Bulletin 4 (41) 2021

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I. Vechtomov E. M., Chermnykh V. V. Main directions of the development of the semiring theory

DOI: 10.34130/1992-2752_2021_4_4

Vechtomov Evgeny Mikhailovich − Doctor of Physical and Mathematical Sciences, Professor, Head of the Department of Fundamental and Computer Mathematics, Vyatka State University, e-mail: vecht@mail.ru

Chermnykh Vasily Vladimirovich − Doctor of Physical and Mathematical Sciences, Pitirim Sorokin Syktyvkar State University, chief scientist, e-mail: vv146@mail.ru

Text

The article highlights and analyzes the main directions of formation and development of Semiring Theory. The first ring-module direction summarizes and extends the theory of rings and modules onto semirings and semimodules over them. The next one is a universal algebraic direction that is based on Universal Algebra and Group Theory. The third direction is connected with study of special classes of semirings and is aimed at using semirings within Mathematics, in Computer Sciences and in applications of Mathematics. The first two directions contain investigating of the general theory of semirings, building structural theories for certain important and interesting classes of abstract semirings. The third direction includes describing of finite semirings with certain conditions.

Keywords: semiring, semifield, semimodule, ring, distributive lattice, development of Theory of Semirings.

References

  1. Golan J. S. Semirings and their Applications. Dordrecht: Kluwer Academic Publishers, 1999. 38 p.
  2. Vandiver H. S. Note on a simple type of algebra in which cancelation law of addition does not hold. Bull. Amer. Math. Soc., 1934, V. 40. Pp. 914–920.
  3. Dedekind R. Uber die Theorie der ganzen algebraischen Zahlen ¨ Supplement XI to P.G. Lejeune Dirichlet: Vorlessungen Uber Zahlentheorie, 4 Anfl., Druck und Verlag, Braunschweig, 1894.
  4. Hilbert D. Uber den Zahlbegriff. ¨ Jahresber. Deutsch. Math. Verein., 1899, V. 8. Pp. 180–184.
  5. Hebisch U., Weinert H. J. Semirings: theory and applications in computer science. Series in Algebra. Vol. V. World Scientific, Singapore, 1998. 361 p.
  6. Golan J. S. The theory of semirings with applications in mathemayics and theoretical computer science. Pitman monographs and syrveys in pure and applied mathematics, V. 54, 1992 (1991).
  7. Glazek K. A Short Guide Through the Literature on Semirings. Preprint No. 39. University of Wroclaw, Math. Inst., Wroclaw, 1985.
  8. Glazek K. A Short Guide to the Literature on Semirings and Their Applications in Mathematics and Computer Science. Technical University Press, 2002.
  9. Maslyaev D. A., Chermnykh V. V. Skew Laurent polynomial semiring. Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical News], 2020, V. 17. Pp. 521–533.
  10. Vechtomov E. M. Vvedenie v polukoltsa: uchebnoe posobie [Introduction to semirings: A Tutorial], Kirov, Izd. Vyatskogo gospeduniversiteta, 2000. 44 p.
  11. Lukin M. A. On universal congruence on semirings. Problemy sovremennogo matematicheskogo obrazovaniya v pedvuzah i shkolah Rossii: interaktivnye formy obucheniya matematike studentov i shkol’nikov. Materialy V Vserossiyskoy nauchno-metodicheskoy
    konferentsyi [Problems of Modern Mathematics Education in Pedagogical Universities and Schools of Russia: Interactive Forms of Teaching Mathematics to Students and Schoolchildren : Proceedings of the V All-Russian Scientific and Methodical Conference], Kirov, Izd. VyatGGU, 2012. Pp. 312–316.
