No abstract

The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

For a general Raman-type model describing the decay of a pumping mode into p components special polynomials are introduced. Asymptotic properties of those polynomials are discussed when the number of excitations tends to infinity. In the investigation of interaction between radiation and media, a number of mathematical complexities arises. The simplest example is the modeling of radiation scattering by excitations of a crystal lattice. Some of the contradictions are caused by the peculiarities of the second quantization method, which is used effectively for quadratic Hamiltonians only. Another problem is the determination of the energy states of the investigated media, which is a problem of eigenvalues. Mathematically we are to examine some system of polynomials connecting each other by a recurrency, which is analogous to the classical case. In this paper we consider the mathematical problems arising therein.

The statistical properties of letters frequencies in European literature texts are investigated. The determination of logarithmic dependence of letters sequence for one-language and twolanguage texts are examined. The pare of languages is suggested for Voynich Manuscript. The internal structure of Manuscript is considered. The spectral portraits of two-letters distribution are constructed.