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The paper considers the adequacy of the model developed earlier by the author for the analysis of income inequality and based on an empirically confirmed hypothesis that the relative (to the income of the richest group) income values of 20% population groups in total income can be represented as a finite functional sequence, each member of which depends on one parameter — a specially defined indicator of inequality. It is shown that in addition to the existing methods of inequality analysis, the model makes it possible to estimate with the help of analytical expressions the income shares of 20%, 10% and smaller groups of the population for different levels of inequality, as well as to identify how they change with the growth of inequality, to estimate the level of inequality for known ratios between the incomes of different groups of the population, etc.

The paper provides a more detailed confirmation of the proposed model adequacy in comparison with the previously obtained results of statistical analysis of empirical data on the distribution of income between the 20% and 10% population groups. It is based on the analysis of certain ratios between the values of quintiles and deciles according to the proposed model. The verification of these ratios was carried out using a set of data for a large number of countries and the estimates obtained confirm the sufficiently high accuracy of the model.

Data are presented that confirm the possibility of using the model to analyze the dependence of income distribution by population groups on the level of inequality, as well as to estimate the inequality indicator for income ratios between different groups, including variants when the income of the richest 20% is equal to the income of the poor 60 %, income of the middle class 40% or income of the rest 80% of the population, as well as when the income of the richest 10% is equal to the income of the poor 40 %, 50% or 60%, to the income of various middle class groups, etc., as well as for cases, when the distribution of income obeys harmonic proportions and when the quintiles and deciles corresponding to the middle class reach a maximum. It is shown that the income shares of the richest middle class groups are relatively stable and have a maximum at certain levels of inequality.

The results obtained with the help of the model can be used to determine the standards for developing a policy of gradually increasing the level of progressive taxation in order to move to the level of inequality typical of countries with social oriented economy.

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.

A mathematical model that reflects the main features of the protests is constructed in this paper. An analytical solution was found with assuming that only excited part of the population involved in protests. The numerical value of the model coefficients was estimated from the real data for the cascade of protests that took place in Leipzig in 1989. The changes of the participants number in the protest action with influence the model coefficients was analysed.

The model of the process of using machinery and equipment is considered, which takes into account the probabilistic nature of the process of their operation and sale. It takes into account the possibility of random hidden failures, after which the condition of the machine deteriorates abruptly, as well as the randomly arising need for premature (before the end of its service life) sale of the machine, which requires, generally speaking, random time. The model is focused on assessing the market value and service life of machines in accordance with International Valuation Standards. Strictly speaking, the market value of a used machine depends on its technical condition, but in practice, appraisers only take into account its age, since generally accepted measures of the technical condition of machines do not yet exist. As a result, the market value of a used machine is assumed to be equal to the average market value of similar machines of the corresponding age. For these purposes, appraisers use coefficients that reflect the influence of the age of machines on their market value. Such coefficients are not always justified and do not take into account either the degradation of the machine or the probabilistic nature of the process of its use. The proposed model is based on the anticipation of benefits principle. In it, we characterize the state of the machine by the intensity of the benefits it brings. The machine is subjected to a complex Poisson failure process, and after failure its condition abruptly worsens and may even reach its limit. Situations also arise that preclude further use of the machine by its owner. In such situations, the owner puts the machine up for sale before the end of its service life (prematurely), and the sale requires a random timing. The model allows us to take into account the influence of such situations and construct an analytical relationship linking the market value of a machine with its condition, and calculate the average coefficients of change in the market value of machines with age. At the same time, it is also possible to take into account the influence of inflation and the scrap cost of the machine. We have found that the rate of prematurely sales has a significant impact on the cost of new and used machines. The model also allows us to take into account the influence of inflation and the scrap value of the machine. We have found that the rate of premature sales has a significant impact on the service life and market value of new and used machines. At the same time, the dependence of the market value of machines on age is largely determined by the coefficient of variation of the service life of the machines. The results obtained allow us to obtain more reasonable estimates of the market value of machines, including for the purposes of the system of national accounts.

