Bulletin 2 (27) 2018

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Issue 2 (27) 2018

I. Belyaeva N. A. The velocity of a stationary pressure flow of a structured liquid

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The pressure flow of a structured liquid with variable viscosity is analyzed. An analytical formula for determining the steady-state flow velocity is obtained from the equation of motion.

Keywords: mathematical modeling, flow, liquid, structured, pressure, stationary, variable viscosity.

References

  1. Belyaeva N. A. Neodnorodnoye techeniye strukturirovannoy zhidkosti (The inhomogeneous flow of a structured liquid), Mathematical modeling, 2006, vol. 18, No. 6, pp. 3–14.
  2. Belyaeva N. A., Stolin A. M., Pugachov D. V., Stelmakh L. S. Neustoychivyye rezhimy deformirovaniya pri tverdofaznoy ekstruzii vyazkouprugikh strukturirovannykh sistem (Unstable modes of deformation during solid-phase extrusion of viscoelastic structured systems), DAN, 2008, vol. 420, No. 6, pp. 777–780.
  3. Belyaeva N. A., Stolin A. M., Stelmakh L. S. Dynamic of Solid-State Extrusion of Viscoelastic Cross-Linked polymeric Materials, Theoretical Foundations of Chemical Engineering, 2008, vol. 42, No. 5, pp. 549–556.
  4. Belyaeva N. A. Osnovy gidrodinamiki v modelyakh : uchebnoye posobiye (Fundamentals of hydrodynamics in models: a manual), Syktyvkar: Publishing house of Syktyvkar State University, 2011, 147 p.
  5. Belyaeva N. A., Yakovleva A.F. Frontal’naya volna napornogo techeniya (Frontal wave of pressure flow), Bulletin of Syktyvkar University. Ser. 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), 2017, pp. 4–12.

For citation: Belyaeva N. A. The velocity of a stationary pressure flow of a structured liquid, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 3–9.

II. Gromov N. А., Kuratov V. V. Harmonic oscillator on Minkowski plane

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The problem of quantum harmonic oscillator on Minkowski plane is discussed. The corresponding Schr¨odinger equation for eigenstates is obtained with the help of Beltrami-Laplas operator of pseudoeuclidean plane. The infifinitely high potential barriers are placed on isotropic lines. The discrete energy eigenvalues of oscillator are obtained.

Keywords:Minkowski plane, Schr¨odinger equation, harmonic oscillator.

References

  1. Gromov N. A., Kuratov V. V. Garmonicheskiy ostsillyator na ploskostyakh Keli-Kleyna s rimanovoy i vyrozhdennoy metrikami (A harmonic oscillator on the Cayley-Klein planes with Riemannian and degenerate metrics), Proceedings of Int. Seminar «Group theoretical methods for studying physical systems» Syktyvkar, 2018, (Bulletin of the Komi Scientifific Center of the Ural Branch of the Russian Academy of Sciences, Issue 33), pp. 21–36.
  2. Gromov N. A., Kuratov V. V. Kvantovaya chastitsa na ploskosti Minkovskogo (Quantum particle on the Minkowski plane), Proceedings of the Komi Scientifific Center of the UrB RAS, 2018, Issue 3 (35), pp. 5–7.
  3. Remnev M. A., Klimov V. A. Metapoverkhnosti: novyy vzglyad na uravneniya Maksvella i novyye metody upravleniya svetom (Metasurfaces: a new view of Maxwell’s equations and new methods of light control), Progress in physical sciences, 2018, vol. 188, No. 2, pp. 169–205.
  4. Smolyaninov I. I. Hyperbolic metamaterials; arXiv: 1510.07137.
  5. Green M. B., Schwartz J., Witten E. Teoriya superstrun (Theory of superstrings), Moscow: Mir, 1990.
  6. Kaku M. Vvedeniye v teoriyu superstrun (Introduction to the theory of superstrings), Moscow: Mir, 1999, 624 p.
  7. Bars I. Relativistic Harmonic Oscillator Revisited, Phys. Rev. D, v. 79, Iss. 4. 045009. 2009, arXiv: 0810.2075.
  8. 8. Betemmen G., Erdei A. Vysshiye transtsendentnyye funktsii (Higher transcendental functions), M.: Mir, 1973, vol. 1.
  9. Shabad A. E. Singulyarnyy tsentr kak negravitatsionnaya chernaya dyra (The singular center as a non-gravitational black hole), Theoretical and Mathematical Physics, 2014, vol. 181, No. 3, pp. 603–613.
  10. Perelomov A. M., Popov V. S. «Padeniye na tsentr» v kvantovoy mekhanike («Falling to the center» in quantum mechanics), Theoretical and Mathematical Physics, 1970, vol. 4, No. 1, pp. 48–65.
  11. Gitman D. M., Tyutin I. V., Voronov B. L. Self-Adjoint Extensions in Quantum Mechanics: General Theory and Applications to Schr¨odinger and Dirac Equations with Singular Potentials, Progress in Mathematical Physics, vol. 62, Birkh¨auser: New York, 2012, 511 p.

