## No 4 (2021)

ABOUT NONLINEAR INTEGRO-DIFFERENTIAL VOLTERRA AND FREDHOLM EQUATIONS

#### Abstract

Two nonlinear problems in terms of abstract operator equations of the form Bx = f are investigated in this paper. In the first problem the operator B contains a linear differential operator A , the Volterra operator K with kernel of convolution type and the inner product of vectors g ( x )Ф( u ) with nonlinear bounded functionals Φ. The first problem is given by equation Bu ( x ) = Аu ( x ) - Ku ( x ) - g ( x )Ф( u ) = f ( x ) with boundary condition D ( B ) = D ( A ). In the second problem the operator B contains a linear differential operator A and the inner product of vectors g ( x )F( Аu ) with nonlinear bounded on C [ a , b ] functionals F , where F ( Аu ) denotes the nonlinear Fredholm integral. The second problem is presented by equation Bu = Аu - gF ( Au ) = f with boundary condition D ( B ) = D ( A ). A direct method for exact solutions of nonlinear integro-differential Volterra and Fredholm equations is proposed. Specifically, the three theorems about existing exact solutions are proved in this paper. The first theorem is mean that for nonzero constant α0 Volterra integro-differential equation Аu ( x ) - Ku ( x ) = 0 is reducing to Volterra integral equation and has a unique zero solution. During it the operator A - K is closed and continuously invertible. Also, if the functions u ( t ), g ( t ) and f ( t ) are of exponential order α then nonhomogeneous equation Аu ( x ) - Ku ( x ) = f ( x ) for each f ( x ) has a unique solution, shown in this paper. The second theorem is mean that for the first investigated problem with an injective operator A - K , for f ( x ) and g ( x ) from C [ a , b ], the exact solution is given by equation u = ( A - K )- 1 f +( A - K )- 1 g b* for every vector b* = Ф( u ) that solves nonlinear algebraic (transcendental) system of n equations b = Ф(( A - K )- 1 f +( A - K )- 1 g b). And if the last algebraic system has no solution, then investigated problem also has no solution. The third theorem is means that exact solution of the second investigated problem is given by u = A - 1( f+g d*) for every vector d* = F ( Au ) that solves nonlinear algebraic (transcendental) system of n equations d = F ( f + g d). In this case we have same property - if the last algebraic system has no solution, then investigated problem also has no solution. Two particular examples for each considered problem are shown for illustration of exact solutions giving by perform the suggested in this paper methods. In the first example was considered integro-differential Volterra and Fredholm equation and in the second case was considered equation with nonlinear Fredholm integral.

**Applied Mathematics and Control Sciences**. 2021;(4):7-20

SOLVING THE PROBLEM OF ONE-DIMENSIONAL THERMAL CONDUCTIVITY ON GRAPHICS PROCESSORS USING CUDA TECHNOLOGY

#### Abstract

A mathematical model for solving the problem of one-dimensional thermal conductivity has been developed and implemented programmatically. The purpose of the simulation is to compare the performance of algorithms on the central and graphics processors. The task of parallelization is relevant, since back in 2015 the number of stream processors in the most powerful video card was 2816, and in 2021 there were video cards with 10 496 stream processors. Applications running on NVIDIA GPUs demonstrate greater performance per dollar of invested funds and per watt of energy consumed compared to implementations built on the basis of central processors alone. This is confirmed by the high demand for video cards from miners, which has led to a 1.5-2.5 times increase in the price of video cards at the moment. The requirements for the hardware and software components necessary for the start of modeling are presented. Three methods of finite difference approximation are implemented: explicit, implicit and Crank-Nicolson on the central and graphics processors. The programming languages chosen are C (CPU) and CUDA C (GPU). For a well-parallelized task, when each thread is executed separately and it does not need data from other threads, the acceleration of calculations on the video card increased up to 60 times (an entry-level video card was used). The CUDA C language appeared relatively recently in 2006 and has a number of features when implementing a parallel algorithm. For the selected schemes: explicit, implicit, Crank-Nicolson, at each iteration, it is necessary to access neighboring threads and synchronize the threads. Synchronization of threads occurs in such a way that all threads wait for the slowest of them at each iteration, so solving problems using finite-difference approximation will be performed slower. A fragment of code on a GPU for implementing the Crank-Nicolson scheme is presented. The implementation of the Crank-Nicolson scheme requires the use of fast shared memory for data exchange between threads. The amount of shared memory is limited and affects the number of cells in the grid. The use of graphics cards gave a significant increase in execution speed even on an entry-level card with a number of 384 stream processors. The article presents a comparative analysis of the computing speed for different grid sizes from 1024 to 4000, as well as for different amounts of computing volumes in one thread.

