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I. Golenev I. I., Yermolenko A. V. Designing a neural network for recognizing handwritten cyrillic symbols

DOI: 10.34130/1992-2752_2021_2_04

Yermolenko Andrey — Ph.D., Associate Professor, Head of the Department of Applied Mathematics and Information Technologies in Education, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Golenev Ilya — student, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

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This paper deals with the modeling of a convolutional neural network (CNN). The model was developed in Python 3.8 using the TensorFlow and Keras.

Keywords: convolutional neural networks, character recognition, deep learning.

References

  1. Goodfellow I., Bengio Y., Courville A. Glubokoe obuchenie [Deep learning] / transl. A. A. Slinkina, M.: DMK Press, 2018, 652 p.
  2. Keras: the Python deep learning API [Electronic resource] / Keras official site. Available at: https://keras.io (Accessed: 28.04.2021).
  3. Deep learning: image recognition with convolutional neural networks [Electronic resource] / Blog kompanii Wunder Fund, algoritmy, mashinnoe obuchenie [Wunder Fund company blog, Algorithms, machine learning]. Available at: https://habr.com/ru/company/wunderfund/ blog/314872/ (Accessed: 05.05.2021).
  4. Babenko V. V., Kotelina N. O., Telnova О. P. Software and information support of the paleopalinological problem, Vestnik Syktyvkarskogo universiteta. Ser. 1: Matematika. Mekhanika. Informatika [Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics], 2021, 1 (38), pp. 26-40.
  5. Gradient descent [Electronic resource] / Svobodnaya enciklopediya Vikipediya [Wikipedia The Free Encyclopedia], Available at: https:// en.wikipedia.org/wiki/Gradient_descent (Accessed: 17.05.2021).
  6. Neural network optimization methods [Electronic resource] / @Siarshai. Available at: https://habr.com/ru/post/318970/ (Accessed:15.05.2021).
  7. Valeev D. I. Development of a system for processing mathematical handwritten formulas with using neural network technologies, VKR. Chelyabinsk, 2018, 43 p.
  8. Kulakova О. A., Voronova L. I. Handwritten letters recognition using neural network, Materialy IX Mezhdunarodnoj studencheskoj nauchnoj konferencii «Studencheskij nauchnyj forum» [Materials of the IX International Student Scientific Conference «Student Scientific Forum»]. Available at: https:// scienceforum.ru/2017/article/2017033009 (Accessed: 10.05.2021).

For citation: Golenev I. I., Yermolenko A. V. Designing a neural network for recognizing handwritten cyrillic symbols, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 4-12. DOI: 10.34130/1992-2752_2021_2_04

II. Yermolenko А. V., Korablev A. Yu., Kotelina N. К., Yurkina M. N. N. К. Popova and her contribution to the development of competitive programming

DOI: 10.34130/1992-2752_2021_2_13

Yermolenko Andrey — Ph.D., Associate Professor, Head of the Department of Applied Mathematics and Information Technologies in Education, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Korablev Anatoly — Lecturer, College of Economics, Law and Informatics, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Kotelina Nadezhda — Ph.D. in Physics and Mathematics, Associate Professor of the Department of Applied Mathematics and Information Technologies in Education, Pitirim Sorokin Syktyvkar State University, e-mail: nkotelina@gmail.com

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The article is devoted to the biography of the associate professor of the Department of Applied Mathematics N. K. Popova, who has worked at Syktyvkar State University for more than 40 years.

Keywords: competitive programming, teaching, biography.

