Applied Mathematics and Control Sciences

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

In this work, using the means of applied mathematics, problems are solved related to the field of automation and control of technological processes, namely, the analytical description of superpositions of rotations that occur during the operation of numerous mechanisms. The practical aspect of the topic is determined by the fact that in mechanisms such as planetary gears, cutter drives in machines for cleaning pipes of large diameters, etc. summation of rotational motions is realized, and the shape of the hodograph is useful information in the design of such devices. The prerequisite for consideration is the principle of summation of rectilinear uniform movements. The aim of the work is to determine how things are in a similar situation when adding rotational synchronous movements. It was found that just as the result of the addition of two uniform rectilinear mechanical movements is also a uniform rectilinear movement, the result of the addition of two uniform unidirectional circular movements is also a uniform circular movement. The hodograph when two uniform oppositely directed circular motions are added is an ellipse. In a particular case, the ellipse can degenerate into a straight line segment. When two asynchronous rotations are added, hodographs in the form of a cochlea are possible, which is similar to Pascal's cochlea.

The work solves the problem of building an intelligent decision support system for servicing oil production equipment. At the first stage - the choice of an intelligent model - it is shown that in the existing conditions it is difficult to obtain a training sample in digital form. On the other hand, there is an opportunity to gain knowledge of subject matter experts - masters and technologists - in the form of a set of linguistic rules. Based on this, a conclusion about the effectiveness of the use of fuzzy logic to solve this problem is made. At the stage of constructing an intelligent model, the use of the matrix approach of fuzzy logic is proposed. To elaborate this approach an algorithm of fuzzy inference based on vector fuzzy predicates is developed. Capabilities and advantages of new algorithm are demonstrated. In particular, it is shown that the matrix representation makes possible reducing computations to solving a system of linear equations. Matrix inference also allows to explicitly determine the range of values of the analyzed parameters at which the knowledge base does not allow making a clear conclusion. A model of a fuzzy logic machine in the form of a fuzzy combinational circuit that analyzes an external memory block is proposed for the analysis of retrospective information on the change in the values of the parameters of technological equipment over time. Specific cases allowing the transition from a state machine to a combinational circuit are shown. Article also shows how this can be done. The main advantage of this approach is the absence of the need to use the difficult to formalize concept of a fuzzy state, which leads to a simplified construction of fuzzy logical devices with memory. At the end work contains brief conclusions about the application of the proposed methods and algorithms for building, testing, implementing a decision support system and about its effectiveness.

In matters of management in social and economic systems, medical issues are becoming increasingly important, and the processes of diagnosis of diseases are becoming the most important task of various health care institutions. Incorrect diagnosis of diseases leads not only to the fact that the patient's treatment becomes ineffective, and sometimes even harmful, but also leads to a significant increase in the costs associated with expensive procedures for further analysis and treatment of the patient. It is for this reason that it is necessary to improve the algorithm of the diagnostic procedure, including two qualitatively new blocks based on the developed methods of latent analysis procedures based on Committee and discriminant analysis. Methods of Committee and discriminant analysis as an auxiliary device are also effective and effective in solving the problem of diagnosing any diseases, including neurological and vascular diseases, as particularly dangerous and significantly impairing the standard of living of the population and leading to increased mortality, as well as worsens the statistical results of health care institutions. The improved algorithm of the diagnostic procedure proposed in the article is supplemented by the authors with two stages, which are added if the picture of symptoms is blurred and the preliminary diagnosis is not presented, or when several diagnoses fit the existing symptoms. This usually occurs due to conflicting, redundant information that corresponds to a variety of diseases. In this case, the method of differential diagnosis is used, which consists in re-interviewing, examining the patient, studying his medical history, prescribing additional clinical, functional and laboratory tests, which in some cases are excessive and even harmful to the patient, as well as increase the total cost of treating the disease, and usually increase the costs of the health care institution as a whole.. To resolve this situation, you can use discriminant analysis methods, which will allow you to find out the necessary diagnosis by assigning factors to a particular disease. However, often there is a situation related to the inconsistency of data that does not allow us to solve the problem of discriminant analysis using standard methods. In this case, you can use current mathematical methods, such as decision trees, the support vector method, and neural networks, but they are poorly interpreted and difficult to understand. therefore, we propose a method of analysis, which is improved by the authors in order to reduce the size of the problem, which helps reduce the number of factors necessary for the diagnosis of diseases. This method allows you to eliminate the inconsistency of data and unambiguously make a diagnosis, some features of which can later be identified by other standard clinical or instrumental methods. Thus, we can conclude that the use of methods of Committee and discriminant analysis in the diagnosis of diseases is not only an auxiliary, but also a mandatory attribute, in order to improve the quality of medical services. Moreover, the main methodological position of the application of these methods is to use the search for latent risk factors for diseases, which allows increasing the proportion of timely diagnosed patients. Also, do not forget about the provision of correct information collection and processing.

Despite the fact that the cluster approach is quite common in scientific works, the issues of the formation, development and evaluation of the effectiveness of cluster-network interactions remain unresolved. The research of the scientific community is based mainly on qualitative methods of cluster analysis (expert analysis, retrospective analysis, comparison method, etc.), however, the need to transform regional development and the transition to neo-economics require the use of economic and mathematical methods of analysis, and their arsenal is relatively small. which necessitates the search for new solutions. An attempt is made in the work to simulate the cluster-network mechanism in the oil and gas industry using neural networks, since the oil sector is one of the key sectors of the Russian economy, which influences the determining rates and paths of the country's socio-economic development, and is subject to the greatest regulation by the government of the country than most other sectors. The most important specific feature of the oil sector is that it is not only able to generate huge monetary resources, but also to accumulate them to solve a large number of socio-economic problems. Based on the results of the trained neural network, using the example of the indicators of the Perm Territory, predicted values of the gross regional product were made and, as a possible core of the oil industry cluster, the profit forecast of the company of the LUKOIL group.