  12. Vechtomov E. M., Petrov A. A. Multiplicatively idempotent semirings. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 2013, V. 18. No. 4. Pp. 41–70.
  13. Vechtomov E. M., Cheraneva A. V. On the theory of semidivision rings). Uspehi matematicheskih nauk [Advances in Mathematical Sciences], 2008, V. 63. No. 2. Pp. 161–162.
  14. Vechtomov E. M., Cheraneva A. V. Semifields and their properties. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 2008, V. 14. No. 5. Pp. 3–54.
  15. Polin S. V. Simple semisfields and semifields. Sibirskiy matematicheskiy zhurnal [Siberian Mathematical Journal], 1974, V. 15. No. 1. Pp. 90–101.
  16. Chermnykh V. V. Funktsional’nye predstavleniya polukolets: monografiya [Functional representations of semirings: monography], Kirov, Izd. VyatGGU, 2010. 224 p.
  17. Vechtomov E. M., Petrov A. A. Polukoltsa s idempotentnym umnozheniem: monografiya [Semirings with idempotent multiplication: monography], Kirov, OOO «Raduga-PRESS»,144 p.
  18. Vechtomov E. M., Lukin M. A. Semirings which are the unions of a ring and a semifield. Uspehi matematicheskih nauk [Advances in Mathematical Sciences], 2008, V. 63. No. 6. Pp. 159–160.
  19. Lukin M. A. Semiring joins of a ring and semifield. Izvestiya vuzov. Matematika [Proceedings of Higher Education Institutions. Mathematics], 2008. No. 12. Pp. 76–80.
  20. Vechtomov E. M., Lubyagina E. N., Chermnykh V. V. Elementy teorii polukolets [Elements of semiring theory], Kirov, OOO «Raduga-PRESS», 2012. 228 p.
  21. Vechtomov E. M., Starostina O. V. Structure of abelian regular positive semirings). Uspehi matematicheskih nauk [Advances in Mathematical Sciences], 2007. V. 62. No. 1. Pp. 199–200.
  22. Vechtomov E. M., Starostina O. V. Generalized abelian regular positive semirings). Vestnik Syktyvkarskogo universiteta. Seriya Matematuka. Mehanika. Informatika [Bulletin of the Syktyvkar University. Series 1. Mathematics. Mechanics. Informatics], 2007, Vol.Pp. 3–16.
  23. Chermnykh V. V., Mikhalev A. V., Vechtomov E. M. Abelianregular positive semirings. Journal of Mathematical Science [New York], 1999, V. 97. Pp. 4162–4176.
  24. Vechtomov E. M. Annihilator characterizations of Boolean rings and Boolean lattices. Matematicheskie zametki [Mathematical Notes], 1993, V. 53. No. 2. Pp. 15–24.
  25. Bogdalov I. F. Invertibility of the Hilbert basis theorem in the class of semirings. Problemy sovremennogo matematicheskogo obrazovaniya v pedvuzah i shkolah Rossii: Tezisy dokladov Mezhregional’noy nauchnoy konferentsyi [Problems of Modern Mathematics Education
    in Pedagogical Universities and Schools of Russia : Abstracts of the Interregional Scientific Conference], Kirov, Izd. Vyatskogo gospeduniversiteta, 1998. Pp. 171–172.
  26. Il’in S. N. Regularity Criterion for Complete Matrix Semirings. Matematicheskie zametki [Mathematical Notes], 2001, V. 70. No. 3. Pp. 366–374.
  27. Il’in S. N. On the applicability to semirings of two theorems from the theory of rings and modules. Matematicheskie zametki [Mathematical Notes], 2008, V. 83. No. 4. Pp. 563–574.
  28. Il’in S. N. On the homological classification of semirings. Itogi nauki i tehniki. Sovremennaya matematika i ee prilozheniya. Tematicheskie obzory [Results of Science and Technology. Modern Mathematics and its Applications. Thematic reviews], 2018, V. 158. Pp. 3–22.
  29. Dale L. Monic and monic free ideals in polynomial semirings. Proc. Amer. Math. Soc., 1976, V. 56. Pp. 45–50.
  30. Babenko M. V., Chermnykh V. V. On skew polynomial semirings on Bezout semiring. Matematicheskie zametki [Mathematical Notes], 2022, V. 111.
