Bulletin 1 (26) 2018

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Issue 1 (26) 2018

I. Makarov P. A., Shcheglov V. I. On the application of the operators formalism to the solution of the electrodynamics problems for bigyrotropic media

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The operator formalism is developed to consider electromagnetic wave processes in stationary, homogeneous, bihyrotropic media. Wave equations are obtained in the general case, and also for waves propagating in paralleland perpendicular to the gyrotropy axis. The solutions of the wave equation and the dispersion relations for the gyroelectric and gyromagnetic waves are analytically obtained. The general method of the solution for waves propagating parallel to the gyrotropy axis is showed.

Keywords: electrodynamics, Maxwell’s equations, bihyrotropic medium, propagation of electromagnetic waves.

References

  1. Shavrov V. G., Shcheglov V. I. Magnitostaticheskiye i elektromagnitnyye volny v slozhnykh strukturakh (Magnetostatic and electromagnetic waves incomplex structures), M.: FIZMATLIT, 2017, 360 p.
  2. Veselago V. G. Elektrodinamika veshchestv s odnov remennootritsatelnymi znacheniyami ε i µ (The electrodynamics of substances with simultaneously negative values of ε µ), Uspekhi Fizicheskikh Nauk, v. 92, № 3, 1967, pp. 517–526.
  3. Vinogradov A. P. Elektrodinamika kompozitnykh materialov (Electrodynamics of composite materials), M.: URSS, 2001, 207 p.
  4. Shcheglov V. I. Raschetdinamicheskoy pronitsayemosti sredy, soderzhashchey magnitnuyu i elektricheskuyukom ponenty (The dynamic permittivity calculation of media having magnetic and electric components), Journal of radio electronics, URL: http://jre.cplire.ru/ win/contents.html: № 7. 2001. URL: http://jre.cplire.ru/win/aug01/ 4/text.html (date of the application: 29.03.2018).
  5. Eritsyan O. S. Opticheskiye zadach ielektrodinamiki girotropnykhsred(Optical problems in the electrodynamics of gyrotropic media), Uspekhi Fizicheskikh Nauk, v. 138, № 4, 1982, pp. 645–674.
  6. Barta O., et al. Magneto-optics in bi-gyrotropic garnet waveguide // Opto-electronics review. Vol. 9. № 3. 2001. Pp. 320–325.
  7. Bukhanko A. F., Sukstanskii A. L. Optics of a ferromagnetic superlattice with noncollinear orientation of equilibrium magnetization vectors in layers // Journal of Magnetism and Magnetic Materials. Vol. 250. 2002. Pp. 338–352.
  8. Dadoenkova N. N., et al. Complex waveguide based on a magnetooptic layer and a dielectric photonic crystal // Superlattices and Microstructures, vol. 100, 2016, pp. 45–56.
  9. Eliseeva S. V., Sannikov D. G., Sementsov D. I. Anisotropy, gyrotropy and dispersion properties of the periodical thin-layer structure of magnetic-semiconductor // Journal of Magnetism and Magnetic Materials. Vol. 322. 2010. Pp. 3807–3816.
  10. Rychly J. et al. Magnonic crystals — Prospective structures for shaping spin waves in nanoscale // Low Temperature Physics. Vol. 41. № 10. 2015. Pp. 745–759
  11. Gurevich A. G., Melkov G. A. Magnitnyye kolebaniyai volny (Magneticoscillations and waves), M.: Nauka, 1994, 464 p.
  12. Landay L. D., Lifhitz E. M. Teoreticheskaya fizika: T. VIII.Elektrodinamika sploshnykh sred (Theoretical physics: Vol. VIII.Electrodynamics of Continuous Media), M.: FIZMATLIT, 2005, 656 p.
  13. Greer J. B., Bertozzi A. L., Sapiro G. Fourth order partial differential equations on general geometries // Journal of Computational Physics. Vol. 216. № 1. 2006. Pp. 216–246.
  14. Elsgolz L. E. Differentsialnyy euravneniya i variatsionnoye ischisleniye(Differential equations and the calculus of variations), M.: URSS,2002, 320p.
  15. Kuznetcov E. A., Shapiro D. A. Metody matematicheskoy fiziki:kurslektsiy (Methods of mathematical physics: Course of lectures),P.I, Novosibirsk State University, 2011, 131 p.

