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)– 1f +(A – K)– 1gb* for every vector b* = Ф(u) that solves nonlinear algebraic (transcendental) system of n equations
b = Ф((A – K)– 1f +(A – K)– 1gb). 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+gd*) for every vector d*= F(Au) that solves nonlinear algebraic (transcendental) system of n equations d = F(f + gd). 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.

Keywords: Boundary value problems, initial problems, Existence Theorem, nonlinear integro-differential equation, Volterra and Fredholm equations, Fredholm integral, exact solutions, direct method, Volterra operator with kernel of convolution type, abstract operator equations.

Merkhasyl M. Baiburin (Nur-Sultan, Republic of Kazakhstan) – Ph.D. in Physics and Mathematics, Associate Professor, Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University (2, Satpayev st., Nur-Sultan, 010008, Republic of Kazakhstan, e-mail: merkhasyl@mail.ru).

1. Bloom F. Ill-posed Problems for Integro-differential Equations in Mechanics and Electromagnetic Theory, SIAM, 1981, 230 p. (series SIAM – Studies in Applied Mathematics)

2. Corduneanu C. Abstract Volterra Equations: A Survey. Mathematical and Computer Modelling, 2000, vol. 32, pp. 1503–1528.

3. Cushing J.M. Integro-differential Equations and Delay Models in Population Dynamics, Springer, 1977, 202 p.

4. Wazwaz A.M. Linear and Nonlinear Integral Equations, Springer, 2011, 639 p.

5. Vlasov V.V., Rautian N.A. Spectral Analysis of Linear Models of Viscoelasticity. Journal of Mathematical Sciences, 2018, vol. 230, iss. 5, pp. 668–672.

6. Volterra V. Theory of functionals and of integral and integro-differential equations. Mineola, New York, Dover Publication Inc, 2005, 288 p.

7. Azbelev N.V., Rahmatullina L.F. Functional-Differential Equations. Differential Equations, 1978, vol. 14, pp. 771–797.

8. Corduneanu C. Integral Equations and Applications, Cambridge, Cambridge University Press, 1991, 366 p.

9. Li Y. Existence and integral representation of solutions of the second kind initial value problem for functional differential equations with abstract Volterra operator. Nonlinear Studies, 1996, vol. 3, pp. 35–48.

10. Mahdavi M. Nonlinear boundary value problems involving abstract Volterra operators. Libertas Mathematica, 1993, vol. XIII, pp. 17–26.

11. Adomian G. Solving Frontier Problems of Physics. The Decomposition Method. – Springer Netherlands, 1994, 354 p.

12. Arqub O.A., Al-Smadi M., Momani Sh. Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations. Abstract and Applied Analysis, 2012, vol. 2012, 16 p, art. 839836. DOI: 10.1155/2012/839836.

13. Baiburin M.M., Providas E. Exact Solution to Systems of Linear First-Order Integro-Differential Equations with Multipoint and Integral Conditions. Mathematical Analysis and Applications, 2019, vol. 154, pp. 1–16.

14. Polyanin A.D., Zhurov A.I. Exact solutions to some classes of nonlinear integral, integro-functional, and integro-differential equations. Doklady Mathematics, 2008, vol. 77, iss. 2, pp. 315–319.

15. Oinarov R.O., Parasidi I.N. Korrektno razreshimye rasshireniia operatorov s konechnymi defektami v Banakhovom prostranstve [Correctly solvable extensions of operators with finite defects in a Banach space]. Journal of the National Academy of Sciences of the Republic of Kazakhstan. Physico-mathematical series, 1988, no. 5, pp. 42–46.

16. Baiburin M.M. Exact solutions to nonlinear integro-differential Volterra and Fredholm Equations. Proceedings of the International Conference dedicated to the 90th anniversary of Academician Azad Khalil oglu Mirzajanzade, 2018, pp. 150–153.

17. Parasidis I.N. Exact solution of some linear Volterra integro-differential equations. Applied Mathematics and Control Sciences, 2019, no. 1, pp. 7–21. – DOI: 10.15593/2499-9873/2019.1.01

18. Parasidis I.N., Providas E. Extension operator method for the exact solution of integro-differential equations. Contributions in Mathematics and Engineering, Springer, 2016, pp. 473–496.

19. Tsilika K.D. An exact solution method for Fredholm integro-differential equations. Informatsionno-upravliaiushchie sistemy. Information and Control Systems, 2019, no. 4, pp. 2–8. – DOI: 10.31799/1684-8853-2019-4-2-8

20. Parasidis I.N., Providas E., Dafopoulos V. Loaded Differential and Fredholm Integro-Differential Equations with nonlocal integral boundary conditions. Applied Mathematics and Control Science, 2018, no. 3, pp. 31–50.

21. Parasidis I.N. and Providas E. On the exact solution of nonlinear integro-differential equations. Applications of Nonlinear Analysis, Springer, 2018,
pp. 591–609.

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.

Keywords: finite difference method, parallel programming, thermal conductivity equation, explicit method, implicit method, Crank – Nicolson method, GPU, Amdahl's law, execution speed.

Pavel A. Sechenov (Novokuznetsk, Russian Federation) – Ph.D. in Engineering, Associate Professor, Department of Applied Information Technologies and Programming, Siberian State Industrial University, Novokuznetsk (42, Kirova st., Novokuznetsk, 654007, e-mail: pavesa89@mail.ru).

