## No 2 (2023)

**Year:**2023**Articles:**10**URL:**https://ered.pstu.ru/index.php/amcs/issue/view/356**DOI:**https://doi.org/10.15593/2499-9873/2023.2

On the expressibility of the functions of the system x'(t) = A∙x(t), the eigenvalues of the matrix of which are non-multiple in the form of linear combinations of derivatives of one function included in this system

#### Abstract

The problem of solving a system of linear ordinary differential equations with constant coefficients is one of the most important problems in both the theory of ordinary differential equations and linear algebra. Therefore, on the one hand, new methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of the most well-known methods for solving a system of linear ordinary differential equations with constant coefficients is the method of reducing a system of linear equations to a single higher-order equation, which makes it possible to find solutions to the original system in the form of linear combinations of derivatives of only one unknown function.In this paper, we consider a refinement of the method for reducing a system of linear ordinary differential equations with constant coefficients to a single higher-order equation, which makes it possible to find a general solution to the original system; namely, we study the expressibility of all functions of the system of linear homogeneous differential equations with constant coefficients x^' (t)= A⋅x(t) in the form of linear combinations of derivatives of only one unknown function x_k (t), which is part of this system. For any matrix A, all of whose eigenvalues are not multiples, a new simple criterion for expressibility in terms of matrix ranks is formulated, and its correctness is proved. The result obtained can also be applied in the study of solutions of the system x^' (t)= A⋅x(t) for periodicity and in the study of linear systems for complete observability.

**Applied Mathematics and Control Sciences**. 2023;(2):8-16

The effect analysis of the finite element approximation order on the protective polymer layer contact deformation

#### Abstract

Problems implementation using the finite element method (FEM) is primarily associated with the proper selection of approximation order of the shape functions, material behavior models, the coupling nature analysis and the implementation techniques choice. The choice of numerical solution parameters affects the computational procedures. The paper solves problem of the FEM parameters selection and its realization methods in order to achieve a reasonable balance between the counting time and the solution quality. Simplex elements have shape functions with linear dependences on the coordinates. The FEM implementation with the use of simplex elements has proven itself during the last decades. However, one can note the description inaccuracy of curvilinear boundaries of research objects, which can introduce additional inaccuracies in the problem numerical solution. Second-order elements will allow describing curvilinear boundaries, but require a large expenditure of computational power especially for nonlinear mechanics problems. The development of numerical application software packages allows us to evaluate the influence of the FEM approximation degree on the problem solution with the use of grid methods and computer resources at the present time. The paper found that an increase in the numerical solution error is observed at the point of initial steel - polymer contact. The study considers the elements of order 1 and 2 to analyze the influence of the FEM approximation order on the problem numerical solution. The study is carried out within the framework of the classical Hertz problem. In addition the paper investigate obtained solutions for modeling different characters of contact surface coupling for the problem of contact between a spherical die and a steel half-space through a protective polymer coating 4 mm to 12 mm thick. The study reveal that the elements order has the least influence on the problem solution at ideal contact and the greatest one at frictional contact.

**Applied Mathematics and Control Sciences**. 2023;(2):17-32

Modeling dependence of normalized difference vegetation index on atmospheric and water indicators in Kirkuk

#### Abstract

Climatic conditions and vegetation cover are among the most important environmental factors on which the existence of life on our planet depends. Climate change and military actions have led to progressive land degradation in the Republic of Iraq. As a result, the area of arable land is reduced, soil fertility is reduced, salinization and desertification of soils occur, and the displacement of herbaceous plants by shrubs. To solve the urgent problem of stopping land degradation, it is necessary to study the causes of this and find appropriate solutions by predicting the state of vegetation cover and create a comprehensive monitoring system for quantifying vegetation degradation. The issues of analysis of the relationship between vegetation cover and climatic conditions in the province of Kirkuk of the Republic of Iraq are considered. The Normalized difference vegetation index was used to assess the condition and distribution of the vegetation cover. The popularity of using this index is explained by the fact that it usually correlates with many indicators of the state of the ecosystem, including temperature, precipitation, humidity, evaporation and mineral or organic saturation of the soil, productivity and biomass of vegetation, etc. The purpose of the article is to study the relationship between the normalized difference vegetation index and atmospheric and water indicators of the local level in the province of Kirkuk in the Republic of Iraq for 12 years from 2010 to 2021, as well as to search for a mathematical model that can be used to predict vegetation cover for the coming years. through some climatic indicators that affect vegetation cover. A regression dependence of the normalized difference vegetation index on climatic conditions was built, its quality indicators were studied. It is shown that the resulting mathematical model can be used to predict the normalized difference vegetation index. The results of forecasting by the model for the next 2022 turned out to be satisfactory.