  31. Rao P. R. Lattice ordered semirings. Math. Sem. Notes, Kobe Univ., 1981, V. 9. Pp. 119–149.
  32. Swamy K. L. N. Duallity residuated lattice ordered semigroups. Math. Ann., 1965, V. 159. Pp. 105–114.
  33. Chermnykh O. V. On drl-semigroups and drl-semirings) Chebyshevskiy sbornik [Chebyshev collection], 2016, V. 17. No. 4. Pp. 167–179.
  34. Burgess W. D., Stephenson W. Pierce sheaves of noncommutative rings. Comm. Algebra, 1976, V. 39. Pp. 512–526.
  35. Grothendieck A., Dieudonne J. El´ements de G´eom´etrie ´ Alg´ebrique 1. I.H.E.S., Publ. Math. 4. — Paris, 1960.
  36. Pierce R. S. Modules over commutative regular rings. Mem. Amer. Math. Soc., 1967, V. 70. Pp. 1–112.
  37. Simmons H. Compact representations — the lattice theory of compact ringed spaces. J. Algebra, 1989, V. 126. Pp. 493–531.
  38. Chermnykh V. V. Sheaf representations of semirings. Uspehi matematicheskih nauk [Advances in mathematical sciences], 1993, V. No. 5. Pp. 185–186.
  39. Chermnykh V. V. Functional representations of semirings. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 2012, V. 17. No. 3. Pp. 111–227.
  40. Markov R. V., Chermnykh V. V. On Pierce stalks of semirings. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 2014, V. 19. No. 2. Pp. 171–186.
  41. Markov R. V., Chermnykh V. V. Semirings close to regular and their Pierce stalks) Trydy IMM UrO RAN [Proceedings of IMM UB RAS], 2015, V. 21. No. 3. Pp. 213–221.
  42. Babenko M. V., Chermnykh V. V. Pierce stalks of semirings of skew polynomials. Trydy IMM UrO RAN [Proceedings of IMM UB RAS], 2021, V. 27. No. 4. Pp. 48–60.
  43. Chermnykh V. V., Chermnykh O. V. Functional representations of lattice-ordered semirings. Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical News], 2017, V. 145. Pp. 946–971.
  44. Chermnykh O. V. Functional representations of lattice-ordered semirings. III. Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical News], 2018, V. 15. Pp. 677–684.
  45. Chermnykh V. V., Chermnykh O. V. Functional representations of lattice-ordered semirings. III. Trydy IMM UrO RAN [Proceedings of IMM UB RAS], 2020, V. 26. No. 3. Pp. 235–248.
  46. Polin S. V. Minimal varieties of semirings. Matematicheskie zametki [Mathematical Notes], 1980, V. 27. No. 4. Pp. 527–537.
  47. Pastijn F. Varieties Generated by Ordered Bands. II. Order., 2005, V. 22. Pp. 129−143.
  48. Guterman A. E. Frobeniusovy endomorfizmy prostranstva matrits: dissertatsiya na soiskanie uchenoy stepeni doktora fiziko-matematicheskih nauk [Frobenius endomorphisms of the matrix space: dissertation for the degree of Doctor of Physics and Mathematics], M.: MSU, 2009. 321 p.
  49. Shitov Ya. N. Lineynaya algebra nad polukoltsami: dissertatsiya na soiskanie uchenoy stepeni doktora fiziko-matematicheskih nauk [Linear algebra over semirings: dissertation for the degree of Doctor of Physics and Mathematics], M.: MSU, 2015. 302 p.
  50. Krivulin N. K. On the solution of generalized linear vector equations in idempotent algebra. Vestnik Sankt-Peterburgskogo universiteta. Seriya 1 [Bulletin of Saint Petersburg University. Series 1], 2006. No.Pp. 23–36.
  51. Krivulin N. K., Romanova E. Yu. Approximate factorization of positive matrices using tropical optimization methods. Vestnik SanktPeterburgskogo universiteta. Seriya 10 [Bulletin of Saint Petersburg University. Series 10], 2020, V. 16. No. 4. Pp. 357–374.
  52. Vorob’ev N. N. Extreme algebra of positive matrices. Elektronische Informatiosverarbeitung und Kybernetik [Electronic information processing and cybernetics], 1967, V. 3. Pp. 39–71.
