The conversion of the poloidal magnetic field energy created by the toroidal plasma current into the kinetic energy of runaway electrons during the termination phase of the plasma discharge disruption in a tokamak is analyzed within the zero-dimensional model of infinitely thin ring conductors. The evolution equations for the plasma current and the induced current in the wall of the vacuum chamber, which are based on the Faraday’s law of electromagnetic induction, are solved analytically. The time dependences of the mentioned currents are obtained under the condition of an exponential decay of the runaway electron current, which makes it possible to determine the efficiency of the specified conversion. When additional conditions are imposed on the parameters of the system, the expressions for the total plasma current and the current in the wall of the tokamak vacuum chamber reproduce similar relations obtained in earlier works. For the extreme cases of an ideal wall (when its resistance is infinitely small) and a wall with infinite resistance, simplified relations for the total plasma current are given, and estimates for the fraction of the magnetic field energy converted into the kinetic energy of runaway electrons are also obtained which are necessary to analyze the thermal load on the elements of the first tokamak walls in the case of disruptions of plasma discharges.

   A device for cumulating plasma clots when testing the surface of dielectric materials for heat shock has been used. The thermal effect on the material can be carried out in a short time interval (no more than 0.1 s) at a sufficiently high temperature (up to 4500 K). The experiments have been performed at atmospheric pressure by an electric explosion of a conducting diaphragm with a current pulse up to 16 kA, a duration of 70 to 100 ms, and an input energy up to 50 kJ. The test device and its appearance are presented. Photos of the effect of the heat shock on the surface of dielectric and ceramic materials on a short-term thermal effect – heat shock during an electric explosion of a conducting diaphragm – are presented. The results obtained on testing of dielectric materials for the short-term heat shock indicate that surface short-term thermal impact (shock) on heat-loaded dielectric materials is promising.

   The problem of the reduction of thermodynamic potentials of a nanomaterial compared to a macromaterial because of a dimensional vacancy-related effect in an ensemble of nanoparticles has been solved. Various particle size distributions such as uniform, linear, exponential, and normal (Gaussian) distribution functions have been considered. The results have been applied to nanostructures based on indium and gold. The reduction of thermodynamic potentials for nanostructures with metallic particles compared to that for massive crystals has been shown to be more considerable for metals with a lower melting temperature since it makes the vacancy concentration higher. The larger the particles in an ensemble, the smaller the reduction of thermodynamic potentials in its absolute value, which is a direct consequence of a greater resemblance between the nanostructure and massive crystal of the same nature. Distribution functions corresponding to an increased number of ultrasmall particles and a decreased number of relatively large particles provide a greater reduction of thermodynamic potentials. It has been shown that the exponential distribution and linearly decreasing functions ensure the maximum and minimum reduction in its absolute value, respectively. The dispersion of a material in the form of nanoparticles is capable of promoting the chemical transformations with the energy barrier (per particle at 300 K) of ~10^–18 and 10^–8 J for indium and gold, respectively.

   A mathematical model of the temperature of electrons of a thermionic converter in an arc mode has been described. Since the physical processes in the arc operation mode of the thermionic converter some-times cannot be accurately described, it is proposed to use reasonable assumptions to analyze the operation of the converter. One of the main characteristics of a low-temperature plasma in the thermionic converter is the temperature of the electrons in the interelectrode gap. The temperature of electrons in the thermionic converter depends to varying degrees on the conditions for converting the thermal energy into the electrical energy, for example, on the interelectrode gap, cesium vapor pressure, and current in the thermionic converter. However, most of the plasma characteristics depend on the ionization and recombination of particles in the interelectrode gap. Approximations in the mathematical model of the thermionic converter are associated with the determination of the physical characteristics of a low-temperature plasma in the interelectrode gap. The dependence of the rate of formation of ions in the interelectrode space of the converter is assumed. The estimate of the formation rate of ions in the interelectrode gap makes it possible to determine the characteristics of thermionic energy conversion. The analysis of the characteristics of the thermionic converter allows the optimization of the design and its operation mode. The presented model makes it possible to obtain the parameters of the thermionic converter in the arc mode of operation.

