Tag Archives: mathematical model


UDC 615.47
doi: 10.26102/2310-6018/2019.24.1.009

M.I. Lukashov, E.V Pismennaya, O.Y. Olisova, L.V. Starodubtseva, L.V. Shulga

The work is devoted to the urgent problem of improving the quality of medical care for the population suffering from genital herpes due to the timely and qualitative assessment of the severity of the disease under study. In the course of the research, a fuzzy mathematical model was obtained, which allows to distinguish four classes of patient’s conditions: patients with clinically undetectable genital herpes; patients with detected traces of herpes; patients with detected herpes; patients with clinically observed herpes. The methodology used in the synthesis of hybrid fuzzy decision rules allows to reliably separate the selected classes of States in the conditions of incomplete and fuzzy representation of the original data with a strongly overlapping structure of classes. The chosen classification system allows to rationalize the choice of treatment regimens depending on the individual condition of patients. As a result of the research it was found that the use of fuzzy logic of decision-making, can improve the quality of decision-making, according to the stages of disease by 10,…, 15% and reduce the duration of treatment by 5,…10%, which allows us to recommend the results for use in medical practice.

Keywords: : mathematical model, genital herpes, fuzzy logic of decision – making, laboratory indicators.

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UDC 004.056

A.A. Gavrishev

Currently, there is a rapid increase in technical equipment and training of persons who commit illegal acts. In this regard, the number of attempts to carry out criminal attacks on objects of high importance has increased dramatically. For perimeter protection of sites of high categories of significance from unlawful attacks using different security system. A great development is a variety of security systems built on the basis of wireless communication lines. At the same time, it is known that wireless security systems themselves are subject to destructive actions aimed at disrupting their performance. Protection against unauthorized access of alarm and service messages in security systems when they are transmitted over a wireless communication channel is an urgent task. One of the main technologies to protect the radio channel of security systems from unauthorized access is the use of noise-like signals. A promising technology for improving the security of information exchange based on noise-like signals is the use of chaotic signals. However, there are very few algorithms for secure information exchange based on chaotic signals for wireless security systems. One of the known algorithms of protected information exchange based on chaotic signals is presented. It is noted that there is no formalized mathematical description of this algorithm of protected information exchange, which allows to understand more clearly the process of its functioning, in the known literature. In this regard, the author, partly on the basis of the well-known literature, for this algorithm of secure information exchange developed a mathematical model, an explanatory block diagram of the developed mathematical model. With the help of a well-known algorithm of secure information exchange and a mathematical model developed on its basis, it is potentially possible to increase the security of transmitted messages from unauthorized access to various wireless security systems. Also, the proposed example on the mathematical description of the algorithm of secure information exchange, due to its simplicity, may be extended to a wider class of algorithms for secure information exchange.

Keywords: : mathematical model, radio channel, security systems, security, information exchange.

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UDC 615.47

A.V. Bykov, N.A. Korenevsky, S.A. Parkhomenko, A.V. Boytsov, E.V. Cymbal, L.V. Starodubtseva

The work is devoted to the actual problem of improving the quality of care for patients suffering from critical ischemia of the lower extremities, turning into gangrene, which may result in amputation and even death. In the course of the research it was shown that the task of predicting the recurrence of lower extremity gangrene belongs to the class of poor results, which was the basis for choosing the method of synthesis of hybrid fuzzy decision rules as the basic method for studying the methodological synthesis. During the synthesis of fuzzy decision rules, the choice of informative features obtained during surveys and inspections, instrumental and laboratory research methods was justified. As the basic elements of fuzzy decision rules applied to a class of high risk of gangrene recurrence, which are aggregated into a fuzzy rule for assessing the confidence that the patient will have a relapse of the lower extremities. : Scale of confidence in the development of gangrene: – confidence in the recurrence of gangrene; II-medium confidence in the recurrence of gangrene; III – high confidence in the recurrence of gangrene; IV – very high confidence in the recurrence of gangrene. The classification decision is made based on the maximum value of the prediction functions. For each of the selected classes, the appropriate scheme for the prevention of lower extremity gangrene recurrence, whose effectiveness was tested using measurement theory, as well as the analyzed mathematical model of their choice depending on the degree of lower extremity gangrene recurrence. In the course of statistical tests, it was shown that, compared with traditional schemes to prevent the use of the proposed technologies, increase the rate of positive results by 2.4 times (58%) and reduce the risk of limb amputation by 2.5 times (73%). The results obtained guarantee the proposed mathematical models for use in the practice of vascular surgeons and angiologists.


Keywords: : gangrene, lower limbs, prediction, mathematical model, fuzzy logic, prevention, model G. Rush.

