Table of contents

A characterization is given of boundary conditions for linear differential operators in a domain with arbitrary boundary, and these conditions are used to state a boundary-value problem in a suitable function class. It is proved that this "generalized Dirichlet problem" has a unique solution which depends continuously on the boundary conditions. Bibliography: 3 titles.

In this paper, the Dirichlet problem is studied for degenerate nonlinear Bellman equations. The main result is an estimate on the second mixed derivative of the solution on the boundary. In some cases this estimate yields estimates on all second derivatives both inside and on the boundary. As an example, the elementary Monge-Ampère equation is studied on a smooth strictly convex domain, and the existence of a solution smooth up to the boundary is established. The method of estimating the second mixed derivatives is based on the reduction to an estimate of the first derivatives of the solution of an auxiliary equation on a suitable closed manifold without boundary. Bibliography: 16 titles.

A theorem is proved on the asymptotic behavior of meromorphic functions of completely regular growth (as previously defined by the author) as outside a set of zero linear density. For entire functions of completely regular growth a uniformity property is established, and some of its applications are presented. An upper bound for the number of deficient values (in the sense of R. Nevanlinna) of such functions is also obtained. Bibliography: 11 titles.

This paper considers multivalued mappings which map a compact metric space into the space of nonempty closed subsets of . A theorem asserting the existence of a continuous branch of such a mapping is proved. This theorem is analogous to a theorem of Michael. As corollaries, theorems on the existence of fixed points of multivalued mappings and on the existence of solutions of differential inclusions are proved. Bibliography: 13 titles.

In this paper conditions are given for the spectrum in an eigenvalue problem of the form to be discrete, where and are operators that are odd-homogeneous of degree (), acting from a reflexive Banach space into the dual. It is proved that the eigenvalues vary monotonically as and vary in the normed linear space of homogeneous operators of degree . Explicit formulas for the eigenvalues and functions are obtained for the case where and are the gradients of the norms in the spaces and ( is a parallelepiped in ). Using these formulas the author obtains estimates for the eigenvalues in homogeneous and asymptotically homogeneous problems with variable coefficients in the space , where is an arbitrary bounded domain in . Bibliography: 12 titles.

The main result of the paper consists in the construction of a model of the full linear group over a finite field, i.e. its representations such that each irreducible representation occurs as a component precisely once. The series of representations thus constructed has the well-known Gel'fand-Graev representation as first term. Bibliography: 12 titles.

The author continues the study of the inverse kinematic problem of diffraction from polycrystalline objects in Sobolev spaces of automorphic functions on the three-dimensional rotation group. An effective intrinsic description is obtained for the orthogonal complement of the subspace of common zeros of a finite family of diffraction operators. Based on this description, a projection method is proposed for constructing an -optimal solution of the diffraction problem with incomplete data. Bibliography: 7 titles.

In this paper the Cauchy problem for the Korteweg-de Vries equation , , , with initial condition is considered in nonlocal formulation. In the case of an arbitrary initial function the existence of a generalized -solution is proved, and its smoothness is studied for . A class of well-posed solutions is distinguished among the generalized solutions under consideration, and within this class theorems concerning existence, uniqueness and continuous dependence of solutions on initial conditions are proved. Bibliography: 28 titles.

This paper continues an earlier one (Math. USSR Sb. 44 (1983), 471-481). A function measuring the extent to which a Riemannian space is nonisotropic at the point is studied. Using , definitions of the notion of correctness of Schur's theorem are given in the multidimensional case. The relations between these definitions are clarified, and sufficient conditions for the correctness of Schur's theorem are given. It is shown that by a small deformation of the given metric it is possible to obtain one in which Schur's theorem is not correct. The methods developed in the paper are applied to study some geometric properties of geodesically parallel surfaces. Bibliography: 11 titles. Figures: 1.

In this paper the author constructs a chain ring (i.e. a ring in which the right and left ideals are linearly ordered by inclusion) with the following properties: 1) is a prime ring; 2) the Jacobson radical of is a simple chain ring (without identity); 3) each element of is a right and left zero divisor. This example gives an answer to one of Brung's questions. In addition, the ring is totally singular, i.e. it coincides with its right (left) singular ideal. The construction is based on a theorem that permits one to assign a chain ring to a right ordered group whose group ring can be imbedded in a division ring. Bibliography: 9 titles.

Results are established on uniform approximation of functions that are continuous on compact subsets of the complex plane and holomorphic in their interiors, by lacunary polynomials whose gaps are of zero or positive density. These generalize and sharpen previous results in this direction by N. U. Arakelyan and the author. Results similar to the Walsh-Lebesgue theorem are given for lacunary polynomials, as well as an inequality for a majorant of the coefficients of the approximating polynomials. Bibliography: 19 titles.

The case is considered of a critical fixed point of a diffeomorphism of codimension 2 whose linear part has the eigenvalues . According to ideas developed by Takens and Arnol'd, to deformations of such diffeomorphisms there correspond families of vector fields invariant with respect to an involution of the plane, namely, a reflection relative to a line passing through the fixed point. Bifurcations in two-parameter families in general position are described. Rigorous proofs are given. Figures: 2. Bibliography: 11 titles.

The connection is considered between the asymptotic behavior of the poles of the mth row in the Padé table of a function f and an m-meromorphic continuation of it. Bibliography: 3 titles.

This article deals with the operator of multiplication by an entire function with indicator when the order is . This operator acts from to , where is a sequence of indicators,

with the standard space of entire functions. It is assumed that the spaces are isomorphic, with respect to a transformation of Borel type, to spaces of functions analytic on many-sheeted closed sets. A criterion is found for the range of to be closed. It is used to derive, in particular, a criterion for an operator of convolution type in a union of -convex domains to be an epimorphism, along with known results about convolution operators and operators of convolution type. The conditions connect the directions of non-completely-regular growth of and of accumulation of its zeros with geometric characteristics of . Bibliography: 26 titles.

This paper studies unary operators in tensor fields on a symplectic supermanifold that are invariant with respect to canonical changes of coordinates. The formal analogue of this problem is considered - operators in formal tensor fields. A complete classification is given for such operators in the case when the image of an operator lies in a tensor field whose fiber is the direct sum of spaces generated by the highest vector. Figures: 5. Bibliography: 5 titles.

The question is considered of the existence of a subsequence of the th row of the Padé table of a function that converges uniformly on compact subsets of the disk ( the radius of -meromorphy of ) which do not contain poles of this function. Bibliography: 8 titles.

The author considers an initial-boundary value problem for the hyperbolic equation

in a rectangle (here is a small parameter and ). It is assumed that the initial and boundary values of the function coincide at the lower corners of the rectangle. A complete asymptotic expansion of the solution in powers of is constructed everywhere in the rectangle. Bibliography: 5 titles.

The author studies basis invariants of finite groups generated by reflections in the real space . Bibliography: 27 titles.

Necessary and sufficient conditions are established for the existence of limits in the sense, , on the boundary of a domain, of solutions of second order elliptic equations in domains with Lyapunov boundaries. Bibliography: 8 titles.