References

  1. Popova N. K. Algoritmy i algoritmicheskiye yazyki [Algorithms and algorithmic languages], Syktyvkar: SSU im. Pitirim Sorokina, 2017, 88 p.
  2. Popova N. K. Modelirovaniye prilozheniy [Modeling applications], Syktyvkar: SSU im. Pitirim Sorokin, 2019, 43 p.
  3. Kotelina N. О., Popova N. К., Yurkina М. N. About the SSU open programming championship, Vestnik Syktyvkarskogo universiteta. Seriya 1: Matematika. Mekhanika. Informatika [Bulletin of Syktyvkar University. Ser. 1: Mathematics. Mechanics. Computer science], 2018, Issue 3 (28), pp. 3-18.
  4. Kotelina N. O., Popova N. K. Organization of programming competitions on the YANDEX.CONTEST platform, Informacionnye tekhnologii v modelirovanii i upravlenii: podhody, metody, resheniya : cbornik nauchnyh statej I Vserossijskoj nauchnoj konferencii [Information technologies in modeling and control: approaches, methods, solutions: Collection ofscientific articles of the I All-Russian Scientific Conference], December 12-14, 2017, Togliatti: Publisher Kachalin Alexander Vasilievich, 2017, pp. 373-377.

For citation: Yermolenko А. V., Korablev A. Yu., Kotelina N. К., Yurkina M. N. N. К. Popova and her contribution to the development of sports programming, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 13-19. DOI: 10.34130/1992­ 2752_2021_2_13

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III. Bannikov A. S. To construction of the reachability set for a fractionalorder linear control system

DOI: 10.34130/1992-2752_2021_2_20

Bannikov Alexander — Ph.D., associate professor, Associate Professor of the Department of Differential Equations, Udmurt State University, e-mail: asbannikov@gmail.com

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A description of the reachability set in space by a phase change is given. An extremal control is constructed that transfers the initial position to the boundary of the reachability set as a solution to the corresponding optimal speed problem. Numerical examples are given. When conducting the numerical experiment, programs in MATLAB and Wolfram Language were used.

Keywords: Caputo derivative, control system, reachability set.

References

  1. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications offractional differential equations, Amsterdam: Elsevier, 2006, 540 p.
  2. Chikrii A. A., Matichin I. I. On linear conflict-controlled processes with fractional derivatives, Trudy Instituta Matematiki i Mekhaniki UrО RAN [Proceedings ofthe Institute ofMathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences], 2011, Vol. 17, no. 2, pp. 256-270.
  3. Matychyn I., Onyshchenko V. On time-optimal control offractional-order systems, Journal of Computational and Applied Mathematics, 2018, Vol. 339, pp. 245-257.
  4. Polovinkin E. S., Balashov M. V. Elementy vypuklogo i siTno vypuklogo analiza [Elements of convex and strongly convex analysis], Moscow: FIZMATLIT, 2004, 416 p.
  5. Garrappa R., Popolizio M. Computing the matrix Mittag-Leffler function with applications to fractional calculus, Journal of Scientific Computing, 2018, Vol. 17, no. 1, pp. 129-153.

For citation: Bannikov A. S. To construction of the reachability set for a fractional-order linear control system, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 20-26. DOI: 10.34130/1992-2752_2021_2_20

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IV. Rogosin S. V., Primachuk L. P., Dubatovskaya M. V. On solution to R-linear conjugation problem with rational coefficients

DOI: 10.34130/1992-2752_2021_2_27

Rogozin Sergey — Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Analytical Economics and Econometrics, Belarusian State University, Minsk, Republic of Belarus, e-mail: rogosin@bsu.by

Primachuk Leonid — Candidate of Physical and Mathematical Sciences, Associate Professor, Belarusian State University, Minsk, Republic of Belarus, e-mail: dubatovska@bsu.by

Dubatovskaya Marina — Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Analytical Economics and Econometrics, Belarusian State University, Minsk, Republic of Belarus, e-mail: dubatovska@bsu.by

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The paper is devoted to an analysis of an effective method of solution to R-linear conjugation problem recently developed bv the authors. The method uses a generalization of G. N. Chebotarev’s algorithm for factorization of the triangular matrix-functions.

Keywords: R-linear conjugation problem, rational coefficients, factorization of matrix-functions, partial indices.

