The channeling effect is considered in the so-called accompanying reference system moving with the velocity equal to the longitudinal component of the channeled particle velocity. In such a system, particle motion is finite and similar to the vibrational motion in a one-dimensional potential (plane channeling) or the two-dimensional finite motion in the central field (axial channeling). For relativistic electrons, such motion can be considered both in quantum and classical approximations. In classical consideration, it is possible to calculate quite simply such important characteristics of motion and electromagnetic radiation as the intensity of the resulting electromagnetic radiation, its spectral characteristics, and characteristic times of electron energy loss. The characteristic lifetimes of quantum channeled states and the probability of transitions between them can be estimated using the results of the classical approach, whereas directly in the quantum approach, this can be done only numerically. The simplified analytical consideration is applied to calculate the spectral characteristics and radiation intensity, which accompanies the axial channeling electrons of GeV energies. It is shown that this mechanism can lead to the conversion of a significant part of the energy of the electron beam into high-energy gamma-ray photons when passing the oriented single crystal target about 1 cm thick.

   The dispersion of the metal surface by a corona discharge is investigated. In particular, the interaction of the corona discharge with the silver surface is considered. A diffusion aerosol spectrometer is used to measure the size distribution of generated silver particles. The diffusion aerosol spectrometer can measure the particle spectrum and concentration from 3 nm to 5 μm. The determined particle size distributions are analyzed. The analysis of the experimental data shows that particles are generated by nucleation from the metal vapor, and sources of this vapor are metal atoms produced in the interaction of the electric discharge with the metal surface. To understand the mechanism of silver vapor generation, the concentration of metal atoms is estimated. These atoms are emitted from the metal surface due to the interaction of charged particles with the metal. The theoretical estimate is based on the induction of the charge in the metal by incident charged particles. The momentum transferred to ions of the crystal lattice by an incident charged particle, the cross section for interaction of the incident particle with metal atoms near the surface, and the dispersion coefficient are calculated in the nonrelativistic approximation disregarding the collective interaction of metal atoms and electrons. The comparison of the experimental data with theoretical results indicates that sublimation of atoms from the metal surface is due to the direct interaction between incident particles and ions of the crystal lattice rather than to the thermal surface effect.

   An estimative hydrodynamic calculation of liquid mixing in a cylindrical tank in a turbulent mode, occurring at pumping of an amine water mixture in and its simultaneous pumping out, has been performed. A high inlet jet velocity provides a turbulence cone practically reaching the opposite wall of the tank. Stirring parameters, including the turbulence cone size and components of the liquid flow velocity inside and outside of the turbulence cone have been determined. It is shown that for given parameters it is impossible to form an area near the inlet and outlet openings where the incoming liquid would immediately be removed by the out-going flow from. Contrary to this, there are circumstances under which the existence of such a region would be possible, namely, inhibiting the mixing “interception” would occur, where the fluid outlet positioned in the area opposite to the inlet opening within the cone of turbulence. Effective interception of the inlet flow at given relative positions of inlet and outlet orifices can take place at certain parameters corresponding to a laminar liquid flow. The advantages of the physical calculation over the purely engineering one are shown to be the ease of scaling and the transparency of factors affecting the final result.

   The fourth-order partial differential equation with power-law nonlinearities is investigated. It is used to describe the propagation of highly dispersed pulses in optical fibers. A group analysis of this equation is performed and the group transformations allowed by the differential equation are constructed in three steps. In the first step, a system of governing equations is obtained. In the second step, the coordinates of a tangent vector field are sought for. In the third step, infinitesimal generators are constructed. As a result, two infinitesimal generators allowed by the equation are found. Thus, it is shown that the considered equation is invariant under space and time translations. This indicates that the dimension of the original partial differential equation can be lowered by considering its reduction in traveling wave variables. As a result, an ordinary differential equation is obtained.

   One of the nonlinear partial differential equations used to describe the propagation of pulses in nonlinear optics is considered. The Cauchy problem for this equation cannot be solved by the method of the inverse scattering problem; for this reason, traveling wave variables are used to find the exact solution of the nonlinear second-order partial differential equation. It is proved that the considered equation has exact solutions in the form of periodic and solitary waves, which are determined using elliptic functions. The values of the parameters for the existence of exact solutions of this equation are given. Exact solutions of differential equations are expressed in terms of the Weierstrass elliptic function. A formula for describing solitary waves is also given. Exact solutions in the form of periodic and solitary waves are illustrated. It is shown that the nonlinearity index has a significant effect on the shape of a pulse propagating in an optical fiber.

   The propagation of solitary waves described by the generalized nonlinear Schrödinger equation with an arbitrary refractive index is investigated. The Cauchy problem for the considered equation cannot be solved; for this reason, traveling wave variables are used to find an analytical solution. These variables reduce the partial differential equation to a system of nonlinear ordinary differential equations. A solution of the resulting system of differential equations in the form of solitary waves is found using the simplest equation method. The numerical solution is constructed taking into account the discretization of the problem in time variable using a split-step method and by means of a finite difference in the space variable. The found analytical solution is used to verify the numerical solution of the solitary wave propagation problem described by the generalized nonlinear Schrödinger equation with periodic boundary conditions. Analytical and numerical solutions are plotted and these plots are analyzed taking into consideration the constraints on the parameters of the mathematical model.