For citation: Makarov P. A., Shcheglov V. I. On the application ofthe operators formalism to the solution of the electrodynamics problems for bigyrotropic media, Bulletin of Syktyvkar University, Series 1: Mathematics.Mechanics. Informatics, 2018, 1 (26), pp. 3–16.

II. Petrakov P. A., Cheredov V. N. The contribution of «hot» phonons to the internal energy of solids

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A mixed thermodynamic model of a solid is constructed, including the interpretation of the energy of acoustic branches of oscillations on the basis of the Debye model, and the branches of optical vibrations and librational rotations based on the Einstein model. In the framework of the development of the theory of thermal oscillations (phonons) of the lattice of solids, the contribution of «hot» phonons, as harmonic oscillators with modes of thermal oscillations with values of indices higher than the predetermined value, to the internal energy of solids is studied. Dependences of the contribution to the internal energy of a molecule caused by acoustic and optical thermal vibrations with modes above the limiting one are studied. The curves of the fraction of the internal energy of solids with a lattice excited by «hot» phonons are obtained, depending on the level of the limiting mode of the oscillator for ice crystals.

Keywords: thermal vibrations, phonons, internal energy, crystal lattice, solid.

References

  1. Cheredov V. N., Kuratova L. A. Dinamika setki mezhmolekulyarnykh svyazey i fazovyye perekhody v kondensirovannykh sredakh (Dynamics of a network of intermolecular bonds and phase transitions in condensed matter), Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2017, 4 (25), pp. 20–32.
  2. Rodnikova M. N., Chumaevskiy N. A. O prostranstvennoysetkevodorodnykhsvyazey v zhidkostyakh i rastvorakh (On the spatial grid of hydrogen bonds in liquids and solutions), Journal of Structural Chemistry, 2006, v. 47, pp. 154–166.
  3. Malenkov G. G. Struktura i dinamika zhidkoyvody (Structure and dynamics of liquid water), Journal of Structural Chemistry, 2006, v. 47, pp. 5–35.
  4. Bushuev Yu. G. Svoystvasetki vodorodnykh svyazey vody (Properties of a network of hydrogen bonds of water), Proceedings of the Russian Academy of Sciences, Chemical series, 1997, № 5, pp. 928–931.
  5. Landau L. D., Lifshitz E. M. Statisticheskaya fizika (Statistical physics), Part 1, Moscow: Fizmatlit, 2010, 616 p.
  6. Wang Kuo-Ting, Brewster M.Q. An Intermolecular Vibration Model for Lattice Ice //International Journal of Thermodynamics. 2010. V. 13. № 2. Pp. 51–57.
  7. Eisenberg D., Kautsman V. Struktura i svoystvavody (Structure and properties of water), Moscow: Direct-Media, 2012, 284 p.
  8. Enochovich A. S. Spravochnik po fizike i tekhnike (Reference book on physics and techniques), Moscow: Prosveshenie, 1989, 224 p.
  9. Zatsepina G. N. Fizicheskiye svoystva i strukturavody (Physical properties and structure of water), Moscow: Moscow State University, 1998, 184 p.
  10. Bertie J. E., Whalley E. Optical Spectra of Orientationally Disordered Crystals. II. Infrared Spectrum of Ice Ih and Ice Ic from 360 to 50 cm−1 //The Journal of Chemical Physics. 1967. V. 46, № 4. Pp. 1271–1281.
  11. Wang Kuo-Ting, Brewster M. Q. An Intermolecular Vibration Model for Lattice Ice //International Journal of Thermodynamics. 2010. V. 13. № 2. Pp. 51–57.

For citation: Petrakov P. A., Cheredov V. N. The contribution of «hot» phononsto the internal energy of solids, Bulletin of Syktyvkar University. Series 1:Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 17–28.

