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I Bazhenov I. I, Extreme points of the range of Liapunov vector measure

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II Zhubr A. V. Calculation of spin bordism groups of some Elenberg-MacLane spaces, II

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III Zhubr A. V. KS-transformations and involutions of normed algebras

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IV Isakov V. N. On the problem of countable addivity of the abstract measures product

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V Poroshkin A. A. On the inclusion of generalised Boolean algebra to Boolean algebra

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VI Samorodnitski A. A. Basic conceptions of Lebesgue-Rohlin space theory. Measure theory on subspaces of generalized Cantor discontinuum

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VII Tichomirov A. N. The rate of convergence in the central limit theorem for weakly dependet random variables

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VIII Antonova N. A. Chaos and order in an integral pulse-width control systems

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IX Belyaeva N. A., Klichnikov L. V. Integral equation method in the volume hardering problem

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X Golovach P. A. Pathwidith and treewidth of joining of two graphs

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XI Kirushev V. A. The quadratic variational problem with nonnegativity condition

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XII Mikhailovskii E. I. The noncoordinate method of obtaining of the conjuctive couples of the tensors

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XIII Nikitenkov V. L. Elastic curve of an axis of multisupport cylindrical vessel of pressure at a thermo-mechanical bend and extreme problems connected with it

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XIV Pevnyi A. B. Discrete periodic splines and solutions of the problem concerning infinite cylindrical shell

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XV Poleshchikov S. M., Kholopov A. A. Generalized KS-transformations of 4-th order

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XVI Sokolov V. Ph. Robust performance of linear controller for linear discrete plant in l1-setting

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XVII Kholopov V. M. , Khudyaev S. I. To asymptotic theory of combustion wave in gases

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XVIII Ermolenko A. V. On the semideformational variant of the boundary values in Karman’s theory of the flexible plates

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XIX Martynov Y. I. The determining equations in the contact problem for bending of plate on the theory of Timoshenko

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XX Teryohin D. E. The stability of cylindrical panel with the inside strengthenings

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XXI Zinchenko I. L. About one classical problem of variational calculs

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XXII Zinchenko I. L., Sangadjieva S. T. Periodicity of a sum of continuous periodic functions

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XXIII Poleshchikov S. M. Proper and improper KS-matrices

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XXIV 25 years of the Faculty of Mathematics

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XXV Evgeny Ilyich Mikhailovsky (on the occasion of his sixtieth birthday)

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I Bazhenov I. I. On some properties of Liapunov vector measure

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II Bobkov S. G. On inequalities of Gross and Talagrand on the discrete cube

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III Yekisheva S. V. A uniform Central Limit Theorem for a set-indexed processes

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IV Lovyagin J. N. On some questions of nonstandart theory of Kantorovich spaces

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VI Poroshkin A. A. On one generalization of the theorem on completeness

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VII Poroshkin A. G. On the metrizability of sequental order topology in ordered groups and vector spaces

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VIII Ryabinin A. A. On rise of Kantor-Fouries measure on imaginary axis

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IX Salnikova T. A. On complete and minimal systems of exponents

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X Samorodnitsky A. A. A Boolean principle of exhaustion and a construction of measure spaces

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XI Saveliyev L. J. Generational functions in the theory of series

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XII Antonova N. A. Chaos and order in pulse-width control systems

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XIII Belyaeva N. A., Belyaev Yu. N. The regulation of strained state of forming cylindrical product

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XIV Gerasin M. L. Stability of cilyndrical shell with ine-sided support

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XV Golovach P. A. On one invariant of graphs defined through optimal numbering of vertices

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XVI Kazakov A. Y. The maximazation of the first eigenvalue for the little displacement equation of a composite

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XVII Kondratieva T. V., Kholopov V. M. The asymptotic of stationary combustion wave for autocatalitic reaction of the first order

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XVIII Mikhailovskii E. I. Nonlinear theory of ridge shells under small trnsversal shears

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XIX Nikitenkov V. L. Nonlinear equations for cylindrical shell with eliptic ovality of the cross section

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XX Tarasov V. N. The problems on eigenvalues for positively homogeneous operators

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XXI Holmogorov D. V. The stability of bar on two elastic surrounding boundary

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XXII Kholopov A. A. Minimal stability losing forms of bar placed between elastic and rigid spaces

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XXIII Khudyaev S. I. To mathematical theory of flame propagation

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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

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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

1. Tuganbaev A. A. Kol’ca Bezu, mnogochleny i distributivnost’ (Bezout rings, polynomials and distributivity) Mathematical notes, 2001, 70:2, pp. 270288.
2. Dale L. Monic and monic free ideals in polynomial semirings, Proc. Amer. Math. Soc., 1976, No 56, pp. 45-50.
3. Dale L. The structure of monic ideals in a noncommutative polynomial semirings, Acta Math. Acad. Sci. Hungar, 1982, 39:1-3, pp. 163-168.
4. Golan J. S. Semirings and their applications, Kluwer Acad. Publ., Dordrecht, 1999.
5. Chermnykh V. V. Functional representations of semirings, J. Math. Sci., New York, 2012, 187:2, pp. 187-267.
6. 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

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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

1. 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.
2. Caianiello E. R. Theory of coupled quantized fields, Supplemento Nuovo Cimento, 1959, V. 14(1), pp. 177191.
3. Caianiello E. R. Regularization and Renormalization, IL Nuovo Cimento, 1959, V. 13(3), pp. 177-191.
4. Mink H. Permanenty (Permanents), M.: Mir, 1982, 216 p.
5. Valiant L. G. The complexity of computing the permanent, Theoretical Computer Science, 1979, V. 8(2), pp. 187-201.
6. 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.
7. Aaronson S., Arkhipov A. The computational complexity of linear optics, Proceedings of the Annual ACM Symposium on Theory of Computing, 2011, pp. 333-342.
8. 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

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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

1. Kostenko I. P. Evolyuciya kachestva matematicheskogo obrazovaniya (1931-2009 gg.) (Evolution of the quality of mathematics education (1931-2009)), Izvestija VGPU, 2013, No 2 (261), pp. 81-87.
2. Muzhikova A. V. Matematicheskaya obrazovannost’ studentov: problemy i perspektivy (Mathematical education of students: problems and prospects), XIX mezhdunarodnaja nauchno-prakticheskaja konferencija Kommunikacii. Obshhestvo. Duhovnost’ 2019, Ukhta: UGTU,2019, Vol. 3, pp. 141-144.
3. 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,.
4. Rozanova S. A. Matematicheskaja kul’tura studentov tehnicheskih universitetov (Mathematical culture of students of technical universities), Moscow: FIZMATLIT Publ., 2003, 176 p.
5. Bogomolova E. P. Diagnoz: matematicheskaya malogramotnost’ (Diagnosis: mathematical illiteracy), Matematika v shkole, 2010, No 4, pp. 3-9.
6. Senashenko V. S., Vostrikova N. A. O preemstvennosti srednego i vysshego matematicheskogo obrazovaniya (On the continuity of secondary and higher mathematics education), Mezhdunarodnaja konferencija Obrazovanie, nauka i jekonomika v vuzah. Integracija v
   mezhdunarodnoe obrazovatel’noe prostranstvo¿, Plock (Poland), 2006, pp. 103-106.
7. 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.
8. Egorova I. P. Proektirovanie i realizacija sistemy professional’nonapravlennogo obuchenija matematike studentov tehnicheskih vuzov (Design and implementation of a system of professional-oriented learning mathematics to students of technical universities:Cand. Sci.
   Thesis), Tolyatti, 2002, 24 p.
9. Verbickij A. A. Aktivnoe obuchenie v vysshej shkole: kontekstnyj podhod (Active learning in higher education: context approach), Moscow: Vysshaja shkola Publ., 1991, 207 p.
10. Grebenkina A. S. Osobennosti kontekstnogo obucheniya vysshej matematike studentov tekhnicheskih special’nostej (Features of context learning of higher mathematics to students of technical specialties), II mezhdunarodnaja nauchno-prakticheskaja konferencija Psihologija
    i pedagogika XXI veka: teorija, praktika i perspektivy, Cheboksary: CNS Interaktiv pljus Publ., 2015, pp. 2430.
11. 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.
12. Janushhik O. V., Sherstnjova A. I., Pahomova E. G. Kontekstnye zadachi kak sredstvo formirovaniya klyuchevyh kompetencij studentov tekhnicheskih special’nostej (Context tasks as a means of forming key competencies of students of technical specialties), Sovremennye problemy nauki i obrazovanija, 2013, No. 6, p. 376.
13. Pidkasistij P. I. Pedagogika (Pedagogics: textbook for students of pedagogical universities and pedagogical colleges), Moscow: Pedagogicheskoe obshhestvo Rossii Publ., 1998, 640 p.
14. Nizhnikov A. I., Rastopchina O. M. Obuchenie vysshej matematike: kontekstnyj podhod (Learning higher mathematics: the context approach), Vestnik Moskovskogo gosudarstvennogo oblastnogo universiteta, 2018, No 3, ðð. 184-193, doi:10.18384/2310-7219-2018-3-184-
    193.
15. Sorokopud Ju. V. Pedagogika vysshej shkoly (Pedagogy of Higher School), Rostov-on-Don: Feniks Publ., 2011, 541 p.
16. Mkrtchjan M. A. Metodiki kollektivnyh uchebnyh zanyatij (Methods of Collective Training), Spravochnik zamestitelja direktora shkoly, 2011, No 1, pp. 5564.
17. Prudnikova O. M., Gabova M. N., Kaneva E. A. K voprosu formirovaniya u studentov kriticheski-refeksivnogo stilya myshleniya (To the Question of Formation of Students Critical-Refiexive Style of Thinking), Nauchno-tehnicheskaja konferencija, Ukhta: UGTU, 2011, Vol. 3, pp. 226-229.
18. 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.
19. 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.
20. Lobos E., Macura J. Mathematical competencies of engineerin students, In ICEE-2010, International Conference on Engineering Education, July 18-22, 2010, Gliwice, Poland, Silestian University of Technology.
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22. Steyn T., Plessis I. D. Competence in mathematics-more than mathematical skills?, International Journal of Mathematical Education in Science and Technology, 2007, Vol. 38, Issue 7, pp. 881-890,doi:10.1080/00207390701579472.
23. Ravn O., Bo Henriksen L. Engineering mathematics in context – learning university mathematics through problem based learning, International Journal of Engineering Education, 2017, Vol. 33, Issue 3, pp. 956-962.
24. Firouzian S., Kashe H., Yusof Y. M., Ismail Z., Rahman R. A. Mathematical competencies as perceived by engineering students, lecturers, and practicing engineers, International Journal of Engineering Education, 2016, Vol. 33, Issue 6, pp. 2434-2445.

