Table of contents

It is shown that various quasilinear elliptic and parabolic differential inequalities and systems of such inequalities defined on bounded domains, and which have point singularities on the boundary do not have solutions. The method of nonlinear capacity is used in the proof. Examples show that the conditions obtained by this method cannot be improved in the class of problems under consideration.

Bibliography: 14 titles.

General inequalities for quadratic forms with coefficients depending on the values of Green's and Robin functions are obtained. These inequalities cover also the reduced moduli of strips and half-strips. Some applications of the results obtained to extremal partitioning problems and related questions of geometric function theory are discussed.

Bibliography: 29 titles.

Several results on the topology of the space of -flats in similar to the Borsuk-Ulam theorem on coverings of a sphere and a ball are proved. Some corollaries on common transversals for families of subsets of  and on measure partitions by hyperplanes are deduced.

Bibliography: 22 titles.

Connections between the boundary behaviour of polyanalytic functions and the structure of the boundary are investigated. In particular, a Jordan domain with Lipschitz boundary is constructed which is regular for the Dirichlet problem in the class of bianalytic functions.

Bibliography: 14 titles.

This paper is concerned with the abstract Cauchy problem , , where is a densely defined linear operator on a Banach space . It is proved that a solution of this problem can be represented as the weak limit , where the function satisfies the equality , , for a natural class of operators. As distinct from Chernoff's theorem, the existence of a global solution to the Cauchy problem is not assumed. Based on this result, necessary and sufficient conditions are found for the linear operator  to be closable and for its closure to be the generator of a -semigroup. Also, we obtain new criteria for the sum of two generators of -semigroups to be the generator of a -semigroup and for the Lie-Trotter formula to hold.

Bibliography: 13 titles.

We compute the automorphism group of the quotient of a generalized Grassmannian G/P by the action of a maximal torus of the semisimple group G. We classify the twisted forms of such quotients, that is, varieties isomorphic to these quotients over an algebraic closure of the base field. It is proved that all such forms are unirational.

Bibliography: 20 titles.

Pure subrings of finite rank in the -adic completion of the ring of integers and in its homomorphic images are considered. Certain properties of these rings are studied (existence of an identity element, decomposability into a direct sum of essentially indecomposable ideals, condition for embeddability into a csp-ring, etc.). Additive groups of these rings and conditions under which these rings are subrings of algebraic number fields are described.

Bibliography: 12 titles.

The behaviour of solutions of the equation with respect to the spectral parameter  is investigated under the assumption that the function does not satisfy the classical conditions. Two types of equations are considered: the Sturm-Liouville equation , whose solutions grow like in the norm of (where is arbitrary), and equations of the form , , whose solutions can grow like in the norm of (where is arbitrary) and even like where .

Bibliography: 3 titles.