For citation:Gromov N. А., Kuratov V. V. Harmonic oscillator on Minkowski plane, Bulletin of Syktyvkar University. Series 1:Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 10–23.

III. Kazakov A. Yu. Exact solution of the heat equation under symmetry conditions

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The paper considers the application of operational calculus to solve two initial-boundary value problems with the equation Tt= a2∆T in areas with cylindrical and spherical symmetry. The solutions are obtained in the form of the traditional Fourier functional series problems for this class.

Keywords: Laplace transformation, heat conduction equation, residues.

References

  1. Aramanovich I. G., Luntz G. L., Elsholz L. E. Funktsii kompleksnogo peremennogo. Operatsyonnoye ischislenie. Teoriya ustoychivosti (Functions of a complex va-riable. Operational calculus. Stability theory). M.: Nauka, 1968. Ed. 2nd, 416 p.
  2. Belyaeva N. A. Matematicheskoe modelirovanie: uchebnoe posobie (Mathematical modeling: tutorial). Syktyvkar: Publishing house of Syktyvkar state University, 2014. 116 p.
  3. Boyarchuk A. K., Golovach G. P. Spravochnoe posobie povysshey matematike. Tom 5. Difffferentsyalnye uravneniya v primerakh I zadachakh. (Handbook on higher mathematics. Volume 5. Difffferential equations in examples and problems). M.: URSS, 1999. 384 p.
  4. Koshlyakov N. S. and others. Uravneniya v chastnykh proizvodnykh matemati-cheskoy fifiziki. Uchebnoe posobie dlya mech.-mat. fak. un-tov (Partial difffferential equations of mathematical physics. Tutorial for mechanics and mathematics faculties of universities). M.: Vysshaya shkola, 1970. 712 p.
  5. Bogolyubov A. N., Kravtsov V. V. Zadachi po matematicheskoy fifizike: Ucheb. posobie (Tasks in mathematical physics: tutorial). M.: Publishing house of Mos-cow state University, 1998. 350 p.
  6. Carslaw H. S., Jaeger J. S. Operatsionnyye metody v prikladnoy matematike (Operational methods in applied mathematics. M.: IL, 1948. 294 p.
  7. Doetsch G. Rukovodstvo k prakticheskomu primeneniyu preobrazovaniya Laplasa i Z-preobrazovaniya (A guide to the practical application of Laplace transform and Z-transform). M.: Nauka, 1971. 288 p.

For citation:Kazakov A. Yu. Exact solution of the heat equation under symmetry conditions, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 24–31.

IV. Kostyakov I. V., Kuratov V. V. Quantum computations and contractions of Lie algebras

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The connection between the nonunitary Kraus transformations of the qubit density matrix with contraction theory of the su(2) Lie algebra is pointed. The use of contraction constructions is demonstrated.

Keywords: contractions of Lie algebras, quantum channels, qubit.

References

  1. Nielsen M. A., Chuang I. L. Kvantovyye vychisleniya i kvantovaya informatsiya (Quantum Computation and Quantum Information), Cambridge University Press, 2010, 676 p.
  2. Preskill J. Kvantovaya informatsiya i kvantovyye vychisleniya (Lecture Notes for Physics 229:Quantum Information and Computation), Izhevsk: RKHD, 2008, 2011, t. 1–2, 464+312 p.
  3. Ruskai M. B., Szarek S., Werner E. An Analysis of CompletelyPositive Trace-Preserving Maps on 2×2 Matrices, Lin. Alg. Appl., vol. 347, 2002, pp. 159–187. ArXiv:quant-ph/0101003.
  4. Gromov N. A. Kontraktsii klassicheskih i kvantovyh grupp (Contractions of classical and quantum groups), M.: Fizmatlit, 2012, 318 p.
  5. In¨on¨u E., Wigner E. P. On the Contraction of Groups and Their Representations, Proc. Nat. Acad. Sci., vol. 39, iss. 6, pp. 510–524, 1953.
  6. Saletan E. J. Contraction of Lie groups, J. Math. Phys., vol. 2, iss. 1, 1961, pp. 1–21.