Inna A. Rybenko (Novokuznetsk, Russian Federation) – Dr. Habil. in Engineering, Associate Professor, Head of Department of Applied Information Technologies and Programming, Siberian State Industrial University, Novokuznetsk (42, Kirova st., Novokuznetsk, 654007, e-mail: rybenkoi@mail.ru).

1. Storti D., Yurtoglu M. CUDA for engineers: an introduction to high-performance parallel computing. Addison-Wesley, New York, 2016, 331 p.

2. Soyata T. GPU Parallel Program Development Using CUDA. CRC Press: Taylor & Francis Group, 2018, 477 p.

3. Sanders J., Kandrot E. CUDA by example: an introduction to general purpose GPU programming. Addison-Wesley, 2011, 232 p.

4. Boreskov A.V., Kharlamov A.A. Osnovy raboty s tekhnologiei CUDA [Basics of working with CUDA technology]. Moscow, DMK Press, 2010, 232 p.

5. Parallel'nye vychisleniia na GPU. Arkhitektura i programmnaia model' CUDA: uchebnoe posobie [Parallel computing on the GPU. Architecture and software model CUDA] / A.V. Boreskov, Kharlamov A.A., Markovskii N.D., Mikushin D.N., Mortikov E.V., Myl'tsev A.A., Sakharnykh N.A., Frolov V.A. Moscow: Izdatel'stvo Moskovskogo universiteta, 2012, 336 p.

6. Sunday F., Edogbanya O.H., Samuel C.Z. Crank Nicolson method for solving parabolic partial differential equations. International Journal of Applied Mathematics and Modeling IJA2M. 2013, Vol. 1, no 3, pp. 8–23.

7. Bobkov S.P., Galiaskarov E.G. Simulation of the heat conduction process using cellular automata systems. Software & Systems, 2020, vol. 33, no. 4, pp. 641–650 (in Russ.). DOI: 10.15827/0236-235X.132.641-650.

8. Afanas'eva E.Iu. Ispol'zovanie tekhnologii parallel'nogo program­mirovaniia CUDA dlia resheniia zadachi teploprovodnosti [Using the CUDA Parallel Programming Technology to Solve the Heat Conduction Problem]. Zavershennye issledovaniia. 2015, no 1, pp. 6–11.

9. Cheng J., Grossman M., McKercher T. Professional CUDA C programming. John Wiley & Sons, 2014, 497 p.

10. Vatutin E.I., Martynov I.A., Titov V.S. Otsenka real'noi proizvo­ditel'nosti sovremennykh videokart s podderzhkoi tekhnologii CUDA v zadache umnozheniia matrits [Assessment of the real performance of modern video cards with support for CUDA technology in the problem of matrix multiplication] Izvestiia Iugo-Zapadnogo gosudarstvennogo universiteta. Seriia: Upravlenie, vychislitel'naia tekhnika, informatika. Meditsinskoe priborostroenie. 2014, no 2, pp. 8–17.

11. Ruzhnikov V.O. Povyshenie proizvoditel'nosti rascheta dinamiki chastits na parallel'nykh sistemakh [Improving the performance of calculating particle dynamics on parallel systems]. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaia matematika. Informatika. Protsessy upravleniia. 2014, no 1, pp. 147–156.

12. Sechenov P.A., Olennikov A.A. Primenenie tekhnologii parallel'nogo programmirovaniia Nvidia CUDA v zadache rasplavleniia sharoobraznoi chastitsy [Application of parallel programming technology Nvidia CUDA in the problem of melting a spherical particle]. Kibernetika i programmirovanie. 2018, no 5, pp. 8–14.

13. Sechenov P.A., Olennikov A.A., Tsymbal V.P. Primenenie tekhnologii parallel'nogo programmirovaniia CUDA v zadache rasplavleniia sharoobraznoi chastitsy [Application of parallel programming technology CUDA in the problem of melting a spherical particle]. Proceedings of the V All-Russian meeting on control sciences, 12–13 May 2016, Ekaterinburg, Ural Federal University named after the first President of Russia B.N.Yeltsin, 2016, pp. 260-263.

14. Omowo B.J., Longe I.O. Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation. International Journal of Applied Mathematics and Theoretical Physics. 2020, Vol 6, no 3, pp. 35–40. DOI: 10.11648/j.ijamtp.20200603.11.

15. Rybenko I.A., Sechenov P.A., Kalashnikov S.N. Razrabotka deter­minirovannoi matematicheskoi modeli nestatsionarnogo teplovogo sostoianiia smerzshegosia v vagone ugol'nogo syr'ia na ustanovke dlia ego razmorozki [Development of a deterministic mathematical model of the non-stationary thermal state of coal raw materials frozen in a car at a plant for its defrosting]. Naukoemkie tekhnologii razrabotki i ispol'zovaniia mineral'nykh resursov. 2021, No 7, pp 243-246.

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.

Keywords: tasks, resources, uniformity of means, target distribution, probability, objective function, absolute and relative error, optimal solution, integer and fractional parts of a number, algorithm

Antonina V. Ganicheva (Tver, Russian Federation) – Ph.D in Physics and Mathematics, Associate Professor, Department of Physical and Mathematical Disciplines and Information Technologies, Tver State Agricultural Academy (7, st. Mar­shal Vasilevsky, Tver, 170904, e-mail: TGAN55@yandex.ru).

