Table of contents

Complex-valued admissible and completely admissible functions on a cone are considered. It is proved that holomorphic functions having nonnegative real part (bounded argument) are admissible. These functions are used as comparison functions in multidimensional Tauberian theorems for generalized functions.

The author discusses classes of periodic functions of variables that are either piecewise monotonic or piecewise monotonic in the sense of Hardy, and clarifies the connections, for such functions, between the property of belonging to space, , and the convergence of series of their trigonometric Fourier coefficients,

We establish the existence, when , of certain results that differ from the one-dimensional case.

This paper describes a class of convex compact sets K in Cⁿ having the following uniqueness property: K is the unique support of every analytic functional for which it is a support.

It is shown that there exists a Levi-flat surface in with boundary on a given two-dimensional sphere that lies in the boundary of a strictly pseudoconvex domain and is totally real everywhere except at a finite number of elliptic and hyperbolic points.

This paper is concerned with the behavior of multiplicative functions on the set , where is a prime and is a nonzero integer. Several results are obtained in which either the average value of a multiplicative function on this set is estimated or its asymptotic behavior is determined. As one application a nontrivial estimate of is found, where , is a sufficiently large prime, and is a character of degree greater than 4.

A theory is developed for superanalytic generalized functions on a superspace over a non-Archimedean Banach superalgebra with trivial annihilator of the odd part. A Gaussian distribution and the Volkenborn distribution are introduced on the non-Archimedean superspace. Existence and uniqueness theorems are proved for the Cauchy problem for linear differential equations with variable coefficients. The Cauchy problem for non-Archimedean superdiffusion, the Schrödinger equation, and the Schrödinger equation for supersymmetric quantum mechanics on a non-Archimedean Riemann surface are considered as applications.

Some exact solutions are found for a system of two nonlinear equations admitting a Lax representation. The dynamics of the scattering data of a fourth-order operator of a special form is indicated, and the system is shown to possess a countable set of first integrals.

An integrable complexification of the hierarchy of the KdV equation is constructed. A countable set of first integrals is found, along with the evolution of the scattering data for the complexification of the KdV equation obtained in [1]. Hirota's method is used to obtain some exact solutions of this equation. A representation of the complexification of the KdV equation as an equation of zero curvature is indicated.

Bibliography: 10 titles.

The author constructs and investigates a fundamental solution of Cauchy's problem for a parabolic equation with a -adic space variable and a real time variable. The question of existence and uniqueness of solutions to Cauchy's problem in classes of bounded and increasing functions is considered, and conditions for nonnegativity of the fundamental solution are found. The problem of determining if the solution stabilizes as is solved for a model equation with constant coefficients.

The author considers Galois group actions on the fundamental groups of curves of hyperbolic type, and proves certain cases of Grothendieck's conjecture about the possibility of recovering a curve from its Galois representation.