01.00.00 Physical-mathematical sciences
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01.00.00 Physical-mathematical sciences
DescriptionAn analysis of the experimental data obtained by the authors, as well as reference books, allowed to hypothesize about the essential role of gravitational convection in electromembrane systems with ampholytes even in underlimiting current regimes. The article is devoted to the development of the mathematical model of ion transport in a flow elecrtomembrane system during electrodialysis of ampholyte-containing solutions with taking into account a possible appearance of gravitational convection, in particular, due to nonisothermal protonation–deprotonation reactions of ampholytes. The article presents the boundary value problem that is the new mathematical model for diffusion, convection and electromigration of four components of the solution (ions of sodium, dihydrogen phosphate and hydrogen, as well as molecules of orthophosphoric acid) in a half of an electrodialysis desalination channel, adjacent to an anion-exchange membrane. The membrane is considered as ideally selective and homogeneous. The system of partial differential equations, that is the base of the model, also includes equations of Navier-Stokes, material balance, convective heat conduction and the electroneutrality condition. The system of equations is supplemented by a number of natural and original boundary conditions. A distinctive feature of this study is the absence of assumptions about the equilibrium of chemical reactions in a diffusion layer. The results of the study can be used for the development of environmentally rational and resource saving membrane technologies for a processing of products of agro-industrial complex
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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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NONPARAMETRIC KERNEL ESTIMATORS OF PROBABILITY DENSITY IN THE DISCRETE SPACES
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 has been started in our articles [3, 4]. This article is a direct continuation of these works [3, 4]. We will regularly use references to conditions and theorems of the articles [3, 4], in which introduced several types of nonparametric estimators of the probability density. We have studied linear estimators. In this article, we consider particular cases - kernel density estimates in discrete spaces. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Under different conditions, we prove the consistency and asymptotic normality of kernel density estimators. We have introduced the concept of "preferred rate differences" and are studied nuclear density estimators based on it. We have introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. Kernel density estimates are considered for sequences of spaces with measures. We give the conditions under which the difference between the densities of probability distributions and of the mathematical expectations of their nuclear estimates uniformly tends to 0. Is established the uniform convergence of the variances. We find the conditions on the kernel functions, in which take place these theorems about uniform convergence. As examples, there are considered the spaces of fuzzy subsets of finite sets and the spaces of all subsets of finite sets. We give the condition to support the use of kernel density estimation in finite spaces. We discuss the counterexample of space of rankings in which the application of kernel density estimators can not be correct
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NON-PARAMETRIC CYCLES ESTIMATORS
01.00.00 Physical-mathematical sciences
DescriptionIn many applications, we study the time series (or a random process), which is the sum of the periodic deterministic function of time and random errors that distort the periodic signal. It is required to estimate the length of the period and the periodic component. It does not assume that the periodic function is included in any parameter family of functions, such as finite sums of sines and cosines. It is obvious that the assumption of occurrence of a periodic function in parametric family does not meet the characteristics of the real world, ie, is conditional, internal mathematical (look for the keys under the lamp because there is a light, not in the bush, where lost, because there are dark). For similar reasons, it is impossible to assume that the distribution function of the random errors is included in any parameter family of distributions. In accordance with the new paradigm of mathematical statistics in this article we studied the problem of nonparametric estimation (minimum) length of the period and the periodic component of the signal. On the basis of natural variation and scope of indicators is suggested a new class of nonparametric estimators of the length of the period and the periodic component in the time series. Based on the general results of statistics of objects of non-numeric nature we proved the consistency of these estimates. From the practical point of view it is necessary to minimize the numerical (one parameter - ability length of period of time) one or more of the 66 functionals, described in the article
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01.00.00 Physical-mathematical sciences
DescriptionIn the article we have considered A. N. Kolmogorov and N. V. Smirnov papers dedicated to one-sided and two-sided goodness-of-fit and homogeneity tests. It has been shown that the term "Kolmogorov - Smirnov test" used incorrectly. We have also given the recommendations on use of the terms "Kolmogorov test", "Smirnov test", "test of Kolmogorov-Smirnov type" and discussed omega-square test (Cramer-von Mises–Smirnov test). Typical errors in the application of these criterions have been considered, in particular to test for normality of the distribution of measurement results
