Bulletin 3 (36) 2020

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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

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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

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  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

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  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.

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