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

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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

1. Mishkevich G. I. Doktor zanimatelnyh nauk (Doctor of Entertaining Sciences), M.: Znanie, 1986, 192 p.
2. Matematika v SSSR za tridcat’ let (The URSS Mathematics for thirty years: 1917-1947), Pod. red. A. G. Kurosha, A. I. Markushevicha, P. K. Rashevskogo, M.-L.: OGIZ, Izd-vo tehn-teor. lit-ry, 1948, 1045 p.
3. Matematika v SSSR za sorok let (The URSS Mathematics for forty years: 19171957), Biobibliografiya, Vol. 2, M.: Fizmatlit, 1959, 819 p.
4. 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.
5. 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.
6. Perelman J. I. Metod Galerkina v variacionnom ischislenii i v teorii uprugosti (Galerkin method in calculus of variations and in the theory of elasticity), Prikladnaya matematika I mehanika, T. V, vyp. 3, 1941, 345-358 p.
7. 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.
8. 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.
9. Blokada 1941-1944. Kniga pamyati, Leningrad, T. 23 (Blockade 1941-1944. The book of Memory, Leningrad, Vol. 23), SPb.:Stella, 2005, 717 p.
10. Odyniec W. P. K 125-letiu reformatora matematicheskogo obrazovaniya O.A. Volberga (1895-1942) (On the 125 anniversary of reformer of the birth of mathematical education O.A .Volberg (1895-1942), Matematika v shkole (Mathematics in School), 4, 2020, p. 54-59.
11. Perelman M. J. O module nepreryvnosti analiticheskih funkcii (On the Continuity Modulus of Analytical Functions), Uchenye zapiski LGU. Seriya mat. nauk, vyp. 12, 1941, 62-82 p.
12. Trudy Pervogo Vsesouznogo s’ezda matematikov (Proceedings of the First All-Union congress of mathematicians), M.-L.: ONTI NKTP SSSR, 1936, 376 p.
13. Fomin D. V. Sankt- Peterburgskie matematicheskie olimpiady (Saint Petersburg mathematics olympiades), SPb.: Politehnika, 1994, 309 p.
14. Matematicheskii enciklopedicheskii slovar’ (The Mathematical Encyclopaedia), M.: Soviet Encyclopaedia, 1988, 848 p.
15. Perelman M. J. Ob odnom svoistve posledovatelnosti polinomov (On one property of sequences of polynomials), Uchenye zapiski LGU. Seriya mat. nauk, vyp. 12, 1941, 83-91 p.

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

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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

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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.

References

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3. Dalinger V. A. Teoreticheskiye osnovy kognitivno-vizual’nogo podkhoda k obucheniyu matematike: monografiya (Theoretical foundations of the cognitive-visual approach to teaching mathematics: monograph), Omsk: Izd-vo OmGPU, 2006, 144 p.
4. Popov N. I. Metodika obucheniya trigonometrii na osnove kognitivnovizual’nogo podhoda (Methods of teaching trigonometry on the basis of cognitive-visual approach), Sibirskiy pedagogicheskiy zhurnal, 2008, no 11, pp. 34-42.
5. Christochevskaya A. S., Christochevsky S. A. Kognitivizatsiya – sleduyushchiy etap informatizatsii obrazovaniya (Ñognitivization – the next stage of informatization of education), Informatika i obrazovaniye, 2018, no 9, pp. 5-11.
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11. Popov N. I. Teoretikometodologicheskiye osnovy obucheniya resheniyu tekstovykh algebraicheskikh zadach (Theoretical and methodological foundations of teaching to solve text-based algebraic problems), Obrazovaniye i nauka. Izvestiya Ural’skogo otdeleniya Rossiyskoy
    akademii obrazovaniya, 2009; no 3(60), pp. 88-96.
12. Popov N. I. Ob efiektivnosti ispol’zovaniya modeli obuchayushchey tekhnologii po trigonometrii pri obuchenii studentov-matematikov
    (Education of students-mathematicians: the effectiveness of implementation of the educational technology when teaching trigonometry),
    Obrazovaniye i nauka, 2013, no 9, pp. 138-153.
13. Bacabac M. A. A., Lomibao L. S. 4S Learning Cycle on Students’ Mathematics Comprehension, American Journal of Educational Research, 2020, Vol. 8, Issue 3, pp. 182-186.
14. Burte H., Gardony A. L., Hutton A., Taylor H. A. Think3d!: Improving mathematics learning through embodied spatial training, Cognitive Research: Principles and Implications, 2017, Vol. 2, Issue 1.
15. Hoogland K., Pepin B., Koning J., Bakker A., Gravemeijer K. Word problems versus image-rich problems: an analysis of effects of task characteristics on students’ performance on contextual
    mathematics problems, Research in Mathematics Education, 2018, Vol. 20, Issue 1, pp. 3752.
16. Bernikova I. K. Skhemy kak sredstva organizatsii myshleniya v protsesse obucheniya matematike (Schemes as means of organizing thinking in the process of teaching mathematics), Vestnik OmGU, 2015, no 1(75), pp. 2327.
17. Rahmawati D., Purwantoa, Subanji, Hidayanto E., Anwar R. B. Process of Mathematical Representation Translation from Verbal into Graphic, International Electronic Journal of Mathematics Education, 2017, Vol. 12, Issue 3, pp. 367-381.
18. Zlotnikov I. V. Psikhologicheskoye i psikho-zicheskoye obespecheniye protsessa obucheniya studentov: metodicheskiye rekomendatsii (Psychological and psychophysical support of the student learning process: guidelines), Riga: Izdatel’stvo RPI, 1988, 36 p.
19. Poya D. Kak reshat’ zadachu / Pod red. YU. M. Gayduka (How to solve a problem / Ed. Yu. M. Gaiduk), M., 1959, 208 p.
20. Kolyagin U. M. Zadachi v obuchenii matematike (Problems in teaching mathematics), M: Prosveschenie, 1977, Ch. 1,113 p.
21. Mordkovich A. G. Besedy s uchitelyami matematiki: ucheb.-metod. Posobiye (Conversations with teachers of mathematics: textbookmethod. allowance), M.: Oniks, 2007, 334 p.
22. Sarantsev G. I. Uprazhneniya v obuchenii matematike (Exercises in teaching mathematics), M., 2005, 254 p.
23. Neshkov K. I., Semushin A. D. Funkcii zadach v obuchenii (Task functions in training), Matematika v shkole, 1971. no 3, pp. 4-7.
24. Popov N. I., Yakovleva E. V. Aktual’nyye problemy obucheniya matematike inostrannykh studentov v vuze (Topical issues of teaching mathematics to international students at a university), Vestnik Moskovskogo gosudarstvennogo oblastnogo universiteta, Series: Pedagogika, 2019, no 3, pp. 144-153.
25. Marasanov A.N. Sistema zadach po trigonometrii v obuchenii matematike uchaschihsya srednih obscheobrazovatelnih uchrejdenii (System of problems in trigonometry in teaching mathematics to students of secondary educational institutions): diss. . . . kand. ped. nauk., Saransk, 2012, 180 p.