  53. Joswig M. Essentials of Tropical Combinatorics. Graduate Studies in Mathematics. V. 219, 2021. 398 p.
  54. Maslov V. P., Kolokol’tsov V. N. Idempotentnyi analiz i ego primenenie v optimal’nom upravlenii [Idempotent analysis and its application in optimal control], M.: Nauka, 1994.
  55. Gondran M., Minoux M. Graphs, Dioids and Semirings. New Models and Algorithms. New York: Springer, 2008. 400 p.
  56. Kolokoltsov V. N., Maslov V. P. Idempotent Analysis and its Applications. Mathematics and its Applications, V. 401, Dordrecht: Kluwer Academic Publishers, 1997.
  57. Litvinov G. L., Maslov V. P., Shpiz G. B. Idempotent functional analysis: an algebraic approach. Matematicheskie zametki [Mathematical Notes], 2001, V. 69. No. 5. Pp. 758–797.
  58. Gillman L., Jerison M. Rings of continuous functions. New York,300 p.
  59. Varankina V. I., Vechtomov E. M., Semenova I. A. Semirings of continuous non-negative functions: divisibility, ideals, congruences. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 1998, V. 4. No. 2. Pp. 493–510.
  60. Vechtomov E. M., Lubyagina E. N., Sidorov V. V., Chuprakov D. V. Elementy funktsionalnoy algebry: monografiya. V 2 tomah [Elements of functional algebra: monography. In 2 volumes] [edited by Vechtomov], Kirov, OOO «Raduga-PRESS», 2016, V. 1, 384 p.; V. 2. 316 p.
  61. Vechtomov E. M., Mikhalev A. V., Sidorov V. V. Semirings of continuous functions Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 2016, V. 21. No. 2. Pp. 53– 131.
  62. Vechtomov E. M., Chuprakov D. V. The principal kernels of semifields of continuous positive functions. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics],
    2008, V. 14. No. 4. Pp. 87–107.
  63. Vechtomov E. M., Chuprakov D. V. Extension of congruences on semirings of continuous functions. Matematicheskie zametki [Mathematical Notes], 2009, V. 85. No. 6. Pp. 803–816.
  64. Bestuzhev A. S., Vechtomov E. M. Cyclic semirings with commutative addition.Vestnik Syktyvkarskogo universiteta. Seriya Matematika. Mehanika. Informatika [Bulletin of the Syktyvkar University. Series 1. Mathematics. Mechanics. Informatics], 2015, V.Pp. 8–39.
  65. Vechtomov E. M., Chuprakov D. V. Finite cyclic semirings with semilattice addition given by the two-generated ideal of natural numbers. Chebyshevskiy sbornik [Chebyshev collection], 2020, V. 21. No. 1. Pp. 82–100.
  66. Vechtomov E. M., Orlova (Lubyagina) I. V. Cyclic semirings with idempotent noncommutative addition. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics],
    2012, V. 17. No. 1. Pp. 33–52.
  67. Vechtomov E. M., Orlova I. V. Cyclic semirings with idempotent commutative addition. Fundamentalnaya i prikladnaya matematika [Fundamental and Applied Mathematics], 2015, V. 20. No. 6. Pp. 17– 41.
  68. Bestuzhev A. S., Vechtomov E. M., Orlova I.V. Structure of cyclic semirings. Sbornik materialov IX nauchnoy konferentsii EKOMOD-2016 “Matematicheskoe modelirovanie razvivayuscheysya
    ekonomiki, ekologii i tehnologii” [Proceedings of the IX Scientific Conference ECOMOD-2016 “Mathematical modeling of developing economy, ecology and technology”], Kirov: Izd. VyatGU, 2016. Pp. 21–30.
  69. Vechtomov E. M., Petrov A. A. Multiplicatively idempotent semirings with three elements. Matematicheskiy vestnik Vyatskogo gosudarstvennogo universiteta [Mathematical Bulletin of Vyatka State University], 2021. No. 2. Pp. 13−23.
  70. Zhao X., Ren M., Crvenkovic S., Shao Y., Dapic P. The varietygenerated by an ai-semiring of order three. Ural Mathematical Journal,2020, V. 6. No. 2. Pp. 117–132.