III. Tarasov V. N. On the elastic line of the rod compressible by longitudinal force located between two rigid walls

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The problem of determining the elastic line compressible by longitudinal force of the rod, located between two rigid walls is considered. The dependence of the elastic line on boundary conditions is studied.

Keywords: elastic line, critical force, boundary conditions, stability, Euler equation.

References

  1. Mihailovskii E. I., Tarasov V. N., Holmogorov D. V. Sakriticheskoepovedeniestersniaprisestkimiogranisheniiaminaprogib (Supercritical thebehavior of longitudinally compressed rod with hard constraints at adeflection of), PMM, 1985, t. 49, vip. 1, pp. 156–160.
  2. Nikolai E. L. Trudi po mexanike (Works on mechanics), M.: Isdatelstvotexniko-teoretiteskoiliteraturi, 1955, 376 p.
  3. Tarasov V. N. Ob ustoichivostiuprugihsistempriodnostoronnihogranicheniyahnaperemescheniya (On stability of elastic systems with one-sided restrictions on the movement), Trudy institute matematiki i mehaniki, Rossiskaya akademiya nauk, Uralskoeotdelenie, Tom 11, No. 1, 2005, pp. 177–188.
  4. Feodosiev V. I. Izbrannyye zadachi i voprosy po soprotivleniyu materialov (Selected problems and questions on the resistance of materials), M.: Nauka, 1967, 376 p.

For citation: Tarasov V. N. On the elastic line of the rod compressible by longitudinal force located between two rigid walls, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 29–46.

IV. Rychkov S. L. Some integral principal values

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A method of evaluation of principal values of integrals  is considered. The plasma dispersion functions of plasmas having quasipower electron energy distribution can be calculated using integrals of such a type. The suggested method differs from previously known ones and results in formulas more convenient for usage. The integrals are represented in terms of Gauss hypergeometric functions for z > 0 and ν > 5/2. Simple asymptotic approximations for values z ≫ 1 are obtained. Graph plots of the results are given.

Keywords: integral principal value, hypergeometric functions, nonequilibrium plasma, quasipower dispersion function, kappa–dispersion.

References

  1. Pierrard V., Lazar M. Kappa distributions: theory and applications in space plasmas // Solar Physics. 2010. V. 267. Pp. 153–174.
  2. Podesta J. J. Plasma dispersion function for the kappa distribution, Report NASA/CR-2004-212770. https://ntrs.nasa.gov/archive/nasa /casi/ntrs.gov/20040161173.pdf (date of the application: 26.03.2018).
  3. Bateman H., Erdelyi A. Vysshiye transtsendentnyye funktsii. Gipergeometricheskayafunktsiya. Funktsii Lezhandra (Higher transcendental functions. Hypergeometric function. Legendre functions), M, Science, 1965, 296 p.

For citation: Rychkov S. L. Some integral principal values, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 47–57.

V. Kotelina N. O. Two-dimensional ternary search and its application in competitive programming

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In this paper the application of ternary search in one problem of competitive programming is considered.

Keywords: two-dimensional ternary search, competitive programming.

References

  1. Distantcionnaya podgotovka po informatike (Distance training in computer science). URL: http://informatics.mccme.ru (date of usage: 29.10.2017).
  2. MAXimal. Sayt M. Ivanova (MAXimal. Site of M. Ivanov) URL: http://e-maxx.ru (date of usage: 12.09.2017).
  3. Knuth D. E. Iskusstvo programmirovaniya (The Art of Computer Programming. Vol. 3. Sorting and Searching) M. :Williams (2007). 832 p.
  4. Konspektystudentovkafedrykomp’yuternykhtekhnologiyUniversiteta ITMO (Conspects of students of ITMO) URL: http://neerc.ifmo.ru/wiki (date of usage: 12.09.2017).
  5. Mathews J. H., Fink K. D. Chislennyyemetody. Ispol’zovaniye MATLAB (Numerical Methods: Using MATLAB) 3rd Edition. Spb.: Williams, 2001, 716 p.
  6. Olimpiady po informatike (Olympiads on informatics) URL: https://neerc.ifmo.ru/school. (date of usage: 12.09.2017).