Alexey V. Ganichev (Tver, Russian Federation) – Associate Professor, Department of Computer Science and Applied Mathematics, Tver State Technical University (22, st. Nikitin nab., Tver, 170026, e-mail: alexej.ganichev@yandex.ru).

1. Abchuk V.A., Matveychuk F.A. Tomashevskiy L.P. Spravochnik po issledovaniyu operatsiy [Handbook of Operations Research]. Moscow, Voyenizdat, 1979, 368 p.

2. Berzin YE.A. Optimal'noye raspredeleniye resursov i teoriya igr [Optimal resource allocation and game theory]. Moscow, Radio i svyaz', 1983, 216 p.

3. Buravlev A.I. Ob otsenke vliyaniya sistemy upravleniya ognem na effektivnost' porazheniya tseley [On the assessment of the impact of the fire control system on the effectiveness of hitting targets]. Armament and Economics, 2012, no. 1 (17). pp. 25-29.

4. Grigor'yev D.A., Maslennikova T.N., Piftankin A.N, Polovinkina A.V. Matematicheskaya model' protsessa upravleniya sredstvami PVO [Mathematical model of the air defense control process]. Automation of Control Processes, 2019, no. 2 (56). pp. 15-22.

5. Kezhayev V.A., Anisimov V.G., Anisimov YE.G. Obosnovaniye resheniy v zadachakh tseleraspredeleniya s ispol'zovaniyem innovatsionnykh tekhnologiy diskretnogo programmirovaniya [Substantiation of solutions in the tasks of target allocation using innovative discrete programming technologies]. Izvestiya Rossiyskoy akademii raketnykh i artilleriyskikh nauk, 2012, no. 2 (72), pp. 97-103.

6. Pis'mennaya V.A., Yakutin A.V. Povysheniye effektivnosti resheniya zadachi tseleraspredeleniya v sistemakh vozdushno-kosmicheskoy oborony [Improving the efficiency of solving the task of target distribution in aerospace defense systems]. Vestnik Kontserna VKO Almaz-Antey, 2017, no. 1. pp. 76-81.

7. Yudin A.V. Tseleraspredeleniye neodnorodnykh sredstv podavleniya metodom dvukh funktsiy [Target distribution of heterogeneous means of suppression by the method of two functions]. Aktual'n•yye problemy nauki i tekhniki: materialy IV Mezhd. konkursa nauch.-issled. rabot. Ufa, 2021. pp. 11-15.

8. Slavyanov A.S. Analiz i prakticheskoye primeneniye modeley raspredeleniya resursov [Analysis and practical application of resource allocation models]. Bulletin of science and practice, 2018, V. 4, no. 9. pp. 228-244.

9. Ganicheva A.V. Optimal'noye resheniye i otsenka poleznosti organizatsionnykh voprosov [Optimal solution and evaluation of the usefulness of organizational issues]. Yaroslavl Pedagogical Bulletin, 2011,no. 3(2). pp. 53.

10. Menshikh V.V., Samorokovsky A.F., Avsentev O.S. Models of resource allocation optimization when solving the control problems in organizational systems. Journal of Physics: Conference Series, 2018, V. 973, no. 1. pp. 12-40.

11. Rezig S., Ezzeddine W., Turki S., Rezg N. Mathematical Model for Production Plan Optimization. A Case Study of Discrete Event Systems Mathematics, 2020, no. 8 (6). P. 955.

12. Filippova A.S. Economic-mathematical modeling of a multi-criteria optimization management problem of a retail unit of a commercial bank Perm University Herald. Economy, 2019, no. 14(1). pp. 93-109.

13. Bakanova A.P., Shikov A.N. The Method of Employee Competencies Management Based on the Ontological Approach. CEUR Workshop Proceedings, 2020, no. 2590. pp. 1-9.

14. Sobolevskiy V.A. K otsenke tochnosti zadachi tseleraspredeleniya sredstv porazheniya [To assess the accuracy of the task of target distribution of means of destruction]. Voyennaya mysl', 2016, no. 8. pp. 32-43.

15. Kataev A.V., Kataeva T.M., Makarova E.L. Project Management: Mathematical Models of Optimal Executors’ Appointment for Project Works. Izv. Sarat. un-ta. Nov. ser. Ser. Ekonomika. Upravleniye. Pravo, 2016, no. 3. pp. 294-299.

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.

Keywords: foam, flotation, flare, filtration, noise, technical vision, frame, video stream, identification, transient process.

Kristina A. Fedoseeva (Berezniki, Russian Federation) – Ph. D. student, Department of Industrial Processes Automation, Perm National Research Polytechnic University, Berezniki branch (7, Telmana str., Berezniki, Perm region, 618404; e-mail: kristya_0103@mail.ru)

1. Shilin A.N., Snitsaruk D.G. Sistema tekhnicheskogo zreniia robota dlia kontrolia geometricheskikh parametrov obechaek [Technical vision system of a robot for control of geometrical parameters of shells]. Pribory i sistemy. Upravlenie, kontrol', diagnostika, 2019, no 8, pp. 36–43.