For citation: Vechtomov E. M., Chermnykh V. V. Main directions of the development of the semiring theory. Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2021. No. 4 (41), pp. 4−40. DOI: 10.34130/1992-2752_2021_4_4

II. Andryukova V. Yu. Variational approach to calculating critical loads in the case of spatial deformation of curved rods

DOI: 10.34130/1992-2752_2021_4_41

Andryukova Veronika Yuryevna − Associate Professor, Komi Science Center, Ural RAS Department, e-mail: veran@list.ru

Text

A detailed derivation of the formulas of elastic energy and work of external forces for rings loaded with central forces is given. xpressions for calculating the critical load are presented in the case of plane deformation of the ring, as well as in the case of the spatial form of buckling.

Keywords: curvilinear bar, critical load, stability, Euler equations, work of external forces, elastic energy

References

  1. Perel’muter A. V., Slivker V. I. Ustoychivost’ ravnovesiya konstruktsiy i rodstvennyye problemy [Stability of the structures equilibrium and related problems]. Vol. 2. Moscow, Izdatel’stvo SKAD SOFT, 672 p.
  2. Nikolai Ye. L. Trudy po mekhanike [Writings on mechanics]. Moscow, Gostekhizdat, 1955. 584 p.
  3. Andryukova V., Tarasov V. Nonsmooth problem of stability for elastic rings. Abstracts of the International Conference “Constructive Nonsmooth Analysis and Related Topics” Dedicated to the Memory of Professor V.F. Demyanov. CNSA-2017. 22-27 may 2017, Part I. SaintPetersburg. Publisher: BBM. Pp. 213–218.
  4. Birger I. A. Prochnost’. Ustoychivost’. Kolebaniya [Strength. Stability. Oscillations]. M.: Mashinostroenie, 1988. 831 p.

For citation: Andryukova V. Yu. Variational approach to calculating critical loads in the case of spatial deformation of curved rods. Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2021, No. 4 (41), pp. 41−49. DOI: 10.34130/1992-2752_2021_4_41

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III. Yermolenko A. V., Melnikov V. A. Solving the problem of abstraction from platform-specific code for iOS and Android applications using the example of SadLion Engine

DOI: 10.34130/1992-2752_2021_4_50

Yermolenko Andrei Vasilievich − PhD in Physics and Mathematics, Associate Professor, Head of Department of Applied Mathematics and Computer Science, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Melnikov Vadim Andreevich − Postgraduate student, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Text

The paper examines existing solutions for cross-platform mobile development, compares their features, advantages and disadvantages. It describes the solution to various problems arising in the development of your own cross-platform engine for development for iOS and Android.
The construction of a system for displaying a visual interface on a user screen using a GPU is considered. The architectural solutions used to write high-performance logic of application behavior in the C ++ programming language are described. The life cycles of applications for the iOS and
Android platforms are considered and a way to abstract from the native life cycle is proposed to generalize the application code on both platforms.The implementation of interlanguage interaction between Java and C ++ using JNI on the Android platform and Objective-C and C ++ is described,
architectural solutions are given for building an abstraction layer that hides such low-level interactions in the engine core.

Keywords: cross-platform development, C ++, Android, iOS.

