Vol 18, No 6 (2012)

The numerical method of finding of meanings of eigenfunctions of discrete semibounded from below operator is developed. Estimates of residuals of sums of series of Reley–Schrodinger of the ”weighed” corrections of the perturbation theory are gained. Approbation of the method on the example of Laplace operator with perturbed function of complex variable is produced.

We consider the Schwarz problem for vector-valued functions analytic according to Douglis. We prove that under certain conditions on the matrix this problem is reduced to the Dirichlet problem for some equivalent system of second-order PDEs. The reversebility of transformations is proved, and on that ground the theorem of uniqueness is established. A special case when reduction is impossible is also viewed. The examples are given.

In the article we consider two disjoint classes of frames, prime and composite Parseval frames, the union of which forms a set of Parseval frames. The main goal is to obtain a description of these two classes. In this article we prove necessary and sufficient conditions for the frame to be a prime Parseval frame.

The uniform exhaustivity criteria are proved for a sequence of exhaustive outer measures defined on the non-sigma-complete class of sets and taking values in an Abelian topological group.

For Barenblatt–Zheltov–Kochina model presentation of filtering of liquids in fissured-porous rocks the explicit solution of the mixed problem in anisotropic semispace with brightly expressed vertical permeability is found by reduction of the considered problem of filtering to the solution of the abstract initial boundary-value problem in Banach space.

The distortion of homeotropic structure of nematic liquid crystal under the influence of periodic shear at ultralow frequencies is theoretically described. It is shown that periodic shear action results in spatially modulated structure in NLC layer. Threshold shear amplitude and spacial periodicity of new molecular orientational structure are found on the base of nonlinear nematic liquid crystal hydrodynamic equations using Galerkin method. Theoretical analysis shows that at low frequencies threshold shear amplitude does not depend on frequency and the spatial period of the order of NLC layer thickness. Theoretical results are compared with experimental data.

A model of oscillation circuit containing a nonlinear fractal component of capacity is proposed. The differential equation of motion of fractional order for forced oscillations under the action of an external signal is obtained. An approximate analytical solution of the equation of motion is conducted by methods of equivalent linearization and slowly varying amplitudes. The amplitude-frequency and phase response of fractional oscillator with cubic nonlinearity are analyzed.

The interaction of silver salt of 2-hydroxy-4,6-bistrinitromethyl-1,3,5-triazine with 1- and 2-bromadamantans was studied. The relative 2-adamantyloxy-4,6-bistrinitromethyl-1,3,5-triazines was synthesized for the first time. It was shown that one of trinitromethyl group in 2-adamantyloxy-4,6-bis(trinitromethyl)-1,3,5-triazines could be substituted selectively under the action of various nucleophiles.

The influence of dynamical Stark shift on the entanglement between two atoms prepared in initial W-like states in Tavis — Cummings model with degenerate two-photon transitions has been investigated. The results show that effect of sudden death of entanglement the effect can be weakened for some initial states with taking into account the Stark shift.