This paper is devoted to mathematical modeling electron motion in collisional ionospheric plasma of the F-layer, which forms in the region of Langmuir turbulence under the influence of a powerful wave. This plasma is characterized by the presence of cavitons, which create a potential energy profile with distinct minima and maxima (potential barriers). Particular attention is given to modeling the passage of electrons through the caviton potential barrier. An expression for calculating the overcoming barrier time is obtained analytically. It is shown that the calculation results within this expression are in good agreement with the results of numerical simulations over a wide range of caviton width and electric field amplitudes. The numerical simulations are performed within the framework of a stochastic approach using Langevin differential equations. Using the elasticity coefficient, the sensitivity of the overcoming barrier time to the caviton width and the electric field amplitude is analyzed. The behavioral features and physical regularities of the overcoming barrier time function obtained through numerical simulations are also discussed.

A stochastic Langevin differential equation approach is proposed to model electron motion in collisionless ionospheric plasma within regions of Langmuir turbulence driven by a powerful wave. The model attributes electron motion to interactions with a broad wave packet excited by this incident wave. The primary mechanisms initiating motion are velocity-space diffusion, characterized by an inhomogeneous alternating electric field, and Landau damping. This paper outlines the model construction principles and analyzes key aspects, including the hierarchy of relaxation timescales, the selection of the simulation time step, the fulfillment of conditions for a broad wave packet, and the stochastic nature of electron-wave interactions (modeled as jumps). Implementation features of the numerical model are also discussed. The model's adequacy is demonstrated by comparing numerical results with those obtained from the widely used Gurevich model for high-energy electrons. Crucially, the proposed approach successfully describes the dynamics across the entire electron distribution, including the bulk population, rather than only the high-energy tail.

The theoretical study of electron acceleration in ionospheric plasma within a Langmuir turbulence layer is presented. Most studies describe electron acceleration using a collisionless model based on a simplified kinetic equation. Our study validates the collisionless approach and proposes a model for accounting for the collisional effects of accelerating electrons with plasma particles. The proposed model is based on stochastic Langevin differential equations. It is shown that, as the size of the caviton region increases, the importance of taking collisions into account for adequately modeling the acceleration processes increases.

The behavior of the energy loss function (ELF) for fast electrons in ionospheric plasma is investigated over a wide energy range. It is demonstrated that at high energies (greater than 2 eV), the results obtained using the widely used Swartz approximation for calculating energy losses deviate significantly from the values obtained using the quantum approach proposed by Larkin. Using the kinetic approach, a universal expression for calculating losses over a wide energy range is derived. This expression yields results that agree well with those obtained using the Swartz approximation for low energies (less than 2 eV) and those calculated taking quantum effects into account using Larkin’s approach.

The article is devoted to a theoretical study of the heating rate of thermal electrons in the superthermal F-region of the ionosphere at altitudes of 150-350 km in a wide range of zenith angle changes of 45 (degrees) -90 (degrees) . A universal formula for calculating the heating rate is obtained. To obtain the formula, the generalized Hoegy theory was used, within the framework of which the heating rate is determined by the flow function of superthermal electrons and their energy loss rate. The dependence of the superthermal electron flow is obtained in two ways: direct numerical simulation and using the bi-exponential function (BiEX-function). An analytical expression for the energy loss function is obtained using the Shkarofsky kinetic approach. It is demonstrated that the obtained relationship is in good agreement with the widely used Swartz approximation. It is demonstrated that the heating rate calculations using the formula obtained in the work are in good agreement with the experimental data

Heating rate of thermal electrons by superthermal electrons in ionospheric F-layer at altitudes of 150-350 km was studied over a wide range of zenith angles 45-90 degrees. As a result, a simple and universal formula for calculating the heating rate was obtained. The generalized Hoegy theory was used to obtain the formula. Within the framework of the Hoegy theory, the heating rate is determined by superthermal electrons flux and energy loss function. The energy dependence of the superthermal electron flux was obtained using the bi-exponential function. The kinetic approach was used to receive an analytical expression for the energy loss function. It has been shown that the obtained expression in good agreement with the widely used Swartz approximation. The applicability limits of the obtained expression for energy losses are discussed. It was demonstrated that the heating rate calculations using the formula obtained in the work has a good agreement with the experiments.

Purpose : the article is aimed at presenting the author’s methodology for evaluating the efficiency of redistribution of the tax base within the consolidated group of taxpayers (hereinafter REAG) the construction of which is based on the determination of relationships and assessment of the degree for influence of tax deductions on the operating profit of the enterprise. Methods : in the part of the study of the tax base and REAG contributions, methods of applying statistical analysis were used, including the study of the structure and dynamics of tax payments of the enterprise. As part of the construction of the model, this article provides the correlation and regression analysis, providing for several stages related to the application of the specific tax regime of REAG as the significant factor of influence on the process of distribution of the taxable base of the organization. The data of public joint-stock company «Oil company «ROSNEFT» were used to assess the influence on the operating profit of the enterprise of the taxes paid. The graphical analysis was used to visualize the data. Results : in the course of the study, the degree of influence of the volume for sales of products, selling prices, decrease of the cost and changes in the sales structure on the volume of tax payments of the enterprise was assessed. These factors determine the effectiveness of the REAG’s tax regime, as they represent the company’s profit and therefore affect the amount of income tax. This dependence is reflected in the developed econometric model, which determines the share of each factor in the total income tax of the enterprise, which allows to optimize the tax base of the organization in the future. Conclusions and Relevance : the analysis revealed the need to improve the efficiency of income tax payment. In response, the methodology for improving the tax regime is proposed. Tax optimization measures include two directions: the process of redistribution of the tax base and the extension of the term for performance of tax obligations within the combined group (from earlier to later). These measures will reduce the risks of taxpayers.