References

  1. Bosnic S., Papp I. The development of hybrid mobile applications with Apache Cordova. 24th Telecommunications Forum. 2016. Pp. 1−4.
  2. Tomozei C. Assessment of the evolution in quality for XamarinAndroid Multimedia Applications. 19th International Conference on Informatics in Economy. Education, Research and Business
    Technologies. 2020. Pp. 47−52.
  3. Melnikov V. A. Development Process of game engine core for 2D games and interfaces Sad Lion Engine. Vestnik Syktyvkarskogo universiteta. Ser. 1: Matematika. Mexanika. Informatika [Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics], 2019, 4
    (33). Pp. 21–37.
  4. Fisz K., Kopniak P., Galan D. A multi-criteria comparison of mobile applications built with the use of Android and Flutter Software Development Kits. Journal of Computer Sciences Institute. Vol. 19.Pp. 107−113.
  5. Dabit N. React Native in action. Shelter Island: Manning Publishing. 320 p.
  6. Java Native Interface Specification [Online]. Oracle. Available: https://docs.oracle.com/javase/7/docs/technotes/guides/jni/spec/ jniTOC.html (Accessed: 22.10.2020).
  7. Rago S., Stevens W. Advanced programming in the UNIX environment, 3rd ed. Upper Saddle River, NJ, Boston, Indianopolis, San Francisco, New York, Toronto, Montreal, London, Munich, Paris, madrid, Capetown, Sydney, Tokyo, Singapore, Mexico City: AddisonWesley, 2013. 1032 p.
  8. Corbet J., Kroah-Hartman G., McKellar J., Rubini A. Linux device drivers, 4th ed. Beijing, Cambridge, Farnham, Koln, Sebastopol, Taipei, Tokyo. 2017. 600 p.
  9. Thornsby J. Android UI design. Birmingham: Packt Publishing. 2016. 356 p.
  10. Anugerah M. A., Sekar G. S. Designing Android User Interface for University Mobile Library. International Conference on Computing, Engineering, and Design (ICCED). 2021. Pp. 224−229.
  11. Neuburg M. Programming iOS 13: Dive Deep into Views, View Controllers, and Frameworks. Sebastopol: O’Reilly. 2020. 1208 p. New York: Oracle.
  12. Nystrom R. Game programming patterns. San Bernardino: Genever Benning, 2018. 345 p.
  13. Ginsburg D., Purnomo B. OpenGL ES 3.0 Programming Guide 2nd Edition. Upper Saddle River, NJ, Boston, Indianopolis, San Francisco, New York, Toronto, Montreal, London, Munich, Paris,
    madrid, Capetown, Sydney, Tokyo, Singapore, Mexico City: AddisonWesley, 2014. 560 p.
  14. Sellers G. Vulkan Programming Guide. The Official Guide to Learning Vulkan. Boston, Columbus, Indianapolis, New York, San Francisco, Amsterdam, Cape Town Dubai, London, Madrid, Milan, Munich, Paris, Montreal, Toronto, Delhi, Mexico City San Paulo, Sydney, Hong
    Kong, Seoul, Singapore, Taipei, Tokyo: Addison-Wesley, 2017. 480 p.
  15. Clayton J. Metal programming guide. Addison-Wesley. 2018. 352 p.
  16. Stroustrup B. The C++ programming languege 4th edition. Uppder Saddle River, Boston, Indianopolis, San Francisco, New York, Toronto, Montral, London, Munich, Paris, Madrid, Capetown, Sydney, Tokyo, Singapore, Mexico city: Addison-Wesley. 2013. 1345 p.
  17. Stroustrup B. Programming: Principles and Practice using C++. 2nd Edition. New Jearsey: Pearson Education. 2015. 1312 p.
  18. Shieldt H. Java: The Complete Reference, Eleventh Edition 11th Edition. 2019. 1248 p.

