01.00.00 Physical-mathematical sciences
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LOGARITHMIC LAW FOR DYNAMICAL SYSTEMS FROM QUARKS TO GALAXIES
01.00.00 Physical-mathematical sciences
DescriptionThe article discusses various examples of dynamical systems in which the motion is determined by the logarithmic law - quark systems, hydrodynamic systems, galaxies. Set the general nature of angular motion on a hypersphere in a space of arbitrary dimension and radial movement 6D in the metric of a logarithmic potential. We investigate the 6D metric describing the case of motion with two centers of symmetry. It is shown that in such a metric exists a class of exact solutions, logarithmically dependent on the gravity center coordinates. It was established that in spiral galaxies the orbital motion is due to the logarithmic potential, which is the exact solution of the field equations of Einstein's theory of gravity. The most well-known and widespread in nature case is turbulent flow over a smooth or rough surface, in which the mean velocity depends logarithmically on the distance from the wall. We derivate the logarithmic velocity profile in turbulent flow from the NavierStokes equations. An analogy of the logarithmic velocity profile and the logarithmic law in the case of erosion of materials under impacts been proposed. In electrodynamics, Ampere's law, which describes the interaction of current-carrying conductors, is a consequence of the logarithmic dependence of the vector potential of the distance from the conductor axis. There is, however, an alternative derivation of Ampere law of the Riemann hypothesis about the currents due to the motion of charges
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01.00.00 Physical-mathematical sciences
DescriptionIn the article, we describe and illustrate a method of mathematical modeling in relation to process of decision-making in the conditions of risk and uncertainty on the example of building of agricultural object
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LOGARITHMIC LAW AND EMERGENCE PARAMETER OF CLASSICAL AND QUANTUM SYSTEMS
01.00.00 Physical-mathematical sciences
DescriptionThe work discusses various examples of physical systems which state is determined by the logarithmic law - quantum and classical statistical systems and relativistic motion in multidimensional spaces. It was established that the Fermi-Dirac statistics and BoseEinstein-Maxwell-Boltzmann distribution could be described by a single equation, which follows from Einstein's equations for systems with central symmetry. We have built the rate of emergence of classical and quantum systems. The interrelation between statistical and dynamic parameters in supergravity theory in spaces of arbitrary dimension was established. It is shown that the description of the motion of a large number of particles can be reduced to the problem of motion on a hypersphere. Radial motion in this model is reduced to the known distributions of quantum and classical statistics. The model of angular movement is reduced to a system of nonlinear equations describing the interaction of a test particle with sources logarithmic type. The HamiltonJacobi equation was integrated under the most general assumptions in the case of centrally-symmetric metric. The dependence of actions on the system parameters and metrics was found out. It is shown that in the case of fermions the action reaches extremum in fourdimensional space. In the case of bosons there is a local extremum of action in spaces of any dimension
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01.00.00 Physical-mathematical sciences
DescriptionIn the article we present a spatial structure of largescale transport systems. The model of a transport network can be presented in the form of a graph, with a set of the nodes corresponding to elements of a network and a set of edges – to sections of roads the connecting these nodes. As the model of a card of roads, it is offered to use prefractal graphs which naturally reflect structure of communications when reviewing a transport network in different scales (the states, regions, areas). Prefractal graphs allow describing structural dynamics of the studied system in the discrete time. One of the most widespread scenarios of structural dynamics is the growth of structure. The statement of tasks of the organization of transport routes contains requirements criteria to finding of optimal solutions. Often these requirements and criteria are contradicting each other. It leads to appearance of a multicriteria problem definition. The multicriteria problem definition on a class of prefractal graphs is considered. The optimum algorithm of separation of the greatest maximum paths by the given criterion is constructed and estimates by remaining criteria are given. In operation computing complexity of the constructed algorithm of separation of the greatest maximum paths on a prefractal graph is calculated and advantage of operation of algorithm on last before algorithm of separation of the greatest maximum paths on normal graphs is justified. The constructed algorithm on prefractal graphs has polynomial complexity
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APPLIED STATISTICS – THE STATE AND THE PROSPECTS
01.00.00 Physical-mathematical sciences
DescriptionApplied Statistics - the science of how to analyze the statistical data. As an independent scientificpractical area it develops very quickly. It includes numerous widely and deeply developed scientific directions. Those who use the applied statistics and other statistical methods, usually focused on specific areas of study, ie, are not specialists in applied statistics. Therefore, it is useful to make a critical analysis of the current state of applied statistics and discuss trends in the development of statistical methods. Most of the practical importance of applied statistics justifies the usefulness of the work on the development of its methodology, in which the field of scientific and applied activities would be considered as a whole. We have given some brief information about the history of applied statistics. Based on Scientometrics of Applied Statistics we state that each expert has only a small part of accumulated knowledge in this area. We discuss five topical areas in which modern applied statistics develops, ie five "points of growth": nonparametric, robustness, bootstrap, statistics of interval data, and statistics of non-numerical data. We discuss some details of the basic ideas of a non-numerical statistics. In the last more than 60 years in Russia, there has been a huge gap between official statistics and the scientific community of experts on statistical methods
