Table of contents

In this paper we study finite 2-groups in which each abelian normal subgroup is metacyclic, i.e. . The main result: a finite 2-group with is an extension of a metacyclic group by a group isomorphic to a subgroup of the dihedral group of order 8.

Let R be a finitely generated prime Pl-algebra over a field F. Z is the center of its ring of fractions. It is proved that Z is the field of fractions of the center of R and that the transcendence degree of Z over F is equal to the maximal length of a chain of prime ideals in R.

In this paper we prove the conjecture of Lissner and Geramita that every noetherian regular OP-ring is a Plücker ring.

Let be a noncompact, locally compact group with an invariant mean, its group algebra, and the ideal of formed by those functions whose Haar integral is zero. In this paper it is shown that the (relative) homological dimension of the Banach -module is infinite. By the same token the (relative) global dimension of the Banach algebra is also infinite. This result is then applied to the study of cohomology groups of a locally compact group with coefficients in Banach -modules.

In this paper inequalities for some characteristics of polynomial mappings of are obtained. These estimates are used to obtain some results on polynomial endomorphisms of with constant nonzero Jacobian.

A direct topological-geometric method is suggested for constructing solutions of partial differential inequalities and equations.

This article is devoted to an examination of the following extremal problem: find the quantity

where is an -dimensional sphere and is the totality of polynomials of degree in variables for which . Here is a measurable set from and the first sup is taken over all measurable having measure . The exact order of growth of which respect to as is found in this article. Various applications of the results are examined as well.

Necessary and sufficient conditions are found for continuity, compactness, and closability of imbedding operators of some function spaces into the space L_p. These results (for p = 2) give criteria for positive definiteness and discreteness of the spectrum of the Dirichlet problem for a selfadjoint elliptic operator of arbitrary order. Some integral inequalities are considered for differentiable functions on a cube.

Necessary and sufficient conditions are found for membership of a function in the class , . A connection is established between the modulus of continuity and, on the one hand, the Fourier coefficients and , and on the other hand the modulus of continuity .

An asymptotic expression as is found for the norms , , where is a Fourier sum of the -periodic function having bounded -variation. Various criteria for the continuity of a function of bounded -variation are obtained as corollaries.

It is established that the spectral functions of the second-order boundary value problem

possess power asymptotics as , when the function possesses power asymptotics as . A partial converse of this fact is also obtained.

For a bounded convex domain in the plane, asymptotic formulas with error tending to zero are constructed for a certain series of eigenvalues of the Laplacian with zero boundary conditions. The boundary of the domain is assumed to be sufficiently smooth. It is proved that

where is the number of eigenvalues (with multiplicities taken into account) less than and is the number of those eigenvalues for which an asymptotic expansion has been found.

In this paper we introduce a measure of deviation of an entire curve from a fixed vector. We establish sharp estimates and other relationships satisfied by this measure of deviation.