Volume 3, Issue 2

Asymptotic properties of proper solutions of a certain class of essentially nonlinear binomial differential equations of the second order are investigated.

The theorem on the tending to zero of coefficients of a trigonometric series is proved when the L 1 -norms of partial sums of this series are bounded. It is shown that the analog of Helson's theorem does not hold for orthogonal series with respect to the bounded orthonormal system. Two facts are given that are similar toWeis' theorem on the existence of a trigonometric series which is not a Fourier series and whose L 1 -norms of partial sums are bounded.

The non-commutative neutrix product of the distributions and is evaluated for r = 0, 1, 2, . . . and s = 1, 2, . . . . Further neutrix products are then deduced.

Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented.

The Skorokhod oblique reflection problem is studied in the case of n -dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to construct in a direct way the reflected (in a convex polyhedral domain) diffusion processes possessing the nice properties.

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.