For citation: Kotelina N. O. Two-dimensional ternary search and its application in competitive programming, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 58–63.

VI. Melnikov V. A. Application of genetic algorithms for finding the optimal nesting sequence

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Keywords: genetic algorithms, optimization, genom, individuals.

References

  1. MacLeod C. An Introduction to Practical Neural Networks and Genetic Algorithms For Engineers and Scientists, p. 85.
  2. He Y., Liu H. Algorithm for 2D irregular-shaped nesting problem based on the NFP algorithm and lowest gravity-center principle, Journal of Zhejiang University, 2006, № 7, pp. 571–574.
  3. Panchenko T. V. Geneticheskie algotritmy (Genetic algorithms), pod red. U. U. Tarasevicha, Astrahan: Izdatelskiydom «Astrahanskiyuniversitet», 2007, p. 16.
  4. Kudryavcev L. D. Matematicheskiyanaliz (Mathematical analysis), 2-e izd, M.: Vyshayashkola, 1973, v. 1, 687 p.
  5. Coordinate Systems, Transformations and Units [Electronic resource] / W3C. 6 мая 2017. URL: https://www.w3.org/TR/SVG/coords.html (date of the application: 25.12.2017).
  6. Melnikov V. A. Metodypredstavleniyafigurobshchegovidadlyazadachidvumernogoraskroya (Methods for representing figures of general kind for a two-dimensional cutting problem), Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2017, 3 (24), pp. 11—24.

For citation: Melnikov V. A. Application of genetic algorithms for finding the optimal nesting sequence, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 64–72.

VII. Kotelina N. O., Popova N. K. The preparation of the online round of the championship on programming on Yandex.Contest platform

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The paper discusses the online round of the open programming championship of Pitirim Sorokin Syktyvkar State University conducted as part of a project «Development of network interaction in the field of mathematics, physics, computer science and robotics between educational organizations of the Finno-Ugric republics of the Russian Federation».Keywords: online round, Yandex.Contest, competitive programming.

References

  1. Arkhiv materialov olimpiad / Olimpiady po informatike (Archive of materials of Olympiads / Computer science Olympiads) URL: https://neerc.ifmo.ru/school/archive/index.html (date of usage: 19.02.2018).
  2. Ofitsial’nyysaytVserossiyskoykomandnoyolimpiadyshkol’nikovpoprogrammirovaniyu / Olimpiady po informatike (Official site of the all-russian team Olympiad in programming / Computer science Olympiads) URL: https://neerc.ifmo.ru/school/russia-team/ index.html (date of usage: 19.02.2018).
  3. Pravila sorevnovaniy / Sorevnovaniya po programmirovaniyu 2.0 (Competition rules / Programming competitions 2.0) URL: http:// codeforces.com/blog/entry/4088?locale=ru (date of usage: 19.02.2018).
  4. Sorevnovaniya po programmirovaniyu 2.0 (Programming competitions 2.0) URL: http://codeforces.com (date of usage: 19.02.2018).
  5. Tablitsa rezul’tatov internet-turaotkrytogochempionataSyktyvkarskogogosudarstvennogouniversitetaim. Pitirima Sorokina po programmirovaniyu / Yandeks.Kontest (Table of results of the online roundof the open programming championship of Pitirim Sorokin Syktyvkar State University / Yandex. Contest) URL: https://contest.yandex. ru/contest/7113/standings/ (date of usage: 19.02.2018).
  6. Timus Online Judge / Arkhivzadach s proveryayushchey sistemoy (Timus Online Judge / The problems’ archive with the testing system) URL: http://acm.timus.ru/ (date of usage: 19.02.2018).

For citation: Kotelina N. O., Popova N. K. The preparation of the online round of the championship on programming on Yandex.Contest platform, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 73–79. 

