Applied Mathematics and Control Sciences

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.

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.

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.

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.

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.