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.

I Kalinin S. I., Leonteva N. V. .(1/2; 1)-convex function. Part 2.

DOI: 10.34130/1992-2752_2020_3_04

Kalinin Sergey — Doctor of Education, Ph.D. in Physics and Mathematics, Professor, Department of Fundamental Mathematics, Vyatka State University, e-mail: kalinin_gu@mail.ru

Leonteva Natalia — Ph.D., Associate Professor of the Department of mathematics and computer science, Glazovsky state pedagogical Institute named after V. G. Korolenko, e-mail: leonteva-natalia-0812@yandex.ru

Text

This article studies the (1/2; 1)-convex functions properties. Especially the paper describes that within the (1/2; 1)-convexity interspace this functions are continuous. Classical Hermite-Hadamard inequality analogue for the convex and concave functions on the segment are introduced. Besides
for discussed functions Jensen’s inequality and his analogue are proved.

Keywords: convex functions, concave functions, Hermite-Hadamard inequality, Jensen’s inequality.

References

1. Kalinin S. I., Leontieva N. V. (1/2; 1)-vypuklyve funktsii ( (1/2; 1) convex functions. Part I)), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, > 1 (26), pp. 97-104.
2. Vinogradov O. L. Matematicheskiy analiz: uchebnik (Mathematical analysis: textbook), SPb.: BHV-Petersburg, 2017, 752 p.
3. Kalinin S. I., Pankratova L. V. Neravenstva Ermita – Adamara: obrazovatel’no-istoricheskiv aspekt (Hermite – Hadamard Inequalities: educational and historical aspect), Mathematical education, 2018, № 3 (87), pp. 17-31.
4. Abramovich S., KlariCic Bakula M., Matic M., PeCaric J. A variant of Jensen-Steffensen’s inequality and quasi-arithmetic means, J. Math. Anal. Applies., 307 (2005), pp. 370-385.

For citation: Kalinin S. I., Leonteva N. V. (1/2; 1)-convex function. Part 2., Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 3 (36), pp. 4-23.

II Komarov I. A., Makarov Р. A., Ustyugov V. A. On the free mechanical vibrations in a drv-friction system

DOI: 10.34130/1992-2752_2020_3_24

Komarov Ilja — Student, Pitirim Sorokin Syktyvkar State University, e-mail: mkrvpa@gmail.com

Makarov Pavel — Ph.D., Assistant Professor of Radio Physics and Electronics, Pitirim Sorokin Syktyvkar State University, e-mail: mkrvpa@gmail.com

Ustyugov Vladimir — Ph.D. in Physics and Mathematics, Department of Radiophysics and Electronics, Pitirim Sorokin Syktyvkar State University, e-mail: ustyugovva@gmail.com

Text

The basic model of free oscillations with dry friction is developed. The classification of free oscillatory systems is performed. The inhomogeneous Lagrange equations of the second kind was solves in the case of a homogenous, static, low-velocity system. The conditions under which the
system is stationary, and also accomplish «stable» and aperiodic oscillations was determined.

Keywords: free oscillations, dry friction, Amonton—Coulomb’s law.

References

1. Jellett J. H. Traktat po teorii treniya (A treatise on the theory of friction), M. – Izhevsk: NIC «Regular and chaotic dynamics», 2009, 264 p.
2. Rosenblat G. M. Sukhoye treniye i odnostoronniye svyazi v mekhanike tvordogo tela (Dry friction and one-sided connections in solid mechanics), M.: URSS, 2010, 205 p.
3. Andronov V. V., Zhuravlev V. F. Sukhoye treniye v zakonakh mekhaniki (Dry friction in the laws of mechanics), M. -Izhevsk: NIC «Regular and chaotic dynamics», 2010, 184 p.
4. Zhuravlev V. F. К istorii zakona sukhogo treniya (To the history of the law of dry friction), Solid mechanics, 2013, > 4, pp. 13-19.
5. Kozlov V. V. Lagranzheva mekhanika i sukhoye treniye (Lagrangian mechanics and dry friction), Nonlinear dynamics, 2010, V. 6, > 4, pp. 855-868.
6. Zhuravlev V. F. Otklik na rabotu V. V. Kozlova «Lagranzheva mekhanika i sukhoye treniye» (Response to the V. V. Kozlov work «Lagrangian mechanics and dry friction»), Nonlinear dynamics, 2011, V. 7, № 1, pp. 147-149.
7. Alekseev A. E. Nelinevnyve zakonv sukhogo treniya v kontaktnvkh zadachakh linevnov teorii uprugosti (Nonlinear laws of dry friction in contact problems of the linear theory of elasticity), Appl. mechanics and tech, phys., 2002, V. 43, > 4, pp. 161-169.
8. Bronovets М. A. et al. Eksperimental’naya ustanovka diva izucheniva treniya i iznashivaniya s imitatsiyey faktorov otkrytogo kosmosa (Experimental simulator of outer-space conditions for the study of friction and wear), Friction and wear, 2009, V. 30, > 6, pp. 529-532.
9. Alexandrov V. M., Bronovets M. A., Soldatenkov I. A. Matematicheskoye modelirovanive iznashivaniya podshipnika skol’zheniva v uslovivakh otkrytogo kosmosa (Mathematical modeling of sliding bearing wear in open space), Friction and wear, 2008, V. 29,№ 3, pp. 238-245.
10. Bronovets M. A., Zhuravlev V. F. Ob avtokolebanivakh v sistemakh izmereniva sil treniya (On self-excited vibrations in friction force measurement systems), Solid mechanics, 2012, > 3, pp. 3-11.
11. Akulenko L. D. et al. Kvazioptimal’nove upravlenive povorotom
    tverdogo tela vokrug nepodvizhnov osi s uchetom treniya (Quasioptimal control of the rotation of a rigid body around a fixed axis, taking into account friction), Izv. RAS. Theory and control systems, 2015, № 3, pp. 3-20.
12. Sviridenok A. I., Mechikov V. V. Trenive skol’zheniva polimernvkh kompozitov v uslovivakh vvsokikh skorostev (Sliding friction of polymer composites at high speeds), Friction and wear, 2005, V. 26, № 1, pp. 38-42.
13. Kolubaev A. V. et al. Generatsiya zvuka pri trenii skol’zheniva (Sound Generation During Slide Friction), JTF Lett., 2005, V. 31,
14. Chernousko F. L., Bolotnik N. N. Mobil’nyye robotv, upravlvayemyve dvizhenivem vnutrennikh tel (Mobile robots controlled by the motion of internal bodies), Tr. IMM URO RAS, 2010, V. 16, # 5, pp. 213-222.
15. Bolotnik N. N., Nunuparov A. M., Chashchukhin V. G. Kapsul’nvv vibratsionnvy robot s elektromagnitnvm privodom i vozvratnov pruzhinoy: dinamika i upravlenive dvizhenivem (Capsuletype vibration-driven robot with an electromagnetic actuator and an opposing spring: Dynamics and control of motion), Izv. RAS. Theory and control systems, 2016, .V”6. pp. 146-160. О свободных механических колебаниях 51
16. Deng Z. et al. Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale, Nature Mater, 2012, V. 11, pp. 1032-1037.
17. Panovko Y. G. Vvedeniye v teoriyu mekhanicheskikh kolebaniy (Introduction to the theory of mechanical oscillations), M.: Nauka, 1991, 256 p.
18. Magnus K. Kolebaniya: Vvedeniye v issledovaniye kolebatel’nykh sistem (Oscillations: An Introduction to the Study of Oscillating Systems), M.: Mir, 1982, 304 p.
19. Lojcjanskij L. G., Lurye A. I. Kurs teoreticheskoy mekhaniki. T. II. Dinamika (Theoretical Mechanics course. T. II. Dinamics), M.: Nauka, 1983, 640 p.
20. Makarov P. A. О variatsionnvkh printsipakh mekhaniki konservativnvkh i nekonservativnvkh sistem (On variational principles of mechanics applied to the motion of conservative and non-conservative systems), Vestnik of Syktyvkar State University. Ser. 1: Mathematics. Mechanics. Informatics, 2017, Rel. 2 (23), pp. 46-59.