VIII. Odyniec W. P. The 1929–1936 Immigration to the USSR: Profiles of Mathematicians. Part 1.

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Keywords: boundary-value problems, singular integral equations, Bessel functions, the Oryol central prison, Fritz Noether, M¨untz theorem, Herman (Chaim) M¨untz, kernel function, Stefan Bergman.

References

  1. Arkhiv Sankt-Peterburgskogo gosudarstvennogo universiteta (St. Petersburg State University Archive), File 7240.14 № 191 (Order № 11 from 14/1–1932 . On assigning of thesis advisor).
  2. Arkhiv Sankt-Peterburgskogo gosudarstvennogo universiteta (St. Petersburg State University Archive), File 7240.14 № 191 (Order № 352а from 20/X–1932).
  3. Bergmann S. Uberdie Kernfunktioneines Bereichs und ihrVerhalten am Rande. Teil 1 // J. furreine und angewandte Math. Bd. 169. Heft 1. 1932. S. 1–42.
  4. Bergmann S. Uberdie Kernfunktioneines Bereichs und ihrVerhalten am Rande. Teil 2 // J. furreine und angewandte Math. Bd. 172. Heft 2. 1934. S. 89–128.
  5. Bergmann S. Zur Theorie von pseudokon formen Abbildungen, Matem. Sbornik, t. 1 (43), No. 1, 1936, pp. 79–96.
  6. Bergman S. O funktsiyakh, udovletvoryayushchikh lineynym differentsial’nym uravneniyam v chastnykh proizvodnykh (Upon the Functions Satisfying Certain Linear Partial Differential Equations), DokladyAkad. ofSci, USSR, vol. 15, No. 5, 1937, pp. 227–230.
  7. Bergmann S. Zur Theorie der linearen Integral — und Funktional gleichun genimcomplexen Gebiet, Izvestiya NIIMM TGU, Tomsk, vol. 1, issue 3, 1937, pp. 242–257.
  8. Bergmann S. The Kernel Function and Conformal Mapping.- Cambridge (Massachusetts): Amer. Math. Society, 1950. 161 p.
  9. Bergman S., Schiffer M. M. Kernel functions and elliptic differential equations in mathematical physics. New York: Academic Press, 1953. 432 p.
  10. Bergmann S. Integral operators in the theory of linear partial differential equations. Berlin-New York: Springer, 1961, 2thed., 1969.
  11. Brewer J. W., Smith M. K. (eds.) Emmy Noether: a tribute to her life and work. New York: Marcel Dekker, Inc., 1981. 237 p.
  12. Del О. А. Nemetskiye emigranty v SSSR v 1930-ye gody. Avtore feratdi ssertatsiinasoiskani yeuchenoy stepenikandi datnauk (German Emigrants in the USSR during the 1930’s. Abstracts of Dissertation for the Degree of a Candidate of Sciences (History)), Moscow: the Russian Academy of State Service., 1995, 22 p.
  13. Juravlev S. V., Tyazhelnikova V. S. Inostrannaya koloniya v SovetskoyRossii v 1920-1930-ye gody (Postanovka problemy i metody issledovaniya) (The Foreign Colony in Soviet Russiaduring the 1920’s– 1930’s), Otechestvennayaistoriya, 1994, No. 1, pp. 179–189.
  14. Lyapunov А. М. Obshchaya zadacha obustoychivosti dvizheniya (General Problem of the Stability of Motion), Ed. H. M. M¨untz, M.- L.: ONTI, 1935, 386 p.
  15. Matematika v SSSR zasorok let. 1917-1957 (Mathematics in the USSR for 1917–1957), vol. 2, Bibliography, M.: State Phiz.-Math. Lit. Publ., 1959, 819 p.
  16. Muntz Ch. Zum Randwertproblem der partiellen Differential gleichung der Minimal flachen // J. fur Reineund Angew. Math. 139. 1911. S. 52–79.
  17. Muntz Ch. Uber den Approximationssatz von Weierstrass / H. A. Schwarz–Festschrift. Berlin: 1914. S. 303–312.
  18. Muntz Ch. Die Losung des Plateauschen Problems uberkonvexen Bereichen // Math. Ann., 94. No. 1–2. 1925. S. 53–96.
  19. Gottfried Noether, 76: Educator in Statistics // New York Times. August 27, 1991. P. 22. (Obituary).
  20. Noether Fr. O rekurrentnykh funktsiyakh Besselya i Ermita (Upon Recurrent Functions of Bessel and Hermite), Izvestiya NIIMM TGU, Tomsk: 1935, Vol. 1, issue 2, pp. 121–125.
  21. Noether Fr. Asymptotische Darstellungen und Geometrische Optik, Izvestiya NIIMM TGU, Tomsk: 1937, t. 1, issue 3, p. 175–189.
  22. NoetherFr. Zur Kinematik des starren Korpers in der Relativtheorie // Annalen der Physik. 336 (5). 1910. S. 914–944.
  23. Noether Fr. Bemerkunguber die Losungszahlzueinan deradjungierten Randwertaufgaben beilinearen Differentialgleichungen // Sitzungsberichte der Heidelberger Akad. der Wissenschaft. Math. Nat. Klasse. 1920, I. Abhandlung. S. 37–52.
  24. Noether Fr. Ubereine Klassesingularer Integralgleichungen // Math. Ann. Bd. 82. 1921. S. 42–63.
  25. Odyniec W. P. Arnol’dVal’fish — zhizn’ voprekistereotipam (k 125-letiyu so dnyarozhdeniya) (Arnold Walfisz – a Life Defying Stereotypes (the 125th anniversary of his birth)), Mathematics in higher education, issue 14, Moskow – Nizhni Novgorod – St. Petersburg, 2016, pp. 105–112.
  26. Ortiz E. L., Pinkus A. Herman Muntz: A Mathematician’s Odyssey //Mathem. Intellig. Berlin. 27. 2005. S. 22–30.
  27. Segal, Sanford L. Mathematicians under the Nazis. Princeton: Princeton University Press, 2003. 536 p.
  28. Siegmund-Schulze R. Mathematiker auf der Fluchtvor Hitler.- Wiesbaden: Vieweg Verlag, 1998. 324 s.
  29. Trudy Vtorogo Vsesoyuznogo matematicheskogo s’yezda. Leningrad. 24-30 iyunya 1934 g. T. 1. Plenarnyye i obzornyye doklady (Proceedings of the Second All-Union Mathematical Congress, Leningrad, June 24- 30, 1934, T. 1, Plenary and overview reports), Moscow-Leningrad: Acad. Sci. USSR Press, 1935, 371 p.