References

  1. Markushevich A. I. On a boundary value problem in the theory of analytic functions, Uch. notes of Moscow University, 1946, I. 100, pp. 20-30.
  2. Mikhailov L. G. Novyy klass osobykh integral’nykh uravneniy i уego primeneniya k differentsial’nym uravneniyam s singulyarnymi koeffitsiyentami [A new class of singular integral equations and its application to differential equations with singular coefficients], Dushanbe: Academy of Sciences of the Tajik SSR, 1963, 1836 p.
  3. Litvinchuk G. S. Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift, Mathematics and its Applications, 2000, V. 523, Dordrecht: Kluwer Academic Publishers, 205 p.
  4. Mityushev V. V. R-linear and Riemann-Hilbert problems for multiply connected domains, Advances in Applied Analysis (Sergei V. Rogosin, Anna A. Koroleva eds.), Springer: Basel, 2012, pp. 147-176.
  5. Mityushev V. V., Rogosin S. V. Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions: Theory and Applications [Monographs and surveys in pure and applied mathematics], Vol. 108, Chapman & Hall / CRC PRESS: Boca Raton – London – New York – Washington, 1999, 296 p.
  6. Litvinchuk G. S. Two theorems on the stability of the quotient indices of the Riemann boundary value problem and their application, Izv. vuzov. Matem. [Izv. universities. Mat.], No. 12, 1967, pp. 47-57.
  7. Litvinchuk G. S., Spitkovsky, I. M. Factorization of measurable matrix functions, Basel-Boston: Birkhauser, 1987, 372 p.
  8. Rogosin S., Mishuris G. Constructive methods for factorization of matrix-functions, IMA J. Appl. Math., 2016, Vol. 81 (2), pp. 365-391.
  9. Sabitov I. Kh. On the general boundary value problem of linear conjugation on a circle, Sib. mat. zh. [Sib. mat. J.], 1964, T. V (1), pp. 124-129.
  10. Primachuk L., Rogosin S., Dubatovskaya M. On R-linear conjugation problem on the unit circle, Eurasian Mathematical Journal, Vol. 11 (3), 2020, p. 79-88.
  11. Chebotarev G. N. Partial indices of the Riemann boundary value problem with a triangular matrix second order, Uspekhi mat. nauk [Advances mat. nauk], 1956, T. XI, Iss. 3, pp. 192-202.
  12. Gakhov F. D. Krayevyye zadachi [Boundary value problems], 3rd ed, M.: Science, 1977, 544 p.
  13. Adukov V. M. Wiener-Hopf factorization of meromorphic matrixfunctions, St. Petersburg Math. J., 1993, V. 4 (1), pp. 51-69.
  14. Muskhelishvili N. I. Singulyarnyye integraTnyye uravneniya [Singular integral equations], 3rd ed., M.: Science, 1968, 511 p.
  15. Primachuk L., Rogosin S. Factorization of triangular matrixfunctions of an arbitrary order, Lobachevsky J. Math., V. 39 (6), 2018,pp. 809-817.

For citation: Rogosin S. V., Primachuk L. P., Dubatovskaya M. V. On solution to R-linear conjugation problem with rational coefficients, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 27-43. DOI: 10.34130/1992-2752_2021_2_27

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V. Shilov S. V. Simulation of explosions of gas-air mixtures taking into account the cloud drift by wind

DOI: 10.34130/1992-2752_2021_2_44

Shilov Sergey — Candidate of Physics and Mathematics, Associate Professor of the Department of Physics and Technology, Pitirim Sorokin Syktyvkar State University, e-mail: shilovsykt@rambler.ru

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The paper simulates the damaging effect of a shock wave during an explosion of a liquefied propane-butane mixture. Calculations were performed using two methods. The first one was used to calculate the zones of destruction of buildings and destruction of people. For the second one, the values of the drift of the gas-air mixture cloud by the wind were determined. With this in mind, the possible danger zones were much wider. Thus, the zones of destruction of buildings and damage to people due to the spread of clouds along the earth’s surface increase by about five to six times. These facts must be taken into account when placing objects that use liquefied gases, as well as when transporting such gases. A probabilistic model was used to determine hazardous areas. As zones of possible destruction of buildings, such distances are conventionally accepted at which the probability of destruction is 90 %. Similarly, for people in dangerous areas, the probability of damage to the eardrums was 90 % or higher.

Keywords: shock wave, defeat, liquefied gas.