2. Balbanov P.V., Iudaev V.A. Sistema tekhnicheskogo zreniia dlia kontrolia kachestva plodoovoshchnoi produktsii [Vision system for quality control of fruits and vegetables]. Promyshlennye ASU i kontrollery, 2020, no 3, pp.10–15. DOI: 10.25791/asu.3.2020.1165

3. Zhao L., Peng T., Xie Y., Gui W., Zhao Y. Froth Stereo Visual Feature Extraction for the Industrial Flotation Process. Industrial & Engineering Chemistry Research, 2019, vol. 58, iss. 31, pp. 14510−14519. DOI: 10.1021/acs.iecr.9b00426

4. Tan J., Liang L., Peng Y., Xie G. The concentrate ash content analysis of coal flotation based on froth images. Minerals Engineering, 2016, vol. 92, pp. 9–20. DOI: 10.1016/j.mineng.2016.02.006

5. Jahedsaravani A., Massinaei M., Marhaban M.H. An Image Segmentation Algorithm for Measurement of Flotation Froth Bubble Size Distributions. Measurement, 2017, vol. 111, pp. 29–37. DOI: 10.1016/j.measurement.2017.07.023

6. Fu Y., Aldrich C.  Flotation froth image recognition with convolutional neural networks. Minerals Engineering, 2019, vol. 132, pp. 183–190. DOI: 10.1016/j.mineng.2018.12.011

7. Zhang J., Tang Z., Liu J., Tan Z., Xu P. Recognition of flotation working conditions through froth image statistical modeling for performance monitoring. Minerals Engineering, 2016, vol. 86, pp. 116–129. DOI: 10.1016/j.mineng.2015.12.008

8. Li J., Cao B., Zhu H. Nie F. Flotation froth image texture extraction method based on deterministic tourist walks. Multimed Tools and Applications, 2017, vol. 76, pp. 15123–15136. DOI: 10.1007/s11042-017-4603-3.

9. Malkov A.V., Gafurov M.N., Logunova O.S. O raspoznavanii mgnovennykh izobrazhenii v videopotoke [About recognizing instant images in a video stream]. Matematicheskoe i programmnoe obespechenie sistem v promyshlennoi i sotsial'noi sferakh, 2018, vol. 6, no. 1, pp. 38.

10. Logunova O.S., Shilov R.E., Lednov A.V. Metodika i algoritmy segmentatsii izobrazheniia pennogo produkta flotatsii [Methods and algorithms for segmentation of the image of the froth flotation product]. Aktual'nye problemy sovremennoi nauki, tekhniki i obrazovaniia, 2018, vol. 9, no. 1, pp. 72–75.

11. Zatonskii A.V., Malysheva A.V. Modernizatsiia algoritmov blikovogo raspoznavaniia parametrov pennogo sloia pri flotatsii kaliinykh rud [Modernization of algorithms for flare detection of froth layer parameters during flotation of potash ores]. Obogashchenie rud, 2018, no. 2 (374), pp. 35–41. DOI: 10.17580/or.2018.02.07

12. Prokhorenkov A.M., Kachala N.M. Tsifrovaia fil'tratsiia signalov v promyshlennykh sistemakh upravleniia [Digital filtering of signals in industrial control systems].Tsifrovaia obrabotka signalov, 2008, no. 3, pp. 32–36.

13. Savinov G.F. O nekotorykh osobennostiakh algoritma optimal'noi fil'tratsii Kalmana-B'iusi [On some features of the Kalman-Bucy optimal filtering algorithm]. Aviakosmicheskoe priborostroenie, 2007, no. 6,
pp. 22–29.

14. Tsyplakov A. Vvedenie v modelirovanie v prostranstve sostoianii [Introduction to State Space Modeling]. Kvantil', 2011, no. 9, pp.1–24.

15. Malysheva A.V. Opredelenie parametrov protsessa flotatsii kaliinoi rudy po videoizobrazheniiu poverkhnostnoi peny flotomashiny [Determination of the parameters of the flotation process of potash ore from the video image of the surface foam of the flotation machine]. Virtual'noe modelirovanie, prototipirovanie i promyshlennyi dizain [Proceedings of the III International scientific and practice conference “Virtual modeling, prototyping and industrial design”]. Tambov, 2017. pp. 148–154.

16. Zatonskii A.V., Fedoseeva K.A., Medvedeva E.S. Vybor metoda identifikatsii tekhnologicheskikh otklonenii po izmeneniiu izobrazheniia peny [The choice of a method for identifying technological deviations by changing the foam image]. Molodezhnaia nauka v razvitii regionov [Proceedings of the All-Russian scientific and practice conference of stidents and new scientists “Youth science in the development of regions”, Berezniki, Russia, 28 April 2021]. Perm, Perm National Research Polytecnic University, 2021. pp. 22–25.

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.

Keywords: data, data analysis methods, social network analysis, target portrait of a social network user, dependencies between the interests of social network users, audience reaction to content, social network analytics systems, analysis of sets of elements, key performance indicators, associative rules.

Timofei A. Shestakov (Bryansk, Russian Federation) – Student, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: alex-rf-32@yandex.ru)

Iurii A. Leonov (Bryansk, Russian Federation) – Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: yorleon@yandex.ru)

Aleksandr A. Kuz'menko (Bryansk, Russian Federation) – Ph. D. in Biology, associate professor in the Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: alex-rf-32@yandex.ru)

Anna S. Sazonova (Bryansk, Russian Federation) – Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: libv88@yandex.ru)

Rodion A. Filippov (Bryansk, Russian Federation) – Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: redfill@mail.ru)

1. Gubanov D.A., Novikov D.A. Chkhartishvili A.G. Sotsial'nye seti. Modeli informatsionnogo vliianiia, upravleniia i protivoborstva: uchebnoe posobie [Social networks. Models of information influence, management and confrontation: study guide]. Moscow, Izdatel'stvo fiziko-matematicheskoi literatury, 2010, 228 p.

