A class of nonlinear operator equations of the first kind is investigated in the Hilbert space. An approximate solution is constructed that is stable with respect to the initial data of the problems. The convergence of the approximate solution to the exact solution of the original equation is proved. The regularization parameters of the errors are selected.

Based on a simple mathematical approach proposed in the paper, we demonstrate a rigorous computation of the Christoffel symbols and the Riemann tensor that obviously have a regular geometric dimension, which is extremely important in solving a huge class of purely physical problems. As examples, we consider four orthogonal coordinate systems, two of which are spherical and cylindrical, i.e. standard for describing any course of tensor analysis, and the other two are parabolic and orthogonal two-dimensional coordinate systems, for which the Christoffel symbols, the Laplace operator, and Riemann and Ricci, whose all components automatically have the correct geometric dimensions, are calculated. A number of physical applications of the described mathematical formalism are demonstrated. An example of a nonorthogonal two-dimensional coordinate system is considered, with the help of which a detailed calculation of the Christoffel symbols of both kinds is given, and an expression is found for the Laplace operator with application to the problems of elasticity theory and hydrodynamics.

An influence of kinetic and quantum properties of degenerate electron plasma on the interference of radiation from metallic and dielectric films is investigated. The reflection and transmission power coefficients as well as the phase differences of interfering beams as functions of the radiation frequency are studied in the cases of P- and S-waves. A difference of the results for the quantum electron plasma from the results for the classical electron plasma and for the classical electron gas is determined in cases of frequencies which are an order and much less than the plasma frequency.

We report the development of a method for producing long aggregates (more than 1 cm) of multi-walled carbon nanotubes (MWCNTs) dispersed in a solvent by means of directed agglomeration with an electric arc in the liquid phase, as well as study the processes accompanying the formation of these aggregates. A field-oriented carbon agglomerate is obtained and it is found that mica with inclusions of amorphous carbon and agglomerates of nanotubes are formed in such a system. By percolation, the condition of MWCNT crosslinking into the agglomerate at different stress values is established; it is determined that at low concentrations the probability of particle crosslinking is no more than 5%. The obtained results may be of practical interest for the enterprises of power industry in the production of modern types of electrical equipment.

The paper considers the development of experimental methods for the study of physical and chemical processes in shock waves. The VISAR laser interferometry method is used to measure the velocity of a free surface of a solid body interacting with the front of a gas shock wave. For the first time, free surface velocity profiles are obtained in interaction with a shock wave in a light and a relatively heavy gas. It is shown that the method provides a higher temporal resolution than the laser schlieren method applied to shock waves, which makes it promising for measuring the structure of the shock wave front in gases.

The influence of the composition of the reagent flow on the formation conditions and the structure of ZnO-based layers is studied. It is shown that a key reserve in increasing the electrical conductivity and mobility of charge carriers of Ga-doped ZnO transparent electrodes is to increase the structural perfection of the layers. Analysis of the obtained results shows that an increase in the partial pressure of metal vapors in the gas phase entails a corresponding increase in the mobility of the components and an increase in the crystalline perfection of the synthesized polycrystalline layers.

Dielectric properties of the 6CHBT nematic liquid crystal with a polar -NCS end group are studied. The principal values of the dielectric permittivity are determined using the methods of dielectric spectroscopy by varying the sample temperature, the angle between the director and electric field strength vector, and the frequency of the test signal. We use a modified dielectric spectroscopy method that takes into account the parasitic contributions of the measuring system. The characteristic dielectric relaxation times τ of rotation of 6CHBT molecules around their short axis in both the nematic and isotropic phases and the corresponding activation enthalpies are calculated. The temperature dependence of the orientational order parameter over the entire range of the nematic phase existence is constructed using the obtained data and the Maier-Saupe theory.

One of the most important educational goals of modern school is to prepare students for the successful completion of final exams. The structure of the Unified Sate Exam in profile-level mathematics includes the proof of increased complexity for geometric problems, requiring students to have a comprehensive knowledge of planimetry. The most important feature is the lack of unified algorithms for solving such problems; the success largely depends on the students’ experience in solving combined planimetric problems. Nevertheless, the practice of solution allowed us to identify some geometric structures that are auxiliary keys to finding the right solution. One of these keys was the auxiliary circle method, which is presented in this paper. The paper describes the essence of the method and the conditions of its application, examines the proofs for the problems taken from real test and measurement materials of the exam, and considers the solutions of the problems in the framework of the described method.

The problem of moving and embedding an object of immutable form in a confined space is analyzed. An algorithm and a program simulating the behavior of such an object are constructed on the example of moving a ‘knight’ on a chessboard. It is shown that there exist optimal ways to move an object of immutable form in the required chessboard cell and a minimum time to overcome all the way and embed it in the desired point.