**Applied Mathematics and Control Sciences**. 2023;(2):33-42

Construction of quantile regression using natural gradient descent

#### Abstract

Building mathematical models is an important part of developing digital products in various industries, medicine, geology, construction, finance and other areas. Modeling allows optimizing production processes, identifying patterns, predicting time series, classifying objects, and constructing regressions. Quantile regression models are a generalization of median regression and can be used to examine data in depth. Quantile analysis involves estimating model parameters and determining quantile values of the dependent variable for given values of the independent variable. This is done by minimizing the loss function based on quantile values. In contrast to the method of least squares, quantile regression allows to predict the values of the dependent variable more accurately when the values of the independent variable change. That is, quantile regression is more robust. It can be used to solve many problems in various fields of science and business, where it is necessary to more accurately predict the values of the dependent variable under changing conditions. The natural gradient descent is an effective method for constructing regression and has a higher rate of convergence than the classical algorithm. However, in practice this method is quite complicated from a computational point of view, since it requires the calculation of the second derivative. This problem is especially acute when training neural networks, where the number of parameters is much higher than when building classical regression models. The study of methods of regression construction and application of numerical methods are of practical and scientific interest. This paper will look at quantile regression, natural gradient descent and their combination to build mathematical models. Gradient descent is one of the most popular optimization methods and is widely used in machine learning. The natural gradient descent is the preferred method because it is more efficient and has a high rate of convergence. In addition, this method is less vulnerable to hitting local minima and provides more accurate estimates of model parameters. In practice, however, this method is computationally difficult, as it requires the calculation of the second derivative. The article presents an algorithm for model building using natural gradient descent. The essence of using quantile regression in a natural gradient descent is to use a quantile estimate of the loss function instead of the usual estimate used in the least squares method. This allows not only the mean value of the dependent variable, but also more extreme values (e.g., median, 25th percentile, 95th percentile, etc.) to be considered when constructing the model. The resulting method has also been compared with other popular quantile regression-supported gradient descent methods on open data sets of different dimensionality, both in terms of the number of factors and the number of observations. In addition, the possibilities of further development and optimization of this method will be discussed.

**Applied Mathematics and Control Sciences**. 2023;(2):43-52

The concept of intelligent control system of small-scale production

#### Abstract

The relevance of the research topic is determined by the rapid development of technology, the variability of the external environment and the beginning of the third industrial revolution, which leads to instability, uncertainty and complexity of the modern world. It is noted that the existing production management tools do not meet new challenges, and that new management tools are needed to ensure the flexibility and effectiveness of production in a volatile environment. The article proposes the concept of creating intelligent control systems for industrial enterprises based on knowledge, fuzzy logic and intelligent information processing. An industrial enterprise is viewed as a complex system consisting of hierarchies of goals, decision making and business processes. A brief analysis of the main types of automated production management systems, such as ERP, APS, MES, SCADA and APCS is given. It is noted that in connection with the transition to industry 4.0, there is a need to create intelligent control systems (ICS) based on knowledge, fuzzy logic and intelligent information processing. It is proposed to develop existing automated control systems by creating built-in intelligent elements capable of automatically solving optimal control problems and influencing production by transferring control actions to the control system. The system is developed on the basis of mutually integrated standard automated control systems and the situational center of the enterprise. The implementation and use of ICS will increase the speed and efficiency of decision-making, allowing enterprises to better adapt to changes and optimize the use of resources.

