Vol. 38 No. 2 (204) (2004)

The paper considers in the space $L^2(0, \infty)$ a Fourier type integral operator U, which arise in the inverse problem of the quantum scattering theory. It is proved, that either operator U or its conjugate U* is invertible.

In this work all distributive and hyperdistributive varieties of semigroups has been found and lattices of subvarieties of them are also described.

The power type estimate of the rate of convergence in Erdӧs–Kac limit theorem is obtained for stationary uniformly strong mixing random sequences.

The paper considers the discrete optimization of the model of the struggle against the pollution of the surrounding environment. The dynamic optimization method, which is based on the stochastic search algorithm “return” method is used. It is proposed to use the optimal spline approximation solution.

It is discussed the problem of an optimal control for the plate's linear vibrations, when the distributive disposed forces influence on it. The problem is solved by the method of Fourie and it is brought to the differential game, which is described by the infinitesimal differential equations of second order. The extremal strategies are constructed by the extreme targeting method. It is shown that if the resources of the first player are more than the resources of the second player and the influencing forces belong to class $L_2$, then the problem of damping of plate's vibrations is solved. In the end of the article a numerical example is given.

Shear vibrations of elastic piezoelectric half-space (piezoelectric of 6 mm class hexogonal symmetry) covered by a dielectric layer is studied. The vibrations in the media are induced due to the periodical force acting on the dielectric layer. Displacements, corresponding electrical potentials as well in vacuum, dielectric layer and piezoelectric half-space are obtained using Fourie’s transformations. An asimptotic formula is obtained for displacement amplitude of points on infinite distance, from the point on which the force acts, where the part of Lyav’s electroelastic waves are separated.

In this paper evolution of thick anisotropic gratings formation in photomonomer–liquid crystal mixture is modeled. The polymer and monomer concentra-tions’ distributions are obtained from modeling of grating formation process. Distributions of dielectric permittivities from Lorentz-Lorenz and more simple approximate formulas are obtained. The comparison of received distributions and appropriate diffraction efficiencies for s- and p-polarized probe beams are done. It appears, that diffraction efficiencies calculated by exact and approximate distributions qualitatively coincide, however there is a quantitative difference and it depends on polarization of the probe beam.

The effect of crystalline disturbances, arising due to dislocations, upon the form of Pendellosung fringes obtained in an X-ray two-block interferometer is considered. It is shown, that for local regions of disloationc Si monocrystals the distribution of Pendellosung fringes coincides with the isostrain lines.

The result of the present work sounds as follows. In Lee’s group, any everywhere dense strongly Liuvilial subgroup generated by finite number of group elements is uniformly distributed in the sense of Arnold-Krilov.