For citation: Komarov I. A., Makarov P. A., Ustyugov V. A. On the free mechanical vibrations in a drv-friction system, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 3 (36), pp. 24-51.

III Suleimanova S.Sh. Dissipation of the energy of an alternating electric field in the half-space of an electron plasma with diffuse boundary conditions

DOI: 10.34130/1992-2752_2020_3_52

Suleymanova Sevda Shirin kyzy — Postgraduate Student, Bauman Moscow state Technical University (national research University), Moscow Polytechnic University, e-mail: sevda-s@yandex.ru

Text

The magnitude of the absorption of the energy of the electromagnetic field in the half-space of the electron plasma is calculated. The case with an arbitrary degree of degeneracy of the electron gas is considered. To determine the absorption, a solution is used of the boundary-value problem
of the behavior (oscillations) of an electron plasma in a half-space with mirror boundary conditions for electrons. The Vlasov — Boltzmann kinetic equation with the collision integral of the BGK type (Bhatnagar, Gross, Kruk) and the Poisson equation for the electric field are applied. The
electron distribution function and the electric field inside the plasma are obtained in the form of expansions in eigen-solutions of the original system of equations. The coefficients of these expansions are found for the case of diffuse boundary conditions. The contribution of the surface to absorption is analyzed. Cases of various degrees of degeneracy of the electron gas are
considered. It is shown that the ratio of the frequency of changes in the electric field and the frequency of bulk electron collisions has a significant effect on the absorption of energy of the electric field near the surface.

Keywords: Vlasov-Boltzmann equation, collision frequency, electric field, Drude, Debye, van Campen modes, dispersion function.

References

1. Keller O. Local fields in the electrodynamics of mesoscopic media, Physics Reports, 1996, Vol. 268, pp. 85-262.
2. Girard C., Joachim C. and Gauthier S. The physics of the nearfield, Rep. Prog. Phys., 2000, Vol. 63, pp. 893—938.
3. Pitarke J. M., Silkin V. M., Chulkov E. V. and Echenique P. M. Theory of surface plasmons and surface-plasmon polaritons, Rep. Prog. Phys., 2007, Vol. 70, pp. 1—87.
4. Bozhevolnyi S. I. Plasmonics Nanoguides and Circuits, Singapore: Pan Stanford Publishing, 2008, 452 p.
5. Latyshev A. V., Suleimanova S. Sh. Analiticheskoye reshenive zadachi о kolebaniyakh plazmy v poluprostranstve s diffuznymi granichnvmi uslovivami (Analytical solution of the problem of plasma oscillations in a half-space with diffuse boundary conditions), Zh. vych. Matem. and math, physics, 2018, Vol. 58, No. 9, pp. 1562-1580.
6. Suleimanova S. Sh., Yushkanov A. A. Dissipatsiva energii peremennogo elektricheskogo polva v poluprostranstve elektronnov plazmy s zerkal’nvmi granichnvmi uslovivami (Dissipation of the energy of an alternating electric held in a half-space electron plasma with mirror boundary conditions), Plasma physics, 2018, Vol. 44, No. 10, pp. 820-831.
7. Lifshits E. M., Pitaevsky L. P. Fizicheskava kinetika (Physical kinetics), M.: Nauka, 1979. 527 p.

For citation: Suleimanova S.Sh. Dissipation of the energy of an alternating electric held in the half-space of an electron plasma with diffuse boundary conditions, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 3 (36), pp. 52-63.

IV Odyniec W. Р. About some mathematicians from the Polytechnic Institute in Leningrad perished in 1941-1943

DOI: 10.34130/1992-2752_2020_3_64

Odyniec Vladimir — Doctor of Physical and Mathematical Sciences, Professor, Pitirim Sorokin Syktyvkar State University, e-mail: W.P.Odyniec@mail.ru

Text

The life and work of several mathematicians from the Polytechnic Institute in prewar Leningrad are described. All of them (N. A. Rosenson, T. N. Blinchikov, A. C. Nvrkova, M. S. Eleckv, V. I. Nikonov, M. A. Gelbcke, N. N. Gernet) perished in 1941-1943.

Keywords: N. A. Rosenson, T. N. Blinchikov, A. G. Nyrkova, M. S. Elecky, V. I. Nikonov, M. A. Gelbcke, N. N. Gernet, Riemanni spaces of the 1st class, Warinng problem, asymptotics of iterated functions, Szasz problem, fractional parts of a function of two variables, calculus of variations, Lagrange series.

