01.00.00 Physical-mathematical sciences
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01.00.00 Physical-mathematical sciences
DescriptionWe have proposed the method for testing of independence of two alternative variables on the basis of statistics of non-numeric data. The method is aimed at application in problems of statistical quality control. Testing of independence is based on set of small samples, i.e., in the Kolmogorov’s asymptotics, when the number of unknown parameters of the distribution increases in proportion to the data size
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01.00.00 Physical-mathematical sciences
DescriptionWithout science it would be impossible to form a full environmental consciousness. To increase the validity and weight of the findings on the impact of environment on quality of life, it is necessary to quantify the strength and direction of the influence of diverse environmental factors. However, it appears that this is quite problematic for a number of reasons. First, it is the lack or inaccessibility of source of data which is necessary for such type of research. The same data, which still can be found cover just small periods of observations (small longitudinal research data), and their completion, including performing experiments, is fundamentally impossible. As a result, it is impossible to require such full data replications, which is a necessary condition for correct applying of factor analysis. Secondly, environmental factors are described with heterogeneous indices measured in different types of measurement scales (nominal, ordinal and numerical) and in different measurement units. Mathematical methods of comparable processing of such data, and the right software tools for these methods, generally speaking, do not exist. Third, these tasks are large-scale problems, i.e. they are not talking about 5 or max 7 factors as it was in factor analysis, but about hundreds and thousands. Fourthly, the original data is noisy and require sustainable methods. Fifthly, environmental factors are interrelated and require nonlinear nonparametric approaches. To solve these problems it is proposed to apply a new innovative intelligent technology: automated system-cognitive analysis and its software tool – a system called "Eidos". We have also given a brief numerical example of assessing the impact of environmental factors on life expectancy and causes of death
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01.00.00 Physical-mathematical sciences
DescriptionThis article is a continuation of the works [1,2], which were devoted to the study of hydrodynamics and transport of salt ions in the experimental electrochemical cell with a rotating disk with a cation exchange membrane of exact current modes, when the condition of local electroneutrality. This article presents a mathematical model of transport of salt ions in a cell with a rotating disk with a cation exchange membrane exorbitant current regimes, taking into account electroconvection. Under these conditions, fluid dynamics depends on the ion transport process salt and described by the system of Navier-Stokes equations in cylindrical coordinate system with the electric forces
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REAL AND NOMINAL SIGNIFICANCE LEVELS IN STATISTICAL HYPOTHESIS TESTING
01.00.00 Physical-mathematical sciences
DescriptionIn the statistical hypothesis testing, critical values often point to a priori fixed (nominal) significance levels. As such, typically researcher uses the values of three numbers 0.01, 0.05, 0.1, to which may be added a few levels: 0.001, 0.005, 0.02, and others. However, for the statistics with discrete distribution functions, which, in particular, include all nonparametric statistical tests, the real significance levels may be different from the nominal, differ at times. Under the real significance level we refer to the highest possible significance level of discrete statistics, not exceeding a given nominal significance level (ie, the transition to the next highest possible value corresponding discrete statistical significance level is greater than a predetermined nominal). In the article, we have discussed the difference between nominal and real significance levels on the example of nonparametric tests for the homogeneity of two independent samples. We have also studied two-sample Wilcoxon test, the criterion of van der Waerden, Smirnov two-sample two-sided test, sign test, runs test (Wolfowitz) and calculated the real significance levels of the criteria for nominal significance level of 0.05. The study of the power of these statistical tests is accomplished by means of Monte Carlo method. The main conclusion: the use of nominal significance levels instead of real significance levels for discrete statistics is inadmissible for small sample sizes
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LIMIT THEOREMS FOR KERNEL DENSITY ESTIMATORS IN SPACES OF ARBITRARY NATURE
01.00.00 Physical-mathematical sciences