For citation: Odyniec W. P. The 1929–1936 Immigration to the USSR: Profiles of Mathematicians. Part 1., Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 80–96.

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

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The article deals with the class of (1/2 ; 1)-convex functions. The authors give a geometric characterization of such functions, derive sufficient conditions for the membership of the function to the class under discussion in terms of derivatives.Keywords:(1/2 ; 1)-convex function, (1/2 ; 1) -concave function, 1/2 -parabolic a

References

  1. Guan Kaizhong. GA-convexity and its applications // Anal. Math. 2013. 39. № 3. Pp. 189–208.
  2. Xiao-Ming Zhang, Yu-Ming Chu, and Xiao-Hui Zhang. The Hermite-Hadamard type inequality of GA-convex functions and its application // J. of Inequal. andApplics. Vol. 2010. Article ID 507560, 11 pages, doi:10.1155/2010/507560.
  3. 3. Kalinin S. I. (α; β)-vypuklyye funktsi i, ikh svoystva i nekotoryye primeneniya ((α; β)-convex functions, their properties and some applications), Ufa International Mathematical Conference. Collection of abstracts, otv. red. R. N. Garifullin, Ufa: RIC BashGU, 2016, pp. 75–76

For citation: Kalinin S. I., Leonteva N. V. (1/2; 1)-convex functions. Part 1., Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2018, 1 (26), pp. 97–104.

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