References

1. Matematika v SSSR za sorok let. Biobibliografiya, T. 2 (The URSS Mathematics for forty years:1917-1957, Biobibliography, Vol. 2), M.: Fizmatlit, 1959, 819 p.
2. Sinkevich G. I. Nikolai Maksimovich Gunter (1871-1941) (Nicholas Maximovich Giinther (1871-1941)), Matematika v vysshem obrazovanii, 17 (2019), pp. 123-146.
3. Nauchnye rabotniki Leningrada (Members of staff of scientific for
   Leningrad), L.: Izd-vo AN SSSR, 1934, 723 p.
4. Alersandrov A. D. Geometriva v Leningradskom universitete (The Geometry at Leningrad University), Vestnik Leningradskogo universiteta, № 11, 1947, pp. 124-148.
5. Odyniec W. P. О leningradskih matematikah, pogibshih v 1941-1944 godah (On some Leningrad based Mathematician perished in 1941-1944), Syktyvkar: Izd-vo SGU im. Pitirima Sorokina, 2020, 122 p.
6. Dissertacii, zashtchishtchennye v Leningradskom ordena Lenina gosudarstvennom universitete im. A. A. Zhdanova v 1934~1954 gg (Bibliograficheskii ukazatel) ( Dissertations defended at the Leningrad State University named after A.A. Zhdanov decorated with to Order of long 1934-1954, (Bibliographical indicator), L.: Izdatelstvo Leningradskogo uni-ta, 1955.
7. Rosenson N. A. Differencialnye invarianty Rimanova prostranstva (Differential invariants of a Riemann spaces), L.: Trudy LIL, Razdel Fiz.-mat., T. 10, > 3, 1936, pp. 57-75.
8. Rosenson N. A. Differencialnye invarianty Rimanova prostranstva (Differential invariants of a Riemann spaces. Part II), L.: Trudy LIL, Razdel Fiz.-mat., T. 4, № 2, 1937, pp. 59-84.
9. Rosenson N. A. Nekotorye neravenstva iz teorii kvadratichnyh form (Some inequality from the theory quadratic forms), L.: Trudy LIL, Razdel Fiz.-mat., T. 4, № 2, 1937, pp. 85-93.
10. Trudy seminara ро vektornomu i tenzornomu analizu s ih prilozheniyami k geometrii, mehanike Ifizike, 6 (Proceedings of the seminar on vector and tensor analysis with its applications to geometry, mechanics and physics. 6), M.: OGIZ, Gos. izd-vo tehn.-teor,lit-rv, 1948, 515 p.
11. Rosenson N. A. О Rimanovvh prostranstvah klassa 1 (Upon Riemann Spaces of the Class 1), Izvestiya AN SSSR, Ser. matem, 4, 1940, pp. 181-192.
12. Rosenson N. A. О Rimanovvh prostranstvah klassa 1. Chast’ II (Upon Riemann Spaces of the Class 1. Part II), Izvestiya AN SSSR, Ser. matem. 5, 1941, pp. 325-351.
13. Rosenson N. A. О Rimanovvh prostranstvah klassa 1. Chast’ III (Upon Riemann Spaces of the Class 1. Part III), Izvestiya AN SSSR, Ser. matem, 7, 1943, pp. 253-284.
14. Kniga pamyati, Leningrad 1941-1945. Frunzenskiy rayon, T. 13 (The book of Memory, Leningrad 1941-1945, the Frunze district. Vol. 13), SPb.: Notabene, 1997. 515 p.
15. Kagan V. F. Nina Arkad’evna Rosenson (1909-1942) (nekrolog), (Nina Arkadyevna Rosenson (1909-1942) (Obituary), Izvestiya AN SSSR, Ser. Matem., T. 7., vvp. 6, 1943, pp. 251-252.
16. Blokada 1941-1944- Kniga pamyati, Leningrad, T. 25, P-R. (Prokof’evRessovskava) (Blockade 1941-1944. The book of Memory, Leningrad, Vol. 25), SPb.: Stella, 2005, 714 p.
17. Dvadcat’ let inzhenerno-fizicheskogo fakulteta LII (The twenty years of the department of engineering and physics of Leningrad Industrial Institute), L.: Izdatelstvo LII, 1939, 68 p.
18. Kniga pamyati (The book of Memory), Sostaviteli: S. A. Sirotkina, E. F. Tarasov, SPb.: Izd-vo SPbGTU, 2000, 90 p.
19. Nyrkova A. G. О polozhitelhyh trigonometricheskih summah (On positive trigonometrical sums), L.: Trudy LIL, Razdel Fiz.-mat. 3:1,1939, pp. 5-10.
20. Nyrkova A. G. Zadacha Szasz’a (Szasz Problem), L.: Trudy Politehnicheskogo instituta, 3, 1941, pp. 50-59.
21. Nikonov V. I. Asimptoticheskie vyrazheniya iterirovannyh funkcyi (Asymptotic expressions to Iterated functions), L.: Trudy LIL Razdel Fiz.-mat, 5:1, 1938, pp. 33-56.
22. Nikonov V. I. Integralnye predstavlenie nekotoryh trigonometricheskih polinomov kak sposob ih izucheniya (The integral representation of some trigonometric polynomials is a method of their study), L.: Trudy LII., Razdel Fiz.-mat., 3:1, 1939, pp. 11-15.
23. Blokada 1941-1944- Kniga pamyati, Leningrad, T. 22, N-P (NikolaevaPavlova) (Blockade 1941-1944. The book of Memory, Leningrad, Vol. 22), SPb.: Stella, 2005, 716 p.
24. Gelbcke M. A. Ob asimptoticheskom vyrazhenii summv drobnvh chastei funkcii dvuh peremennvh (On an asymptotic expression of the sum of fractional parts of a two-variable function), Zhurnal Leninqradskoqo Fiziko-matematicheskoqo obshtchestva, T. 1, Vyp. 2,
    1927, pp. 281-298.
25. Gelbcke M. A. Otnositelno g(k) v problemie Varinga (Relating g(k) to Waring problem), Izvestiya AN SSSR, VII seriva, Otd. matematicheskih I estestvennvh nauk, 1933, vyp. 5, pp. 631-640.
26. Trudy 2-go Vsesouznogo s’ezda matematikov (Leningrad, 24-30 iunya 1934), T. 1 Plenarnye i obzornye doklady (Proceedings of the 2nd All Union Congress of Mathematicians. Leningrad, June 24-30, 1934, Vol.l , Plenary and survey reports), L.-M.: Izd-vo AN SSSR, 1935, 469 p.
27. Vachromeeva О. B. Professor matematiki Nadezhda Nikolaevna Gernet (1877-1943), (Professor of Mathematics Nadezhda Nikolaevna Gernet (1877-1943)), Vestnik Nizhegorodskogo gosudarstvennogo universiteta, № 4 (116), 2019, pp. 105-109.

For citation: Odvniec W. P. About some mathematicians from the Polytechnic Institute in Leningrad perished in 1941-1943, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 3 (36), pp. 64-86.

V Yermolenko А. V., Ladanova S. V. Contact problem for two plates with different fixing

DOI: 10.34130/1992-2752_2020_3_87

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

Ladanova Svetlana — Student, Pitirim Sorokin Syktyvkar State University, e-mail: ea74@list.ru

Text

An analytical solution for two plates is given using the classical theory of bending of flat plates. In this case, one plate is hinged, the second has a rigid fastening. It is shown that when using the Sophie Germain-Lagrange equation, contact reactions contain concentrated forces.

Keywords: plate, contact problem, Sophie Germain-Lagrange equation, analytical solution.

References

1. Mikhailovskii E. I. Shkola mekhaniki obolochek akademika Novozhilova (School of Shell Mechanics Academician Novozhilov), Syktyvkar: Izd-vo Svkt. un-ta, 2005, 172 p.92 Ермоленко А. В., Ладанова С. В.
2. Mikhailovskii Е. I., Tarasov V. N. О sxodimosti metoda obobshhennoj reakcii v kontaktny’x zadachax so svobodnoj granicej (On the convergence of the generalized reaction method in contact problems with a free boundary), Journal ofApplied Mathematics and Mechanics, 1993, v. 57, № 1, pp. 128-136.
3. Yermolenko A. V., Mikhailovskii E. I. Granichnye uslovija dlja podkreplennogo kraja v teorii izgiba ploskih plastin Karmana (Boundary conditions for the reinforced edge in the Karman theory of bending of flat plates), MTT. 1998, > 3, pp. 73-85.
4. Mikhailovskii E. I., Badokin К. V., Yermolenko A. V. Teorija izgiba plastin tipa Karmana bez gipotez Kirhgofa (The theory of bending of Karman-tvpe plates without the Kirchhoff’s hypotheses), Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 1999, 3, pp. 181-202.
5. Mikhailovskii E. I., Yermolenko A. V., Mironov V. V., Tulubenskaya Ye. V. Utochnennyye nelineynyye uravneniya v neklassicheskikh zadachakh mekhaniki obolochek : Uchebnoye posobiye (Refined nonlinear equations in nonclassical problems of shell mechanics: Textbook), Syktyvkar: Izd-vo Syktvvkarskogo un-ta, 2009, 141 p.
6. Mikhailovskii E. I., Toropov A. V. Matematicheskiye modeli teorii uprugosti (Mathematical models of the theory of elasticity), Syktyvkar: Svkt Publishing House. University, 1995, 251 p.
7. Grigolyuk E. I., Tolkachev V. M. Kontaktnyye zadachi teorii plastin i obolochek (Contact problems in the theory of plates and shells), M.: Mashinostroyeniye, 1980, 411 p.

For citation: Yermolenko А. V., Ladanova S. V. Contact problem for two plates with different fixing, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2020, 3 (36), pp. 87-92.

I. Antonova N.A. Dynamics in pulse-frequency-modulated control systems

Text

Conditions are obtained for existence of mT-periodic modes (m = 1, 2, 3) in one dimensional control systems employing pulse-frequency modulation of the first and second kinds.

II. Bazhenov I.I. The property of nonatomicity for set families and vector measures

Text

We introduce the new concept of atom for family of subsets of some set. This notion coincides with the notion of atom of vector measure if the family in question contains only sets of zero measure. Sufficient conditions of nonatomicity of family of sets are given for one special case. We also establish sufficient conditions of nonatomicity of the vector measure n(E) = φ(m(E)) constructed by means of linear and continuous operator φ and vector measure m with values in topological vector space.

III. Veksler A.I., Koldunov A.V. On the normed lattice and its normed completion

Text

Here (negative) answers on two problems conserning the relations between properties of a normed lattice and its normed completion are presented.

IV. Vorobyova E.V. Some ergodic properties of the homogeneous Markov chain with the continuous parameter

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We prove for any Markov chain with finite space of states that the final probabilities vector is orthogonal to the columns of generator. And in the case of any discrete space of states we find an explicit formula for final probabilities in terms of the generator’s resolvent.

V. Zvonilov V.I. Rigid Isotopies of Trinomial Curves with the Maximal Number of Ovals

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Let l be the number of ovals of nonsingular real trinomial curve уⁿ + b(x)у + w(x) = 0. In this paper the sharp upper bound for l is found for any n. The rigid isotopy is understood as a path in the space of nonsingular real trinomial curves with n fixed. The rigid isotopy classification of such curves with the maximal l is given. In particular case n=3 the rigid isotopy classification of trigonal M-curves is obtained.

