The problems in question consist in recovering an unknown source in the boundary condition and an unknown potential at the smallest term of the hyperbolic-type equation in linearized and exact statements. We consider the dependences of the desired source and given information on different variables. We study the properties of the required functions. The main result of the investigation is the theorems of uniqueness of the potential q(y) on the semi-axis у > 0 and of the sources f(x), f(t) and which belong to special classes of the functions.

We have shown the necessity of accounting for the medium-range order and relaxation times in the statistical theory of liquid crystals and, more generally, in the condensed state theory. Simple models of nematic and smectic liquid crystals are considered.

We consider nematic liquid-crystal (LC) domains, the central part (core) of which has an orienting coating providing a predetermined angle of the director pretilt and the outer side of the domain does not exert any significant orienting influence. We propose a calculation method and construct a theoretical and computational model of the dependence of the phase delay difference ДФ on the coordinates of the light beam propagation through the domain. The methods are proposed for determining the geometric parameters of the domain and its birefringence in the observed optical pattern.

We consider the kinetic problem on high-frequency conductivity and Hall constant of a thin metal film placed in a transverse stationary magnetic field and a longitudinal alternative electric field. The electron reflection from the upper and lower film boundaries is assumed to be diffusive. The relationship between the film thickness and the electron mean free path is arbitrary. The dependences of the conductivity and Hall constant on such dimensionless parameters as the electric field frequency, magnetic field induction and film thickness are investigated. The obtained results are compared with the experimental data.

We have obtained calculation data on the effect of the spatial distribution of the liquid-crystal director on the optical properties of a regular microrelief element. We have proposed orientation distributions suitable for use in the composition of diffraction gratings.

We have calculated matrix elements of the Boltzmann collision operator for nonequilibrium gas mixtures in the framework of the generalized torque method. The value of the partial change in the momentum due to collisions is obtained, which is expressed through the matrix elements for an arbitrary law of interaction between particles of a nonequilibrium mixture with a large temperature difference between the component and the relative velocity comparable to the thermal speed, when the function of the zero approximation is the Maxwell function. Decomposition for a partial change in the momentum due to collisions includes terms regarding hydrodynamic variables up to third degree and, therefore, the scope of its applicability applies to small, but finite values. Moreover, the contribution of higher moments strongly depends on the law of interaction between colliding particles and is evaluated for the power-law interaction. We have derived an equation for the evolution of the partial chaotic energy. We have estimated the contribution of nonlinear terms of the hydrodynamic variables to the desired expression.

Our generators of tasks created in the MATLAB environment provide the formation of a fund of assessment tools for current and intermediate certification in teaching the subject ‘Linear algebra’. We demonstrate the use of the MATLAB-created generator of matrices with integer eigenvalues for the formation of the bank of test tasks. We describe methods for visual geometric representation of the action of a linear operator in two-dimensional and three-dimensional spaces. Examples are presented.

We consider theoretical studies of thermal radiation by German physicist W. Wien. As a result of these studies, W. Wien discovered laws of thermal radiation in 1893-1896, and together with German physicist O. Lummer he designed a laboratory source of thermal radiation in 1895.