**Applied Mathematics and Control Sciences**. 2021;(4):23-41

A METHOD FOR SOLVING THE PROBLEM OF OPTIMIZING THE DISTRIBUTION OF AN INTEGER RESOURCE

#### Abstract

The problem of optimizing the distribution of an integer resource (funds) by tasks (activities, goals) is investigated. The methods investigating this problem relate to the field of combinatorial optimization, namely, to the tasks of assigning goals. The known methods for solving this problem are numerical, selective, approximate, require a large number of iterations, do not involve checking the conditions for the existence of an integer solution, in some cases they can produce a solution not only far from optimal, but also violating the range of acceptable values of variables. The purpose of this work is to develop a new analytical method for solving the problem of the distribution of integer resources by the method of indefinite Lagrange multipliers. To do this, the allocated resources are represented as the sum of the integer and fractional parts of the number. The conditions are formulated and proved when the fractional parts of the variables of the solution of the problem are zero, that is, it is an integer. A theorem (criterion for the existence of an integer solution) is proved, which determines the necessary and sufficient conditions under which the solution of the problem exists and is found according to the algorithm developed in the article. Such conditions include the homogeneity of resources, as well as additional conditions (restrictions on integers and positivity of additional derived formula conditions of the problem). It is shown that the obtained solution of the problem corresponds to the maximum of the objective function. An algorithm for finding an integer solution to the problem of resource allocation by the method of indeterminate Lagrange multipliers is developed and a specific example is analyzed. The method described in this article can be used for the allocation of resources in industrial production, agriculture, organizational management systems, educational process, solving issues of target allocation in military affairs, building information systems, techno sphere security, emergency response, creating systems for the protection of objects and alarm systems. In this case, it is necessary to adapt it to the problems and tasks under consideration. It can also be used for the distribution of life-supporting resources: food, clothing, heat, electricity, gas, water supply.

**Applied Mathematics and Control Sciences**. 2021;(4):42-55

IMPROVING THE ADAPTIVE FILTRATION OF THE FOAM SURFACE GLARE TREND

#### Abstract

One of the important processes in the production of potash fertilizers is the froth flotation process. The quality of the final product depends significantly on the quality of the flotation. Technical vision is successfully used to control the flotation process. However, the existing methods of processing the video stream are inapplicable for controlling the process of flotation of potash ore due to the large scatter of statistical characteristics from one processed frame to another. This article discusses the use of nonblind filters to process streaming data. It is concluded that their application causes problems in identifying the moment of the beginning of the deviation. Based on this, the aim of the work is to reduce the noise level without affecting the identification of the transient, in other words, to improve the identification of the beginning of the transient by means of tunable blind filtering. It is proposed to recognize sets of N consecutive frames instead of single ones. For this, for each N frame, the number of bubbles, the average and median distances between them, and the average values of illumination and color components were calculated. From these calculations, it was concluded that the use of the arithmetic mean number of flares from N frames did not lead to an effective, significant reduction in the noise level. Therefore, it was proposed to use a different vector norm. As a result, an effective method for adaptive filtering of the trend of the number of highlights has been developed. On the materials of real video filming, a study was made of the change in noise from the number of frames. The results obtained show that the proposed method can reduce the standard deviation by 10-25% for different surveys. This proves the possibility of using the developed method for processing video streams both in laboratory and in industrial conditions.

**Applied Mathematics and Control Sciences**. 2021;(4):59-71

INTELLECTUAL ANALYSIS OF INFORMATION ABOUT USERS OF SOCIAL NETWORKS

#### Abstract

Social networks began to play an important role in the informatization of society. Experts from all over the world are researching social network data to solve various tasks, such as creating popular content, conducting advertising campaigns, meeting the information needs of society, ensuring state security, etc. The analysis of social networks is understood as the solution of such tasks as determining the tonality of the text, determining the target portrait of the audience, searching for associative rules, calculating community performance indicators and data visualization. The article considers the relevance of solving the problem, analyzes the results of previous work, examines the audience's reaction to content, builds a target portrait of subscribers of various communities, examines the relationship between user interests. The initial data of the study are social networks, or rather informational messages, opinions, subnets and communities, individual users, external nodes.The paper considers the classification of social network analysis systems (such as Brand Analytics, IQBuzz, Agorapulse, Semantic Force, Talkwalker) according to the following criteria: users, analysis methods, objects of analysis, data sources, features.To determine the audience's reaction to the content, the method of determining the tonality of the text was applied by analyzing comments to the content. The cluster analysis method was used to determine the target profile of users in a particular community. To find patterns between the user's interests in the work, the frequency analysis of sets of elements was considered. The search for associative rules was carried out using the Apriori algorithm. As a result, the works are presented in the form of graphs and diagrams. In the course of the work, an integrated approach to solving problems was used, which made it possible to create an automated information and analytical system that can be used as analytical tools in this area.

