Table of contents

In this paper we study the properties of the spectrum of the boundary-value problem

Let be the points of the spectrum of this problem, arranged in order of increasing absolute value. Our main result isTheorem.Let satisfy the conditions

Then for any

    for 

Bibliography: 2 items.

The main result is a description of homogeneous (i. e. invariant relative to left and right shifts) algebras with uniform convergence on a compact group. As a corollary we obtain a generalization of a theorem of Rider: let the real annihilater of a homogeneous antisymmetric algebra be separable in the topology of the definite norm in the conjugate space. Then the connected component of the identity of the group is commutative and . Rider proved that if , then is commutative and connected. Bibliography: 3 references.

We construct elements of the matrix which connects different bases for class I representations of the group SO(2,1). These matrix elements are expressed in terms of Whittaker functions. In this way integral relations are obtained for these and orhte special functions. Bibliography: 5 items.

In his book Problèmes concrets d'analyse fonctionnelle, Paul Lévy introduced the concept of the mean of the function on Hilbert space over the ball of radius with center at the point , and investigated the properties of the Laplacian

but he did not determine which functions have means. Moreover, the mean and the Laplacian are not invariant, in general, under rotation about the point . In the present paper we give a class of functions with invariant means on Hilbert space. An example of such a class is the set of functions for which , where the function is uniformly continuous and has invariant means, is the identity operator, and is a symmetric, completely continuous operator whose eigenvalues, arranged in decreasing order of absolute value , have the property that uniformly in (§3). The invariant mean of such a function exists and is given by the formula

and its Laplacian is . In §4 we consider the Dirichlet problem and the Poisson problem for the ball and give sufficient conditions for the solution to be expressed by the Lévy formulas. Bibliography: 7 entries.

In this paper we generalize the concepts of wave operators and scattering operators. We find sufficient conditions for the existence of generalized wave operators for the Friedrichs model with discontinuous kernel, and for differential operators with potential of Coulomb type. Bibliography: 16 items.

In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations

in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the "vanishing viscosity method"; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity. Bibliography: 22 items.

In this article we consider three-dimensional PM manifolds with positive curvature which are homeomorphic to a ball and have convex boundary, For these PM manifolds there is defined in a natural way the radius of the inscribed sphere and the integral mean curvature of the boundary. The new results consist of a proof of the estimates

where is the volume of the PM manifold, is the diameter, is the area of the boundary and is the intrinsic diameter of the boundary. Incidentally, properties of geodesics and the construction of their boundaries are investigated. The results obtained are completely analogous to the two-dimensional case. In particular, a construction is investigated similar to the special case of cutting out lunes from a two-dimensional PM manifold: it is shown that the union of the geodesics joining an interior point of the PM manifold to a point on the boundary form a finite collection of tetrahedra which are glued together into a "three-dimensional cone" after cutting out from the PM manifold the "remaining material". Bibliography: 9 items. 11 figures.

It is shown that each locally unknotted simple arc in three-dimensional euclidean space E³ lies on a disc D⊂E³, whence it follows that there exists a pseudo-isotopy of the space E³ which carries a line segment into the locally unknotted simple arc. Bibliography: 16 items.