   A fourth-order nonlinear equation, which is the second member of the K₂ hierarchy equations, has been considered. The K₂ hierarchy, as well as the K₁ hierarchy of equations, was introduced more than twenty years ago. A distinctive feature of these hierarchies is that all the equations belonging to this hierarchy do not have any first integrals in a polynomial form and, apparently, as well as the six Painlevé equations, they do not have solutions expressed in terms of classical functions. The equations of the K₂ hierarchy, as well as the equations of the first and second Painlevé hierarchies, are used to describe physical processes, and their investigation is of interest in this regard. It has been shown that this fourth-order equation from the K₂ hierarchy can be represented as a special dynamical system for which a formal representation of the general solution can be obtained.

   A sixth order partial differential equation with nonlocal and power nonlinearities has been discussed. The equation is used to describe optical pulses. Since the Cauchy problem for this partial differential equation cannot be solved by the method of the inverse scattering transform, the traveling wave reduction of this equation has been considered. The use of the traveling wave variables separates the imaginary and real parts of the equation; as a result, a system of ordinary differential equations (ODEs) has been obtained. The Painlevé test is used to check the integrability of this system. The system of ODEs does not pass the Painlevé test, since complex Fuchs indices exist. Conditions for some parameters of the equation, under which the system of ODEs becomes consistent, have been obtained. Taking into account these conditions, one ODE of the sixth order has been evaluated. The method of the simplest equations has been used to construct solutions. The constructed solutions have been expressed in terms of the exponential function and the Jacobi elliptic function and have the form of solitary and periodic waves. The found wave solutions of the equation have been illustrated for some values of the parameters.

   The generalized Kuramoto–Sivashinsky equation is used to describe a number of nonlinear physical processes.

   The aim of this work is to study the influence of the dispersion term of the generalized Kuramoto–Sivashinsky equation on the nature of the dynamics of the system at three different degrees of nonlinearity.

   The system dynamical regime can be qualitatively determined by calculating the largest Lyapunov exponent. If the largest Lyapunov exponent is positive, then the nearby trajectories diverge exponentially and the behavior of the system is chaotic. To find the largest Lyapunov exponent, one needs to repeatedly perturb the trajectory of the dynamical system and find the distance between the perturbed and unperturbed trajectories. To do this for a partial differential equation, it is necessary to transform it into the finite-dimensional system of ordinary differential equations. In this work, for the Kuramoto–Sivashinsky equation, this transition is carried out using the approximation of spatial derivatives on a discrete grid. The boundary conditions are chosen to be periodic. For the resulting system of ordinary differential equations, the largest Lyapunov exponent has been calculated as a function of the dispersion parameter for three degrees of nonlinearity of the equation. It has been found that an increase in the bifurcation parameter leads to a decrease in the largest Lyapunov exponent, and, consequently, to the suppression of chaotic motion. Using the method of lines, the dependences of the solution of the generalized Kuramoto–Sivashinsky equation on the spatial and time variables for several values of the dispersion parameter are constructed. The resulting plots illustrate the transition from a chaotic to a regular regime with an increase in the bifurcation parameter.

   A new scheme has been presented to select the optimal option for decommissioning of a typical nuclear facility and to analyze its stability using the Monte Carlo method. According to the proposed scheme, the optimal option is chosen by multiple Monte Carlo realizations of the particular indicators for each considered option of decommissioning, ranking the options according to the complex indicator in descending order, and calculating the total number of places for each option. The option with the minimum total number of seats is the best. Similarly, the analysis of the stability of the selected decommissioning variant of the nuclear power plant is carried out, for which the entered stability indicator is calculated. The numerical experiments have shown the efficiency of the developed scheme for choosing the optimal variant of the decommissioning of a nuclear facility and its stability under conditions of uncertainty in the initial data. The optimal variant of decommissioning and its stability for a conditionally real object have been numerically calculated.