**Applied Mathematics and Control Sciences**. 2021;(4):72-91

MODELS AND METHODS FOR AUTOMATING THE ANALYSIS OF LOGGING FOR THE TASKS OF GEOGRAPHICALLY DISTRIBUTED SOCIO-ECONOMIC SYSTEMS

#### Abstract

Forests play a crucial role in maintaining the Earth's global biodiversity and preserving the ecological balance. In general, forest cover around the world is crucial and is an important indicator of the overall level of health on the planet. It is well known that forests properly purify the air, preserve watersheds, prevent erosion, improve water quality and provide natural resources. In addition, forests help in the face of global warming and absorb a lot of carbon dioxide, which is the main greenhouse gas, thus helping to protect the globe from climate change. In many cases, the range or extent of illegal logging cannot be accurately calculated, mainly due to the nature of the activity. It is estimated that illegal forest activities worldwide lead to the loss of approximately 10-15 billion US dollars in annual government revenues. In the mid-1990s, illicit trade accounted for almost 15% of world trade. In addition, it was pointed out that in the most vulnerable forest regions, more than half of all logging operations were carried out illegally. Despite recent work on environmental initiatives and the development of various tools for monitoring export forest products, more than ever before, it is necessary to use systems to detect illegal logging. Over the past decades, the development of remote sensing technologies, as well as advances in information and communication technologies (ICT), have made it possible to use automated or semi-automatic surveillance solutions in vast areas such as forests. Technologies such as video surveillance, wireless surveillance systems, aerial photographs and satellite images are used. The article discusses the main approaches for analyzing changes in the area of logging. These methods can be used in real time by studying and comparing changes in the areas of forest stands.

**Applied Mathematics and Control Sciences**. 2021;(4):92-115

IDENTIFICATION OF THE POTENTIAL OF CLUSTER-NETWORK INTERACTION OF ORGANIZATIONS BASED ON ESTABLISHING A LINK BETWEEN GRP AND MARKET CONCENTRATION INDICATORS

#### Abstract

This paper examines the main generally accepted market concentration indices adapted to the sub-industry structure. These indicators can be metrics for determining the dominant sub-sectors, which can be used to analyze cluster interactions. In fact, indicators can serve as an indicator of the potential for the development of cluster interaction. The current study hypothesizes that if there is one dominant sub-sector in the region, then enterprises that are "representatives" of such a sub-sector, having the most significant weight in the formation of this sub-sector and the industry as a whole, will influence the change in the GRP of the region much more than other enterprises not from the dominant industry. Thus, the paper exploresthe relationship between these metrics and the rate of gross regional product per capita, as one of the key indicators of regional development. At the first stage, a neural network is built and trained to identify a pattern between one of the metrics and the rate of gross regional product. Further, approximating by n-th order polynomials, various specifications of regression equations are considered, between all metrics and the change in the rate of gross regional product. The assumption is made that only the diversification of production does not lead to the socio-economic development of the region, but the creation and development of cluster interaction allows to increase the rate of gross regional product.

**Applied Mathematics and Control Sciences**. 2021;(4):119-135

ASSESSMENT OF PROFESSIONAL SKILLS, GAME THINKING, AND TRAINING OF A HIGH-CLASS FOOTBALL GOALKEEPER BASED ON A STRUCTURED FRACTAL APPROACH

#### Abstract

In the article structuring possibility of training elements content up to the level of the professional football player (goalkeeper) skill using a fractal approach to form an objective assessment of the sportsman's game actions during a match, competition, training are considered. In the introduction, the need for an objective assessment of the tactical skills and game thinking of the sportsman is actualized by developing assessment tools available for children schools and sections. The component composition of training structure content, including the target-oriented, conceptual, substantive and procedural sections are considered in the main part of the article. A method of fractal analysis of game episodes for assessing the professional skill and game thinking of the sportsman in the dynamic aspect is proposed. Based on a modified entropy analysis of expert assessments of the personal game of goalkeepers, the level of determinism of the state the sportsmen tactical skill was studied

**Applied Mathematics and Control Sciences**. 2021;(4):136-152