2. Belov V.S. Informatsionno-analiticheskie sistemy. Osnovy proek­tirovaniia i primeneniia: uchebnoe posobie [Information and analytical systems. Fundamentals of design and application: tutorial]. Moscow, Evraziiskii otkrytyi institut, 2010, 112 p.

3. Leonov Yu.A., Leonov E.A., Kuzmenko A.A., Martynenko A.A., Averchenkova E.E., Filippov R.A.  Selection of rational schemes automation based on working synthesis instruments for technological processes. Yelm, WA, USA, Science Book Publishing House LLC, 2019, 192 p.

4. Chubukova I.A. Data Mining. Moscow, Internet-Universitet Informatsionnykh Tekhnologii (INTUIT), 2016, 470 p.

5. Kotel'nikov E.V., Klekovkina M. V. Avtomaticheskii analiz tonal'nosti tekstov na osnove metodov mashinnogo obucheniia [Sentiment analysis of texts based on machine learning methods]. Komp'juternaja lingvistika i intellektual'nye tehnologii, 2012, iss. 11 (18), pp. 7–10.

6. Osipova Iu.A., Lavrov D.N. Primenenie klasternogo analiza metodom k-srednikh dlia klassifikatsii tekstov nauchnoi napravlennosti [Application of cluster analysis by the k-means method for classification of scientific texts]. MSiM Publ., 2017, pp. 108–121.

7. Hipp J., Guntzer U., Nakaeizadeh G. Algorithms for Association Rule Mining – A General Survey and Comparison. In Proc. ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Tübingen, Germany, 2000, pp. 58–64.

8. Leonov E.A., Leonov Yu.A., Leonov, Kazakov Yu.M., Filippova L.B.  Intellectual subsystems for collecting information from the internet to create knowledge bases for self-learning systems. Advances in Intelligent Systems and Computing, 2017, vol 679, pp. 95-103 – DOI:10.1007/978-3-319-68321-8_10

9. Saraee M., Nematbakhsh M.A., Ramezani R. MRAR: Mining Multi-Relation Association Rules. Journal of Computing and Security, 2014, vol 1, no. 2, pp. 133–158.

10. Ivanichev I. KPI v SMM. 30+ metrik jeffektivnosti marketinga v social'nyh setjah [KPIs in SMM. 30+ metrics of social media marketing effectiveness]. Internet-agentstvo «Teksterra». – Jelektron. dan. – 2018. – URL: https://texterra.ru/blog/kpi-v-smm-metriki-effektivnosti-marketinga-v-sotsialnykhsetyakh.html (Accessed 25 November 2020).

11. Senatorov A.A. Kontent-marketing: strategii prodvizheniia v sotsial'nykh setiakh [Content Marketing: Social media promotion strategies]. Moscow, Al'pina Pablisher, 2020, 160 p.

12. Rubtsova Iu. Avtomaticheskoe postroenie i analiz korpusa korotkikh tekstov (postov mikroblogov) dlia zadachi razrabotki i trenirovki tonovogo klassifikatora [Automatic construction and analysis of the corpus of short texts (microblogging posts) for the task of developing and training a tone classifier]. Inzheneriia znanii i tekhnologii semanticheskogo veba, 2012, vol. 1,  pp. 109–116.

13. Kuzmenko A.A., Filippova L.B., Sazonova A.S., Filippov R.A. Intelligent System of Classification and Clusterization of Environmental Media for Economic Systems. Advances in Economics, Business and Management Research, 2020, vol. 139. pp. 583–586.

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.

Keywords: remote sensing of logging, models and methods of analysis of forest stands, intelligent systems.

Jurii A. Leonov (Bryansk, Russian Federation) – Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: yorleon@yandex.ru).

Aleksandr A. Kuz'menko (Bryansk, Russian Federation)– Ph. D. in Biology, associate professor in the Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: alex-rf-32@yandex.ru).

Rodion A. Filippov (Bryansk, Russian Federation) – Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: redfil@mail.ru).

Liudmila B. Filippova (Bryansk, Russian Federation)– Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: libv88@mail.ru).

Anna S. Sazonova (Bryansk, Russian Federation) – Ph. D. in Engineering, associate professor, associate professor, Department of Computer Technologies and Systems, Bryansk State Technical University (7, b-r 50th anniversary of October, Bryansk, 241035, e-mail: libv88@yandex.ru)

1. Kuz'menko A.A., Kondrashov D.E. Modelirovanie izmenenija granic lesnyh nasazhdenij v zadachah raspredelennyh jekonomicheskih sistem. / Avto­matizacija i modelirovanie v proektirovanii i upravlenii. – Brjansk. – 2020. – S. 12–20.

2. Kuz'menko A.A., Parinov A.V. Intellektual'naja sistema raspoznavanija ob#ektov okruzhajushhej sredy dlja zadach upravlenija territorial'no raspredelennymi jekonomicheskimi sistemami. Vestnik voronezhskogo instituta FSIN Rossii. – Voronezh. – 2020. – S. 105-114 Tacconi, L.; Boscolo, M.; Brack, D. National and International Policies to Control Illegal Forest Activities: A Report Prepared for the Ministry of Foreign Affairs of the Government of Japan, July 2003; Government of Japan: Tokyo, Japan, 2003.