For citation:Kostyakov I. V., Kuratov V. V. Quantum computations and contractions of Lie algebras, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 32–39.

V. Pimenov R. R. The geometry of perpendicularity: the axiomatic multidimensional space and de Morgan’s laws

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We propose the short axiomatic for fifinite dimensional geometrical structure, using only perpendicularity relation. This structure appear projective space in which hold De Morgan’s laws. We show the connection work with the Veblen’s axiom and with partition four elements to pairs various modes. The research is connected with ortholattice, matroids, Galois connections and quantum logic.

Keywords: the foundation of geometry, perpendicularity, ortholattice,Galois connections, logic, projective space.

References

  1. Cameron P. J. Projective and Polar Spaces, second edition, Sep 2000. http://www.maths.qmul.ac.uk/ pjc/pps/
  2. Birkhoffff G. Teoriya reshetok (Lattice Theory), М.: Nauka, 1984, 565 p. Aigner M. Kombinatornaya teoriya (Combinatorial Theory), M .: Mir, 1982, 556 p.
  3. Bachmann F. Postroenie geometrii na osnove ponyatiya simmetrii (Aufbau der Geometrie aus dem Spiegelungsbegriffff), М.: Nauka, 1969, 380 p.
  4. Pimenov R. I. Yedinaya aksiomatika prostranstv s maksimal’noy gruppoy dvizheniy (Unifified axiomatics of spaces with the maximum group of motions), Litovsk. Mat. Sb., vol. 5, No. 3, 1965, pp. 457–486.
  5. Maclaren M. D. Atomic orthocomlemented lattices, Pacifific Journal of Mathematics, vol. 14, June 1964, pp. 697–612 (site https://msp.org/pjm/1964/14-2/pjm-v14-n2-p18-p.pdf)
  6. Norman D. Megill and Mladen Pavicˆ´i, Hilbert Lattice Equations, Ann. Henri Poincare 99 (9999), 1–24 1424-0637/99000-0, DOI 10.1007/ s00023-003-0000 ©2009 Birkhauser Verlag Basel/Switzerland (site https: //bib.irb.hr/datoteka/413891.megill-pavicic-a-henri-p-09r.pdf)
  7. Odyniec W. P. Ob istorii nekotorykh matematicheskikh metodov, ispol’zuyemykh pri prinyatii upravlencheskikh resheniy (Upon the history of some mathematical methods, which use for the taking a steering decision), Syktywkar: SGU, 2015, 107 p.
  8. Vasukov V. L. Kvantovaya logika (Quantum logic), M.: Per Se, 2005, 191 p.
  9. Tabachnikov S. Skewers, Arnold Mathematical Journal, 2, 2016, pp. 171–193.
  10. Pimenov R. R. Obobshcheniya teoremy Dezarga: geometriya perpendikulyarnogo (The generalization the Desargues’s theorem and geometry of perpendicularity), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2016, 1 (21), pp. 28–43.
  11. Pimenov R. R. Traktovki teorem Pappa: perpendikulyarnost’ i involyutivnost’ (The interpretation and generalizations the Pappus’s theorems: involutions and perpendicularity), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 29–45.
  12. Pimenov R. R. Geometriya perpendikulyarnogo: tupyye i ostryye ugly v izvestnykh teoremakh (The geometry of perpendicularity: obtuse and acutes angles in known theorems), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2017, 3 (24), pp. 56–73.

For citation:Pimenov R. R. The geometry of perpendicularity: the axiomatic multidimensional space and de Morgan’s laws, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics,2018, 2 (27), pp. 40–70.

VI. Odyniec W. P. About a Wien-born Mathematician Who Immigrated to the USSR for the Development of a «New Society»

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The life and work of Felix Frankl (1905–1961), a prominent mathematician from Wien, who immigrated to the USSR in 1929 for the development of a «new society» is presented.

Keywords: the border of oriented manifold, prime end, L. Pontryagin, Zhukovsky’s propeller, M. V. Keldysh, Frankl’s problem, Frankl-Laval nozzle, L. Euler, model of bora.