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UDC 616.31

N.A. Korenevskiy, T.I. Subbotina, I.I. Khripina, S.A. Parkhomenko,
S.N. Rodionova

The work is devoted to the urgent problem of improving the quality of care for patients suffering from critical ischemia of the lower limbs, passing into gangrene, which can result in amputation and even death. In the course of the studies, it was shown that the task of predicting the origin and development of gangrene of the lower limbs belongs to the class of poorly formalizable problems with an indistinct data structure, which served as the basis for choosing as a basic research apparatus the methodology for the synthesis of hybrid fuzzy decision rules. During the synthesis of fuzzy decisive rules, the selection of informative signs obtained during surveys and examinations, instrumental and laboratory methods of investigation was justified. As the basic elements of fuzzy decision rules, the functions of belonging to the class of high risk of gangrene development are obtained, which are aggregated into an unclear rule of confidence assessment that the patient will develop gangrene of the lower limbs. On a scale of confidence in the development of gangrene, experts determined the secondary functions of belonging to such predictable classes of patient conditions as: I – low confidence in gangrene development; II – moderate confidence in the development of gangrene; III – high confidence in the development of gangrene; IV – very high confidence in the development of gangrene. The decision to classify is taken based on the maximum value of prognostic membership functions.The performed mathematical modeling and expert evaluation of the fuzzy models obtained showed that their predictive confidence is at least 0.9. The same quality of forecasting was confirmed during statistical tests on control samples of 100 persons per class for such indicators as diagnostic specificity, sensitivity and efficiency, as well as prognostic significance of positive and negative results.
The results obtained make it possible to recommend the proposed mathematical models for use in the practice of cardiovascular surgeons and angiologists.

Keywords: : gangrene, lower extremities, prediction, mathematical model, fuzzy logic.

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UDC 519.97, 519.6, 007.681.5

E.A. Andreeva, V.M. Tsiruleva

Currently, an important technical and theoretical task is to develop methods and methods for managing complex dynamic objects that use both traditional methods for controlling dynamic systems (the Pontryagin maximum principle, the Bellman control synthesis method, the theory of automatic control), and methods based on the training of artificial neural networks, such as methods with a reference model, predictive neural control, method for back propagation of an error, etc. Neuropravlenie can be used in the management of fighters, asynchronous electric drives and computers. To develop intelligent control systems, methods of artificial intelligence can be combined with the achievements of the classical theory of optimal control. The article shows the possibility of combining classical methods of optimal control and optimization methods, such as the Pontryagin maximum principle for delayed argument systems, dynamic programming methods, etc., with methods using artificial neural networks.. The use of neural control technologies is caused by the existence of uncontrolled noises and interference. The advantage of neural networks is the possibility of their training, with the right choice of the activation function, accounting for delay in signal transmission between neurons and the formation of an input signal. The aim of the article is the development and construction of a generalized mathematical model for controlling a complex dynamic automatic control system using methods of optimal control theory, optimization methods and neural networks; developing a general hybrid algorithm for obtaining optimal values of control functions and weighting coefficients of a neural network that optimize a given functional. The created model can be used for various activation functions, taking into account the lag and limitations on the control parameters. An algorithm for constructing a numerical solution is developed depending on the values of the parameters of the model, the method, and the type of activation functions. At the end of the article the results of the computational experiment are shown.

Keywords: optimal control, multilayer artificial neural network, neuron ensemble, activation function, mathematical model, system of differential equations with delayed argument, multicriteria problem, maximum principle with delayed argument, discrete optimal control problem.

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UDC 004.942

I.S. Maksyutov, A.B. Migranov

This article describes the generation of a mathematical model of the servo electric drive, which is designed to control the angle of rotation of a pneumatic gun, which is part of an automatic fire extinguishing system using artificial intelligence. At the moment, none of the existing automatic fire extinguishing systems provides fast and effective fire elimination with minimal losses. This topic is very relevant, since the topic of firefighting is one of the most problematic, including in the field. The obtained mathematical model will allow the fire suppression system to react more quickly to the readings of the temperature sensors that determine the core of the flame in order to turn the gun barrel towards the flame. To obtain the final mathematical model, differential equations and transfer functions are drawn up, characterizing the behavior of each member entering the system. The differential equation for the DC motor with voltage regulation in the armature circuit is derived taking into account the moment of inertia (JH) and the load resistance moment (Mn), which are applied to the motor shaft. As the results of the work, the experiment was carried out with the help of the MATLAB software and the curves of the transient process were obtained, showing the speed of the system.

Keywords: pneumatic gun, servomotor, angle of rotation, transfer function, mathematical model, speed of the system.

