Volume 4, Issue 5

Solutions are obtained for the boundary value problem, y ( n ) + f ( x , y ) = 0, y ( i ) (0) = y (1) = 0, 0 ≤ i ≤ n – 2, where f ( x , y ) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.

The non-commutative neutrix convolution product of the functions x r cos – (λ x ) and x s cos + ( μx ) is evaluated. Further similar non-commutative neutrix convolution products are evaluated and deduced.

The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.

An effective factorization and partial indices are found for a class of unitary matrix functions.

For Lizorkin–Triebel spaces the family of extension operators is constructed which yield a minimal (in order) value of the norm among all possible extensions of a given function defined initially on the interval of an arbitrary small length. The techniques used restrict us to the one-dimensional case and spaces defined via differences of first order.

A new simple proof of the well-known theorem of McEliece about the complete weight enumerators of ternary self-dual codes is given.

In the class of Hölder functions we give the necessary and sufficient condition for solvability of a Fredholm integral equation whose kernel has fixed singularity in the segment of variation of an independent variable. Finding a solution is reduced to solving a regular integral equation of second kind.

It is shown that the convergence of convolution products of probability measures on certain non-locally compact topological abelian groups can be verified by means of characteristic functionals. Analogous results are obtained also for almost everywhere convergence of series of independent random elements in the considered groups. A connection with the Sazonov property of the groups is discussed.