Table of contents

For an arbitrary Γ-extension of an algebraic number field the projective limit of the S-units with respect to the norm is treated. Questions connected with the definition of its structure are studied. Bibliography: 6 titles.

The action of the group G₀ of fixed points of a semisimple automorphism θ of a reductive algebraic group G on an eigenspace V of this automorphism in the Lie algebra g of the group G is considered. The linear groups which are obtained in this manner are called θ-groups in this paper; they have certain properties which are analogous to properties of the adjoint group. In particular, the notions of Cartan subgroup and Weyl group can be introduced for θ-groups. It is shown that the Weyl group is generated by complex reflections; from this it follows that the algebra of invariants of any θ-group is free. Bibliography: 30 titles.

In this work a criterion is given for the finiteness of the number of indecomposable representations of partially ordered sets over an arbitrary commutative ring. Determination of finiteness is reduced to solving a matrix problem. Bibliography: 8 titles.

Sufficient conditions are obtained for the decomposability of a homogeneous representation of a locally compact group into a direct integral of its irreducible representations in terms of the associated algebra of the representation. These results are then applied to homogeneous representations of connected complex semisimple Lie groups. Bibliography: 11 titles.

Harmonic analysis on a Riemannian symmetric space can be connected with the study of a nonstationary system of equations that has been constructed with respect to the ring of Laplace operators. The scattering theory for this system generalizes the scattering theory for hyperbolic equations constructed by Lax and Phillips. The paper contains a series of new spectral theorems generalizing the Harish-Chandra theorem and a formulation of a causality principle for scattering operators. Bibliography: 23 items.

In this work there is an investigation of a family of invertible (i.e. ) analytic matrix-valued functions () which are -contractive and which have monotonically increasing -forms as . Invariance with respect to of the rank of the matrix is established, and conditions for convergence of are investigated. As a special case a theorem is obtained on the invariance of ranks of limiting radii of Weyl disks, which is fundamentally of significance in the theory of classical problems (the moment problem, the Nevanlinna-Pick problem, the Weyl problem on the number of square-integrable solutions of a system of differential equations, and so forth). Bibliography: 17 titles.

Imbedding theorems for Sobolev spaces with a symmetric norm are established. In particular, the results obtained lead to new imbedding theorems for Orlicz-Sobolev spaces. The proof of the basic results is based on geometric inequalities of the type of the classical Brunn-Minkowski inequality. Bibliography: 30 titles.

Using a maximum principle a solution is found of Ulam's problem of determining a continuous motion of a unit segment from the position AB to A'B' under which the total path traversed by the ends of this segment is as small as possible. Bibliography: 3 titles.

In this work there are constructed a function such that the difference between the Fourier series expansion and the Fourier integral expansion for summation over squares diverges almost everywhere on , and a function , , , for which the difference diverges at a point. Bibliography: 5 titles.