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In this paper we consider a new modification of the discrete version of Mikhailov’s ‘power–society’ model, previously proposed by the author. This modification includes social-economical dynamics and corruption of the system similarly to continuous ‘power–society–economics–corruption’ model but is based on a stochastic cellular automaton describing the dynamics of power distribution in a hierarchy. This new version is founded on previously proposed ‘power–society’ system modeling cellular automaton, its cell state space enriched with variables corresponding to population, economic production, production assets volume and corruption level. The social-economical structure of the model is inherited from Solow and deterministic continuous ‘power–society–economics–corruption’ models. At the same time the new model is flexible, allowing to consider regional differentiation in all social and economical dynamics parameters, to use various production and demography models and to account for goods transit between the regions. A simulation system was built, including three power hierarchy levels, five regions and 100 municipalities. and a number of numerical experiments were carried out. This research yielded results showing specific changes of the dynamics in power distribution in hierarchy when corruption level increases. While corruption is zero (similar to the previous version of the model) the power distribution in hierarchy asymptotically tends to one of stationary states. If the corruption level increases substantially, volume of power in the system is subjected to irregular oscillations, and only much later tends to a stationary value. The meaning of these results can be interpreted as the fact that the stability of power hierarchy decreases when corruption level goes up.

Technical diagnosis’ current trends are connected to application of FEM computer simulation, which allows, to some extent, replace real experiments, reduce costs for investigation and minimize risks. Computer simulation, just at the stage of research and development, allows carrying out of diagnostics of equipment to detect permissible fluctuations of parameters of equipment’s work. Peculiarity of diagnosis of rolling equipment is that functioning of rolling equipment is directly tied with manufacturing of product with required quality, including accuracy. At that design of techniques of technical diagnosis and diagnostical modelling is very important. Computer simulation of cold rolling of strip was carried out. At that upper working roll was doing vibrations in horizontal direction according with published data of experiments on continuous 1700 rolling mill. Vibration of working roll in a stand appeared due to gap between roll’s craft and guide in a stand and led to periodical fluctuations of strip’s thickness. After computer simulation with the help of DEFORM software strip with longitudinal and transversal thickness variation was gotten. Visualization of strip’s geometrical parameters, according with simulation data, corresponded to type of inhomogeneity of surface of strip rolled in real. Further analysis of thickness variation was done in order to identify, on the basis of simulation, sources of periodical components of strip’s thickness, whose reasons are malfunctions of equipment. Advantage of computer simulation while searching the sources of forming of thickness variation is that different hypothesis concerning thickness formations may be tested without conducting real experiments and costs of different types may be reduced. Moreover, while simulation, initial strip’s thickness will not have fluctuations as opposed to industrial or laboratorial experiments. On the basis of spectral analysis of random process, it was established that frequency of changing of strip’s thickness after rolling in one stand coincides with frequency of working roll’s vibration. Results of computer simulation correlate with results of the researches for 1700 mill. Therefore, opportunity to apply computer simulation to find reasons of formation of thickness variation of strip on the industrial rolling mill is shown.

To study nonlinear effects of biological species interactions numerical-analytical approach is being developed. The approach is based on the cosymmetry theory accounting for the phenomenon of the emergence of a continuous family of solutions to differential equations where each solution can be obtained from the appropriate initial state. In problems of mathematical ecology the onset of cosymmetry is usually connected with a number of relationships between the parameters of the system. When the relationships collapse families vanish, we get a finite number of isolated solutions instead of a continuum of solutions and transient process can be long-term, dynamics taking place in a neighborhood of a family that has vanished due to cosymmetry collapse.

We consider a model for spatiotemporal competition of predators or prey with an account for directed migration, Holling type II functional response and nonlinear prey growth function permitting Alley effect. We found out the conditions on system parameters under which there is linear with respect to population densities cosymmetry. It is demonstated that cosymmetry exists for any resource function in case of heterogeneous habitat. Numerical experiment in MATLAB is applied to compute steady states and oscillatory regimes in case of spatial heterogeneity.