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01.00.00 Physical-mathematical sciences
DescriptionFuzzy sets are the special form of objects of nonnumeric nature. Therefore, in the processing of the sample, the elements of which are fuzzy sets, a variety of methods for the analysis of statistical data of any nature can be used - the calculation of the average, non-parametric density estimators, construction of diagnostic rules, etc. We have told about the development of our work on the theory of fuzziness (1975 - 2015). In the first of our work on fuzzy sets (1975), the theory of random sets is regarded as a generalization of the theory of fuzzy sets. In non-fiction series "Mathematics. Cybernetics" (publishing house "Knowledge") in 1980 the first book by a Soviet author fuzzy sets is published - our brochure "Optimization problems and fuzzy variables". This book is essentially a "squeeze" our research of 70-ies, ie, the research on the theory of stability and in particular on the statistics of objects of non-numeric nature, with a bias in the methodology. The book includes the main results of the fuzzy theory and its note to the random set theory, as well as new results (first publication!) of statistics of fuzzy sets. On the basis of further experience, you can expect that the theory of fuzzy sets will be more actively applied in organizational and economic modeling of industry management processes. We discuss the concept of the average value of a fuzzy set. We have considered a number of statements of problems of testing statistical hypotheses on fuzzy sets. We have also proposed and justified some algorithms for restore relationships between fuzzy variables; we have given the representation of various variants of fuzzy cluster analysis of data and variables and described some methods of collection and description of fuzzy data
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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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THE DEVELOPMENT OF COUNTRIES' CREDIT RATING ASSESSMENT SYSTEM
01.00.00 Physical-mathematical sciences
DescriptionThis work presents a new approach to the countries’ credit rating definition, based on the advanced mathematical models, such as neural network model, multiple regression, cluster analysis and discriminant analysis. A range of the analyses such as discriminant, cluster, multiple regression models and a neural network were performed on the following economic figures: GDP per capita, GDP value, annual growth rate of GDP, FDI - foreign investment, rate of unemployment, consumer price inflation index, the size of government debt in percentage of GDP. The results, obtained for each model were combined in the countries’ credit rating estimation system called "7M"
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MATHEMATICAL MODELS OF MEDICALECONOMIC CONTROL OF DRUGS PRESCRIPTION
01.00.00 Physical-mathematical sciences
DescriptionThe federal program on essential drugs provision (EDP) is one of the most significant and socially important state projects; it is directed to the reduction of morbidity and mortality together with the improvement of life quality of the society and its social climate. In accordance with the federal law “On social state assistance” from 17.07.1999 №178- FL, the essence of the program is that medical recipes are dispensed for preferential medicines to be received by federal program participants. The medical-economic control (MEC) of the drugs designation and provision of federal benefit recipients is performed basing on the automated registries examination of released drugs. The number of passed and failed examination recipes is determined according to the registers processing results. A certain percentage of the accepted for payment prescriptions is a subject for MEC. For the purpose of the recipes selection for testing, the paper proposes the mathematical models of criteria application and MEC-planning. The game model of organization and MEC performance in health care organizations is build basing on the theory of games. The considered play model suggests that the health services quality examination need to be adjusted and some strategies are to be improved. The solution on the planning of checked recipes number allows to perform the inspection of all the health care organizations, involved in EDP program
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SIMULATION OF TURBULENT FLOW IN A CAVITY ON THE NAVIER-STOKES EQUATIONS
01.00.00 Physical-mathematical sciences
DescriptionThe article deals with the numerical solution of the Navier-Stokes equations describing turbulent flow in a rectangle cavity or in a cuboid with one open face at high Reynolds numbers. It is known, that there is a mechanism of turbulent mixing in natural systems, leading to an increase in the viscosity of the continuous medium. In this regard, we suggest methods of regularization of the Navier-Stokes equations, similar to the natural mechanisms of mixing. We proposed the models based on the properties of the turbulent environment. For this we modified the continuity equation taking into account the pressure fluctuations. It is shown that the incompressibility condition is can be violated due to pressure fluctuation even for flows with low Mach numbers. Modification of continuity equation by the introduction of turbulent viscosity allows the regularization of the Navier-Stokes equations to solve the problems with rapidly changing dynamic parameters. It was shown that the modification of the continuity equation taking into account turbulent fluctuations leads to a system of nonlinear equations of parabolic type. A numerical model of turbulent flow in the cavity with the rapid change in the parameters of the main flow developed. Discovered type of instability of the turbulent flow associated with the rapid changes in the main flow velocity. In numerical simulations found that due to the acceleration of the main flow there is the unsteady vortex flow in the cavity, which is characterized by the integral of energy not vanishing with time, vibrations that have a certain period, depending on the turbulent viscosity