**I Babenko M. V. On the polynomial semiring over a Bezout semiring**

DOI: 10.34130/1992-2752_2020_4_05

Babenko Marina − Senior Lecturer of the Department of applied mathematics and computer science, Vyatka State University, e-mail: usr11391@vyatsu.ru

The article examines a polynomial semiring over a Bezout Rickart semiring. Namely, let all left annihilator ideals of the semiring S be ideals. Then the semiring of polynomials R = S[x] is a semiring without nilpotent elements and every finitely generated left monic ideal from R is principal iff S is a left Rickart left Bezout semiring and any non-zero divisor of the semiring S is convertible to S. This result is analogous to the statement for rings, if the condition each finitely generated left monic ideal of R is principal replaced by R is left Bezout ring. The left monic ideal of a polynomial semiring is a left ideal that contains each monomial of its polynomial. The principal left monic ideals over a left Rickart left Bezout semiring are described.

**Keywords:** polynomial semiring, Rickart semiring, Bezout semiring, monic ideal.

**References**

- Tuganbaev A. A. Kol’ca Bezu, mnogochleny i distributivnost’ (Bezout rings, polynomials and distributivity) Mathematical notes, 2001, 70:2, pp. 270288.
- Dale L. Monic and monic free ideals in polynomial semirings, Proc. Amer. Math. Soc., 1976, No 56, pp. 45-50.
- Dale L. The structure of monic ideals in a noncommutative polynomial semirings, Acta Math. Acad. Sci. Hungar, 1982, 39:1-3, pp. 163-168.
- Golan J. S. Semirings and their applications, Kluwer Acad. Publ., Dordrecht, 1999.
- Chermnykh V. V. Functional representations of semirings, J. Math. Sci., New York, 2012, 187:2, pp. 187-267.
- Maslyaev D. A., Chermnykh V. V. Polukol’ca kosyh mnogochlenov Lorana (Semirings of skew Laurent polynomials), Siberian electronic mat. reports., 2020, Vol. 17, pp. 512-533.

**For citation:** Babenko M. V. On the polynomial semiring over a Bezout semiring, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 5-15.

**II Efimov D. B. A method for computing the hafnian**

DOI: 10.34130/1992-2752_2020_4_16

Efimov Dmitry − Ph. D., research associate, Institute of Physics and mathematics of the Komi national research center of the Ural branch of the Russian Academy of Sciences, e-mail: dmefim@mail.ru

The hafnian was initially introduced by E.R. Caianiello, by analogy with the Pfaffian, as a convenient mathematical apparatus for working with certain quantum-mechanical quantities. From a combinatorial point of view, the hafnian of a symmetric matrix is equal to the sum of weights of perfect matchings of a graph with the given incidence matrix. In contrast to the Pfaffian, the hafnian has a smaller set of good properties, and determining its value is an example of a complex computational problem. We consider a new method for calculating hafnian of a matrix in terms of permanents of its submatrices. We also give a comparison with other methods in terms of computational complexity. The property underlying the method could also be used outside the context of the computation speed, for example, to estimate the hafnian of a nonnegative matrix based on known estimates of the permanent.

**Keywords:** hafnian, permanent, computational complexity

**References**

- Caianiello E. R. On quantum field theory – I: Explicit solution of Dyson’s equation in electrodynamics without use of Feynman graphs, IL Nuovo Cimento, 1953, V. 10(12), pp. 1634-1652.
- Caianiello E. R. Theory of coupled quantized fields, Supplemento Nuovo Cimento, 1959, V. 14(1), pp. 177191.
- Caianiello E. R. Regularization and Renormalization, IL Nuovo Cimento, 1959, V. 13(3), pp. 177-191.
- Mink H. Permanenty (Permanents), M.: Mir, 1982, 216 p.
- Valiant L. G. The complexity of computing the permanent, Theoretical Computer Science, 1979, V. 8(2), pp. 187-201.
- Bjorklund A., Gupt B., Quesada N. A faster hafnian formula for complex matrices and its benchmarking on a supercomputer, ACM Journal of Experimental Algorithmics, 2019, V. 24(1), 17 p.
- Aaronson S., Arkhipov A. The computational complexity of linear optics, Proceedings of the Annual ACM Symposium on Theory of Computing, 2011, pp. 333-342.
- Kruse R., Hamilton C. S., Sansoni L., Barkhofen S., Silberhorn C., Jex I. Detailed study of Gaussian boson sampling, Physical Review A, 2019, V. 100(3), 032326.