References

  1. Rachevsky B. S. Szhizhennyye uglevodorodnyye gazy [Liquefied petroleum gases], Moscow, Oil and gas Publ, 2009, 164 p.
  2. Gazovozy. Avtotsisterny SUG [Gas carrier. Tankers. Liquefied petroleum gas]. Available at: https://rodisgroup.ru (Accessed 14 October 2020).
  3. Hramov G. N. Goreniye i vzryv [Burning and explosion], SaintPetersburg, St. Petersburg State Technical University Publ, 2007. 278 p.
  4. Vaidogas, ER (Vaidogas, Egidijus Rytas); Kisezauskiene, L (Kisezauskiene, Lina); Girniene, I (Girniene, Ingrida). The risk to structures built near roads and rails used for moving hazardous materials, Journal of civil engineering and management. Volume: 22, Issue: 3, Pages: 442-455. DOL10.3846/13923730.2015.1120773. Published: APR 2 2016. Document Type: Article.
  5. Barilla, N (Barilla, Nilambar); Mishra, IM (Mishra, Indra Mani); Srivastava, VC (Srivastava, Vimal Chandra). The risk to structures built near roads and rails used for moving hazardous materials, Journal of civil engineering and management. Volume: 40, Pages: 449-460. DOI: 10.1016/j.jlp.2016.01.020. Published: MARDocument Type: Article.
  6. Rukovodstvo po bezopasnosti «Metodika otsenki posledstviy avariynykh vzryvov toplivno-vozdushnykh smesey» [Safety Guide «Methods for assessing the effects of emergency explosions of fuel-air mixtures»]. Series 27. Issue 15. Moscow, Closed Joint Stock Company «Scientific and Technical Center for the Study of Industrial Safety Problems»,44 p.
  7. RB G-05-039-96. «Rukovodstvo po analizu opasnosti avariynykh vzryvov i opredeleniyu parametrov ikh mekhanicheskogo deystviya» [RB G-05-039-96. «Guidelines for analyzing the danger of emergency explosions and determining the parameters oftheir mechanical action»]
    (approved. By the resolution of Gosatomnadzor of Russia of 31.12.1996 N 100).
  8. Golovataya O. S., Petrakov A. P., Shilov S. V. Modeling of explosion hazards of liquefied gas tankers, Matematicheskoye modelirovaniye i informatsionnyye tekhnologii: NatsionaVnaya (Vserossiyskaya) nauchnaya konferentsiya (6-8 dekabrya 2018 g., g. Syktyvkar) : sbornik materialov [Mathematical modeling and information technologies: national (all-Russian) scientific conference (December 6-8, 2018, Syktyvkar): collection of materials], Syktyvkar: publishing house of SSU. Pitirima Sorokina, 2018, pp. 49-51.

For citation: Shilov S. V. Simulation of explosions of gas-air mixtures taking into account the cloud drift by wind, Bulletin ofSyktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 44-57. DOI: 10.34130/1992-2752_2021_2_44

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VI. Gubar L. N., Popov N. I. Implementation of the technology of guaranteed learning when students study the course of probability theory and mathematical statistics

DOI: 10.34130/1992-2752_2021_2_58

Gubar Lyudmila — Senior Lecturer, Department of Physics, Mathematics and Information Education, Pitirim Sorokin Syktyvkar State University, e-mail: lyudmila.336878@yandex.ru

Popov Nikolay — Doctor of Education, Ph.D. in Physics and Mathematics, Professor, Head of the Department of physics, mathematics and information education, Pitirim Sorokin Syktyvkar State University, e-mail: popovnikolay@yandex.ru

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The article presents the results of a pedagogical experiment related to the study of the use of guaranteed learning technology in the educational process when students study the section of mathematics “Probability theory and mathematical statistics”. The conducted research has confirmed that the application of this technology can significantly increase the level of mathematical knowledge and the effectiveness of teaching students.

Keywords: guaranteed learning technology, technological map, probability
theory and mathematical statistics, algorithm for solving mathematical
problems, pedagogical experiment.

