weyl matrix is investigated. A solution of this problem is given for arbitrary behavior of the spectrum, along with necessary and sufficient conditions and a uniqueness theorem.

Rationality of the moduli variety of curves of genus 5 is proved.

Elliptic problems of the form

are considered under appropriate conditions. This class of problems includes the inhomogeneous Emden-Fowler problem

with . The first part of this article is concerned with radial solutions, where and . The second part considers solvability in classes of functions with prescribed bound on decay at infinity, but without assumptions on radial symmetry.

This article presents a solution of the problem of one-sided holomorphic extension of a function defined on a smooth hypersurface dividing the unit ball in into two domains. In addition, it discusses the possibility of replacing the ball by another domain.

Solutions of differential equations of first and arbitrary order in Hilbert space are investigated; they arise in the study of elliptic problems in cylindrical domains and in domains with singular points. Existence theorems are obtained for a broad class of right sides, and the asymptotics of a solution as t→∞ is constructed under "minimal" conditions on the coefficients. The results make considerable progress possible in the study of qualitative properties of solutions of elliptic equations of higher order.

Let G be a group given by a free presentation G = F/N, and N' the commutator subgroup of N. The quotient F/N' is called a free abelianized extension of G. We study the homology of F/N' with trivial coefficients. In particular, for torsion-free G our main result yields a complete description of the odd torsion in the integral homology of F/N' in terms of the mod p homology of G.

It is proved that for any noncyclic hyperbolic torsion-free group there exists an integer such that the factor group is infinite for any odd . In addition, . (Here is the subgroup generated by the th powers of all elements of the groups .)

A solution is given of the restricted Burnside problem for 2-groups.

The Dirichlet kernel is defined for periodic functions of several variables; it consists of harmonics and has minimal order of the norm with respect to the choice of harmonics of the mixed Weyl derivative in the space . A similar problem on the minimal order of the norm is solved for the Favard kernel. Both problems generalize to the case of several derivatives.