On the basis of the Gellmann–Feynman theorem for an isolated Wigner–Zeitz cell, the dependence of the binding energy on the electron deficit has been calculated. The critical parameter of the average number of electrons the removal of which leads to the destruction of a given element has been estimated for 38 metals. The stability of metals is violated when removing about 13.1–13.4% of the available conduction electrons, and the critical charge depends nonmonotonically on the atomic number of the element. The calculations have been performed within the local density model and compared to the results obtained using the free electron model. It has been shown that the calculations within these models give close results, which allows us to use the free electron gas approximation for further estimates. Analysis of the results has showed that the critical parameter determining the stability of the cell depends on the classification among transition or non-transition metals and on its average size. It has been found that the main parameters affecting the stability of the metal are the enthalpy of atomization and the energy of the lowest state of the valence electron in an isolated atom. Calculations have shown that the critical parameter determining the stability of the metal depends almost linearly on the enthalpy of atomization, which allows us to extrapolate experimental data obtained by measuring the enthalpy in a small electron deficit to its critical value.

   Consider the propagation of pressure waves in emergency situations on the equipment of the circuit for reactor VVER-1000. Here we discuss two situations: an instant rupture (10 –4 ) in the main circulation pipelines of the primary circuit with a double End Break (DEB) and instantaneous stop (10 –4 ) of the main circulating pump. The considered emergency situations are included in the list of different types of reports necessary for VVER safety justification [1]. As a model for investigation we chose the 3 rd unit of Kalinin NPP (VVER-1000, model 320). All thermohydraulic and physics data for this are taken from the international stander problem Kalinin-3 [2, 3]. The first analysis for those emergency situations was made in works [4, 5]. For the calculations, the code of improved evaluation ATHLET [6] was used, which is included in the AC2 software package, officially obtained by the national research Nuclear University of MEPhI on the basis of a license agreement with Gesellschaft fur Anlagen-und Reaktorsicherheit (GRS) GmbH, Germany [7]. The ATHLET code is certified in Russia for calculations of stationary and transient regimes at reactors with water coolant [8]. We consider in detail the initial period of the accidents, because only at this stage we can observe the strongest amplitude and frequency of pressure fluctuations on NPP elements, which can lead to significant dynamic loads on the structural elements of these objects. This can be estimated either by: joint strength and hydrodynamic calculations, or it is possible to use the results obtained in this work as boundary conditions for the calculation of dynamic loads. The basic reason for the pressure waves in case of instant pump stop was the instant stop of the MCP. But in the case of instant rupture in the pipelines of the first circuit, the main reason is the instant boiling (superheated) of the coolant.

   Various classes of nonlinear reaction–diffusion equations with variable coefficients c(x)u_t = [a(x)f(u)u_x]_x + b(x)g(u), which is based on the representation of the solution in the implicit form ∫ h(u)du = ξ(x)w(t) + η(x), x(u),where the functions h(u), ξ(x), and w(t) are determined during the study of arising functional differential equations. Examples of specific reaction–diffusion equations and their exact solutions are given. The main attention is paid to nonlinear equations of a sufficiently general form, which contain several arbitrary functions that depend on the desired function u and the spatial variable x. Many new generalized traveling-wave solutions and functional separable solutions are described. It is important to note that solutions of such types are usually noninvariant (i. e., they cannot be obtained by classical Lie group analysis of differential equations).

   A mathematical model of the insulin–glucose balance has been proposed to describe the time dynamics of the insulin and glucose concentrations. The actuality of such models is that the analysis of these concentrations in human blood helps to study serious diseases treatment, e.g., diabetes. The numerical and analytical properties of the model, which is the system of two ordinary differential equations of the first order with the initial conditions, have been presented. The system has been examined for the Painlevé test in two special cases, suggesting that the glucose concentration is outside the allowable range: hypoglycemia and hyperglycemia. Particular analytical solutions have been found taking into account the conditions under which the system passes the Painlevé test. Asymptotic solutions have been obtained for the cases of hypoglycemia and hyperglycemia. In the case of hyperglycemia, the solution has been described by hypergeometric functions of Kummer and Tricomi. Graphs of analytical solutions have been constructed and analyzed. Numerical solutions of the system have been found by the fourth order Runge–Kutta method taking into account various external sources of glucose. Plots of numerical solutions have been obtained, showing fluctuations in the glucose concentration corresponding to the results of medical research in diabetes treatment.

   A CMOS ternary majority element of 65 nm bulk technology has been simulated with measures to improve the resistance to noise pulses arising under the influence of single ionizing particles. The impact on the transistors of the element has been modeled with TCAD tools in the form of the collection of the charge from the track of the particle. Charge collection from tracks with normal direction to the crystal surface is simulated. The linear energy transfer to the track is 60 MeV cm²/mg. The CMOS ternary majority element based only on NAND logical elements has been studied. Its topology is based on the interleaving transistors of the cascade connection of NAND logical elements, which provides increased noise immunity to the impacts of single particles. This ternary majority element has fewer transistors than the standard elements based on AND and OR logical gates. It can be useful to design the majority logic of the elements of 28–65 nm CMOS bulk systems that are resistant to the effects of single nuclear particles.

   An increase in the efficiency of algorithms of object tracking in video streams through the parametric optimization of these algorithms has been discussed. In contrast to most of such studies, accuracy and performance parameters are taken into account in one objective optimization function. There are five different results of object tracking in video streams: correct tracking, false object tracking, premature object loss, finding an object with a delay, and double object detection after premature loss. It has been shown that the parametric optimization makes it possible to increase the accuracy and performance of template matching tracking by factors of 4.7 and 2.2, respectively. At the same time, it has been found that the absence of parametric optimization can lead in practice to the refusal of efficient modifications. For example, template adaptation modification increases the template tracking quality only in 10% of the cases and decreases it in other 90 % of the cases. Particular parameters of the algorithm for template object tracking in video streams have been recommended for computer vision systems.

   The process of plastic flow localization in nonpolar elastic–plastic materials subjected to fast shear deformations has been considered. The oxygen free copper and low carbon steel are materials under study. A mathematical model of the processes of plastic flow localization in these materials has been formulated taking into account thermal softening effects. The processes of plastic flow localization are initiated through the uneven initial heating of material. A numerical algorithm based on the finite-difference method has been proposed to perform the numerical simulation of the processed under consideration. In order to optimize the operation of the proposed algorithm, an adaptive mesh refinement technique has been developed. This technique makes it possible to increase the grid resolution in regions of plastic flow localization during the calculation, keeping the accuracy of a numerical simulation on a smaller grid. The algorithm is tested on the problem suggested by other authors and is applied to numerical simulation of multiply ASB formation. The algorithm has demonstrated high performance and accuracy for both problems.

   Two numerical approaches for approximating measurements of the network traffic recorded in a trunk channel based on the traditional least squares method and the coefficient of determination R² . To additionally estimate the accuracy of the approximation of the analyzed data by a lognormal distribution, the dynamics of the dependence of the maximum intensity of the network traffic on the size of the aggregation window has been analyzed. A high level of compliance of the observation data with a lognormal law has been achieved in both approaches. At the same time, the accuracy of approximation increases noticeably when additional terms are included in the approximating function. It has been shown that the dependence of the determination coefficient on the size of the aggregation window for the analyzed network packets allows one to control the accuracy of the observation data by a lognormal law.