References

  1. Popov N. I., Gubar L. N. Interdisciplinary relations as the basis of formation of students’ professional competences corresponding to the standards of WorldSkills, in study of probability theory and mathematical statistics by students, Vestnik MGPU. Seriya «Informatika i informatizaciya obrazovaniya» [Vestnik MGPU … Series Informatics and informatization of education], 2019, no 4 (50), pp. 73-80.
  2. Popov N. I., Gubar L. N. About the interdisciplinary relations of the course of probability theory and mathematical statistics in teaching college students, Vostochno-evropejskij nauchnyj zhurnal [East European Scientific Journal], no 9 (61), Vol. 1, 2020, pp. 42-48.
  3. Choshanov M. A. E-Didactics: a new look at learning theory in the digital age, Obrazovatel’nye tekhnologii i obshchestvo [Educational technologies and society], 2013, no 3, pp. 684-696.
  4. Monakhov V. M. Vvedenie v teoriyu pedagogicheskih tekhnologij [Introduction to the theory of pedagogical technologies: monografiya]. Volgograd: Peremena, 2006, 319 p.
  5. Gefan G. D., Kuz’min О. V. Comparative analysis of the effectiveness of educational methods on the example of teaching the probability theory and mathematical statistics, Vestnik Tomskogo gosudarstvennogo pedagogicheskogo universiteta [Bulletin ofthe Tomsk State Pedagogical University], 2017, no 4 (181), pp. 49-56.
  6. Safuanov I. S., Atanasyan S. L. Mathematical education in Singapore: traditions and innovations, Nauka i shkola [Science and school], 2016, no 3, pp. 38-44.
  7. Krivshenko L. P., Vajndorf-Sysoeva M. E. Pedagogika [Pedagogy]: Uchebnik. M.: Iz-vo Prospekt, 2010, 432 p.
  8. Monakhov V. М., Silchenko А. Р., Tikhomirov S. A. Genesis and Function of Professional Pedagogical Activity in Terms of IEE, Yaroslavskij pedagogicheskij vestnik [Yaroslavl Pedagogical Bulletin], 2017, no 6, pp. 112-122.
  9. Monakhov V. M. Pedagogical aspects of the integration of pedagogical technologies and information technologies as a qualitatively new stage of informatization of mathematical education, Informatizaciya obucheniya matematike i informatike: pedagogicheskie aspekty: Materialy mezhdunarodnoj nauchnoj konferencii, posvyashchennoj 85- letiyu Belorusskogo gosudarstvennogo universiteta [Informatization of teaching mathematics and computer science: pedagogical aspects : materials of the international scientific conference dedicated to the 85th anniversary of the Belarusian State University], Minsk, 2006, pp. 287-291.
  10. Popov N. I. Rukovodstvo к resheniyu zadach po teorii veroyatnostej i matematicheskoj statistike dlya psihologov [Guide to solving problems in probability theory and mathematical statistics for psychologists], Uchebnoe posobie, Joshkar-Ola: Izd-vo Mar. gos. un-t, 2006, 76 p.
  11. Gmurman V. E. Rukovodstvo к resheniyu zadach po teorii veroyatnostej i matematicheskoj statistike [A Guide to problem solving in probability theory and mathematical statistics], Moscow: Higher school, 1979, 400 p.
  12. Yilmaz R., Argun Z. Role of visualization in mathematical abstraction: The case of congruence concept, International Journal of Education in Mathematics, Science and Technology (IJEMST), 2018, 6(1), pp. 41-57.

For citation: Gubar L. N., Popov N. I. Implementation of the technology of guaranteed learning when students study the course of probability theory and mathematical statistics, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 58-77. DOI: 10.34130/1992-2752_2021_2_58

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VII. Pevnyi А. В., Kozhagel’diev N. V. New equations for a reservoir of equal resistance

DOI: 10.34130/1992-2752_2021_2_78

Pevny Alexander — Doctor of Physical and Mathematical Sciences, Professor, Department of Applied Mathematics and Information Technologies in Education, Syktyvkar State University named after Pitirim Sorokin,University e-mail: pevnyi@syktsu.ru

Kozhageldiev Nikita — student, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Text

New equations for the shape of equal resistance reservoir are obtained. The reservoir has the shape of a drop. The results of computer experiments are given.

