01.00.00 Physical-mathematical sciences
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ANALYTICAL METHOD OF THE RESEARCH OF KINETICS FOR THE PROCESS OF DROPLET DISPERSION
01.00.00 Physical-mathematical sciences
DescriptionIn the article we have concerned the researched mathematical method for the process of droplet dispersion on a horizontal surface. On the basis of the asymptotical method and the method of the division of variables we have found the approximate analytical solution of the problem
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ORBIT ABNORMAL MOVEMENT IN GENERAL RELATIVITY
01.00.00 Physical-mathematical sciences
DescriptionThe solution of the problem of the displacement of the perihelion of Mercury in Einstein's theory of gravitation, taking into account the mechanism of formation of matter from dark energy, has been obtained. It is shown that the observed value of the precession imposes a restriction on the equation of state of dark energy in the case of static fields
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ON THE NUMERATIONS OF THE FINITE PARTIALLY ORDERED SETS
01.00.00 Physical-mathematical sciences
DescriptionIn this article, we discuss various issues related to the formulas approximating the distribution function of prime numbers pi(x). This question has occupied many scholars, but the exact function is well approximated function pi(x) over the number of positive integers not. Based on certain hypotheses, we present a new function s(x) is very well approximated pi(x). The above article hypotheses are so important that their numerical validation and refinement for the lengths of the segments more in 1014 - one of the main areas related to the problem of approximation of the function pi(x) throughout the series of natural numbers. After analyzing the behaviors and constructs many functions, we are building the basis of the function s(x), which is well approximates the function pi(x) throughout the series of natural numbers. We also present a table of values for x, less or equal 1022 for the difference of s(x) - pi(x)
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01.00.00 Physical-mathematical sciences
DescriptionWe consider an approach to the transition from continuous to discrete scale which was defined by means of step of quantization (i.e. interval of grouping). Applied purpose is selecting the number of gradations in sociological questionnaires. In accordance with the methodology of the general stability theory, we offer to choose a step so that the errors, generated by the quantization, were of the same order as the errors inherent in the answers of respondents. At a finite length of interval of the measured value change of the scale this step of quantization uniquely determines the number of gradations. It turns out that for many issues gated it is enough to point 3 - 6 answers gradations (hints). On the basis of the probabilistic model we have proved three theorems of quantization. They are allowed to develop recommendations on the choice of the number of gradations in sociological questionnaires. The idea of "quantization" has applications not only in sociology. We have noted, that it can be used not only to select the number of gradations. So, there are two very interesting applications of the idea of "quantization" in inventory management theory - in the two-level model and in the classical Wilson model taking into account deviations from it (shows that "quantization" can use as a way to improve stability). For the two-level inventory management model we proved three theorems. We have abandoned the assumption of Poisson demand, which is rarely carried out in practice, and we give generally fairly simple formulas for finding the optimal values of the control parameters, simultaneously correcting the mistakes of predecessors. Once again we see the interpenetration of statistical methods that have arisen to analyze data from a variety of subject areas, in this case, from sociology and logistics. We have another proof that the statistical methods - single scientificpractical area that is inappropriate to share by areas of applications
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ASYMPTOTICS OF ESTIMATES OF PROBABILITY DISTRIBUTION DENSITY
01.00.00 Physical-mathematical sciences
DescriptionNonparametric estimates of the probability distribution density in spaces of arbitrary nature are one of the main tools of non-numerical statistics. Their particular cases are considered - kernel density estimates in spaces of arbitrary nature, histogram estimations and Fix-Hodges-type estimates. The purpose of this article is the completion of a series of papers devoted to the mathematical study of the asymptotic properties of various types of nonparametric estimates of the probability distribution density in spaces of general nature. Thus, a mathematical foundation is applied to the application of such estimates in non-numerical statistics. We begin by considering the mean square error of the kernel density estimate and, in order to maximize the order of its decrease, the choice of the kernel function and the sequence of the blur indicators. The basic concepts are the circular distribution function and the circular density. The order of convergence in the general case is the same as in estimating the density of a numerical random variable, but the main conditions are imposed not on the density of a random variable, but on the circular density. Next, we consider other types of nonparametric density estimates - histogram estimates and Fix-Hodges-type estimates. Then we study nonparametric regression estimates and their application to solve discriminant analysis problems in a general nature space
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ASYMPTOTIC METHODS OF STATISTICAL CONTROL
01.00.00 Physical-mathematical sciences
DescriptionStatistical control is a sampling control based on the probability theory and mathematical statistics. The article presents the development of the methods of statistical control in our country. It discussed the basics of the theory of statistical control – the plans of statistical control and their operational characteristics, the risks of the supplier and the consumer, the acceptance level of defectiveness and the rejection level of defectiveness. We have obtained the asymptotic method of synthesis of control plans based on the limit average output level of defectiveness. We have also developed the asymptotic theory of single sampling plans and formulated some unsolved mathematical problems of the theory of statistical control
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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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EINSTEIN, SCHROEDINGER, AND ATOM
01.00.00 Physical-mathematical sciences
DescriptionIn this paper, we consider gravitation theory in multidimensional space. The model of the metric satisfying the basic requirements of quantum theory is proposed. It is shown that gravitational waves are described by the Liouville equation and the Schrodinger equation as well. The solutions of the Einstein equations describing the stationary states of arbitrary quantum and classical systems with central symmetry have been obtained. Einstein’s atom model has been developed, and proved that atoms and atomic nuclei can be represented as standing gravitational waves
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BASIC RESULTS OF THE MATHEMATICAL THEORY OF CLASSIFICATION
01.00.00 Physical-mathematical sciences
DescriptionThe mathematical theory of classification contains a large number of approaches, models, methods, algorithms. This theory is very diverse. We distinguish three basic results in it - the best method of diagnosis (discriminant analysis), an adequate indicator of the quality of discriminant analysis algorithm, the statement about stopping after a finite number of steps iterative algorithms of cluster analysis. Namely, on the basis of Neyman - Pearson Lemma we have shown that the optimal method of diagnosis exists and can be expressed through probability densities corresponding to the classes. If the densities are unknown, one should use non-parametric estimators of training samples. Often, we use the quality indicator of diagnostic algorithm as "the probability (or share) the correct classification (diagnosis)" - the more the figure is the better algorithm is. It is shown that widespread use of this indicator is unreasonable, and we have offered the other - "predictive power", obtained by the conversion in the model of linear discriminant analysis. A stop after a finite number of steps of iterative algorithms of cluster analysis method is demonstrated by the example of k-means. In our opinion, these results are fundamental to the theory of classification and every specialist should be familiar with them for developing and applying the theory of classification
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BINDING ENERGY BIFURCATION AND CHAOS IN ATOMIC NUCLEI
01.00.00 Physical-mathematical sciences
DescriptionThe model of chaotic behavior of nucleons in nuclei, based on the model of nuclear interactions and the Fermi-Dirac statistics is discussed