3. Tacconi, L.; Boscolo, M.; Brack, D. National and International Policies to Control Illegal Forest Activities: A Report Prepared for the Ministry of Foreign Affairs of the Government of Japan, July 2003; Government of Japan: Tokyo, Japan, 2003.

4. Hoare, A. Energy, Environment and Resources. In Illegal Logging and Related Trade–The Response in Ghana: A Chatham House Assessment; Chatman House: London, UK, 2014. [Google Scholar]

5. Brack, D. Briefing Paper: Illegal Logging; Chatham House: London, UK, 2006. [Google Scholar]

6. Brack, D.; Hayman, G. Intergovernmental Actions on Illegal Logging, Options for Intergovernmental Action to Help Combat Illegal Logging and Illegal Trade in Timber and Forest Products; The Royal Institute of International Affairs: London, UK, 2001. [Google Scholar]

7. Lawson, S.; MacFaul, L. Illegal Logging and Related Trade–Indicators of the Global Response; Chatham House: London, UK, 2010. [Google Scholar]

8. Babis, M.; Duricek, M.; Harvanova, V.; Vojtko, M. Forest Guardian–Monitoring System for Detecting Logging Activities Based on Sound Recognition, Researching Solutions in Artificial Intelligence, Computer Graphics and Multimedia. In Proceedings of the IIT.SRC 2011, Bratislava, Slovakia, 4 May 2011; pp. 1–6. [Google Scholar]

9. Kuzmenko A. A., Filippova L.B., Sazonova A.S., Filippov R.A. Intelligent System of Classification and Clusterization of Environmental Media for Economic Systems // Proceedings of the International Conference on Economics, Management and Technologies 2020 (ICEMT 2020). – Advances in Economics, Business and Management Research, volume 139. – pp. 583–586.

10. Leonov E.A., Intellectual subsystems for collecting information from the internet to create knowledge bases for self-learning systems / E.A. Leonov,
Y.A. Leonov, Y.M. Kazakov, L.B. Filippova / In: Abraham A., Kovalev S., Tarassov V., Snasel V., Vasileva M., Sukhanov A. (eds) – Text : electronic // Proceedings of the Second International Scientific Conference “Intelligent Information Technologies for Industry” (IITI’17). IITI 2017. Advances in Intelligent Systems and Computing. – 2017 – vol. 679. – Springer, Cham, p. 95-103 – DOI: 10.1007/978-3-319-68321-8_10

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.

Keywords: cluster-network interaction, regional development, gross regional product, market concentration indicators, metric, Hall–Taidman index, cluster load index, neural networks, specification of regression equations, production diversification.

Leonid V. Kozhemyakin (Perm, Russian Federation) – Ph.D. Student, Department of Applied Mathematics, Perm National Research Polytechnic University (29, Komsomolsky av., Perm, 614990, e-mail: lvkozhemyakin@yandex.ru).

Leonid N. Yasnitsky (Perm, Russian Federation) – Dr. Habil in Engineering, Professor, Department of Applied Mathematics and Computer Science, Perm State University (15, Bukireva st., Perm, 614990, e-mail: yasn@psu.ru); Professor, Department of Information Technologies in Business, National Research University “Higher School of Economics” (38, Studencheskaya st., Perm, 614070).

Sergey V. Rusakov (Perm, Russian Federation) – Dr. Habil in Physics and Mathematics, Professor, Head of Department of Applied Mathematics and Informatics, Perm State University (15, Bukireva st., Perm, 614990, e-mail: rusakov@psu.ru).

1. Boschma R. Relatedness as Driver of Regional Diversification: A Research Agenda. Regional Studies, 2017, vol. 51 (3), pp. 351–364. DOI: 10.1080/00343404.2016.1254767

2. Hausmann R., Hidalgo C.A. The Network Structure of Economic Output. Journal of Economic Growth, 2011, vol. 16 (4), pp. 309–342.

3. Hidalgo C.A., Klinger B., Barabasi A.L., Hausmann R. The Product Space Conditions the Development of Nations. Science, 2007, vol. 317, pp.482–487.

4. Pinheiro F.L., Alshamsi A., Hartmann D., Boschma R., Hidalgo C.A. Shooting High or Low: Do Countries Benefit from Entering Unrelated Activities?, available at: https://arxiv.org/abs/1801.05352 (accessed 17 June 2020).

5. Kutsenko E., Eferin Y. “Whirlpools” and “SafeHarbors” in the Dynamics of Industrial Specialization in Russian Regions. Foresight and STI Governance, 2019, vol. 13 (3), pp.24–40. DOI:org/10.17323/2500-2597.2019.3.24.40.

6. Vasil'ev A.N. O pokazateliakh spetsializatsii regionov [About the indicators of specialization of regions]. Problemy sovremennoi ekonomiki, 2009, vol. 2 (30), available at: http://www.m-economy.ru/art.php?nArtId=2559 (accessed 22 January 2021).

7. Uskova T.V. et al. Proizvodstvennye klastery i konkurentosposobnost' regiona [Industrial clusters and regional competitiveness]. Vologda, Institut sotsial'no-ekonomicheskogo razvitiia territorii RAN, 2010, 246 p.

8. Francisco D., Upadhyay V. Productive Specialization and Regional Development at State Level in India. Regional Science Inquiry Journal, 2010, vol. 2 (2), pp. 105–118.