VI. Golovach P.A. Distance constrained labelings of trees

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An assignment of nonnegative integers to the vertices of a graph G is L(p₁, p₂, … , p_k) – labeling (or coloring) if for every two vertices at distance at most i <= k, the difference of the integers (labels) assigned to them is at least p_i. An interest to such labelings is motivated by their usage in models of telecommunication networks. We prove that the existence problem of L(p, 1, 1) – labeling with labels, that are at most λ, is NP-complete for trees.

VII. Yermolenko A.V. The calculation of round plates with refined theories

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The Karman-Timoshenko-Naghdi type equations are used for the calculation of round plates with rigid boundary. The problem is solved not using additional condition about transversal shears on boudary.

VIII. Zheludev V.A., Pevnyi А.В. Discrete periodic frames

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We construct the filter bank of perfect reconstruction for the discrete N-periodic signals. This bank generates the wavelet tight frames in the spaces C^N and R^N.

IX. Malozemov V.N., Pevnyi А.В., Selyaninova N.A. Primal lifting scheme

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We present the analysis of the primal lifting scheme for the constructing of the wavelet decomposition of the discrete periodic signals. A description of the set of all control functions β_ν(k) is given.

X. Mironov V.V. The account of transversal shears in a problem about a bend of the cylindrical panel

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In this paper the task about a bend of cylindrical panel under effect of normal load is considiered. The normal load are distributed on field, simular to middle surface of a panel. The bend of panel on register of transversal shears by S.P. Timoshenko’s model is described. The mechanism of dependence of a momemts for transversal shears is confirmed — the graphics of moments for change of curvature of middle surface and for change of transversal shears are be in anti-phases in fields of a maximal absolute values.

XI. Mikhailovskii E.I. The classical linear theory of shells

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The equations and the boundary values of the modern classical linear theory are consecutively obtained. On equations’ deduction the Novozhilov-Finkel’shtein criterion was used to estimate the Kirchhoff-Love hypotheses. The final variant of shells’ theory includes the deformation boundary values, which were obtained by K.F. Chernykh for one-related middle surface and were generalized by the author of article for multirelated middle surface. It is shown that the compatibility conditions can be obtained directly from the equilibrium equations of shells’ theory; initially condition was obtained by A.L. Gol’denweiser from the Gauss-Petersson-Kodacci equations for deformed middle surface. The author’s operation form is used for recording the general equations of the linear shells’ theory.

XII. Nikitenkov V.L. On the integer-valued solving of the linear cutting problem

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It was proved that optimal value of the target function for the integer-valued problem of linear cutting just slightly differs from the corresponding value for the linear cutting problem. On this basis it was offered the effective complex algorithm of integer-valued problem solution.

XIII. Ezovskih V.E. Color sampling algorithms

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Three algorithms of color sampling suitable for color conversion are considered. Some practical features of realization are briefly discussed.

XIV. Kotyrlo E.S. Methods of labor market demand prediction in professional skills structure

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The problem of labor demand prediction in professional skills structure is one of the main problems that impact on the labor market equilibrium and human capital efficiency. In this article prediction methods and their efficiency to using in labor market are analyzed; an analytic model of labor demand prediction in professional skills structure is constructed; a version of statistical survey execution is suggested.

XV. Mironov V.V., Kuznetsova N.V. The task about axis-symmetrical eigen-oscillations of round rigidly fixed plates

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In this paper the task about axis-symmetrical eigen-oscillation of round rigidly fixed plates is considiered. The analytical solution is given.

XVI. Nikitenkov V.L., Sakovnich D.J. Realization of complex algorithm for integer-valued problem of linear cutting

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Complex method for solving of integer-valued problem of linear cutting are considered. The results of testing on enterprize are describe.

XVII. Nikitenkov V.L., Yasinsky V.I. Web-services of complex algorithm for integer-valued problem of linear cutting

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The problems of working over a network and the Internet, the superiority of Web-services over the other server-applications, a short description of the applied Web-service and the examples of Web-service query are discussed.

XVIII. Pevnyi А.В., Istomina M.N. Mercedes-Benz frame in n-dimensional space

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We construct an equal-norm tight frame in Rn consisting of n+1 vectors. The angles between any different vectors are equal π/2 + arcsin(1/n).

XIX. Poroshkin A.G., Gabova M.N., Grelya E.N. On the Arzela – Borel theorem.

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Let X be a topological space, (Y, V) — uniform space and (f_α) be a sequence or directedness of functions f_α : X —> Y. In this paper autors prove the generalization of Arzela — Borel theorem: when f_α ϵ C(X) ∀α and f_α(x) —> f(x) ∀x ϵ X, then f ϵ C(X) if and only if f_α converges to f on X quasi-uniformly.

XX. Tarasov V.N., Pavlova L.A. The proof of geometric theorems by means of computer algebra

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Some geometric theorems can be stated in coordinate form as polynomials in algebra and can be proved by algorithmic methods. In article the theorems of Pascal and Pappe Alexandrinian are prooved by means of computer algebra. Also some properties of Torricellian point for arbitrary tetrahedron are stated.

I. Vechtomov E.M., Starostina O.V. Generalized Abelian-regular positive semirings

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Notions of generalized Abelian-regular positive semirings (arp-semirings) and generalized Boolean semirings are introduced. It is showed that the research of generalized arp-semirings reduces to arp-semirings. Functional representation of generalized arp-semirings and generalized Boolean semirings are obtained. Dauns-Hofmann’s theorem about representation of biregular rings is extended on biregular semirings.

II. Lovyagin Yuri N. Hyperrational numbers as the basis of analysis

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Theory of hyperrational numbers as the basis of analysis is considered. The main aspects of the theory are adduced. Analogs of the classical differential and integral calculus of one-variable function theorems are stated.

III. Odyniec W.P., Prophet M.P. On three forgotten results of S.Krein, N.Bogolyubov and V.Gurari with applications to bernstein operators

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The results of M. Frechet in 1934 about the largesteigenvalue of a stochastic matrix [6] attracted attentionto positive linear operators with norm 1. The study of compact linearoperators with stochastic kernel by N.M. Krylov and N.N.Bogolyubov during the 1930’s ([9], [10]) was generalized by S. Krein and N.N. Bogolyubov a decade later in [2]. These results contributed to the 1968 paper [8] of M. Krasnoselski in which the problem of determining minimal-normshape-preserving projections was present. Unfortunately, many of these papers are practically unknown, as they were published in the Ukrainian language. On the other hand, recent developments in the theory of minimal shape-preserving projections have been made using methodsthat are independent of Krasnoselski’s work (see [3] and [11]). In this paper, we attempt to connect these two directions by studying the (so called) Bernstein operators.

IV. Prazdnikova El. Wl. Modelling the real analysis in the framework of axiomatic of hypernatural numbers

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In the paper formalized theory of Non-Standard theory of numbers and formalized theory of hyperrational numbers are stated. The main theorems of classical differential calculus are modelled.

V. Savelev L.J., Ogorodnikov V.A., Sereseva O.V. Stochastic model of piecewise linear process

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The stochastic process with piecewise linear trajectories is considered. The process is based on models of stochastic walk on a straight line. The distributions of stochastic variables, forming this process, and, in particular, distribution of relative time expectations for Poisson flow of points are investigated. The appropriate mathematical expressions for these distributions, and also expressions for mean of process as time-varying function are received.

VI. Mikhailovskii E.I. A non-linear theory of flexible shells Zhuravsky-type

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A non-linear theory of flexible shells has been built, taking into account transversal shears by D. I. Zhuravsky model, By introducing generalized forces and moments the equations are reduced to the kind formally coinciding whith equations in the theory of shells, based on Kirchhoff’s hypotheses. This allows to formulate an effective algorithm of accounting transversal deformations in some of Kirchhoff’s variations of the theory of shells and planes, The algorithm is illustrated by more exact specification of K. Margyerre’s non-linear theory of shallow shells.

VII. Duriagin A.M. Real harmonic frames, toughness and redundancy

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It was proved that real harmonic frames possess maximal redundancy, i.e. if any m-n vectors are deleted then remaining n vectors form frame in Rⁿ (in the general case, it is not tight frame). The fast frame expansion algorithm is offered.

VIII. Kholmogorov D.V., Tarasov V.N. Influence of boundary conditions on the stability of a cylindrical shell

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In the work the problem of stability of the cylindrical shell, loaded with external normal pressure or undergoing axial compression, is examined. With a numerical study the tasks of displacement are approximated by splines. In contrast to the traditional trigonometric series the application of splines makes it possible to consider the real boundary condition, in particular, the displacement of the ends of shell as rigid whole is allowed.