References

  1. Gutman L. N., Frankl F. Termo-gidrodinamicheskaya model’ bory (Thermo-hydrodynamic Model of Bora), Doklady AN USSR, vol. 130, No. 3, 1960, pp. 533–536.
  2. Kazakov A. In Commemoration of the Late Professor Lev N.Gutman, Ukrainskii gidrometeorologichnyi zhurnal, No. 4, 2009, pp. 11–12.
  3. Keldysh M. V., Frankl F. Vneshnyaya zadacha Neymana dlya nelineynykh ellipticheskikh uravneniy v szhimayemom gaze (Neuman’s Exterior Problem for Nonlinear Elliptic Equation of Compressed Gas), Izvestiya AN USSR, VII Ser, 1934, No. 4, pp. 561–601.
  4. Keldysh M. V., Frankl F. Strogoye obosnovaniye teorii vinta Zhukovskogo (Strict Foundating the Theory of Zhukovsky Propeller), Mat. Sbornik, 42, No. 2, 1935, pp. 241–273.
  5. Matematika v SSSR za 40 let 1917–1957 (Mathematics in the USSR during the Forty Years 1917–1957), vl. 2, Biobibliography, Moscow: Fizmatgiz, 1959, 819 p.
  6. Frankl F. Upon the theory of prime ends (Doctorate Thesis), Wien, University, 1927, 25 Bl, Verbund-ID-Nr.AC06513142.
  7. Frankl F., Pontryagin L. A Knoth Theorem with the application to the dimension theory, Mathem. Annalen, v. 102, No.1, 1930, pp. 785–789.
  8. Frankl F. Characterizing of (n-1)-dimension closed set of Rn, Mathem. Annalen, vol. 103, No. 1, 1930, pp. 784–787.
  9. Frankl F. Upon the theory of prime ends, Mat. Sbornik, 38, No. 3–4, 1931, pp. 66–69.
  10. Frankl F. Upon the topology of the three-dimensional space, Monats-hefte f¨ur Mathem. und Physik, 38, 1931, p. 357–364.
  11. Frankl F. O ploskoparallel’nykh vozdushnykh techeniyakh cherez kanaly pri okolozvuchnykh skorostyakh (Upon the plane-parallel air flow through the cannels by near sound speed), Mat. Sbornik, 40, No. 1, 1933, pp. 59–72.
  12. Frankl F., Alekseeva R. Dve krayevyye zadachi iz teorii giperbolicheskikh uravneniy v chastnykh proizvodnykh s prilozheniyem k sverkhzvukovym gazovym techeniyam (Two boundary-value problem from the theory of hyperbolic partial differential equations with the application to the supersonic gas flow), Mat. Sbornik, 41, No. 3, 1934, pp. 483–502.
  13. Frankl F. O zadache Koshi dlya lineynykh i nelineynykh uravneniy v chastnykh proizvodnykh vtorogo poryadka giperbolicheskogo tipa (Upon the Cauchy problem for the hyperbolic-type linear and nonlinearpartial differential equations of the second order), Mat. Sbornik, v. 2, 44, No. 5, 1937, pp. 793–814.
  14. Frankl F. I., Khristianovich S. N., Alekseeva R. N. Osnovy gazovoy dinamiki (Foundation of Gas Dynamics), Issue 364, Moscow: CAGI, 1938, 111 p.
  15. Frankl F. I., Karpovich E. A. Gazodinamika tonkikh tel (Gas dynamics of thin bodies), Moscow-Leningrad: GTTL, 1948, 175 p.
  16. Frankl F. I., Il’ina A. A., Karpovich E. A. Kurs aerodinamiki v primenenii k artilleriyskim snaryadam (The course of air dynamics with application to artillery projectiles) (ed. by L.I. Sedov), Moscow: Oborongiz, 1952, 684 p.
  17. Frankl F. I., Sukhomlinov G. A. Vvedeniye v mekhaniku deformiruyemykh tel (Introduction to the mechanics of deformed bodies), Frunze: 1954, 204 p.
  18. Frankl F. I. O pryamoy zadache teorii sopla Lavalya (On the direct problem of the Laval nozzle), Uchenyezapiski Kabardino-Balkarskogo universiteta, Issue 3, 1959, pp. 35–61.
  19. Frankl F. I. Izbrannyye trudy po gazovoy dinamike (Selected works of gas dynamics), Moscow: Nauka, 1973,711 p.
  20. Frankl F. I. O sisteme uravneniy dvizheniya vzveshennykh potokov (On a system of equations of the motion of suspended flow), Issledovanie maksimalnogo stoka, volnovogo vozdeistviya i dwizheniya nanosov, Moscow: AN USSR, 1960, pp. 85–91.
  21. Euler L. Integral’noye ischisleniye (Integral Calculus), vol. III (Transl. and comment. by F. Frankl), Moscow: Fizmatgiz, 1958, 447 p.

