Volume 1, Issue 4

The sufficient conditions are established for the correctness of the linear boundary value problem dx ( t ) = dA ( t ) · x ( t ) + df ( t ); l ( x ) = c 0 , where and are matrix- and vector-functions of bounded variation, , and l is a linear continuous operator from the space of n -dimentional vector-functions of bounded variation into .

Using the methodology and results of the theory of filtering of conditionally Gaussian processes, the optimal schemes of transmission of Gaussian signals through the noisy feedback channel are constructed under the new power conditions.

Some sufficient conditions are found for a pair of weight functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.

This paper deals with singular integral operators that are bounded, completely continuous, and Noetherian on manifolds with a boundary in weighted Hölder spaces.

It is proved that for a Cauchy type singular operator, given by equality (Khvedelidze, Current problems of mathematics 7: 5-162, 1975), to be bounded from the Lebesgue space L p (Γ) to L q (Γ), as , it is necessary and sufficient that either condition (Calderon, Proc. Nat. Acad. Sci. USA 4: 1324-1327, 1977) or (David, L'integrale de Cauchy sur le courbes rectifiables, Prepublications Univ. Paris–Sud; Dept. Math., 1982) be fulfilled.

A multiple Vandermonde matrix which, besides the powers of variables, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. For the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler beta-function.

For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied.

The sufficient conditions of the existence, uniqueness, and correctness of the solution of the modified boundary value problem of de la Vallée-Poussin have been found for a nonlinear ordinary differential equation u ( n ) = f ( t , u , u ′, … , u ( n –1) ), where the function f has nonitegrable singularities with respect to the first argument.