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In this work, the problem of constructing an electronic analogue of heterogeneous DNA is solved with the help of the methods of mathematical modeling. Electronic analogs of that type, along with other physical models of living systems, are widely used as a tool for studying the dynamic and functional properties of these systems. The solution to the problem is based on an algorithm previously developed for homogeneous (synthetic) DNA and modified in such a way that it can be used for the case of inhomogeneous (native) DNA. The algorithm includes the following steps: selection of a model that simulates the internal mobility of DNA; construction of a transformation that allows you to move from the DNA model to its electronic analogue; search for conditions that provide an analogy of DNA equations and electronic analogue equations; calculation of the parameters of the equivalent electrical circuit. To describe inhomogeneous DNA, the model was chosen that is a system of discrete nonlinear differential equations simulating the angular deviations of nitrogenous bases, and Hamiltonian corresponding to these equations. The values of the coefficients in the model equations are completely determined by the dynamic parameters of the DNA molecule, including the moments of inertia of nitrous bases, the rigidity of the sugar-phosphate chain, and the constants characterizing the interactions between complementary bases in pairs. The inhomogeneous Josephson line was used as a basis for constructing an electronic model, the equivalent circuit of which contains four types of cells: A-, T-, G-, and C-cells. Each cell, in turn, consists of three elements: capacitance, inductance, and Josephson junction. It is important that the A-, T-, G- and C-cells of the Josephson line are arranged in a specific order, which is similar to the order of the nitrogenous bases (A, T, G and C) in the DNA sequence. The transition from DNA to an electronic analog was carried out with the help of the A-transformation which made it possible to calculate the values of the capacitance, inductance, and Josephson junction in the A-cells. The parameter values for the T-, G-, and C-cells of the equivalent electrical circuit were obtained from the conditions imposed on the coefficients of the model equations and providing an analogy between DNA and the electronic model.

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.

In this paper, compared fundamentally different turbulence models for calculating a strongly swirling flow in an abrupt expanding pipe. This task is not only of great importance in practice, but also in theoretical terms. Because in such a flow a very complex anisotropic turbulence with recirculation zones arises and the study of the ongoing processes allows us to find an answer to many questions about turbulence. The flow under consideration has been well studied experimentally. Therefore, it is a very complex and interesting test problem for turbulence models. In the paper compared the numerical results of the one-parameter v_t-92 model, the SSG/LRR-RSMw2012 Reynolds stress method and the new two-fluid model. These models are very different from each other. Because the Boussinesq hypothesis is used in the one-parameter v_t-92 model, in the SSG/LRR-RSM-w2012 model, its own equation is written for each stress, and for the new two-fluid model, the basis is a completely different approach to turbulence. A feature of the approach to turbulence for the new two-fluid model is that it allows one to obtain a closed system of equations. Comparison of these models is carried out not only by the correspondence of their results with experimental data, but also by the computational resources expended on the numerical implementation of these models. Therefore, in this work, for all models, the same technique was used to numerically calculate the turbulent swirling flow at the Reynolds number $Re=3\cdot 10^4$ and the swirl parameter $S_w=0.6$. In the paper showed that the new two-fluid model is effective for the study of turbulent flows, because has good accuracy in describing complex anisotropic turbulent flows and is simple enough for numerical implementation.

Laser damage to transparent solids is a major limiting factor output power of laser systems. For laser rangefinders, the most likely destruction cause of elements of the optical system (lenses, mirrors) actually, as a rule, somewhat dusty, is not an optical breakdown as a result of avalanche, but such a thermal effect on the dust speck deposited on an element of the optical system (EOS), which leads to its ignition. It is the ignition of a speck of dust that initiates the process of EOS damage.

The corresponding model of this process leading to the ignition of a speck of dust takes into account the nonlinear Stefan –Boltzmann law of thermal radiation and the infinite thermal effect of periodic radiation on the EOS and the speck of dust. This model is described by a nonlinear system of differential equations for two functions: the EOS temperature and the dust particle temperature. It is proved that due to the accumulating effect of periodic thermal action, the process of reaching the dust speck ignition temperature occurs almost at any a priori possible changes in this process of the thermophysical parameters of the EOS and the dust speck, as well as the heat exchange coefficients between them and the surrounding air. Averaging these parameters over the variables related to both the volume and the surfaces of the dust speck and the EOS is correct under the natural constraints specified in the paper. The entire really significant spectrum of thermophysical parameters is covered thanks to the use of dimensionless units in the problem (including numerical results).

