Aim. We construct a computer model of a neural network for object recognition. Methodology. Based on the ideas underlying the theory of recognition and the theory of neural networks, we have constructed a model of neural network operation designed to recognize the studied mappings with a given accuracy. For the successful operation of a neural network, we have used open access databases from remote servers, which makes it possible to use neural networks in cramped conditions (in the absence of powerful computers). A program in Python has been developed to organize and manage the neural network. Results. A neural network has been constructed that recognizes the studied mappings with a given accuracy. To manage the built neural network and attract database arrays from remote servers, a program in Python has been developed. The principle of neural network operation is demonstrated in practice, using the example of recognition of the studied mappings. Research implications. The model provides a real recipe for constructing a neural network and using it in practice in the absence of a powerful computer.

Aim. Based on the principles of correlation theory, calculate the dependence of the dry friction sliding factor k in the form of function from the Poisson coefficients of both bodies σ, σ′ and temperature T. Methodology. The research method is based on the application of Green’s temperature functions for unbalanced processes. Results. the dependence of the dry friction factor on the Poisson coefficients of both bodies and temperature has been found. Research implications. A mathematical description of the dry friction factor depending on the parameters, meaning the function k = k(σ, σ′, T); this is extremely important for the practical application of the resulting dependency.

Aim. We study experimentally the dispersion of the permittivity of a 5CB nematic liquid crystal in a wide temperature range and numerically approximate the obtained dependences. Methodology. The longitudinal and transverse components of the complex permittivity of a 5CB liquid crystal are measured in a wide temperature range from 263 to 318 K and in the frequency range from 100 Hz to 1 MHz by the capacitive method and at a frequency of 39.25 GHz by the waveguide method. Results. The supercooled liquid crystal phase of the substance in question, which is a combination of nematic molecules and molecular associations, is found and studied. The numerical approximation of the dielectric spectra of the nematic and supercooled liquid crystal phases is performed. Research implications. It is shown that dispersion of the longitudinal and transverse components of the permittivity of a 5CB liquid crystal in the nematic phase is satisfactorily described by the Debye equations, and in the supercooled phase by the Cole-Cole equations. The relaxation times of the dipole polarization are calculated.

Aim. The purpose of this work is to obtain an analytical estimate of the highest value of the effect of high-speed overlap in a shock wave. The ground state of the considered shock-compressed binary gas mixture is highly dispersed in the concentrations of light (significant predominance) and heavy components. Methodology. The paper uses analytical methods of theoretical physics, and first of all, higher algebra and mathematical analysis, which allow one, due to their great versatility, to set the limits of the maximum value of the effect under consideration, which would be much more difficult to do by numerical methods. Results. Analytical estimates are obtained that allow one, first of all, to determine the fundamental conditions for the existence of the high-speed effect. In addition, an analytical assessment of the highest value of this effect is presented. Research implications. The results obtained are certainly essential for reproducing the effect of high-speed overlap under optimal conditions of its greatest intensity.

Aim. We determine the relationship between the parameters of a materials structure and metrics of their thermal protection properties. Methodology. Use is made of a theoretical analysis and comparison of theory and experiment based on the results of two methods of practical measurements of the thermal conductivity components of a random scattering medium. Results. A method for measuring the radiant, conductive and total heat transfer coefficient of the medium using screens is described. A new method for determining the components of heat transfer with use the compression of the canvas is found. Metrics of heat-shielding properties of materials are measured. Research implications. It is found that the conductive component of the thermal conductivity of light insulation materials is almost equal to the thermal conductivity of air, and the radiant component is inversely proportional to the density of the structure. The conductive and radiation parameters of the medium are determined.

Aim. We consider the main ideas of post-optimal analysis in linear programming problems on the example of the production planning in the case of two types of products and three types of resources. Methodology. Optimal solutions of linear programming problems depend on the parameters of mathematical models of real processes under consideration. In this paper, the model of the production planning problem is selected. The parameters of this model are the profits from the sale of a unit of each product type and the volumes of each resource type. When considering the general problem of producing n types of products using m types of resources, there is a problem associated with the inability to see mathematical objects. In order to visualize mathematical objects, the problem of producing two types of products using three types of resources is considered. This approach allows one to reduce post-optimal analysis to graphical sensitivity analysis of the optimal solution to changes in the parameters of the linear programming problem under consideration. Results. The main ideas of post-optimal analysis are considered in two cases. Graphical analysis of stability of the optimal solution to changes in the profit from the sale of a unit of each product type with the fixed profit from the sale of a unit of another product type is carried out. Graphical analysis of sensitivity of the total profit to the changes in the volume of each fully used resource type with the fixed volumes of the other resource types is performed. Research implications. The work is of practical significance since it can be used by lecturers at the initial stage of studying linear programming. In general, post-optimal analysis is a rather complicated task and is performed after studying the simplex method and duality theory. The described approach allows one to consider the main ideas of post-optimal analysis on the coordinate plane immediately after considering the graphical method. It can be used by lecturers both in the classroom and in self-study of students, as it uses the simplest linear programming tools.