   An efficient finite-volumetric scheme has been developed to calculate the processes of formation of adiabatic shear bands. The formation of adiabatic shear bands occurs at high-rate deformations of elastic–plastic materials. Numerical simulation of such processes using Lagrangian methods is associated with a number of problems, the main of which is a strong distortion of the grid when simulating high deformations. Large mesh distortions are decrease accuracy of finite element methods. To keep accuracy it is necessary to make operations, such as re-meshing and re-interpolation of data, which are expensive in terms of performance. To avoid this problem we use Euler's description of motion of an elastic-plastic material. A modification of the well-known hypoelastic Wilkins model has been considered. In this article, we propose a numerical method for modeling high-speed shear deformations on two-dimensional grids. The method is based on a finite-volume, while the Courant–Isaacson–Rees method is used to approximate the solution of the Riemann problem. Then, this method is tested on several test problems suggested by other authors.

   The influence of structural features of the text on the author gender classification has been studied. Considered structural features include a linear structure, a syntactic structure, and a structure calculated in the hidden layers of the language model. Existing syntax-accounting methods that have high computational complexity and working time have been develop by analyzing syntactic paths for each word of each sentence, or sequential analysis of sentences structures. The proposed development is based on the use of attention graph layers (GAT) within the neural network architecture, whose input is the matrix of syntax connectivity of all words of the text. An artificially created vector is added to the input feature matrix of each text, which accumulates the activities of all words in the text and is used to characterize the text and classify it. For the proposed network architecture, the method is implemented for evolutionary selection of hyperparameters based on the tree parzen estimator. The results obtained show that the syntax structure of the text for the considered task of author’s gender identification on the open corpora RusPersonality and Gender Imitation Crowdsource “a” increases the accuracy by 2 and 5 %, respectively, according to the f1-score metric with weighted averaging over classes.

   The susceptible–infected–removed (SIR) model, which is a compartmental mathematical model of epidemic outbreak, is considered in the form of the recently proposed one-parameter model. For the particular case of Moscow, the parameters of the model are found that describe the first and second waves of the COVID-19 epidemic. We have analyzed the parameter δ = β/(αN) that determines the behavior of the reduced SIR model dimensionless compartment variables and which is equal to the peak proportion of the infected persons. The results show that both waves can be fitted with the SIR model with satisfactory accuracy. The parameter δ, as well as the infected-to-removed transition rate β, can be asserted equal for the two waves. On the contrary, the susceptible-to-infected transition rate α and the size N of the population potentially exposed to the infection proved to have changed in the second peak compared to the first one. Thus, the parameter δ can be used as an unambiguous and robust characteristic of the dynamics of the outbreak in a particular region.

   The screening of pregnant women in the Moscow region has been performed jointly by mathematicians and doctors to detect prognostic signs of pre-eclampsia, which is a very serious complication of pregnancy, often resulting in the death of the mother and fetus. The mathematical analysis of preliminary experimental data is carried out. As signs (markers) of a possible complication, the angio- and anti-angiogenic proteins PlGF (placental vascular growth factor) and sFlt-1 (a factor inhibiting vascular growth), as well as their ratio, have been considered. Depending on the gestational age, sliding standards have been obtained for patients with a successful pregnancy for each of the factors. The method of logical symptoms has been used to seek prognostic rules. The symptom list contains the concentrations of the indicated proteins in the blood serum and their ratios, expressed on a percentile scale and obtained at different pregnancy periods. A mask of “the class of successful patients versus the class with the most severe form of preeclampsia” has been constructed. The indistinguishability values of logical vectors from the mask have been calculated for each patient. Two boundaries have been obtained for the indistinguishability u. It has been shown that u ≲ 0.348 indicates a risk of developing preeclampsia, and the patients require further monitoring. At u ≲ 0.18, preeclampsia will almost certainly develop.