Modeling the fight against terrorist, pirate and robbery acts at sea is an urgent scientific task due to the prevalence of force acts and the insufficient number of works on this issue. The actions of pirates and terrorists are diverse. Using a base ship, they can attack ships up to 450–500 miles from the coast. Having chosen the target, they pursue it and use the weapons to board the ship. Actions to free a ship captured by pirates or terrorists include: blocking the ship, predicting where pirates might be on the ship, penetrating (from board to board, by air or from under water) and cleaning up the ship’s premises. An analysis of the special literature on the actions of pirates and terrorists showed that the act of force (and actions to neutralize it) consists of two stages: firstly, blocking the vessel, which consists in forcing it to stop, and secondly, neutralizing the team (terrorist groups, pirates), including penetration of a ship (ship) and its cleaning. The stages of the cycle are matched by indicators — the probability of blocking and the probability of neutralization. The variables of the act of force model are the number of ships (ships, boats) of the attackers and defenders, as well as the strength of the capture group of the attackers and the crew of the ship - the victim of the attack. Model parameters (indicators of naval and combat superiority) were estimated using the maximum likelihood method using an international database of incidents at sea. The values of these parameters are 7.6–8.5. Such high values of superiority parameters reflect the parties' ability to act in force acts. An analytical method for calculating excellence parameters is proposed and statistically substantiated. The following indicators are taken into account in the model: the ability of the parties to detect the enemy, the speed and maneuverability characteristics of the vessels, the height of the vessel and the characteristics of the boarding equipment, the characteristics of weapons and protective equipment, etc. Using the Becker model and the theory of discrete choice, the probability of failure of the force act is estimated. The significance of the obtained models for combating acts of force in the sea space lies in the possibility of quantitative substantiation of measures to protect the ship from pirate and terrorist attacks and deterrence measures aimed at preventing attacks (the presence of armed guards on board the ship, assistance from warships and helicopters).

The study of the mechanism for the development of mass panic in view of its extreme importance and social danger is an important scientific task. Available information about the mechanism of her development is based mainly on the work of psychologists and belongs to the category of inaccurate. Therefore, the theory of fuzzy sets has been chosen as a tool for developing a mathematical model of a person's susceptibility to panic situations. As a result of the study, an fuzzy model was developed, consisting of blocks: “Fuzzyfication”, where the degree of belonging of the values of the input parameters to fuzzy sets is calculated; “Inference” where, based on the degree of belonging of the input parameters, the resulting function of belonging of the output value to an odd model is calculated; “Defuzzyfication”, where using the center of gravity method, the only quantitative value of the output variable characterizing a person's susceptibility to panic situations is determined Since the real quantitative values for linguistic variables mental properties of a person are unknown, then to assess the quality of the developed model, without endangering people, it is not possible. Therefore, the quality of the results of fuzzy modeling was estimated by the calculated value of the determination coefficient R2, which showed that the developed fuzzy model belongs to the category of good quality models $(R^2 = 0.93)$, which confirms the legitimacy of the assumptions made during her development. In accordance with to the results of the simulation, human susceptibility to panic situations for sanguinics and cholerics can be attributed to “increased” (0.88), and for phlegmatics and melancholics — to “moderate” (0.38). This means that cholerics and sanguinics can become epicenters of panic and the initiators of stampede, and phlegmatics and melancholics — obstacles to evacuation routes. What should be taken into account when developing effective evacuation measures, the main task of which is to quickly and safely evacuate people from adverse conditions. In the approved methods, the calculation of normative values of safety parameters is based on simplified analytical models of human flow movement, because a large number of factors have to be taken into account, some of which are quantitatively uncertain. The obtained result in the form of quantitative estimates of a person's susceptibility to panic situations will increase the accuracy of calculations.

The paper considers the problem of transverse impact on a thin preloaded fiber. The commonly accepted theory of transverse impact on a thin fiber is based on the classical works of Rakhmatulin and Smith. The simple relations obtained from the Rakhmatulin – Smith theory are widely used in engineering practice. However, there are numerous evidences that experimental results may differ significantly from estimations based on these relations. A brief overview of the factors that cause the differences is given in this article.

This paper focuses on the shear wave velocity, as it is the only feature that can be directly observed and measured using high-speed cameras or similar methods. The influence of the fiber preload on the wave speed is considered. This factor is important, since it inevitably arises in the experimental results. The reliable fastening and precise positioning of the fiber during the experiments requires its preload. This work shows that the preload significantly affects the shear wave velocity in the impacted fiber.

Numerical calculations were performed for Kevlar 29 and Spectra 1000 yarns. Shear wave velocities are obtained for different levels of initial tension. A direct comparison of numerical results and analytical estimations with experimental data is presented. The speed of the transverse wave in free and preloaded fibers differed by a factor of two for the setup parameters considered. This fact demonstrates that measurements based on high-speed imaging and analysis of the observed shear waves should take into account the preload of the fibers.

This paper proposes a formula for a quick estimation of the shear wave velocity in preloaded fibers. The formula is obtained from the basic relations of the Rakhmatulin – Smith theory under the assumption of a large initial deformation of the fiber. The formula can give significantly better results than the classical approximation, this fact is demonstrated using the data for preloaded Kevlar 29 and Spectra 1000. The paper also shows that direct numerical calculation has better corresponding with the experimental data than any of the considered analytical estimations.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.

In this article in the frameworks of the sine-Gordon mode we have calculated the dynamical characteristics of kinks and antikinks activated in the homogeneous polynucleotide chains each if them contains only one of the types of the bases: adenines, thymines, guanines or cytosines. We have obtained analytical formulas and constructed the graphs for the kink and antikink profiles and for their energy density in the 2D- and 3D-dimension. Mass of kinks and antikinks, their energy of rest and their size have been estimated. The trajectories of kink and antikink motion in the phase space have been calculated in the 2D- and 3D-dimension.