   The system of Maxwell’s equations is considered in the Darwin approximation. The study of the system is based on the reduction of systems of partial differential equations (linear and nonlinear) to systems of ordinary differential equations. The variable ψ, where ψ(x, y, z, t) = const is the level surface of some functions, is chosen as an independent variable in systems of ordinary differential equations. The reduction is based on constructing an extended system of equations of characteristics (basic system) for a partial differential equation of the first order (basic equation), which is satisfied by the level surface of the selected functions. All the necessary relations are added to the basic system as the first integrals to obtain a system of ordinary differential equations for the system of partial differential equations under consideration. To search for level surfaces for solutions of the system of equations under consideration, both the approaches previously described in a number of our articles and the newly proposed variants of our method are used. Three systems of ordinary differential equations with different independent variables (different functions ψ(x, y, z, t)) are presented. It is shown that obtaining a level surface in each of the considered approaches has a functional arbitrariness. Some exact solutions of the considered system of partial differential equations are obtained. As an example, for one of the systems of ordinary differential equations, the set of solutions of which depends on the choice of an arbitrary function g(ψ) in the basic equation t = g(ψ), a solution is written for the case where g(ψ) = ψ. The solution of the problem of determining a vortex-free electromagnetic field by a given charge
distribution is given.

   Reliability is one of the most important quality parameters. Since the software is an integral part of the process control system, the failure of software components causes a serious decrease in the product reliability and can cause the loss of the required function. The main sources of control system software and hardware failures in the nuclear industry are described and methods for preventing malfunctions are considered. Nevertheless, experience in software development allows us to highlight the most common errors in its design. The consolidated reasons for software failures include imperfection of the technical task for programming, errors of the programmer developing software modules, errors of operating personnel using software, changes in software operating conditions, the possibility of the appearance of unacceptable arithmetic operations, and an insufficiently complete analysis of the properties of the system in which the software is functioning. Errors made during programming cannot always be quickly noticed when analyzing the text of a program module. This is especially true of program modules having complex logical operations. Therefore, during verification, one should be guided by the documents intended to ensure the quality of the program module, which contain recommendations for usage typical software designs, sizes of software modules, etc.

   Nuclear reactors are widely used in various fields of human activity, for example, for the production of electricity, the production of isotopes, for educational and research purposes, as well as in space engines. Despite their useful purpose, they pose a danger to humans and society due to hazardous radionuclides generated during operation. Therefore, it is imperative to perform actions that reduce the release of radionuclides into the environment. These activities include operating a nuclear reactor under stable conditions and complete control over the system. In this study, the control of the nuclear reactor using a linear-quadratic regulator of the optimal control method is carried out using a nonlinear model of the rigid-point kinetic equation with one group delayed neutron. First, it is shown that the nuclear reactor is asymptotically unstable. Moreover, there are unlimited input and limited output. As a result, a proportional gain compensator is applied to the system to form a closed loop system that stabilizes the system. In addition, the system takes into account reactivity feedback by combining the equations of point kinetics and additional equations of thermal hydraulics. The simulated equation is linearized and a linear quadratic control strategy is applied to achieve performance specifications such as minimum overshoot, settling time, and system stabilization. The simulation results have been confirmed by previous studies.

   Gas turbine engines are complex devices that convert air and fuel into a gas–air mixture, which is used to create jet thrust. Gas turbine engine parts undergo increased loads, in particular, temperature loads. High-strength materials are used to manufacture these parts, and a class of heat-resistant nickel-based alloys are used for parts operating at high temperatures. Turbine discs for gas turbine engines with operating temperatures up to 800°C are one of the most heavily loaded parts. High demands are placed on disk heat-resistant alloys and their manufacturing process, including heat treatment. It is necessary to strictly control the technological parameters of the heat treatment mode, including the heat treatment temperature and the cooling rate of semi-finished products. The software for a control system for a given mode of a variable thermal profile is developed in this work. The structure of the system, its organization, and development and debugging stages are described. Algorithms have been developed to control heating of a muffle furnace and a cooling unit. The developed program allows the heat treatment of the stamping of the disk of a gas turbine engine according to a preselected mode. This makes it possible to obtain the required structure of a nickel-based deformable superalloy, which in turn significantly increases the operation characteristics.

   The world economy is experiencing the turbulence of the end of a long wave of development that precedes its entry into the new cycle, on which the 21st century will leave its mark. Two opposite economic logics are proposed to solve the challenges faced by the Earth: The first one is the conventional one based on the management of scarcity which corresponds to Malthusian ecologists. The second, the management of abundance, is a very non-academic answer with the Prigogine’s thermodynamical approach of economics, and the inclusion of knowledge in the production function. The proof of the analogy between Shannon’s definition of entropy and the Clausius’s one opens the way to a new vision of Economy, abandoning the so-called politicaly correct approach of shortage management for the benefit of an Economy of abundance. The apply of Prigogine’s concept of thermodynamic open system allows to break the deadlock of the green neo-Malthusianism ideology which is only able to make the poor even poorer and more numerous and to prophesy the end of the world.