Entrepreneurship, embodying the process of funds formation in the form of small independent firms — small and medium enterprises is an integral element of the economic system. These subjects carry out a whole range of important socio-economic functions, which to a large extent can influence the overall level of the region’s wealth. These include the creation of additional jobs, an increase in the level of the competitive environment, the alignment of social disproportions, the introduction and promotion of innovative technologies. These factors directly affect the socio-economic development of both the country as a whole and a particular territory. Accordingly, small and medium enterprises are one of the levers of influence on the welfare of the region and the quality of life in it. The article provides a statistical analysis of the small and medium modern businesses development in the Russian Federation, reflecting the mutual influence of income from activities in the regions on financial activities in the territory and the impact on a broad impact. Also, the paper reflects the results of a conceptual approach to the study of previously presented studies of economists devoted to identifying the dependencies of the small and medium businesses development and the level of welfare of the region. As a result of the study, it was found that at the level of the constituent entities of the Russian Federation, the main reasons for the development of small and medium enterprises are the formation of a business support infrastructure, represented by a whole range of measures that provide significant support to small and medium businesses. Namely, the implementation of programs for the financing of small and medium enterprises, the provision of information, consulting and educational support measures. Also, the list of ongoing activities includes property and innovative support for small and medium-sized businesses.

The work is devoted to the development of a mathematical model for studying the probability distribution of money of an agent. The model is based on the Fokker–Planck equation. To calculate the diffusion term, we used the quadratic dependence of the money balance of an agent in the Yakovenko model. To calculate the drift term, we propose to use a function (potential) that takes into account the income (i.e. the influx of money) and expenditures (i.e. the outflow of money) for an agent. For an analytical description of the income of an agent, a linear dependence on money balance was used. Expenditures were characterized by the demand for essential goods, long-term and luxury goods. Tornquist functions were used to describe the demand functions. The construction of the potential made it possible to identify atypical conditions for the formation of the probability distribution of money.

The paper presents a model of fission of excited atomic nuclei. The model is based on the stochastic Langevin differential equation. The amplitude of the Langevin source is characterized by white noise. The adequacy of modeling is tested in two ways: by comparing the analytical solution of the Langevin equation with the numerical one in the harmonic oscillator mode and by comparing the results of numerical modeling with the results obtained by other authors. It is shown that in both cases the results of dynamic modeling provide sufficient accuracy.

At present, the level of economic well-being of each country is subject to crisis trends. A state’s well-being is defined as the support of the population and the provision of material, social, and spiritual necessities of life. The level of well-being depends directly on the nature of economic relations and the degree of improvement of productive forces. For the most part, this relationship is examined from the point of view of the effectiveness of a country’s social and economic policies. It is therefore important to maintain a decent standard of living and quality of life. This article conducts a statistical analysis of living standards in the Russian Federation and its current state, using a system of indicators, which reflect the mutual influence of key indicators of economic development of the country on the quality of life of the population. Additionally, this article employs a conceptual approach to studying previous economic research on identification of dependencies of improvement of population’s quality of life and the level of the country’s economic well-being. This, in turn, leads to the analysis of the already formed state of welfare of the state and to determining the dynamics of the development of welfare in comparison with previous years. The results show that forecasting further dynamics of the state’s welfare development helps in studying the factors that influence the changes in the level and quality of life of the state’s citizens. These include the creation of additional jobs, the improvement of the educational environment, the equalization of social disparities, and the dynamics of labour productivity, among others. These factors have a direct impact on the socio-economic development of the country as a whole, and a single territory.

A model of the air-steam flow rising above a locally heated water surface is developed. The model is based on the system of free convection equations in the Boussinesq approxima-tion and is implemented numerically. The adequacy of the numerical model is supported by the order of magnitude agreement between numerical predictions and experimental data, and with the analytical calculation of the slip velocity of water droplets based on Stokes drag. The temperature and velocity fields in the air-steam mixture are calculated. Based on the velocity field, a relation is proposed for calculating the potential field (potential) caused by the region of lowered dynamic pressure in the centre of the heated spot. The calculated potentials are compared with the potentials in other works using empirical approaches based on the processing of experimental data. (c) 2020 Published by Elsevier Inc.

