A method of a small parameter is proposed for the regularization of a new class of solutions of nonlinear operator equations of the first kind in a Hilbert space. The convergence of the regularized solutions to the exact solution of the original equation is proved. The regularization parameter, depending on the error in the right-hand side, is selected. The rate of convergence of the regularized solutions to the exact solution is estimated.

We have defined three classes of Riemannian spaces whose affine connections are functionally Abelian. We have determined and proved a necessary and sufficient condition for a three-dimensional Riemann space to exhibit a functionally Abelian property.

The possibility of using an ‘extraordinary’ iight wave for the implementation of linear acceieration in the plasma [1] is discussed. It is shown that a wave necessary for acceleration exists in the plasma resonance conditions when the phase velocity is equal to the speed of light and the drift of the electrons across the external magnetic field under the influence of the Lorentz force is compensated by the electric field of the wave. The deviation from the resonance frequency of the wave is accompanied by the emergence of nonlinear plasma dispersion. It is shown that in this case the solution of ‘slow’ equations for the amplitudes, obtained by averaging the initial equations for a high-frequency wave period, is a solitary pulse (soliton).

The energy of a threshold breakdown of through holes in metal foiis of different thicknesses by high-power laser radiation is investigated experimentally. Properties of the foil materials, which are typical of ‘liquid metal-gas’ phase transitions of the second kind, are determined. A controlling parameter of the threshold breakdown is a through (threshold) hole on the shady side of the target, which appears at a minimum pulse energy needed only for the emergence of an outlet for a given foil thickness, and the laser pulse energy.

One of classical boundary problems of the kinetic theory (the problem of thermal sliding) of a rarefied gas along a flat solid surface is considered. A model kinetic Boltzmann equation with a model integral of collisions in the BGK (Bhatnagar, Gross, Krook) form is used. As boundary conditions use is made of the boundary Maxwell conditions (mirror- diffusion). The generalized method of a source is applied to solve the problem. Comparison with previously obtained results is performed.

We have obtained data on the effect of the periodic microrelief profile on optical diffraction properties. A method is proposed for calculating a diffraction pattern, which makes it possible to estimate the relief parameters for specified optical properties of a spatially nonuniform anisotropic structure.

We consider the classical formulation of the problem of the direct current in a cylindrical conductor with an embedded superconductor of finite length. We present the solution obtained without boundary current spreading. It is shown that the authors' model has a fundamentally different solution. The direct current, as was shown in the previous work of the authors, is pushed in any type of a conductor to the surface together with the magnetic field, which leads to the so-called surface current. This current can be regarded as a volume one but flowing in a thin layer of finite thickness. Because this thickness does not depend on the thickness of the material and the nature of the conductor, it can be assumed that it is on the order of the characteristic depth of penetration of the magnetic field into the superconductor according to London’s theory.

Phosphate glasses (in mol%) 40Na₂O₃-20Al₂O₃-40P₂O₅(серия I) and 40Na₂O-15Al₂O₃-5Fe₂O₃-40P₂O₅ (series II) were irradiated with 8 MeV electrons to doses equivalent to 0.5 and 1.0 MGy. ESR spectra of all the glasses were recorded at room temperature. It was found that irradiation leads to the formation of phosphorus-oxygen hole centers PO4^2- (D5), PO3^2- ion-radicals (D2) and centers (D₆) associated with the Q³ units. ESR spectra of these radiation paramagnetic centers were observed in series-I glasses that do not contain Fe2O3. In electron irradiated series-II glasses with 5 mol% Fe2O3, phosphorus-oxygen centers almost disappear and ESR spectra exhibit intense broad lines caused by Fe (III) ions.

We report a complex study of cloud technologies in the educational process. The investigation is carried out by the example of Microsoft Office 365. The study was performed using components of the cloud-based technology in education, and the obtained results made it possible to determine the frequency of their use in the educational process. On the basis of these data, we drew conclusions on the need to explore the least popular components by all the participants in the educational process.