**For citation:** Efimov D. B. A method for computing the hafnian, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 16-25.

**III Gabova M. N., Muzhikova A. V. Ñontext approach in the teaching of mathematics future engineers**

DOI: 10.34130/1992-2752_2020_4_26

Gabova Maria − Senior Lecturer of the Department of the Department of higher mathematics, Ukhta state technical University, e-mail: amuzhikova@mail.ru

Muzhikova Alexandra − Ph. D., associate Professor of the Department of the Department of higher mathematics, Ukhta state technical University, e-mail: amuzhikova@mail.ru

There is a problem of reducing the mathematical education of school graduates, and as a result, the lack of motivation and cognitive activity of first-year students when studying mathematics in higher school. Mathematics, devoid of professional direction, is not of interest to most students of a technical higher school. The efectiveness of the teaching process can be achieved by using a context approach. Context teaching is teaching in which the subject and social content of students’ professional activity is modeled in the language of science and with the help of the entire system of forms, methods and means of teaching. Considering context teaching as an integral system that meets the corresponding principles, the article presents the developed methodological and organizational support for educational activities.

The main idea in developing the content is a gradual transition from abstract mathematical concepts to their applied meaning in related sciences, and then to their application in professional fields. The principles of context teaching are best implemented when using active and interactive forms of teaching and their corresponding methods. The most efective methods in terms of achieving the goals of teaching, development and education were shown by such methods as problem-based lecture format, swapping of topics in pairs and partners rotation, paragraph-by-paragraph study of theoretical material in small groups, task swapping in practical classes, etc.

The use of a context approach allows students to develop social interaction, motivation and cognitive activity, mathematical literacy, the ability to apply mathematics in their educational and professional activities and contribute to the formation of a modern engineer capable of creative activity and self-realization.

**Keywords: **mathematics for engineer, context approach, active and interactive methods of teaching.

**References**

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- Muzhikova A. V., Gabova M. N. Razvitie gramotnoj matematicheskoj rechi studentov v tekhnicheskom vuze (Development of Competent Mathematical Speech of Students in a Technical University), Vysshee obrazovanie v Rossii, 2019, Vol. 28, no. 12, pp. 66-75,.
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mezhdunarodnoe obrazovatel’noe prostranstvo¿, Plock (Poland), 2006, pp. 103-106. - Zajniev R. M. Preemstvennost’ matematicheskoj podgotovki v inzhenerno-tehnicheskom obrazovanii (Continuity of mathematical training in engineering and technical education), Kazan: Kazan State University Publ., 2009, 366 p.
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i pedagogika XXI veka: teorija, praktika i perspektivy, Cheboksary: CNS Interaktiv pljus Publ., 2015, pp. 2430. - Kolbina E. V. Metodika formirovanija matematicheskoj kompetentnosti studentov tehnicheskih vuzov v problemno-prikladnom kontekste obuchenija. Kand. Diss. (Methods of forming mathematical competence of students of technical universities in the problem-applied context of learning: Cand. Diss.), Barnaul, 2016, 221 p.
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193. - Sorokopud Ju. V. Pedagogika vysshej shkoly (Pedagogy of Higher School), Rostov-on-Don: Feniks Publ., 2011, 541 p.
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- Muzhikova A. V. Interaktivnoe obuchenie matematike v VUZe (Interactive Teaching of Mathematics in Higher School), Vestnik Syktyvkarskogo universiteta. Serija 1: Matematika. Mehanika. Informatika, 2015, Vol. 1 (20), pp. 74-90.
- Muzhikova A. V. Issledovanie efektivnosti kollektivnyh uchebnyh zanyatij po vysshej matematike (Study the Interactive Teaching Efiectiveness in Higher Mathematics), Vestnik Tomskogo gosudarstvennogo pedagogicheskogo universiteta, 2018, No 7 (197), pp. 174-181.
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**For citation:** Gabova M. N., Muzhikova A. V. Ñontext approach in the teaching of mathematics future engineers, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 26-50.