This paper presents a mathematical model of an air steam flow arising above a locally heated water surface. The model is based on the system of equations of free convection in the Boussinesq approximation and is implemented as a computer program in C language. Numerical simulation aided in obtaining the velocity fields of the jet are obtained at various values of the water surface temperature. The values of the flow velocities obtained in the framework of the Stokes approximation are compared with the calculated values based on the results of experiments on the levitation of water droplets. As a result of the comparison, the condition of the applicability of the approximation of Stokes to estimate the velocity of an air steam flow is formulated.

A mathematical model for calculating the optimal tax path for the effective functioning of companies is developed. The model is based on the principle of the distribution of fixed assets between consumption, investment and taxes. The governing principle of distribution is the maximization of output, which is a key criterion for the desired tax path. As a result of numerical modelling, an approach is proposed for formulating the optimal tax rate, which allows, on the one hand, to increase tax collection, and on the other, to ensure the continuous development of companies. Since the model is obtained on the basis of the fundamental principles of economics and the physics of evolutionary processes, it is expected to expand to foreign practices.

The condensational growth of spherical water microdroplets is studied in a laboratory setup and with a mathematical model. In the experiment, droplet clusters are kept in a freely levitated state within an upward-oriented flow of water vapor. In the presence of an electrostatic field of 1.5.10(5) Vm(-1), droplet growth is accelerated by factors 1.5 to 2.0 as compared to conditions without any external electric field. Presumably water molecules in the ambient air are accelerated through the presence of the electric field. A kinetic model to predict the acceleration of condensational growth confirms this hypothesis to be feasible. The droplets themselves are polarized so that the deposition of steam molecules is facilitated in the electric field. The simplifications and limitations of the model are discussed.

Horizontal oscillations of small droplet clusters (from one to four droplets) levitating over a locally heated water layer in upward vapor-air flow are investigated experimentally. These oscillations are caused by a complex dynamic interaction between the droplets and the nonsteady gas flow. The path of the center of the droplet cluster is similar to a random walk in a potential well. The vibrational spectra of clusters' centers obtained by Fourier analysis showed several frequency peaks between f = 1.61 and 5.96 Hz found in all clusters, which shows that the cluster tends to oscillate as a whole. The possibility of decoupling of the aerodynamic interaction between the gas flow and the droplets and the interaction between individual droplets is discussed.

Digital technologies have changed the relationship between the society and business entities, taxpayers and the state. Ceteris paribus, the ability to effectively manage financial flows and make administrative decisions depends on the correct and established interaction between the state and taxpayers. This study aims to form and develop a taxpayer’s understanding of the digital age with all its features and opportunities for information and communication technologies, including mathematical modeling methods that form the basis of the digital economy for building and sustaining business development, improving the systemic vision of business processes. The research hypothesis is that the further development of economic entities management in the digital context, as well as the coordination of these entities’ interests, is possible only in the partnership of the key economic participants, with the taxpayer at the forefront. That will allow identifying the areas for improving tax trajectories. Using polynomial approximation, the authors have obtained the models of tax trajectories of companies that allow predicting tax burden. The data for approximations are obtained using the previously constructed mathematical model of the optimal tax path. The main input data of the model are fixed assets and human resources, the totality of which form the production function. The analysis of the transformation of tax paths shows ways for achieving a balance of interests between both the state and the taxpayers. Finding this balance will help to overcome the crisis of confidence in the authorities, the development of adaptability and creativity of Russian society to new tax changes. A number of parameters determines the scale of this task. They include the complexity of the object of study, the long-term and multi-aspect nature of the impact which modeling the digital economy has on adaptation to the new digital realities of the state and taxpayers, as well as the absence of significant analogues of the solution to this problem in global and Russian economics.

The relatively stable clusters of regularly positioned small droplets formed over the locally heated water surface have been studied starting from the early paper by the first author. The life of a cluster appeared to be not long because of growing of droplets due to condensation of steam and the final coalescence of large droplets with the substrate layer of water. However, further laboratory studies of biochemical processes in single droplets will be possible only in the case of their longer life. One way to stabilize the cluster suggested recently by the authors is an external infrared heating which prevents growing of droplets. The present paper is focused on another way to stabilize the cluster. This is an expected rebuilding of droplet clusters at periodic changes in power of the heating laser beam. The frequency of these oscillations is more important than their amplitude because of the resonance nature of the phenomenon. The experiments showed that in the simplest variant of alternating constant power values with the same duration of heating, the droplet growth rate is approximately twice reduced with a power modulation period of about 0.9 s. It means that such a "rejuvenescence" procedure can be used at laboratory conditions to prolong the life of levitating droplet clusters. (C) 2018 Elsevier Ltd. All rights reserved.

A self-assembled cluster of microdroplets levitating over a heated water surface is a fascinating phenomenon with potential applications for microreactors and for chemical and biological analysis of small volumes of liquids. Recently, we suggested a method to synthesize a cluster with an arbitrary number of small monodisperse droplets. However, the interactions, which control the structure of the cluster, are still not well understood. Here we propose a Langevin computational model considering the aerodynamic forces between the droplets and random diffusion-like fluctuations. Characteristic length and time scales and scaling relationships of interactions are discussed. The model shows excellent agreement with experimental observations for a small number of droplets.