**Applied Mathematics and Control Sciences**. 2023;(2):53-60

Decision support system for power cable insulation vulcanization production line management

#### Abstract

The paper proposes a structural model of a decision support system for controlling a cable insulation continuous vulcanization line. The proposed decision support system is based on a mathematical model of the cable insulation continuous vulcanization technological process, a database, and a mode correction algorithm. The DSS will make it possible to quickly develop a new regime in case of using new materials or structures, as well as to correct the current regime in case of unplanned changes during the production process. The core of the DSS is the proposed mathematical model, which is based on conservation laws and is presented as a system of differential equations closed by boundary conditions. The vulcanization process is described taking into account the temperature-time dependence of the kinetic parameters, which made it possible to take into account the uneven heating of the cable, both in the radial direction and along the length. The numerical implementation of the differential mathematical model made it possible to carry out a systematic analysis of the nature of the processes in the vulcanization pipe and to evaluate the influence of various technological, structural and material parameters on the cable insulation vulcanization completion degree. As a result of the analysis, significant parameters were identified that significantly affect the vulcanization process completion degree. Based on the obtained numerical results, technological parameters dependence technological surfaces on the geometry of the product and the properties of the materials used were constructed, a regression mathematical model was proposed that allows determining the values of the process control parameters without resorting to the use of a differential mathematical model. Based on the results of the analysis, an algorithm for correcting the technological regime is proposed, taking into account only significant process parameters. The proposed algorithm allows you to adjust the value of the isolation rate, depending on external influences and deviations from the specified parameters. The results of the study can be used in the production of cable and wire products with vulcanizable insulation, when it is necessary to quickly select the optimal technological mode, take into account changes in the cable design, properties of the insulation material, as well as possible pressure deviations inside the vulcanization pipe.

**Applied Mathematics and Control Sciences**. 2023;(2):61-71

Criterion of solvability of the discriminant analysis problem underlying the medical organization management model

#### Abstract

The article considers the task of building an effective and efficient organization of the management processes of a medical organization. A distinctive feature is noted - social orientation as the main function, and not just profit maximization with limited resources. This approach requires taking into account not only financial and resource components, but also the impact of many other external factors. However, the authors, realizing the difficulty of describing them, propose to take into account the effect of the "social orientation of the institution" in an implicit form when describing the financial condition of the organization. Taking into account implicit, including external, factors is possible when implementing models not only of statistical and financial analysis, as is usually accepted, but also their extensions to some image recognition methods, which often have a simple computer implementation. In particular, the formalization of the financial condition model of the organization is proposed, which reduces to the problem of mathematical programming with state vectors of both internal and external factors. The possibility of considering a decision-making model that would take into account the influence of possible external factors is also considered, while the "informality" of the problem in the sense of the absence of classical methods of solution is indicated. In this regard, it is proposed to reduce the model to the task of discriminant analysis, which is essentially a system of inequalities. Methods for solving systems of linear inequalities are known, but they are difficult to implement. For this reason, it is proposed to add an optimality criterion (objective function) to the discriminant analysis model, which allows to obtain a linear programming problem that already has simpler computer implementations of solutions. However, this possibility is present only in the case of linear constraints, which is a somewhat special case of the general problem. To solve the problem, in its broadest sense, it is proposed not only to reduce it to mathematical programming, and then to discriminant analysis, but also an effective recognition algorithm indicating the solvability of the problem. The process of constructive problem solving often requires a lot of time and effort, even with the powerful development of modern technologies. With the help of the given recognition algorithm, the discriminant analysis model in some formulation is output in a matrix form, which allows us to conclude when the model has the existence of a separating functional. The authors also propose the use of majority committee methods. The derived algorithm is quite simple to implement, but at the same time effective and efficient in practice, which makes it universal for this kind of tasks.

