01.00.00 Physical-mathematical sciences
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INVERSE PROBLEM MODELS OF THE SAMUELSON–HICKS
01.00.00 Physical-mathematical sciences
DescriptionThe article continues the cycle of their studies associated with the formulation and development of methods of construction of nonnegative solutions of inverse problems for dynamic systems. In this article the authors formulated and investigated inverse problems for dynamic systems: model of Samuelsson– Hicks. The technique of constructing non-negative solutions of the studied inverse problems. This method is based on the following scheme of the solution. First, we have to identify the formulation of the direct problem, then the formulation of the inverse. This work investigates how correct the mathematical models describing the dynamic economic system are. Further, in the specified tabular solutions of the direct problem, we have built a system of algebraic equations containing the unknown estimated parameters of the studied model. Then posed inverse problem is reduced to solution of a problem of quadratic programming, the solutions of which are defined in MS Excel. The theoretical material is accompanied by the specific example
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METHODS OF REDUCING SPACE DIMENSION OF STATISTICAL DATA
01.00.00 Physical-mathematical sciences
DescriptionOne of the "points of growth" of applied statistics is methods of reducing the dimension of statistical data. They are increasingly used in the analysis of data in specific applied research, such as sociology. We investigate the most promising methods to reduce the dimensionality. The principal components are one of the most commonly used methods to reduce the dimensionality. For visual analysis of data are often used the projections of original vectors on the plane of the first two principal components. Usually the data structure is clearly visible, highlighted compact clusters of objects and separately allocated vectors. The principal components are one method of factor analysis. The new idea of factor analysis in comparison with the method of principal components is that, based on loads, the factors breaks up into groups. In one group of factors, new factor is combined with a similar impact on the elements of the new basis. Then each group is recommended to leave one representative. Sometimes, instead of the choice of representative by calculation, a new factor that is central to the group in question. Reduced dimension occurs during the transition to the system factors, which are representatives of groups. Other factors are discarded. On the use of distance (proximity measures, indicators of differences) between features and extensive class are based methods of multidimensional scaling. The basic idea of this class of methods is to present each object as point of the geometric space (usually of dimension 1, 2, or 3) whose coordinates are the values of the hidden (latent) factors which combine to adequately describe the object. As an example of the application of probabilistic and statistical modeling and the results of statistics of non-numeric data, we justify the consistency of estimators of the dimension of the data in multidimensional scaling, which are proposed previously by Kruskal from heuristic considerations. We have considered a number of consistent estimations of dimension of models (in regression analysis and in theory of classification). We also give some information about the algorithms for reduce the dimensionality in the automated system-cognitive analysis
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SIMULATION OF HEXAGONAL TURBULENT FLOW IN THE NORTH POLAR REGION OF SATURN
01.00.00 Physical-mathematical sciences
DescriptionAs we know, currently, around the north pole of Saturn there is a large-scale hexagonal flow, with characteristic scales of length and speed - 120 m / s and 14,500 km respectively. This trend observed for more than 35 years, is the subject of many experimental and theoretical studies. In this study, we propose a model and discuss the numerical solutions of the equations describing turbulent flow in the planetary boundary layer around the north pole of Saturn. It has been shown that a small violation of the axial symmetry in geostrophic shear leads to the development of hexagonal patterns in a turbulent boundary layer. In addition, under the influence of Coriolis forces and turbulent eddy viscosity gradient in a turbulent boundary layer formed jet pressed to the bottom edge of the layer. These results are used to simulate the observed hexagonal flow around the north pole of Saturn. It is assumed that the small amplitude geostrophic flow is described by a sum of zero and the sixth current harmonic functions, which leads to the excitation current at the upper boundary of the planetary boundary layer. It is found that such excitation enhanced in the boundary layer and reaches a maximum in the jet pressed to the bottom border. This jet, circulating on the hexagon coincides with the region of origin of the cloud cover, which is registered in the experiments. This excitation mechanism hexagonal flow around the north pole of Saturn is confirmed by numerical calculations of three-dimensional non-stationary planetary boundary layer
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PROPAGATION AND BRANCHING OF STRAIMERS IN CONDUCTING ENVIRONMENT
01.00.00 Physical-mathematical sciences