**IV Odyniec W. P. The Fate of two mathematicians: Perelman and Perelman Jr.**

DOI: 10.34130/1992-2752_2020_4_51

Odyniec Vladimir − Doctor of Physical and Mathematical Sciences, Professor, Syktyvkar state University named after Pitirim Sorokin, e-mail: W.P.Odyniec@mail.ru

In the article the work of Jacob I. Perelman (1882-1942) in the area of mathematics and its application to the theory of elasticity is described for the first time at the ever of the Grand Patriotic War. Also described the life and work of his son Michael J. Perelman (1919-1942).

**Keywords: **J. I. Perelman, M. J. Perelman, Galerkin method, continuity modulus, the least power and pseudo-power of a topological space.

**References**

- Mishkevich G. I. Doktor zanimatelnyh nauk (Doctor of Entertaining Sciences), M.: Znanie, 1986, 192 p.
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- Leibenson L. S. Variacionnye metody resheniya sadach teorii uprugosti (Variational methods of solving the problems of the theory of elasticity), M.-L.: 1943, 287 p.
- Kniga pamyati. Leningrad. 1941-1945. Primorskii raion T. 12 (The Book of Memory. Leningrad. 1941-1945. The Primorsky District. Vol. 12), SPb: Notabene, 1997, 557 p.
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- Galerkin B. G., Perelman J. I. Napryazheniya i peremeshtcheniya v krugovom zylindricheskom truboprovode (Tensions and Displacements in Cylindrical Pipe Line), Izvestiya nauchno- issledovatelskogo instituta gidrotehniki, T. 27, 1940, pp. 160-192.
- Odyniec W. P. O leningradskih matematikah, pogibshih v 1941-1944 godah (On some Leningrad based Mathematicians perished in 1941 -1944), Syktyvkar: Izd- vo SGU im. Pitirima Sorokina, 2020, 122 p.
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- Trudy Pervogo Vsesouznogo s’ezda matematikov (Proceedings of the First All-Union congress of mathematicians), M.-L.: ONTI NKTP SSSR, 1936, 376 p.
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**For citation: **Odyniec W. P. The Fate of two mathematicians: Perelman and Perelman Jr., Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 51-65.

**V Pevnyi A. B., Yurkina M. N. Sieve of Eratosphenes complexity and distribution of primes**

DOI: 10.34130/1992-2752_2020_4_66

Pevny Alexander − Doctor of Physics and Mathematics, Professor, Department of Applied Mathematics and Information Technologies in Education, Pitirim Sorokin Syktyvkar State University, e-mail: pevnyi@syktsu.ru

Yurkina Marina − Senior Lecturer, Department of Applied Mathematics and Information Technologies in Education, Syktyvkar State University named after Pitirim Sorokin, e-mail: yurkinamn@gmail.com

Primes are widely used not only in pure mathematics, but also in related disciplines. And although they have been known for a long time, many problems concerning prime numbers are still open and the questions of their study do not lose their relevance. One of the well-known algorithms for finding all primes not exceeding a given N is the sieve of Eratosthenes. To estimate the number of operations required to execute this algorithm, the authors used one result of P. L. Chebyshev. In 1849 P. L. Chebyshev proved a two-sided estimate for the number of primes not exceeding a given N. Based on these estimates, the article establishes that the number of operations in the Eratosthenes algorithm is estimated as O (N ln ln N).

**Keywords:** sieve of Eratosthenes, primes, Chebyshev.

**References**

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**For citation:** Pevnyi A. B., Yurkina M. N. Sieve of Eratosphenes complexity and distribution of primes, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 66-72.

**VI Popov N. I., Yakovleva E. V. Use of the schematization method in teaching students and pupils in math**

DOI: 10.34130/1992-2752_2020_4_74

The publication objective is to highlight and generalize the features of using the schematization method in teaching mathematics as a means of developing thinking and mathematical abilities of learners. The research is based on analyzing scientific and methodical works of Russian and foreign scientists on both activity theory, pedagogy, and author’s researches on applying the schematization method in teaching mathematics. The article proposes a schematic model to teach pupils and students solve mathematical problems. The methodological approaches developed in the research can be used in teaching mathematics at diferent levels of education. We believe that the method described in this paper can be successfully applied in studying natural sciences.

**Keywords:** a method of schematization; teaching mathematics; stages of mathematical problems solving; schematic model.

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**For citation:** Popov N.I., Yakovleva E.V. Use of the schematization method in teaching students and pupils in math, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 4 (37), pp. 74-87.