**Applied Mathematics and Control Sciences**. 2023;(2):72-82

Optimal labor market management under constraints in the form of a finite-dimensional system of nonlinear differential equations integrated by quadratures

#### Abstract

The paper separates a class of finite-dimensional systems of nonlinear differential equations, the exact analytical solution of which can be represented in the form of quadratures. The paper uses a particular case of the system of the separated class as a set of equality constraints for the problem of optimal management of a closed finite-dimensional labor market with a common selection coefficient - the management parameter for the system under study. The paper specifies the definitions of the qualification categories for labor market subjects, with allowance for the physical meaning of their behavior in the system under study. It introduces quality factors for meeting the demand for labor, which are the averaged difference between the remuneration of labor and proceeds of the activities of subjects at each of the three qualification categories. It introduces a quality function in respect of the management of the labor market system, which is a sum of the products of the functions of the shares owned by the subjects at each of the three qualification categories by their quality coefficients. It considers labor markets with different ratios of quality factors. The case where the quality factor of subjects of a low qualification category is higher than that of a high qualification category has been shown to contradict the physical meaning of the model. Quality function vs management parameter curves are plotted for each labor market system under study. The paper gives examples of real-life labor markets for every physically admissible ratio of quality factors. The optimal management of the labor market system is shown to not necessarily imply that the management parameter tends to its extreme values. The paper plots a management quality function for a real-life labor market with a city-forming enterprise exemplified by the labor market of the village of Sylva, the Perm Territory, and determines the optimal values of management parameters.

**Applied Mathematics and Control Sciences**. 2023;(2):83-92

Mathematical and instrumental means of providing an intelligent decision support system for cargo traffic management

#### Abstract

In the current geopolitical and economic conditions, the issue of making operational management decisions for the effective functioning of the country's enterprises is particularly acute, taking into account organizational and technological features, as well as increasing levels of risk and uncertainty associated, among other things, with the need to process flows of specific rapidly changing information. In this regard, intelligent decision support systems are becoming increasingly popular, containing modules for collecting, processing and modeling formalized data and having both a universal character and adaptation to a specific range of issues relevant to a certain group or even a single enterprise. The purpose of the presented study was to develop such a system for managing cargo flows. Based on the analysis, the developed intelligent decision support system is proposed, which is used to improve the efficiency of the cargo flow management process. Its main blocks and their logical relationship are described. The mathematical support consists of a complex of classical models adapted to individual modules, as well as developed generalized models of cargo planning and placement. When forming a set of alternative routes, Yandex Maps are used. To automate the selection of the cargo transportation route, algorithms of fuzzy logic rules are used, with the implementation of calculations in the Yandex DataLens environment. The main attention is paid to the subsystem of cargo transportation process management. A generalized functional scheme of digitalization of the formation of waybills and control is presented, as well as a block diagram of the algorithm based on a software product developed and implemented at a number of enterprises that automates the process of filling out waybills, taking into account the specifics of the fleet and selected routes. Within the framework of the synthesis of the structure and algorithm of the intelligent system, which allows to optimize the process of organizing cargo transportation, the approbation of individual proposed modules and approaches was carried out, which showed an increase in the efficiency of forming and choosing routes for organizing cargo flows, reducing the time to find an operational solution, as well as the costs of transportation and control of cargo movement in the conditions of emerging problems associated with accidental factors. The intelligent decision support system is aimed at small firms, as well as enterprises engaged, among other things, also in the organization of cargo transportation by means of their vehicles, since they are in a greater risk zone than large companies whose main economic activity is the provision of cargo transportation services and having a wide network of representative offices, logistics centers and the composition of vehicles.

**Applied Mathematics and Control Sciences**. 2023;(2):93-107

Diagnosing stock market bubbles with advanced ADF tests

#### Abstract

A full-fledged empirical test of the latest methodology in the field of recursive procedures for identifying and dating market bubbles - the GSADF test - has been performed. The use of previous iterations of recursive tests did not allow to accurately determine the presence of a market bubble at the early stages of its formation. The results of this study prove that the GSADF test using a sliding window significantly improves the discriminatory ability of recursive tests, at the early stages of formation it allowed to detect such episodes of stock market collapse as the Japanese economic bubble of 1986-1991 and the dot-com bubble in the USA in 1995-2001. Testing based on current market data demonstrates the formation of a market bubble in the NASDAQ stock index since August 2020.

**Applied Mathematics and Control Sciences**. 2023;(2):108-119