The dynamics of three population interactions (two predators and a prey, two prey and a predator) are considered. The onset of families of stationary distributions and limit cycle branching out of equlibria of a family that lose stability are investigated in case of homogeneous habitat. The study of the system for two prey and a predator gave a wonderful result of species coexistence. We have found out parameter regions where three families of stable solutions can be realized: coexistence of two prey in absence of a predator, stationary and oscillatory distributions of three coexisting species. Cosymmetry collapse is analyzed and long-term transient dynamics leading to solutions with the exclusion of one of prey or extinction of a predator is established in the numerical experiment.

In 2007 Kasman conducted a series of original computer experiments with sine-Gordon kinks moving along artificial DNA sequences. Two sequences were considered. Each consisted of two parts separated by a boundary. The left part of the first sequence contained repeating TTA triplets that encode leucines, and the right part contained repeating CGC triplets that encode arginines. In the second sequence, the left part contained repeating CTG triplets encoding leucines, and the right part contained repeating AGA triplets encoding arginines. When modeling the kink movement, an interesting effect was discovered. It turned out that the kink, moving in one of the sequences, stopped without reaching the end of the sequence, and then “bounced off” as if he had hit a wall. At the same time, the kink movement in the other sequence did not stop during the entire time of the experiment. In these computer experiments, however, a simple DNA model proposed by Salerno was used. It takes into account differences in the interactions of complementary bases within pairs, but does not take into account differences in the moments of inertia of nitrogenous bases and in the distances between the centers of mass of the bases and the sugar-phosphate chain. The question of whether the Kasman effect will continue with the use of more accurate DNA models is still open. In this paper, we investigate the Kasman effect on the basis of a more accurate DNA model that takes both of these differences into account. We obtained the energy profiles of Kasman's sequences and constructed the trajectories of the motion of kinks launched in these sequences with different initial values of the energy. The results of our investigations confirmed the existence of the Kasman effect, but only in a limited interval of initial values of the kink energy and with a certain direction of the kinks movement. In other cases, this effect did not observe. We discussed which of the studied sequences were energetically preferable for the excitation and propagation of kinks.

Problems of multiple covering ($k$-covering) of a bounded set $G$ with equal circles of a given radius are well known. They are thoroughly studied under the assumption that $G$ is a finite set. There are several papers concerned with studying this problem in the case where $G$ is a connected set. In this paper, we study the problem of minimizing the number of circles that form a $k$-covering, $k \geqslant 1$, provided that $G$ is a bounded convex plane domain.

For the above-mentioned problem, we state a 0-1 linear model, a general integer linear model, and a nonlinear model, imposing a constraint on the minimum distance between the centers of covering circles. The latter constraint is due to the fact that in practice one can place at most one device at each point. We establish necessary and sufficient solvability conditions for the linear models and describe one (easily realizable) variant of these conditions in the case where the covered set $G$ is a rectangle.

We propose some methods for finding an approximate number of circles of a given radius that provide the desired $k$-covering of the set $G$, both with and without constraints on distances between the circles’ centers. We treat the calculated values as approximate upper bounds for the number of circles. We also propose a technique that allows one to get approximate lower bounds for the number of circles that is necessary for providing a $k$-covering of the set $G$. In the general linear model, as distinct from the 0-1 linear model, we require no additional constraint. The difference between the upper and lower bounds for the number of circles characterizes the quality (acceptability) of the constructed $k$-covering.

We state a nonlinear mathematical model for the $k$-covering problem with the above-mentioned constraints imposed on distances between the centers of covering circles. For this model, we propose an algorithm which (in certain cases) allows one to find more exact solutions to covering problems than those calculated from linear models.

For implementing the proposed approach, we have developed computer programs and performed numerical experiments. Results of numerical experiments demonstrate the effectiveness of the method.

