Table of contents

Actions of reductive groups on normal algebraic varieties with one-parameter families of spherical orbits of maximal dimension are studied under the assumption that the categorical quotient for the action is one-dimensional. As an application, the classification of the actions of the group on three-dimensional normal affine varieties is completed. The ground field is assumed to be algebraically closed and of characteristic zero.

The Sturm-Liouville operator on an interval with zero boundary conditions is considered; here is a strictly convex function of class on the real line and is a numerical parameter. The dependence of the eigenvalues of on is studied. The spectral analysis of the Schrödinger operator with magnetic field in a strip with Dirichlet boundary conditions on the boundary of the strip reduces to this problem. As a consequence of the main result the following theorem is obtained. Let be the restriction of to the interval and let be the periodic extension of on the entire axis (with period ). Then all the gaps in the spectrum of the Schrödinger operator are non-trivial.

The Hermite-Pade approximants are studied for systems of Markov functions (introduced in this paper) with structure described by a graph. Results of an asymptotic nature are stated in terms of certain equilibrium problems of potential theory concerning vector potentials.

Explicit constructions are given for certain residues and symbols from differentials and -groups of two-dimensional local fields to differentials and multiplicative groups of one-dimensional local fields. The maps obtained are used to construct the Gysin morphisms between the cohomology of the sheaves of regular differential forms and between the Chow groups in the case of a projective morphism of an algebraic surface onto an algebraic curve.

Hyperbolic systems of conservation laws with a functional-calculus operator on the right-hand side are considered in the space of second-order symmetric matrices. The entropies of such systems are described. The concept of a generalized entropy solution (g.e.s.) of the corresponding Cauchy problem is introduced, the properties of g.e.s.'s are analyzed, and the lack of their uniqueness in the general case is demonstrated. Using a stronger version of the defining entropy condition, the class of strong g.e.s.'s is distinguished. The Cauchy problem under discussion is shown to be uniquely soluble in this class.

On an arbitrary Riemannian symmetric space of rank 1 the Nikol'skii classes are defined by considering differences along geodesics. These spaces are described in terms of the best approximations by polynomials in spherical harmonics on , that is, by linear combinations of the eigenfunctions of the Laplace-Beltrami operator on . The results of Nikol'skii and Lizorkin on the approximation of functions on the sphere are generalized.

The present paper is devoted to the problem of contact equivalence of the Monge-Ampere equations with two independent variables. When the Monge-Ampere equation is in general position an affine connection can be associated with it in a natural manner. This association enables us to formulate and prove a number of criteria for the contact equivalence of the Monge-Ampere equations in general position that make use of the corresponding properties of affine connections.