DescriptionIn this work, we develop a model describing the propagation and branching of a streamer in a conducting medium in external electric field. To describe the contribution of the conductivity currents, we modified the standard electrostatic equation taking into account the vortex component of the electric field. As a result of this generalization, the streamer model is formulated in the form of nonlinear equations of parabolic type. In the framework of the proposed model, the problem of the propagation of a streamer in the form of a traveling wave is considered, which leads to the emergence of SaffmanTaylor streamers. For streamers of this type, the branching problem is formulated, which has a unique solution. The dependence of the branch point on the parameters of the problem-the speed of the streamer, the diffusion coefficient of the electrons and the strength of the external electric field, is found. The branching mechanism of the streamer head by dividing it into two parts has been well studied and several alternative models have been formulated for its description. The novelty of the problem in question is that the streamer splits into two three-dimensional channels that are symmetric with respect to the given plane. Numerical experiments also revealed the mechanism of branching of the streamer in the cathode region, connected with the separation of the main channel into several lateral branches. It is noted, that in nature both branching mechanisms are realized, whereas in theory the instability of the surface of the streamer head is investigated
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GRAVITATIONAL WAVES IN THE RICCI FLOW FROM SINGULARITIES MERGER
01.00.00 Physical-mathematical sciences
DescriptionIn this study, we investigate the problem of the emission of gravitational waves produced in collisions of particles submitted to the singularities of the gravitational field. A system of non-linear parabolic equations describing the evolution of the axially symmetric metrics in the Ricci flow derived. A model describing the emission of gravitational waves in the collision and merger of the particles in the Ricci flow proposed. It is shown that the theory of the Ricci flow describes the problem of black holes merge, consistent with Einstein-Infeld theory, which describes the dynamics of the material particles provided by the singularities of the gravitational field. As an example, we consider the metric having axial symmetry and comprising two singularities simulating particles of finite mass. We have numerically investigated the change of the metric in the collision and merger of the particles. The initial and boundary conditions using the exact solution of the static problem, so the collision persist particularly metrics caused by the presence of particles. In numerical experiments determined that the collision of the particles in the Ricci flow leads to the formation of gravitational waves, similar in structure to the waves, registered in the LIGO experiment. Consequently, we can assume that the observed gravity waves caused mainly by transients associated with the change in the metric of a system. The influence of the parameters of the problem - the speed and mass of the particles, on the amplitude and intensity of the emission of gravitational waves was numerically simulated. We have found chaotic behavior of gravitational potentials at the merger of the singularities in the Ricci flow
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MATHEMATICAL METHODS OF RESEARCH OF INVERSE DYNAMIC ECONOMIC SYSTEMS
01.00.00 Physical-mathematical sciences
DescriptionThe article continues the cycle of their studies associated with the formulation and development of methods of construction of nonnegative solutions of inverse problems for dynamic systems. In practice, we have developed and tested mathematical models of dynamic systems. The basis of these models was based on the apparatus of linear algebra, mathematical analysis, mathematical programming, differential equations, optimization methods, optimal control theory, probability theory, stochastic processes, operations research, game theory, statistical analysis. The inverse problem in various models of mathematical Economics was considered rare. These tasks were sufficiently well investigated in the study of physical processes. As shown by the analysis of the theoretical and applied studies of economic processes they represent considerable interest for practice. Therefore, the article considered the inverse problem of the mathematical model, as shown already introduced the results of other mathematical models, are of considerable interest in applied and theoretical research. In this article the authors formulated and investigated the inverse problem for dynamical systems zero-order and the model of Keynes. For their solution, the authors propose to build a system of algebraic equations, then, using methods of quadratic programming, to find the best average of mean square estimation of the model parameter, which are defined in MS Excel
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01.00.00 Physical-mathematical sciences
DescriptionThis article is devoted to the asymptotic analysis of boundary value problem for a system of equations of Nernst-Planck and Poisson for a singularly perturbed system of ordinary differential equations [1], based on two parameters. This boundary value problem simulates electrodiffusion of four kinds of ions at the same time in the diffusion layer in electro-membrane systems with perfectly selective membrane, taling into consideration the reaction of recombination of two ions. Meanwhile the other two ions represent ions of a binary salt. As a simple example, we consider the transport of ions sodium, chlorine, hydrogen and hydroxide, moreover, hydrogen and hydroxyl ions recombine in the diffusion layer. A more complex case is the transfer of the products of dissociation of the dihydrogen phosphate of sodium, namely, ions of sodium and dihydrogen phosphate, the latter dissociate at the interface, in turn, hydrogen ions and hydrogen phosphate. Thus, in the solution can simultaneously store three different types of ions: sodium, hydrogen, phosphate. During the transfer, hydrogen ions and ions of hydrogen phosphate recombine to produce phosphoric acid. The article has revealed the structure of the Nernst diffusion layer at currents above Harkatsa current. It is shown, that in the diffusion layer, there are two types of boundary layers: the inner (reaction) boundary layer and boundary layer at the interface solution / membrane