9. Chelnokova O.Iu. Modelirovanie ispol'zovaniia indeksa Kherfindalia-Khirshmana pri analize stepeni kontsentratsii firm na otraslevom rynke [Modeling of use of Herfindal-Hirshmam Index in the Analysis of the Degree of Concentration of Firms on the Industry Market]. Professional'naia orientatsiia, 2018, vol. 2, pp. 54–58.

10. Plotnikova T.N., Shibaeva T.A. Klasterno-setevaia model' regional'nogo razvitiia [Cluster-network Modeling of Regional Development]. Fundamental'nye issledovaniia, 2016, vol. 2, pp. 193–196.

11. Mikhalev D.A. Modelirovanie protsessov formirovaniia i razvitiia regional'nykh promyshlennykh klasterov [Modeling of Processes of Formation and Development of Regional Industrial Clusters]. Ph. D. thesis. Ivanovo, 2015, 223 p.

12. Kozhemyakin L.V., Ospiova M.U., Nikitin V.N. Organization Behavior Control of Spatial Cluster-Network Integration of The Oil And Gas Complex. SHS Web Conferences, 2021. vol. 116, art. 00014. DOI: 10.1051/shsconf/­202111600014.

13. Iasnitskii L.N. Iskusstvennyi intellekt. Elektivnyi kurs [Artificial Intelligence. Elective Course]. Moscow, Binom, Laboratoriia znanii, 2011, 197p.

14. Borovikov V.B. Neironnye seti. Statistica Neural Networks: metodologiia i tekhnologii sovremennogo analiza dannykh [Neural Networks. Statistica Neural Networks: Methodology and Technologies of Modern Data Analysis, 2ndedition] Eds.V. B. Borovikov. Moscow, Telekom, 2008, 392 p.

15. Lengyel B., Iwasaki I., Szanyi M. Industry Cluster and Regional Economic Growth: Evidence from Hungary. Hitotsubashi Journal of Economics, 2010, vol. 51, no. 2, pp. 149–167. DOI: 10.15057/18777.

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

Keywords: goalkeeper, sport, mathematical model, analysis, training, coach, entropy, skill, fractal, training structure.

Aleksei Stepanov В. (Perm, Russian Federation) – Postgraduate Student, Department of Theory & Methods of Physical Culture and Tourism, Perm State Humanitarian Pedagogical University (24, Sibirskaya st., Perm, 614045).

Viacheslav V. Belykh (Izhevsk, Russian Federation) – Ph.D. in Engineering, Associate Professor, Department of Physics and Optotechnics, Kalashnikov Izhevsk State Technical University (7, Studencheskaya st., Izhevsk, 426069,
e-mail: bil_ha@mail.ru).

Vladimir A. Stepanov (Izhevsk, Russian Federation) – Ph.D. in Engineering, Associate Professor, Department of Software, Kalashnikov Izhevsk State Technical University (7, Studencheskaya st., Izhevsk, 426069, e-mail: Vladimir1@udm.ru).

Anastasiia P. Kaliagina (Moscow, Russian Federation) – Student, Russian State University for the Humanities (korp. 5, b. 6, Miusskaja pl., Moscow, 125047, e-mail: kap0319@gmail.com).

Valeriia A. Stepanova (Moscow, Russian Federation) – Student, Institute for Leadership and Healthcare Management, I.M. Sechenov First Moscow State Medical University (Sechenov University) (st. 1, b. 28, Aleksandra Solzhenitsyna st., Moscow, 109004).

1. Stepanov A.V. Matematicheskoe modelirovanie pri professional'nom orientirovanii futbolista i progresse razvitiia navykov v dostizhenii top-urovnia [Mathematical modeling with professional orientation of the football player and progress of skills development in reaching the top level]. Scientific notes of the P. F. Lesgaft University, 2019, no. 8 (174). pp. 210–215.

2. Kirillova G.D. Protsess razvivaiushchego obucheniia kak tselostnaia sistema [The process of developing learning as an integral system]. Saint Petersburg, Obrazovanie, 1996, 135 p.

3. Makarenko N., Beliaev F.P., Belitskaia L.A., Zueva M.V., Karankevich A.I. Vliianie zritel'noi fraktal'noi stimuliatsii na psikhofiziologicheskie kharakteristiki i tekhniko-takticheskoe umenie sportsmenov, zanimaiushchikhsia nastol'nym tennisom [The influence of visual optical fractal stimulation on the psychophysiological characteristics and technical-tactical skills of table tennis athletes]. Sports Science Bulletin, 2021, no. 1, pp. 34–40.

4. Dvoriatkina S.N. Tekhnologiia fraktal'nogo predstavleniia uchebnykh elementov pri variativnom strukturirovanii soderzhaniia obucheniia matematike v vuze [The technology of fractal representation of educational elements in the variability of structuring of the math training content in high school]. Yaroslavl Pedagogical Bulletin, 2015, no. 5, pp. 128–133.

5. Dodonov B.I. Test-anketa: emotsional'naia napravlennost' [Test questionnaire: emotional focus]. Available at: https://vsetesti.ru/330 (Accessed: 20 June 2019).

6. Siniavskii V.V. Oprosnik «Kommunikativnye i organizatorskie sposobnosti» [Questionnaire "Communication and organizational skills"]. Available at: http://testoteka.narod.ru/lichn/1/17.html (Accessed: 20 June 2019).