IX. Mikhailovskii E.I., Mironov V.V., Kuznetsova J.L. About one solution algorithm for solving of nonlinear boundary problem Karman-type

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On the basis of Green’s well-known function, an algorithm of edge problems with Karman-type non-linearity to the corresponding system of algebraic equations. The algorithm is realized on tne examples of simply supported opened cylindrical shell.

X. Pevnyi А.В., Istomina M.N. Recovery of the signal in the case when one frame coefficient is erased

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The authors consider Mercedes-Benz frame and frame expansions for every vector x ϵ Rⁿ. In the case when one frame coefficient is erased the authors suggest a fast algorithm for the recovery of the vector x.

XI. Malozemov V.N., Pevnyi A.B. Mercedes-Benz systems and tight frames

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The paper is for the section “Easy Reading for Professionals”. The authors study the tight frames in the space in Rⁿ. A complete description of the tight frames in Rⁿ consisting of n+1 vectors is given. An exact lover bound for the frame potential is proved.

I. Balakin S.V. Runs characteristics in a ternary Marcov Chain

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A ternary Markov chain is considered. There are functionals concerned with a number of events and a number of runs. Probability generating functions for joint distributions of this functionals are obtained. Precise formulas for expectations, variances and covariations are founded.

II. Vechtomov E.M., Chuprakov D.V. Congruences on semirings of continious functions and F-spaces

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The congruences of semirings of non-negative continuous functions on topological space are investigated. In terms such congruences new algebraic characterizations of F-spaces and P-spaces are received.

III. Mekler A.A. On positive integer characteriyation of regular variyng quasi-concave modulars

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The initiated in [1] study of the correspondence between quasi-concave modulars and sequences of positive integers is continued. In the presented paper are examenated especially regularly variying quasi-concave modulars. It is stated in Theorems 9, that a sequence of positive integers corresponds to a fast variyng quasi-concave modular (that means the case of regularity index α=0) iff it is equivalent to a integer sequence which tends to +∞. In Theorem 8 is proved that the sequence of positive integers corresponds to a slow variyng modular (the case of regularity index α=1) iff it is equivalent to a sequence of the form 211•••1211•••121••• •••••• where the lengths of blocks of units tends to +∞. In Theorem 5 the case of intermediate value of regularity index α: 0<α<1 is investigated.

IV. Savelev L.J. Extension of a measure up to integral

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In this paper the task about a bend of cylindrical panel under effect of normal load is considiered. The normal load are distributed on field, simular to middle surface of a panel. The bend of panel on register of transversal shears by S.P.Timoshenko’s model is described. The mechanism of dependence of a moments for transversal shears is confirmed – the graphics of moments for change of curvature of middle surface and for change of transversal shears are be in anti-phases in fields of a maximal absolute values.

V. Belyaeva N.A. The structural models of deformation processes

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Mathematical models of flow and deformation processes of materials with evolutionarizating structure are presented. The problems spread to wide sphere of structure-sensitive objects – from powder systems to polymer materials and composites. This models allow to define change of deformation, temperature and structure characteristics of varried systems in the various processing-hardening, non-Newtonian flow, solid-state extrusion.

VI. Nikitenkov V.L. Basic property of the optimal integer-valued solving of the linear cutting problem

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It was proved that basic solution of the some modification linear cutting problem is optimal integer-valued solving of the integer cutting problem.

VII. Sakovnich D.Y. Singularity in the format cutting problem

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In scope of format cutting problem the problem of minimization winder adjustments is considered. It was offered the method for getting the most singular solution if the problem.

VIII. Kluchnikov E.A. SOA data store and search system

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The author consider SOA data store and search system written in Java. SOA approach makes such systems scalable and easy to integrate as basis of enterprise infrastructure. Java technology makes it easier to use the full power of modern servers.

IX. Simakov A.V. Parallel compression of huge images

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In this paper we describe an efficient algorithm for parallel compression of huge images. The algorithm is based on wavelet transform. The article is focused on practical implementation and evaluation of this compression method. As a result, all described features are implemented in software library written in C language. This library and compression program is freely available through the Internet and distributed with full source code under the terms of Open Source GNU GPL license.

X. Poroshkin A.G., Popova L.A. Remark about the outer measures geneated by measures

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It is proved that outer measures generated by measures in sense of Halmos, Vuklikh, Poroshkin are lower continuous, however the outer measure of Lebesque in R is not continuous from above and not exhaustive.

XI. Tarasov V.N., Andryukova V.Yu. On stability and supercritical behavior of a spherical shell

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The problem of the spherical shell experiencing external normal pressure is considered. A precise formula is used for the calculation of the work of external forces. In the work a variational approach is applied. Cubic splines are used for the finite-dimensional approximation of displacements. The influence of nonlinear terms on the amount of the critical force is investigated.

XII. Tulubenskaya E.V., Kargin R.V. The stability of the longitudinally compressed shank with unconstant rigidity at the border of two Winkler’s ambiences

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The stability of the longitudinally compressed shank with unconstant rigidity at the border of two Winkler’s ambiences is investigated. The algorithm is based on the local search variants.

XIII. Jakovlev V.D. On one way of introducing integrals in the high school mathematics

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I. Vechtomov E.M., Chuprakov D.V. Pseudo-complements in a lattice of congruence of semirings of continuous functions

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We investigate the properties of lattice of congruencies of semirings and semifields of continuous nonnegative (positive) functions, set on an arbitrary topological space X. It’s proved that these lattices are lattices with pseudocomplements and every its element has not more than one complement. In terms of lattices of congruencies of semirings of continuous functions over X, the characterizations of some properties of space X are obtained.

II. Eltsov N.P., Ogorodnikov V.A., Prigarin S.M. Analysis of cascade models of random fields

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The stochastic process with piecewise linear trajectories is considered. The process is based on models of stochastic walk on a sraight line. The distributions of stochastic variables forming this process, and, in particular, distribution of relative time expectations for Poisson flow of points are investigated. The appropriate mathematical expressions for these distributions, and also expressions for mean of process as time-varying function are received.

III. Lovyagin Yu.N., Prazdnikova E.Wl. Elementar functions on the set of complex hyperrational numbers

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The basic elementary functions are enterd within the limits of the model-theoretic approach to the concept of complex hyperrational numbers. Their basic properties are resulted.

IV. Odyniec W.P. One differential inequality and its application to the Pearson’s distribution densities

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We prove that for some large set of functions including the densities of Pearson distributions the condition 0<(f/f')'<1 is sufficient for differential inequality f''(t)*f(t)<(f'(t))² (Theorem 1). The application of Theorem 1 to the densities of Reason distributions is given.

V. Belyaeva N.A. Influence of the typical timeses on modes hardphase extrusion

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There are difined qualitative different extrusion regimes – stationary, quasistationary, transient modes – on base of the comparison of the typical times of the structuring, compactions and extrusion processes.

VI. Bayborodina O.V., Sakovnich D.Y. Application of the complex algorithm for solving of size cutting problem in order to increase the efficiency of business processes of core production

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The possibility to increase the efficiency of business processes of spiral winding cores production by implementing the complex algorithm for solving of integer- valued problem of linear cutting is investigated. The results of the comparison analysis of MSY used calculation and the complex algorithm on the actual data of MSY are given.

VII. Mikhailovskii E.I., Tulubenskaya E.V. The account of the transverse strain in the problem about stability of the cylindrical cover in the conditions constructive nonlinearity

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In work is considered the problem about stability of longitudinal compressed simply supported the cylindrical cover located at the border of two Winkler’s ambiences. The specified theory of Margera-Timoshenko type is used. The problem is solved by the means of the combined algorithm variant search.

VIII. Tarasov V.N. Modeling the cell membrane potential of the sinoatrial node

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The work of the heart associated with a regular contractions of muscle forming the heart wall. The heart rate caused by the spread of the electrical action potential (AP) on heart fibers. Most of these fibers can self-excite. This is in contrast to AP in the nerve cells. For self-excitation AP in neuronal cells requires a signal (current initialization). Sector of the heart, giving the excitation maximum value – it is sinoatrial node. In this paper a mathematical model of electrical activity of cells of the sinotrial node is proposed.