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01.00.00 Physical-mathematical sciences
DescriptionThis article is a continuation of the previous works of the authors [The influence of reaction dissociation / recombination of molecules of water on transportation of electrolyte 1:1 in the membrane systems in the diffusion layer. Part 1. Mathematical model // Scientific journal of Kuban State Agrarian University, 2016. No. 07(121) and The influence of the reaction of dissociation / recombination of molecules of water on transportation of electrolyte 1: 1 in membrane systems in the diffusion layer. Part 2. Asymptotic analysis // Scientific journal of Kuban State Agrarian University, 2016. – №08(122)] and devoted to assessing the possibility of gravitational convection due to the recombination of hydrogen and hydroxyl ions. The article presents the solution of a boundary-value problem, which is a mathematical model of electrodiffusion for the four types of ions at the same time (two ions of salts and hydrogen and hydroxyl ions) in the diffusion layer in electro-membrane systems with ideal selective membrane, with the heat transfer equation and the Navier-Stokes equation. The article shows the possibility of the emergence of gravitational convection due to the exothermic reaction of recombination of water molecules in the depth of the solution. The article considered the reaction of recombination of hydrogen ions and hydroxyl, although the main results can be applied, after appropriate modifications, and to amfolit-containing solutions, such as wine, juices, dairy products, microbiological processing of biomass (amino acids, anions of polybasic carboxylic acids), municipal effluent (anions of phosphoric acid), etc.
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ABOUT THE NEW PARADIGM OF MATHEMATICAL METHODS OF RESEARCH
01.00.00 Physical-mathematical sciences
DescriptionIn 2011 – 2015, the scientific community was represented by a new paradigm of mathematical methods of research in the field of organizational and economic modeling, econometrics and statistics. There was a talk about a new paradigm of applied statistics, mathematical statistics, mathematical methods of economics, the analysis of statistical and expert data in problems of economics and management. We consider it necessary to develop organizational and economic support for solving specific application area, such as the space industry, start with a new paradigm of mathematical methods. The same requirements apply to the teaching of the respective disciplines. In the development of curricula and working programs, we must be based on a new paradigm of mathematical methods of research. In this study, we present the basic information about a new paradigm of mathematical methods of research. We start with a brief formulation of a new paradigm. The presentation in this article focuses primarily on the scientific field of "Mathematical and instrumental methods of economy", including organizational and economic and economic-mathematical modeling, econometrics and statistics, and decision theory, systems analysis, cybernetics, operations research. We discuss the basic concepts. We talk about the development of a new paradigm. We carry out a detailed comparison of the old and the new paradigms of mathematical methods of research. We give information about the educational literature, prepared in accordance with the new paradigm of mathematical methods of researches
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THE PROBLEM OF RESEARCH OF FINAL RANKING FOR GROUP OF EXPERTS BY MEANS OF KEMENY MEDIAN
01.00.00 Physical-mathematical sciences
DescriptionIn various applications, it is necessary to analyze several expert orderings, i.e. clustered rankings objects of examination. These areas include technical studies, ecology, management, economics, sociology, forecasting, etc. The objects can be some samples of products, technologies, mathematical models, projects, job applicants and others. In the construction of the final opinion of the commission of experts, it is important to find clustered ranking that averages responses of experts. This article describes a number of methods for clustered rankings averaging, among which there is the method of Kemeny median calculation, based on the use of Kemeny distance. This article focuses on the computing side of the final ranking among the expert opinions problem by means of median Kemeny calculation. There are currently no exact algorithms for finding the set of all Kemeny medians for a given number of permutations (rankings without connections), only exhaustive search. However, there are various approaches to search for a part or all medians, which are analyzed in this study. Zhikharev's heuristic algorithms serve as a good tool to study the set of all Kemeny medians: identifying any connections in mutual locations of the medians in relation to the aggregated expert opinions set (a variety of expert answers permutations). Litvak offers one precise and one heuristic approaches to calculate the median among all possible sets of solutions. This article introduces the necessary concepts, analyzes the advantages of median Kemeny among other possible searches of expert orderings. It identifies the comparative strengths and weaknesses of examined computational ways