Metal hydrides are an interesting class of chemical compounds that can reversibly bind a large amount of hydrogen and are, therefore, of interest for energy applications. Understanding the factors affecting the kinetics of hydride formation and decomposition is especially important. Features of the material, experimental setup and conditions affect the mathematical description of the processes, which can undergo significant changes during the processing of experimental data. The article proposes a general approach to numerical modeling of the formation and decomposition of metal hydrides and solving inverse problems of estimating material parameters from measurement data. The models are divided into two classes: diffusive ones, that take into account the gradient of hydrogen concentration in the metal lattice, and models with fast diffusion. The former are more complex and take the form of non-classical boundary value problems of parabolic type. A rather general approach to the grid solution of such problems is described. The second ones are solved relatively simply, but can change greatly when model assumptions change. Our experience in processing experimental data shows that a flexible software tool is needed; a tool that allows, on the one hand, building models from standard blocks, freely changing them if necessary, and, on the other hand, avoiding the implementation of routine algorithms. It also should be adapted for high-performance systems of different paradigms. These conditions are satisfied by the HIMICOS library presented in the paper, which has been tested on a large number of experimental data. It allows simulating the kinetics of formation and decomposition of metal hydrides, as well as related tasks, at three levels of abstraction. At the low level, the user defines the interface procedures, such as calculating the time layer based on the previous layer or the entire history, calculating the observed value and the independent variable from the task variables, comparing the curve with the reference. Special algorithms can be used for solving quite general parabolic-type boundary value problems with free boundaries and with various quasilinear (i.e., linear with respect to the derivative only) boundary conditions, as well as calculating the distance between the curves in different metric spaces and with different normalization. This is the middle level of abstraction. At the high level, it is enough to choose a ready tested model for a particular material and modify it in relation to the experimental conditions.

The mathematical model based on the linear integro-differential Boltzmann equation is considered in this article. The model describes the radiation transfer in the scattering medium irradiated by a point source. The inverse problem for the transfer equation is defined. This problem consists of determining the scattering coefficient from the time-angular distribution of the radiation flux density at a given point in space. The Neumann series representation for solving the radiation transfer equation is analyzed in the study of the inverse problem. The zero member of the series describes the unscattered radiation, the first member of the series describes a single-scattered field, the remaining members of the series describe a multiple-scattered field. When calculating the approximate solution of the radiation transfer equation, the single scattering approximation is widespread to calculated an approximate solution of the equation for regions with a small optical thickness and a low level of scattering. An analytical formula is obtained for finding the scattering coefficient by using this approximation for problem with additional restrictions on the initial data. To verify the adequacy of the obtained formula the Monte Carlo weighted method for solving the transfer equation is constructed and software implemented taking into account multiple scattering in the medium and the space-time singularity of the radiation source. As applied to the problems of high-frequency acoustic sensing in the ocean, computational experiments were carried out. The application of the single scattering approximation is justified, at least, at a sensing range of about one hundred meters and the double and triple scattered fields make the main impact on the formula error. For larger regions, the single scattering approximation gives at the best only a qualitative evaluation of the medium structure, sometimes it even does not allow to determine the order of the parameters quantitative characteristics of the interaction of radiation with matter.

The article is devoted to the construction and investigation of an averaged mathematical model of an aluminum-cobalt-molybdenum hydrocracking catalyst oxidative regeneration. The oxidative regeneration is an effective means of restoring the activity of the catalyst when its granules are coating with coke scurf.

The mathematical model of this process is a nonlinear system of ordinary differential equations, which includes kinetic equations for reagents’ concentrations and equations for changes in the temperature of the catalyst granule and the reaction mixture as a result of isothermal reactions and heat transfer between the gas and the catalyst layer. Due to the heterogeneity of the oxidative regeneration process, some of the equations differ from the standard kinetic ones and are based on empirical data. The article discusses the scheme of chemical interaction in the regeneration process, which the material balance equations are compiled on the basis of. It reflects the direct interaction of coke and oxygen, taking into account the degree of coverage of the coke granule with carbon-hydrogen and carbon-oxygen complexes, the release of carbon monoxide and carbon dioxide during combustion, as well as the release of oxygen and hydrogen inside the catalyst granule. The change of the radius and, consequently, the surface area of coke pellets is taken into account. The adequacy of the developed averaged model is confirmed by an analysis of the dynamics of the concentrations of substances and temperature.