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UDC 519.97, 519.6, 007.681.5

E.A. Andreeva, V.M. Tsiruleva

Currently, the world is actively developing a new applied area of mathematics, related to the study of artificial neural networks. Interest in them is caused both by theoretical and applied achievements: the possibilities of using computations in spheres previously related only to the field of human intelligence were opened. The relevance of research in this direction is confirmed by numerous examples of the use of neural networks in automation systems [1], robotics of image recognition processes [2], adaptive control [3], forecasting and creating expert systems [4], research of associative memory [5], etc. In complex practical tasks, the trained neural network acts as an expert. An example is medical diagnostics, where a neural network can take into account a large number of numerical parameters (electrical impulses of the nerve cells of the brain and its parts, recorded by means of encephalograms, pressure, weight, etc.). The aim of the work is to construct an artificial oscillatory neural network that can be used to model the activity of the brain: associative memory and attention. The model is formalized as a multicriteria optimal control problem with delay. The purpose of neural network management is its training, which includes the construction of an optimal process that meets the specified criteria. One of the criteria is the terminal criterion determining the state of the neural network at the final moment of time. The optimality conditions in the continuous model are obtained with the help of the Maximum principle for problems with delayed argument [6], [7], [8]. The boundary value problem of the maximum principle is constructed [9]. To obtain optimal conditions in a discrete model that approximates a continuous model, the method of rapid automatic differentiation and numerical methods for solving extremal problems are used [9], [10], [11]. The results of a numerical experiment are presented

Keywords: :optimal control, oscillatory neural network, neuron ensemble, mathematical model, multicriteria problem, maximum principle with delayed argument, discrete optimal control problem.

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UDC 519.676

V.I. Novoseltsev, A.N. Noev, D.E. Orlova

The mathematical model allowing in quantitative expression to establish influence of mutual cyberattacks to economic efficiency of contestant firms is considered. The basis of model is worked out by Lotke-Voltaire’s made in the assumption the modified equations that change of economic efficiency of each firm in the absence of the competitor and, accordingly, cyberattacks, is described by the logistical equation. The qualitative method of differential calculus defines conditions at which observance, despite mutual attacks, competitors do not undergo economic bankruptcy, and continue to function in a normal mode. As the integrated indicator characterizing economic efficiency of contestant firms, the volume of the goods realized by them or the rendered services is applied. The model can be used for a substantiation of requirements to maintenance of information security of competing subjects of the modern market in the conditions of mutual cyberattacks.

Keywords: : cyberattack, mathematical model, economic efficiency, information security, stability.

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UDC 623.681

V. N. Nadtochiy

Currently one of the urgent tasks in radar is the task of the recognition of air targets. At the decision of tasks of synthesis and analysis of radar recognition systems it is necessary to use a mathematical model of the reflected signal. Therefore, developed and justified mathematical model of the signal reflected from air targets with a turbojet engine based on the geometric features of the engine, taking into account the spectral components of the secondary modulation caused by reflection from the steps of compressor (turbine) low-pressure engine. Implementation of the mathematical model obtained based on the effect the secondary modulation when using modulation features. The mathematical model of the reflected signal allows to represent the spectral-Doppler portrait of the aerial target, which takes into account the distribution of the amplitude and phase matching of all components present in the spectrum of the reflected signal. Also allows us to investigate design features of turbojet engines due to the number of blades in the turbine stages and their geometric dimensions, shaft speed, compressor as well as engine size. In the developed mathematical model shows the dependence of the effect of different wavelength of radiation the possibility of its application to radar recognition of air targets such as «aircraft with turbojet engine».

Keywords: : mathematical model, spectral-Doppler portrait, matching components, secondary modulation.

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UDC 519.977.5

E.A. Andreeva, V.M. Tsiruleva, L.G. Kozheko

At the present stage of development of science, technology and economics, much attention is paid to the development of the mathematical theory of optimal control, since it combines fundamental mathematical developments with actual applied problems. One of such urgent tasks is the conservation and use of natural resources [1]. The aim of the work is to build a mathematical model for fisheries management and to determine the optimal control of this process. The model takes into account the factor of natural birth rate, mortality and other parameters. With the advent of new information, the model is improved and supplemented by new conditions, constraints on the parameters of the problem [2], [3], [4]. The fisheries management is carried out by monitoring the intensity of capture. The goal of management is to maximize profits and preserve the population at a given level [5], [6]. The paper considers a continuous model that takes into account the size (weight) of the population, so that the entire fish population is divided into three age classes, differing in weight and size. In addition, the restriction on market demand is taken into account. The model of fisheries management allows to maximize profit from sale of the catch and to keep a level of a population necessary for the further development. To obtain optimality conditions in the continuous model, the Pontryagin Maximum Principle [5], [7] is used, and in the discrete model approximating the continuous one, the method of rapid automatic differentiation and numerical methods for solving extremal problems [5], [7] are used.

Keywords: : optimal control, fishing, structured age population, mathematical model, equilibrium state, Pontryagin maximum principle, discrete optimal control problem.

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