For citation: Yermolenko A. V., Melnikov V. A. Solving the problem of abstraction from platform-specific code for iOS and Android applications using the example of SadLion Engine. Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2021, No. 4 (41), pp. 50−69.
DOI: 10.34130/1992-2752_2021_4_50

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IV. Dorofeev S. N., Esetov E. N., Nazemnova N. V. Analogy as the basis for teaching students the vector method of geometric problem solving

DOI: 10.34130/1992-2752_2021_4_70

Dorofeev Sergey Nikolaevich – Doctor of Pedagogy, Professor of the Department of Higher Mathematics and Mathematical Education, Togliatti State University (Russia, 445020, Samara Region, Tolyatti, Belorusskaya St., 14)

Esetov Yelzhan Nurlykhanovich – postgraduate student of the department “Higher Mathematics and Mathematical Education” Togliatti State University (Russia, 445020, Samara region, Tolyatti, Belorusskaya st., 14)

Nazemnova Natalia Vladimirovna − Candidate of Pedagogical Sciences, Senior Lecturer, Department of Higher Mathematics, Penza State University (Russia, 440020, Penza region, Penza, Krasnaya st., 40

Text

This article examines the ways and the methods that contribute to improving the quality of teaching students the basics of vector algebra and methods of their application to solving geometric problems. For this purpose, the necessary knowledge of the basics of vector algebra, which students should learn in the process of studying the topic “Fundamentals of
vector algebra”, is highlighted and systematized. The paper substantiates the fact that such a method of cognition as analogy plays an important role in the effectiveness of the process of
teaching high school students to apply the basics of vector algebra to solving geometric problems. Some examples of interrelated tasks that contribute to improving the quality of teaching students the use of the vector method are given.

Keywords: Vector method, training in solving geometric problems, analogy.

References

  1. Boltyanskij V. G. Analogy — commonality of axiomatics. Sovetskaya pedagogika [Soviet pedagogy], 1975. No. 1. Pp. 83−93.
  2. Dorofeev S. N. Teoriya i praktika formirovaniya tvorcheskoj aktivnosti budushhix uchitelej matematiki v pedagogicheskom vuze, dissertaciya na soiskanie uchenoj stepeni doktora pedagogicheskix nauk [The theory and practice of forming the creative activity of future
    teachers of mathematics in a pedagogical university], Penza, 2000. 410 p.
  3. Atanasyan L. S., Butuzov V. F., Kadomcev S. B. et al. Geometriya. 7−9 klassy [Geometry. 7-9th grade]. M.: Prosveshhenie, 384 p.
  4. Aleksandrov A. D., Verner A. L., Ry‘zhik V. I. Geometriya. 9 klass [Geometry. 9th grade]. M.: Prosveshhenie, 2015. 175 p.
  5. Atanasyan L. S., Butuzov V. F., Kadomcev S. B. et al. Geometriya. 7−9 klassy [Geometry. 7-9th grade]. M.: Prosveshhenie, 255 p.
  6. Dorofeev S. N., Zhuravleva O. N., Ry‘bina T. M., Sarvanova Zh. A. Formation of research competencies of students in the mathematics classroom. Sovremenny‘e naukoemkie texnologii
    [Modern knowledge-intensive technologies]. 2018. No. 10. Pp. 181−185.
  7. Uteeva R. A. Teoreticheskie osnovy‘ organizacii uchebnoj deyatel‘nosti uchashhixsya pri differencirovannom obuchenii matematike v srednej shkole. Dissertaciya doktora ped. nauk [Theoretical foundations of the organization of students’ learning activities in differentiated learning of mathematics in high school]. Moscow, 1998. 363 p.
  8. Kudryavcev L. D. Mysli o sovremennoj matematike i ee izuchenii [Thoughts on Modern Mathematics and its Study]. M.: Nauka, 1977. 123 p.
  9. Dorofeev S. N. UDE as a method of preparing future bachelors of teacher education for professional activities. Gumanitarny‘e nauki i obrazovanie. MordGPI im. M. E. Evsev‘eva [Humanities and Education / M. E. Evsevyev Mordovian State Pedagogical University].
    No. 1, 2013. Pp. 14−17.
  10. Sarancev G. I. Kak sdelat‘ obuchenie matematike interesny‘m [How to make learning math interesting]. M.: Prosveshhenie. 2011. 160 p.
  11. Dorofeev S., Pavlov I., Shichiyakh R., Prikhodko A. Differentiated Training as a Form of Organization of Education and Cognitive Activity of Future Masters of Pedagogical Education.
    Applied Lingvistics Research Jounal, 2021, 5(3), Pp. 216−222.