The article presents a numerical experiment for a mathematical model of oxidative regeneration of an aluminum-cobalt-molybdenum hydrocracking catalyst. The experiment was carried out using the Kutta–Merson method. This method belongs to the methods of the Runge–Kutta family, but is designed to solve stiff systems of ordinary differential equations. The results of a computational experiment are visualized.

The paper presents the dynamics of the concentrations of substances involved in the oxidative regeneration process. A conclusion on the adequacy of the constructed mathematical model is drawn on the basis of the correspondence of the obtained results to physicochemical laws. The heating of the catalyst granule and the release of carbon monoxide with a change in the radius of the granule for various degrees of initial coking are analyzed. There are a description of the results.

In conclusion, the main results and examples of problems which can be solved using the developed mathematical model are noted.

The paper presents a physical and numerical analysis of the dynamics and radiation of explosion products formed during the Russian-American experiment in the ionosphere using an explosive generator based on hexogen (RDX) and trinitrotoluene (TNT). The main attention is paid to the radiation of the perturbed region and the dynamics of the products of explosion (PE). The detailed chemical composition of the explosion products is analyzed and the initial concentrations of the most important molecules capable of emitting in the infrared range of the spectrum are determined, and their radiative constants are given. The initial temperature of the explosion products and the adiabatic exponent are determined. The nature of the interpenetration of atoms and molecules of a highly rarefied ionosphere into a spherically expanding cloud of products is analyzed. An approximate mathematical model of the dynamics of explosion products under conditions of mixing rarefied ionospheric air with them has been developed and the main thermodynamic characteristics of the system have been calculated. It is shown that for a time of 0,3–3 sec there is a significant increase in the temperature of the scattering mixture as a result of its deceleration. In the problem under consideration the explosion products and the background gas are separated by a contact boundary. To solve this two-region gas dynamic problem a numerical algorithm based on the Lagrangian approach was developed. It was necessary to fulfill special conditions at the contact boundary during its movement in a stationary gas. In this case there are certain difficulties in describing the parameters of the explosion products near the contact boundary which is associated with a large difference in the size of the mass cells of the explosion products and the background due to a density difference of 13 orders of magnitude. To reduce the calculation time of this problem an irregular calculation grid was used in the area of explosion products. Calculations were performed with different adiabatic exponents. The most important result is temperature. It is in good agreement with the results obtained by the method that approximately takes into account interpenetration. The time behavior of the IR emission coefficients of active molecules in a wide range of the spectrum is obtained. This behavior is qualitatively consistent with experiments for the IR glow of flying explosion products.

The dynamics of public attention to COVID-19 epidemic is studied. The level of public attention is described by the daily number of search requests in Google made by users from a given country. In the empirical part of the work, data on the number of requests and the number of infected cases for a number of countries are considered. It is shown that in all cases the maximum of public attention occurs earlier than the maximum daily number of newly infected individuals. Thus, for a certain period of time, the growth of the epidemics occurs in parallel with the decline in public attention to it. It is also shown that the decline in the number of requests is described by an exponential function of time. In order to describe the revealed empirical pattern, a mathematical model is proposed, which is a modification of the model of the decline in attention after a one-time political event. The model develops the approach that considers decision-making by an individual as a member of the society in which the information process takes place. This approach assumes that an individual’s decision about whether or not to make a request on a given day about COVID is based on two factors. One of them is an attitude that reflects the individual’s long-term interest in a given topic and accumulates the individual’s previous experience, cultural preferences, social and economic status. The second is the dynamic factor of public attention to the epidemic, which changes during the process under consideration under the influence of informational stimuli. With regard to the subject under consideration, information stimuli are related to epidemic dynamics. The behavioral hypothesis is that if on some day the sum of the attitude and the dynamic factor exceeds a certain threshold value, then on that day the individual in question makes a search request on the topic of COVID. The general logic is that the higher the rate of infection growth, the higher the information stimulus, the slower decreases public attention to the pandemic. Thus, the constructed model made it possible to correlate the rate of exponential decrease in the number of requests with the rate of growth in the number of cases. The regularity found with the help of the model was tested on empirical data. It was found that the Student’s statistic is 4.56, which allows us to reject the hypothesis of the absence of a correlation with a significance level of 0.01.