Keywords: shell, droplet form, reservoir.

References

  1. Yermolenko A. V., Kozhagel’diev N. V. Graphoanalytical method for calculating an equal resistance reservoir, Vestnik Syktyvkarskogo universiteta Ser. 1: Matematika. Mekhanika. Informatika [Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics], 2020, 2 (35), pp. 85-91.
  2. Novozhilov V. V., Chernyh K. F., Mihajlovskij E. I. Linejnaya teoriya tonkih oboochek [Linear theory of thin shells], L: Politekhnika, 1991, 656 p.
  3. Gordon J. Konstrukcii, Hi Pochemu ne lomayutsya veshchi [Structure, or why things do not break], M: Mir, 1980, 390 p.
  4. Yermolenko A. V., Osipov K. S. On using Python libraries to calculate plates, Vestnik Syktyvkarskogo universiteta Ser. 1: Matematika. Mekhanika. Informatika [Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics], 2019, 4 (33), pp. 86-95

For citation: Pevnvi А. В., Kozhagel’diev N. V. New equations for a reservoir of equal resistance, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 78-84. DOI: 10.34130/1992-2752_2021_2_78

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VIII. Popov N. I., Arihin E. M., Yermolenko I. A. The use of an electronic course when students study the basics of mathematical analysis

DOI: 10.34130/1992-2752_2021_2_85

Popov Nikolay — Doctor of Education, Ph.D. in Physics and Mathematics, Professor, Head of the Department of physics, mathematics and information education, Pitirim Sorokin Syktyvkar State University, e-mail: popovnikolay@yandex.ru

Arihin Eduard — student, Pitirim Sorokin Syktyvkar State University, e-mail: popovnikolay@yandex.ru

Yermolenko Ilya — student, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Text

The transition to educational standards of a new generation in a higher educational institution involves the renewal of technologies, means and forms of training for future teachers of mathematics, physics and computer science. When designing an electronic course in the educational environment of the university «Short Course in Differential Calculus», the problem of modular training is considered.

Keywords: fundamentals of mathematical analysis, modular training, electronic course.

References

  1. Popov N. I., Nikiforova E. N. On the effectiveness of the use of the electronic course «Mathematics» in teaching students in agroengineering areas of training, Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Seriya Informatika i informatizatsiya obrazovaniya [Bulletin of the Moscow City Pedagogical University. Series Informatics and informatization of education], 2017, No (40), pp. 45-50.
  2. Suvorova T. N. Analysis of approaches to the typology of electronic educational resources, Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta. Seriya Informatika i informatizatsiya obrazovaniya [Bulletin of the Moscow City Pedagogical University. Series
    Informatics and informatization of education], 2015, >1 (31), pp. 70-84.
  3. Dikov A. V., Rodionov M. A., Chernetskaya T. A. The educational blogosphere as an effective means of organizing the educational process, Informatika i obrazovaniye [Computer science and education], 2018, No 1 (290), pp. 38-46. I. Kedraka K., Rotidi G. University Pedagogy: A New Culture is Emerging in Greek Higher Education, International Journal of Higher Education, 2017, Vol. 6, No 3, pp. 147-153.
  4. Popov N. I. Fundamentalizaciya universitetskogo matematiche-skogo obrazovaniya [Fundamentalization of university mathematics education] : monograph, Yoshkar-Ola: MarSU, 2012, 135 p.
  5. Popov N. I., Nikiforova E. N. Kratkij kurs differencial’подо ischisleniya : uchebnoe posobie [Differential Calculus Short Course: A Study Guide], Syktyvkar: Publishing house of SSU named after Pitirim Sorokin, 2019, 85 p.

For citation: Popov N. I., Arihin E. M., Yermolenko I. A. The use of an electronic course when students study the basics of mathematical analysis, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2021, 2 (39), pp. 85-89. DOI: 10.34130/1992 2752_2021_2_85

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