A thorough mathematical study of the corresponding nonlinear system of differential equations made it possible for the first time for the general case of thermophysical parameters and characteristics of the thermal effect of periodic laser radiation to find a formula for the value of the permissible radiation intensity that does not lead to the destruction of the EOS as a result of the ignition of a speck of dust deposited on the EOS. The theoretical value of the permissible intensity found in the general case in the special case of the data from the Grasse laser ranging station (south of France) almost matches that experimentally observed in the observatory.

In parallel with the solution of the main problem, we derive a formula for the power absorption coefficient of laser radiation by an EOS expressed in terms of four dimensionless parameters: the relative intensity of laser radiation, the relative illumination of the EOS, the relative heat transfer coefficient from the EOS to the surrounding air, and the relative steady-state temperature of the EOS.

In the last quarter of the twentieth century, the nature of global demographic and economic development began to change rapidly: the continuously accelerating growth of the main characteristics that took place over the previous two hundred years was replaced by a sharp slowdown. In the context of these changes, the role of a long-term forecast of global dynamics is increasing. At the same time, the forecast should be based not on inertial projection of past trends into future periods, but on mathematical modeling of fundamental patterns of historical development. The article presents preliminary results of research on mathematical modeling and forecasting of global demographic and economic dynamics based on this approach. The basic dynamic equations reflecting this dynamics are proposed, the modification of these equations in relation to different historical epochs is justified. For each historical epoch, based on the analysis of the corresponding system of equations, a phase portrait was determined and its features were analyzed. Based on this analysis, conclusions were drawn about the patterns of world development in the period under review.

It is shown that mathematical description of technology development is important for modeling historical dynamics. A method for describing technological dynamics is proposed, on the basis of which the corresponding mathematical equations are proposed.

Three stages of historical development are considered: the stage of agrarian society (before the beginning of the XIX century), the stage of industrial society (XIX–XX centuries) and the modern era. The proposed mathematical model shows that an agrarian society is characterized by cyclical demographic and economic dynamics, while an industrial society is characterized by an increase in demographic and economic characteristics close to hyperbolic.

The results of mathematical modeling have shown that humanity is currently moving to a fundamentally new phase of historical development. There is a slowdown in growth and the transition of human society into a new phase state, the shape of which has not yet been determined. Various options for further development are considered.

Certifying a transport airplane for the flights under icing conditions in Russia was carried out within the framework of the requirements of Annex С to the AP-25 Aviation Rules. In force since 2023 to replace AP-25 the new Russian certification document “Airworthiness Standards” (NLG-25) proposes the introduction of Appendix O. A feature of Appendix O is the need to carry out calculations in conditions of high liquid water content and with large water drops (500 microns or more). With such parameters of the dispersed flow, such physical processes as the disruption and splashing of a water film when large drops enter it become decisive. The flow of a dispersed medium under such conditions is essentially polydisperse. This paper describes the modifications of the IceVision technique implemented on the basis of the FlowVision software package for the ice accretion calculations within the framework of Appendix O.

The main difference between the IceVision method and the known approaches is the use of the Volume of fluid (VOF) technology to the shape of ice changes tracking. The external flow around the aircraft is calculated simultaneously with the growth of ice and its heating. Ice is explicitly incorporated in the computational domain; the heat transfer equation is solved in it. Unlike the Lagrangian approaches, the Euler computational grid is not completely rebuilt in the IceVision technique: only the cells containing the contact surface are changed.