DescriptionSome estimators of the probability density function in spaces of arbitrary nature are used for various tasks in statistics of non-numerical data. Systematic exposition of the theory of such estimators had a start in our work [2]. This article is a direct continuation of the article [2]. We will regularly use references to conditions and theorems of the article [2], in which we introduced several types of nonparametric estimators of the probability density. We studied more linear estimators. In this article we consider particular cases - kernel density estimates in spaces of arbitrary nature. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Asymptotic behavior of kernel density estimators in the general case of an arbitrary nature spaces are devoted to Theorem 1 - 8. Under different conditions we prove the consistency and asymptotic normality of kernel density estimators. We have studied uniform convergence. We have introduced the concept of "preferred rate differences" and studied nuclear density estimators based on it. We have also introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. We have found the asymptotic behavior of dispersions of kernel density estimators and considered the examples including kernel density estimators in finite-dimensional spaces and in the space of square-integrable functions
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NONPARAMETRIC ESTIMATION OF CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS
01.00.00 Physical-mathematical sciences
DescriptionThe article is devoted to the nonparametric point and interval estimation of the characteristics of the probabilistic distribution (the expectation, median, variance, standard deviation, variation coefficient) of the sample results. Sample values are regarded as the implementation of independent and identically distributed random variables with an arbitrary distribution function having the desired number of moments. Nonparametric analysis procedures are compared with the parametric procedures, based on the assumption that the sample values have a normal distribution. Point estimators are constructed in the obvious way - using sample analogs of the theoretical characteristics. Interval estimators are based on asymptotic normality of sample moments and functions from them. Nonparametric asymptotic confidence intervals are obtained through the use of special output technology of the asymptotic relations of Applied Statistics. In the first step this technology uses the multidimensional central limit theorem, applied to the sums of vectors whose coordinates are the degrees of initial random variables. The second step is the conversion limit multivariate normal vector to obtain the interest of researcher vector. At the same considerations we have used linearization and discarded infinitesimal quantities. The third step - a rigorous justification of the results on the asymptotic standard for mathematical and statistical reasoning level. It is usually necessary to use the necessary and sufficient conditions for the inheritance of convergence. This article contains 10 numerical examples. Initial data - information about an operating time of 50 cutting tools to the limit state. Using the methods developed on the assumption of normal distribution, it can lead to noticeably distorted conclusions in a situation where the normality hypothesis failed. Practical recommendations are: for the analysis of real data we should use nonparametric confidence limits
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QUANTUM GRAVITY AND YANG-MILLS THEORY
01.00.00 Physical-mathematical sciences
DescriptionIn this paper, we consider Einstein's theory of gravitation in connection with Yang-Mills theory. The model of the metric satisfying the basic requirements of quantum theory is proposed. The mechanism of generation of baryonic matter of dark energy is discussed
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01.00.00 Physical-mathematical sciences
DescriptionThe article presents a new approach to 2D modeling of transport of salt ions in EMC (electro systems: electrodialysis devices, electro-cells, etc.) under the condition of electrical neutrality with limiting and overlimiting current density. For definiteness as seen half of EMS channel EDA desalting (electrodialysis apparatus), the right border, which serves as a CEM (cation exchange membrane). The new approach in the use of partial differential equations of the first order, instead of equations of convective diffusion. A common method of transport modeling binary electrolyte in the EMS under the condition of electrical neutrality, is to use the equation of convective diffusion (partial differential equations of the second order). The article presents a new approach to modeling 2D transfer binary electrolyte in EMS under the same conditions, using partial differential equation of the first order for the decision, which does not require a boundary condition for concentration on the membrane surface. This allows you to simulate the transport of salt ions, as in prelimit and exorbitant current density and to determine the boundaries of the field of electrical neutrality
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THEORETICAL TOOLS OF STATISTICAL METHODS
01.00.00 Physical-mathematical sciences
DescriptionWe have considered the basic mathematical tools (theorems, methods) which are used regularly in the justification of new results in the field of statistical methods: rules of large numbers, central limit theorems, the necessary and sufficient conditions for the inheritance of convergence, the linearization method, the invariance principle
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GENERAL RELATIVITY AND METRICS OF INHOMOGENEOUS ROTATING UNIVERSE
01.00.00 Physical-mathematical sciences
DescriptionThe metric of inhomogeneous rotating Universe is discussed. There are examples of universal metrics obtained in Einstein's theory of gravitation. On the basis of solutions of Einstein’s equation we have proposed universal metric describing the properties of galaxies, groups and clusters of galaxies in inhomogeneous rotating Universe