7. Briling E.E. Test na stressoustoichivost' [Stress tolerance test]. Available at: http://www.psi.lib.ru/test/test7.htm (Accessed: 20 June 2019).

8. Liri T. Diagnostika mezhlichnostnykh otnoshenii [Diagnostics of interpersonal relationships]. Available at: http://testoteka.narod.ru/mlo/1/26.html (Accessed: 20 June 2019).

9. Orlov Iu.M. Test – oprosnik «Potrebnost' v dostizhenii tseli. Shkala otsenki potrebnosti v dostizhenii uspekha» [Test - questionnaire “The need to achieve
a goal. Scale for assessing the need to achieve success”]. Available
at: https://psycabi.net/testy/475-metodika-orlova-yu-mtest-oprosnik-potrebnost-v-dostizhenii-tseli-shkala-otsenki-potrebnosti-v-dostizhenii-uspekha (Accessed:
20 June 2019).

10. Gao J., Xu B. Complex Systems, Emergence, and Multiscale Analysis: A  Tutorial and Brief Survey. Applied Science, 2021, vol. 11 (12), art. 5736, 62 p. DOI: 10.3390/APP11125736.

11. Hemelrijk C.K., Hildenbrandt H., Some Causes of the Variable Shape of Flocks of Birds. PLoS ONE 2011, vol. 6, iss. 8 pp. 1–13. DOI: 10.1371/­journal.pone.0022479 · Source: PubMed.

12. Hildenbrandt H., Carere C., Hemelrijk C.K. Self-organized aerial displays of thousands of starlings: A model. Behavioral Ecology, 2010, vol. 21, pp. 1349–1359. DOI: 10.1093/beheco/arq149

13. Shaw E. Schooling fishes. American Scientist. 1978, vol.66, pp.166–175.

14. Reynolds C.W., Flocks, herds and schools: A distributed behavioral model. Association for Computing Machinery, New York, NY, USA, 1998, pp. 273–282. DOI: 10.1145/280811.281008

15. D’Orsogna M.R., Chuang Y.L., Bertozzi A.L.; Chayes L.S. Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse. Physics Review Letters, 2006, vol. 96, iss. 10. Art. 104302, DOI: 10.1103/PhysRevLett.96.104302.

16. Hemelrijk C.K., Hildenbrandt H. Self-Organized Shape and Frontal Density of Fish Schools. Ethology, 2008, vol. 114, pp. 245–254 DOI: 10.1111/j.1439-0310.2007.01459.x

17. Blagin A.V., Blagina L.V., Popova I.G., Sakharova Iu.V. Entropiinyi analiz slozhnykh sistem kak instrument inzhenernoi deiatel'nosti [Entropy analysis of complex systems as an engineering tool]. Inzhenernyj vestnik Dona, 2018, no. 4 available at: ivdon.ru/ru/magazint/archive/n4y2018/5364 (Accessed: 20 June 2019)

18. Khaken G. Informatsiia i samoorganizatsiia. Makroskopicheskikh podkhod k slozhnym iavleniiam [Information and self-organization. Macroscopic approach to complex phenomena]. Moscow, Mir, 1991, 240 p.

19. Shennon K. Raboty po teorii informatsii i kibernetika [Works on information theory and cybernetics]. Moscow, Inostrannaja literatura, 1963, 830 p.

20. Gromenko V.M., Fattakhov F.T, Trudovaia I.V., Ivashov A.V., Fattakhov A.F. Opyt primeneniia entropiinogo koeffitsienta Shennona k analizu fizicheskoi podgotovlennosti dvukh grupp uchenikov chetvertykh klassov [The experience of applying the Shannon entropy coefficient to the analysis of physical fitness of two groups of fourth grade students]. Scientific notes of the V.I. Vernadsky Crimean Federal University Biology. Chemistry, 2017, vol. 3 iss.69, no. 4, pp. 55–69.

21. Gerashchenko I.G., Shamardin A.I., Zubarev Iu.A., Kulikov A.A.
O printsipe neopredelennosti v sportivnoi pedagogike [On the uncertainty principle in sports pedagogy]. Teoriia i praktika fizicheskoi kul'tury, 1998 no. 9, pp. 2–6.

22. Saptsin V.M., Tsipoviaz A.T. Printsip neopredelennosti i problema izmerimosti v sportivnoi pedagogike i sorevnovaniiakh [The principle of uncertainty and the problem of measurability competitions]. Physical education of students, 2009, no.3, pp. 95–99.

23. Khakimova E.G., Gerasimov M.K. Innovatsii v obrazovatel'noi srede s ispol'zovaniem informatsionnoi entropii [Innovations in the educational environment using information entropy]. Bulletin of Kazan Technological University, 2014, vol.17, no. 1, pp. 305-307

24. Kramarenko S.S. Metod ispol'zovaniia entropiino-informatsionnogo analiza dlia kolichestvennykh priznakov [Method of using entropy-information analysis for quantitative features]. Proceedings of the Samara Scientific Center of the Russian Academy of Sciences, 2005, vol. 7, no. 1, pp. 242-247

25. Plokhinskii N.A. Algoritm biometrii [Algorithm of biometrics]. Moscow, Publishing House of Moscow State University,1980, 150 p.

26. Bir S. Kibernetika i menedzhment [Cybernetics and management]. Moscow, Dom Kniga, 2010, 280 p.