The IceVision 2.0 version accounts for stripping the film, as well as bouncing and splashing of falling drops at the surfaces of the aircraft and ice. The diameter of secondary droplets is calculated using known empirical correlations. The speed of the water film flow over the surface is determined taking into account the action of aerodynamic forces, gravity, hydrostatic pressure gradient and surface tension force. The result of taking into account surface tension is the effect of contraction of the film, which leads to the formation of water flows in the form of rivulets and ice deposits in the form of comb-like growths. An energy balance relation is fulfilled on the ice surface that takes into account the energy of falling drops, heat exchange between ice and air, the heat of crystallization, evaporation, sublimation and condensation. The paper presents the results of solving benchmark and model problems, demonstrating the effectiveness of the IceVision technique and the reliability of the obtained results.

Instrument of numerical-analytical modeling of characteristics of propagation of electromagnetic waves in chaotic space plasma with taking into account effects of gravitation is developed for interpretation of data of measurements of astrophysical precision instruments of new education. The task of propagation of waves in curved (Riemann’s) space is solved in Euclid’s space by introducing of the effective index of refraction of vacuum. The gravitational potential can be calculated for various model of distribution of mass of astrophysical objects and at solution of Poisson’s equation. As a result the effective index of refraction of vacuum can be evaluated. Approximate model of the effective index of refraction is suggested with condition that various objects additively contribute in total gravitational field. Calculation of the characteristics of electromagnetic waves in the gravitational field of astrophysical objects is performed by the approximation of geometrical optics with condition that spatial scales of index of refraction a lot more wavelength. Light differential equations in Euler’s form are formed the basis of numerical-analytical instrument of modeling of trajectory characteristic of waves. Chaotic inhomogeneities of space plasma are introduced by model of spatial correlation function of index of refraction. Calculations of refraction scattering of waves are performed by the approximation of geometrical optics. Integral equations for statistic moments of lateral deviations of beams in picture plane of observer are obtained. Integrals for moments are reduced to system of ordinary differential equations the firsts order with using analytical transformations for cooperative numerical calculation of arrange and meansquare deviations of light. Results of numerical-analytical modeling of trajectory picture of propagation of electromagnetic waves in interstellar space with taking into account impact of gravitational fields of space objects and refractive scattering of waves on inhomogeneities of index of refraction of surrounding plasma are shown. Based on the results of modeling quantitative estimation of conditions of stochastic blurring of the effect of gravitational lensing of electromagnetic waves at various frequency ranges is performed. It’s shown that operating frequencies of meter range of wavelengths represent conditional low-frequency limit for observational of the effect of gravitational lensing in stochastic space plasma. The offered instrument of numerical-analytical modeling can be used for analyze of structure of electromagnetic radiation of quasar propagating through group of galactic.

Calcium is a major nutrient regulating metabolism in a plant. Deficiency of calcium results in a growth decline of plant tissues. Ca may be lost from forest soils due to acidic atmospheric deposition and tree harvesting. Plant-available calcium compounds are in the soil cation exchange complex and soil waters. Model of soil calcium dynamics linking it with the model of soil organic matter dynamics ROMUL in forest ecosystems is developed. ROMUL describes the mineralization and humification of the fraction of fresh litter which is further transformed into complex of partially humified substance (CHS) and then to stable humus (H) in dependence on temperature, soil moisture and chemical composition of the fraction (nitrogen, lignin and ash contents, pH). Rates of decomposition and humification being coefficients in the system of ordinary differential equations are evaluated using laboratory experiments and verified on a set of field experiments. Model of soil calcium dynamics describes calcium flows between pools of soil organic matter. Outputs are plant nutrition, leaching, synthesis of secondary minerals. The model describes transformation and mineralization of forest floor in detail. Experimental data for calibration model was used from spruсe forest of Bulgaria.

A mathematical model for the production of biogas from animal waste was developed. An algorithm for identification of model parameters was developed. The accuracy of model identification was performed. The result of simulation for batch and continuous modes of supply of substrate was shown. The optimum flow rate of the substrate for continuous operation was found.

The work aims to study the admissibility of the quasi-steady approach application in fluid flow modeling inside of evaporating drops placed on a solid horizontal substrate. Non-steady model has been developed to compare results with a quasi-steady model. For the first time one-dimensional motion equation of fluid in a drop is proposed from a momentum conservation law. We have shown that inward flow is possible on the edge of drop in one-dimensional models. It may be explained by existence of stagnation points.