IX. Belyaeva N.A., Osipova V.V. The structural compaction model of viscoelastic composite material

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The structural model of solidphase plunger compaction of porous viscoelastic material is presented. Decisions of problems for case of the set speed and pressure on plunger are received. The program complex in the programming environment Delphi is developed. Results of numerical experiment are presented.

X. Mikhailovskii E.I., Gintner V.V., Mironov V.V. Auxiliary operator method for solving linear boundary-value problem

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Linear boundary-value problem with homogeneous boundary condition under certain Green function may be considered as solved. But analytical construction of Green function is very problematical even for a homogeneous linear boundary-value problem with variable coefficients. At the same time one can select (auxiliary) differential operator with the same order, as operator of considered problem, the one for which Green function of the original problem with the boundary condition is fairly simply computed. This article shows that in these cases the original boundary-value problem is reduced to a Fredholm integral equation of the first or second kind. For its solution, for example, method of mechanical quadrature can be used.

XI. Pevnyi A.B., Duriagin A.M. Spherical designs

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Definition of spherical t-design is made. The example of 4-design R³ is presented. The designs on a plane is in detail investigated, it is possible to name them designs.

XII. Pevnyi A.B., Istomina M.N., Maksimenko V.V. Complementary tight frames

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For any tight frame in the space Rⁿ consisting of m vectors, m>n, the authors construct a complimentary tight frame in the space of dimension n’=m-n.

XIII. Vechtomov E.M. About problems of high professional education in Russia

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We are analyzing the state of affairs and exploring the contradictions in our high professional education. Our high education needs a modernization, but it should rely on reality in Russia and the traditions of fundamental education, not following western examples. Real education, moral upbringing and true science should be priorities of our government and society.

I. Vechtomov E.M. Structure of semifields

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This work is an analytic scientific review over the theory of semifields, sponsored by grant RFBR № 08-01-11000-ано.

II. Efimov D.B., Kostyakov I.V., Kuratov V.V. On exact representations for the group motions of Galilean plane
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The Pimenov algebra with two generators D₂ is defined and some of its properties are shown. Some exact two- and three-dimensional matrix D₂-representations for the group motions of Galilean plane (the Galilean group) are considered. A geometric interpretation of them is giving. We consider also a exact representation of the Galilean group by elements of Grassmann algebra.

III. Kostyakov I.V., Kuratov V.V. Massive Yang-Mills fields, translation and nonsemisimple gauge symmetry

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Gauge fields of semisimple groups of internal symmetries are massless and require special techniques for their mass. Massive mechanisms usually contain translational transformations specific to nonsemisimple groups. We show that under the localization nonsemisimple internal symmetry gauge fields corresponding to generators of translation, are massive. In addition, we introduce nonlinear generalizations of well-known models, with local translational symmetry and as a result, the massive gauge fields. Thus, the local Galilean symmetry is realized on a special pair of scalar fields, leading to massive electrodynamics, and the localization of the Euclidean group leads to massive non-Abelian theory without matter fields. We propose a simple interpretation of the Stueckelberg mechanism.

IV. Tikhomirov A.N. On the circular law of random matrices

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In review the new results on proving the circular law for random matrices are given. Among them are the results obtained by the author together with F.Gotze for rared non-Hermitian matrices of large dimension.

V. Malozemov V.N., Solov’eva N.A. On the frame matrix

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The authors consider the following problem. Given positive-definite Hermitian matrix S and positive numbers a₁, … , a_m, m ≥ n, find a frame {φ₁, … , φ_m} in Cⁿ such that S is frame matrix and equations IIφ₁II = a₁, … , IIφ_mII = a_m hold. The authors give new proof of the theorem on necessary and sufficient conditions for existence of such frame.

VI. Podorov A.E., Sakovnich D.Y. Platform for testing methods of solving linear cutting problem

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Common ideas of building applications for solving cutting problems were developed. Using these ideas, platform for testing algorithms was built. This platform was applied for compare some algorithms.

VII. Belyaeva N.A., Nikonova N.N. Structural model of extrusion with usage of the generalised model of Newton

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The mathematical model of Solid-Phase is presented extrusion of a porous viscoelastic material with a condition of constancy of speed plunger the press. The results confirming rightfulness of replacement of the equation of movement on the equation of equilibrium are received. For the description of considered stream Lagrangian (mass) coordinates are used.

VIII. Belyaeva N.A., Spiridonov A.V. The structural models of deformation processes

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The mathematical model of tverdofaznoy extrusion of porous viscoelastic material is presented with the condition of constancy of speed of plunzhera the press. Got results, confirm legitimacy of replacement of equalization of motion on equalization of equilibrium in the works before executed on this subject. For a chosen type of flow specification lagranzh(mass) coordinates are used. 

IX. Vasilyev А.А., Koroleva А.N. Some applications of computational geometry to linear programming problems

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The work considers one of the approaches to the solution of linear-programming problem with two variables: the one of computational geometry and its generalization on the problem of the least covering circle. Numerical approbation of the given method is carried out, the algorithm of the decision is constructed.

X. Nikitenkov V.L., Podorov A.E. Modifications of waste cutting problem

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The problem of cutting materials with various length and limited stores is considered as the waste cutting problem. Five it’s modifications are offered. In some of them methods of a finding admissible inverse matrix are given. Results of numerical experiments are discussed.

XI. Yakovlev V.D., Afonin R.E. The hunting on numbers

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In this article the history of search of amicable pairs since times of ancient Greeks and up to now is shown. Also current outcomes of search of aliquot sequences and Mersenne numbers are given.

I. Mikhailovskii E.I. Word about Valentin Valentinovich Novozhilov

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II. Mikhailovskii E.I., Tarasov V.N. The constructive – nonlinear mechanics of plates and shells

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The review of the basic results obtained by authors and their disciples for last 20 years in the field of constructive-nonlinear mechanics of plates and shells is given. The general course of the proof proposed by authors of a method of the generalized reaction for the solving of contact problems with free boundary and a method of local search of eigenvalues of positively homogeneous operators for the solving of essentially nonlinear spectral problems is explained. Algorithms of local search of variants in a combination to their full search on a rare grid and with movement on parameter of rigidity of one of elastic environments are illustrated. Effect of accounting transversal deformations in the equations of mechanics of plates and shells is investigated.

III. Belyaeva N.A. Deformation of viscoelastic materials with changing structure

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Deformation processes of materials with changeable structure is presentated. Worked out mathematical models are including wide field of objects beginning with systems-powder to polymer and composite materials. This models give a chance to define deformation, temperature and structure characteristicses during hardening, solid state extrusion and non-Newtonian flow.

IV. Tarasov V.N., Andryukova V.Yu. Of nonlinear fluctuations of rectangular plates

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Linear and nonlinear fluctuations of rectangular plates are researched. The difference scheme for solving the dynamic equations of Karman is considered. The results of numerical experiments are analyzed, solutions obtained on the basis of a linear equations of vibrations of plates and the solutions obtained by numerical analysis of nonlinear Karman equations are compared.

V. Belyaev Y.N. Characteristic matrix of laered-periodic structure

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The method of recurrent calculation of the matrix characterising distribution of elastic waves in periodic layered structure is offered. The estimation of efficiency of this method is made.

VI. Sergey G. Bobkov. The growth of L^p-norms in presence of logarithmic sobolev inequalities

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The growth of L^p-norms is considered under various hypotheses, including LS_q (logarithmic Sobolev) inequalities.

VII. Vechtomov E.M., Sidorov V.V. On difinability of semirings of continuous functions by their subalgebra lattice

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It is solved the problem of semirings of continuous functions difinability by their subalgebra lattice. Namely, it is proved that an isomorphism of lattices of all subalgebras of semirings of continuous nonnegative functions over arbitrary topological spaces implies an isomorphism of semirings of continuous functions.

VIII. Petr A. Golovach, Pinar Heggernes. Choosability of P₅-free graphs

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A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k > 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we prove that the problem is fixed parameter tractable on P₅-free graphs when parameterized by k. This graph class contains the well known and widely studied class of cographs. Our result is surprising since the parameterized complexity of k-coloring is still open on P₅-free graphs. To give a complete picture, we show that the problem remains NP-hard on P₅-free graphs when k is a part of the input.

IX. Kostyakov I.V., Kuratov V.V. Limit transitions in gauge theories

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We show how to obtain Lagrangians with nonsemisimple gauge symmetry, using contractions. The limit transitions of SO(2) and SU(2) gauge theories are considered