Table of contents

The asymptotics is found for a solution of the system of equations

of steady-state elastic vibrations of an isotropic medium. Here , is a small parameter, is a bounded closed surface given in spheroidal coordinates by the equation , and is the exterior of . The vector-valued function satisfies a radiation condition. The asymptotics of the solution of the problem is found up to , arbitrary, in the case where the boundary condition does not depend on the polar angle , and up to in the case of boundary conditions which are not axially symmetric. The formulas obtained are valid everywhere near the body (including neighborhoods of the end points) and far from it. Bibliography: 12 titles.

The author obtains an unimprovable estimate of the averaging method for a two-frequency problem with analytic right-hand sides under condition , which means a nonzero rate of change of the frequency ratio along trajectories of the averaged system. It turns out to be of order for initial data outside a set of measure of order , where is a small parameter of the problem and is an upper bound for the maximal multiplicity of the roots of a certain finite set of equations (it is assumed that ). Bibliography: 8 titles.

The main theorem asserts that the multiplicative group of a free product of rings, all of which satisfy the condition , with the amalgamated skew field , is a free product of a certain family of its subgroups with an amalgamated subgroup . As an application a ring is indicated for which the group is a nontrivial free factor of ( being any natural number greater than one). Bibliography: 12 titles.

Nondegeneracy and compatibility is proved for the period map on the strata of the discriminant of a versal deformation of a critical point of a function (under hypotheses indicated in the paper). The weight foliation on the discriminant of a versal deformation of is described. The singularities of the foliation are connected with the singularities of the degeneration of the symplectic structure on a four-dimensional manifold. Figures: 10. Bibliography: 30 titles.

The author obtains a description of the structure of a representation of the symmetric group in the space of -linear elements of the variety of Lie algebras generated by the Lie algebra of vector fields on the line. It is proved that this space, as an -module, is isomorphic to the space of homogeneous harmonic polynomials of degree in variables. Bibliography: 16 titles.

The orders of the upper bounds over certain classes of functions of two variables for approximations of these functions by sums of products of functions of the individual variables are obtained. As a corollary, optimal estimates are obtained for the singular numbers of integral operators and the Kolmogorov widths of classes of functions having an integral representation with kernels from the classes under consideration. Bibliography: 20 titles.

This article is devoted to the locally polynomially convex hull of a CR-manifold. 1) An "edge of the wedge" type theorem is obtained for piecewise smooth CR-manifolds in . 2) It is shown that a CR-manifold of class is locally polynomially convex if and only if in a neighborhood of each point it foliates into complex analytic submanifolds of maximal possible dimension. 3) It is shown that only locally polynomially convex CR-manifolds are examples of manifolds on which the tangential Cauchy-Riemann equations are solvable locally for any -closed form . Bibliography: 16 titles.

A wide class of unbounded domains is considered in complete Riemannian manifolds of negative curvature that are homeomorphic to planes, and the possibility of immersing them regularly and isometrically in three-dimensional Euclidean space is investigated. Let a metric of the manifold under consideration be given on the parameter plane by a line element of the form , where . The set is considered, where and is twice continuously differentiable. Let denote the corresponding domain on the manifold. Then the domain can be isometrically immersed in by means of a surface of class . This result is proved by constructing a smooth solution of a special form of the Gauss-Peterson-Codazzi system of equations in the domain . Figures: 2. Bibliography: 11 titles.

A unit vector field is considered, defined on some neighborhood in -dimensional Euclidean space , for which a formula is established that generalizes the Gauss-Bonnet formula. For this purpose, using the vector field , a map is constructed from an arbitrary hypersurface onto the -dimensional unit sphere . It is proved that the volume element of the sphere and the volume element of the hypersurface are connected under this map by the relation , where is the unit normal to and is a vector of the curvature of the field :

Here the are symmetric functions of the principal curvatures of the second kind of the field , . The flux of the vector field through a closed hypersurface , divided by the volume of the -dimensional unit sphere , equals the degree of the map of to determined by the vector field . For a field , given on all of , including the point at infinity, the Hopf invariant is calculated by use of the vector field . Bibliography: 5 titles.

Properties of convergence and uniform boundedness are studied for partial sums of a special trigonometric series with an algebraic polynomial in the exponent of the imaginary exponential. Bibliography: 9 titles.

On the normal bundle of a submanifold in a Riemannian space a natural Riemannian metric is introduced. The structure of surfaces with strongly parabolic normal bundle metric is determined. It is shown that the Sasaki metric of the normal bundle of vectors of fixed length of a two-dimensional Veronese surface has constant sectional curvature. Bibliography: 16 titles.

It is proved in this paper that, for every , each integral unimodular quadratic form is the intersection index form of some smooth closed manifold of dimension . The question is also studied of the realizability of such forms by manifolds with additional structures on the stable normal bundle and, as a consequence, of the realizability of forms by highly connected manifolds. Bibliography: 10 titles.

A solution of the problem of exact determination of a time optimal control for the equation , in the open loop form as well as in the closed loop form is presented in this paper. A system of special polynomials, called canonical variables, is introduced. The solution is obtained in terms of Hankel determinants and a sequence of Markov determinants in the canonical variables. A connection between the solution obtained and the power moment problem is investigated. Bibliography: 6 titles.

The index is estimated for invariant domains preserving the orientation of homeomorphisms of two-dimensional manifolds. Figures: 4. Bibliography: 5 titles.

A method asymptotic with respect to a small parameter is presented for solving Cauchy problems for the evolution equations

where is a linear operator and is a nonlinear operator. It is assumed that the method of regular expansion in powers of leads to secular terms. Such terms can be removed by suitably defining how the terms of the asymptotic solution depend on the slow variable . The proposed method is modified for equations of second order in . The possibility of getting rid of the terms secular with respect to , and of applying the asymptotic methods in the case of problems with strong forced resonance, is indicated. Examples are given which illustrate possibilities for the proposed methods. Bibliography: 16 titles.

The "dissipativity-conservativity" alternative is proved and an ergodic decomposition for solvable flows is obtained. Figures: 1. Bibliography: 14 titles.

The author proves theorems on the continuity of imbedding operators for abstract anisotropic spaces, which theorems arise in the investigation of the theory of boundary-value problems for partial differential-operator equations. Bibliography: 19 titles.