For citation: Dorofeev S. N., Esetov E. N., Nazemnova N. V. Analogy as the basis for teaching students the vector method of geometric problem solving. Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2021, No. 4 (41), pp. 70−82. DOI: 10.34130/1992-2752_2021_4_70

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V. Yermolenko A. V., Belyaev E. A., Turkova O. I. One contact problem for two plates

DOI: 10.34130/1992-2752_2021_4_83

Yermolenko Andrei Vasilievich − PhD in Physics and Mathematics, Associate Professor, Head of Department of Applied Mathematics and Computer Science, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Belyaev Evgeniy Anatolievich − Postgraduate student, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Text

Using the generalized reaction method, a numerical solution of the contact problem for two plates is given. One plate is hinged, the other one is rigidly fixed. It is shown that the distribution of contact reactions significantly depends on the relative position of the plates. In this case, the contact zone is either a segment or a point.

Keywords: plate, contact problem, generalized reaction method, numerical solution.

References

  1. Yermolenko А. V., Ladanova S. V. Contact problem for two plates with different fixing. Vestnik Syktyvkarskogo universiteta. Ser. 1: Matematika. Mexanika. Informatika [Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics], 2020, 3 (36). Pp. 87- 92.
  2. Ермоленко А. В. Kontaktnye zadachi so svobodnoj granicej [Free Boundary Contact Problems]. Syktyvkar: Izd-vo SGU im. Pitirima Sorokina, 2020. (CD-ROM). 105 p.
  3. Yermolenko A. V., Osipov K. S. On using Python libraries to calculate plates. Vestnik Syktyvkarskogo universiteta. Ser. 1: Matematika. Mexanika. Informatika [Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics], 2019, 4 (33). Pp. 86–95.
  4. Mihajlovskii E. I., Toropov A. V. Matematicheskiye modeli teorii uprugosti [Mathematical models of the theory of elasticity]. Syktyvkar: Sykt Publishing House. University, 1995. 251 p.
  5. Mikhailovskii E. I., Tarasov V. N. On the convergence of the generalized reaction method in contact problems with a free boundary. Jurnal prikladnoy matematiki i mekhaniki [Journal of Applied Mathematics and Mechanics], 1993, v. 57, No. 1. Pp. 128–136.

For citation: Yermolenko A. V., Belyaev E. A., Turkova O. I. One contact problem for two plates . Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2021. No. 4 (41), pp. 83−89. DOI: 10.34130/1992-2752_2021_4_83

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VI. Rogosin S. V. Remark to the paper

DOI: 10.34130/1992-2752_2021_4_90

Rogozin Sergey Vasilyevich − PhD in Physics and Mathematics, Associate Professor at the Department of Analytical Economics and Econometrics, Belarusian State University, Minsk, Belarus, e-mail: rogosin@bsu.by

Text

An assertion on p. 31 “Note that X(z) is a rational matrix which is analytic outside of the unit disc (but not necessary analytic at infinity) since. . . ” is imprecise. This assertion including the expression after it be omitted since on the first stage of factorization the corresponding
transformation is performed only on the unit circle and does not involve any analyticity properties of the matrix X(z).

For citation: Rogosin S. V. Remark to the paper. Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2021. No. 4 (41), pp. 90−91. DOI: 10.34130/1992-2752_2021_4_90

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