For citation:Odyniec W. P. About a Wien-born Mathematician Who Immigrated to the USSR for the Development of a «New Society», Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 71–85.

VII. Yermolenko A. V., Melnikov V. A. Calculation of the contact interaction of a rectangular plate and a base by the Karman theory

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This article is about contact interaction of rectangular plate with base by Karman theory with usage of fifinite difffference method under the constant normal load. Wanted functions were discovered with method of generalized reactions developed in Syktyvkar state university. The obtained graphs arequalitatively consistent with the calculations of a cylindrically bent plate.

Keywords: plate, method of generalized reaction, contact problem, the Karman theory.

References

  1. Mikhailovskii E. I., Toropov A. V. Matematicheskie modeli teorii uprugosti (Mathematical models of the theory of elasticity). Syktyvkar: Syltyvkar University, 1995. 251 p.
  2. Yermolenko A.V. Chislennye metody v reshenii kontaktnyh zadach so svobodnoj granicej (Numerical methods in solving contact problems with a free boundary), Problems of the development of the transport
  3. infrastructure of the northern territories: Proceedings of the All-Russian Scientifific and Practical Conference on April 25–26, 2014. SPb.: Publishing House GUMRF them. adm. S.O. Makarova, 2015, pp. 29–35.
  4. Mikhailovskii E. I., Tarasov V. N. O 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 of Applied Mathematics and Mechanics, 1993, v. 57, No. 1, pp. 128–136.
  5. Yermolenko A. V. Utochnennye sootnosheniya teorii plastin, orientirovannye na reshenie kontaktnyh zadach (Refifined relations of the theory of plates, oriented to the solution of contact problems), Bulletin of Syktyvkar University. Ser. 1. Mathematics. Mechanics. Informatics, 2014, 19, pp. 25–32.

For citation:Yermolenko A. V., Melnikov V. A. Calculation of the contact interaction of a rectangular plate and a base by the Karman theory, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 86–92.

VIII. Uvarovskaya O. V., Mikhailov A. V. Use of modern pedagogical technologies in high school (on the example of linear and vector algebras)

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The processes taking place in the higher school now, predetermine the new requirements for the teaching of disciplines. To implement the competency approach in higher education, a transition from a one-way interaction process — monologue (in the broadcast mode), to an active process of two-way communication — is necessary for dialogue (fifirst in communication and then communication) to facilitate more effffective student learning. The application of interactive forms of teaching in teaching, which are realized through modern pedagogical technologies, allow to form the competences defifined in GEF. The article presents and substantiates theproject of the lesson on the topic «Complex numbers» using the integration of technologies for developing critical thinking and learning in cooperation.

Keywords: modern pedagogical technologies, complex numbers.

References

  1. Zagashev I. O., Zair-Bek S. I. Kriticheskoye myshleniye: tekhnologii razvitiya (Critical thinking: development technologies), St. Petersburg, 2003. 284 p.
  2. Pedagogika vysshey shkoly (Pedagogy of Higher School), Textbook. allowance, under the general ed. O. B. Uvarovskaya, Syktyvkar: Publishing House of Syktyvkar State University, 2013.
  3. Polat E. S. Novyye pedagogicheskiye i informatsionnyye tekhnologii v sisteme obrazovaniya (New pedagogical and information technologies in the education system), Moscow: Academy, 2000.
  4. Uvarovskaya O. V. Pedagogika professional’nogo obrazovaniya (Pedagogy of Vocational Education) [Electronic resource]: textbook: text of the manual Electronic book on CD-ROM. Feder. state. budget. a higher education institution is established. education Syktyv. Gos. University of. Pitirima Sorokina: Izd-vo SSU im. Pitirima Sorokina, 2017.
  5. Kurosh A. G. Kurs vysshey algebry, devyatoye izdaniye (Course of Higher Algebra. Ninth edition), Moscow: Nauka, 1968.
  6. Entsiklopediya dlya detey (Encyclopedia for Children). Vol. 11. Mathematics. Ed. M. D. Aksenova; method. and otv. Ed. V. A. Volodin. M.: Avanta+. 2003, 688 p.: Ill.
  7. Bergelson M. Yazykovyye aspekty virtual’noy kommunikatsii (Language aspects of virtual communication), Vestn. Moscow State University, 2002, S. 19, No. 1, 54 p.

For citation:Uvarovskaya O. V., Mikhailov A. V. Use of modern pedagogical technologies in high school (on the example of linear and vector algebras), Bulletin of Syktyvkar University. Series 1:Mathematics. Mechanics. Informatics, 2018, 2 (27), pp. 93–106.

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