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NONPARAMETRIC AND APPLIED STATISTICS IN OUR COUNTRY
01.00.00 Physical-mathematical sciences
DescriptionWe continue the series of articles about the history of statistics. We discuss the development of nonparametric and applied statistics in our country in 1930 - 1980 years. We have presented the studies of the great statisticians of the twentieth century, such as N.V. Smirnov, L.N. Bolshev, V.V. Nalimov. American statistics show Russian debate about applied statistics. We have briefly listed the process of creation of the All-Union Statistical Association (1990) and its further developments
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MODELS OF NEURONET INFLATION IN RUSSIA
01.00.00 Physical-mathematical sciences
DescriptionThe article’s conclusion is that for adequate and effective inflation modeling in Russia by means of modern neuronet technologies it is necessary to consider tendencies of economic development. For training and forecast, it is necessary to use only those periods of time within the limits of which identical economic tendencies work. The article uses modern tool means, such as neuronet, which is offered to technology, for approximation and forecasting of rates of inflation
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THE HIGHER ASYMPTOTIC EXPANSIONS FINDING FOR BOUNDARY VALUE PROBLEM OF THE ZOM MODEL
01.00.00 Physical-mathematical sciences
DescriptionIn this article authors propose the asymptotic solution of the boundary value problem modeling the transport of salt ions in the cell electrodialysis desalination unit. The domain of the camera desalting broken into two subdomains: electroneutrality and space charge. Subdomains has own asymptotic expansion. The subdomain of the space charge has unique solvability of the current approach used by the solvability condition of the next approximation
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01.00.00 Physical-mathematical sciences
DescriptionAdequate and effective assessment of the efficiency, effectiveness and the quality of scientific activities of specific scientists and research teams is crucial for any information society and a society based on knowledge. The solution to this problem is the subject of scientometrics and its purpose. The current stage of development scientometrics differs greatly from his previous appearance in the open as well as paid on-line access to huge amount of detailed data on a large number of indicators on individual authors and on scientific organizations and universities. The world has well-known bibliographic databases: Web of Science, Scopus, Astrophysics Data System, PubMed, MathSciNet, zbMATH, Chemical Abstracts, Springer, Agris, or GeoRef. In Russia, it is primarily the Russian scientific citing index (RSCI). RSCI is a national information-analytical system, accumulating more than 9 million publications of Russian scientists, as well as the information about citation of these publications from more than 6,000 Russian journals. There is too much information; it is so-called "Big data". But the problem is how to make sense of these large data, more precisely, to identify the meaning of scientometric indicators) and thus to convert them into great information ("great information"), and then apply this information to achieve the objective of scientometrics, i.e. to transform it into a lot of knowledge ("great knowledge") about the specific scientists and research teams. The solution to this problem is creating a "Scientific smart metering system" based on the use of the automated system-cognitive analysis and its software tools – an intellectual system called "Eidos". The article provides a numerical example of the creation and application of Scientometric intelligent measurement system based on a small amount of real scientific data that are publicly available using free on-line access to the RSCI
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01.00.00 Physical-mathematical sciences
DescriptionThis article briefly discusses the mathematical nature of the author's proposed modification of the weighted least squares, in which the amount of the data is used as the weights of observations. There are two variants of this modification. In the first one, the weighting of the observations was made by replacing one observation with a certain amount of the information in it by the corresponding number of observations for unit weight, and then we applied the standard method of least squares. In the second method, the weighting of the observations was performed for each value of the argument by replacing all observations with a certain amount of information in one observation of unit weight which had been obtained as a weighted average of them, and then we applied the standard method of least squares. We have described in detail the technique of numerical calculations of the amount of information in the observations, based on the theory of automated system-cognitive analysis (ASC-analysis) and implemented it with a help of software tools - intelligent system called "Eidos". The article provides an illustration of the proposed approach on a simple numerical example. In the future, we are planning to give more detailed mathematical basis of the method of weighted least squares, which is modified by using the amount of information as weights, but also to explore its properties