I. Belyaeva N. A., Yakovleva A. F. Frontal wave of pressure flow
The model of a pressure flow of a structured liquid is analyzed. An inhomogeneous solution of the diffusion-kinetic equation is constructed in the region of nonmonotonicity of the discharge-pressure characteristic. This solution corresponds to a heteroclinic trajectory connecting two stable homogeneous states.
Keywords: pressure flow, homogeneous equilibrium states, heteroclinic trajectory, traveling wave.
References:
1. Belyaeva N. A., Sazhina A. N. Analizusrednennogonapornogotecheniya (Analysis of the averaged pressure flow), Twenty-third annual session of the Academic Council of Syktyvkar State University named after Pitirim Sorokin (February readings): a collection of materials / Otv.red. N. S. Sergiev, Syktyvkar: Publishing House of SSU named after Pitirim Sorokin, 2016, pp. 60–69.
2. Kolmogorov A. N, Petrovsky I. G, Piskunov N. S. Issledovanieuravneniya diffuzii, soedinennoj s vozrastaniemkolichestvaveshchestva, i ego primenenie k odnojbiologicheskoj problem (An investigation of the diffusion equation, coupled with the increase in the amount of matter, and its application to a single biological problem), Bul. Moscow State University. Section A, 1937, 633 p.
3. Kholodnik M., Klich A., Kubichek M., Marek M. Metodyanalizanelinejnyhdinamicheskihmodelej (Methods of analysis of nonlinear dynamic models), Moscow: Peace, 1991, 368 p.
4. Khudyaev S. I. Porogovyeyavleniya v nelinejnyhuravneniyah (Threshold phenomena in nonlinear equations), M.: Fizmatlit, 2003, 272 p.
For citation:Belyaeva N. A., Yakovleva A. F. Frontal wave of pressure flow, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 3–12.
II. Mikhailov A. V. The fluctuations of the ring supported with threads
Problems of fluctuations of the elastic rings supported with elastic threads; problems of the stability of elastic rings under the action of a pulsating load are considered.
Keywords: ring, fluctuation, stability, natural frequency, Euler-Ostrogradsky equation, monodromy matrix, Mathieu equation.
References:
1. Abramowitz M., Stegun I. A. Spravochnikpospecial’nymfunkciyam (Handbook of mathimatical functions with formulas, graphs and mathematical tables, National bureau of standards, applied mathematics series„ 1964, 1046 p.
2. Vol’mir A. S. Ustojchivost’ deformiruemyhsistem (Stability of deformable systems), Moscow: Nauka, 1967, 984 p.
3. Gelfand I. M., Fomin S. V. Variacionnoeischislenie (Calculus of Variations), Moscow: Gos. izd-vofiz.-matem. literatury, 1961, 228 p.
4. Lerman L. M. Linejnye differencial’nye uravneniyaisistemy (The linear differential equations and systems), Nizhny Novgorod: Nizhegorodsliyuniversitet, 2012, 89 p.
5. Mathews J., Walker R. L. Matematicheskiemetody v fizike (Mathematical methods of physics), New York – Amsterdam: W. A. Benjamin INC., 1964, 475 p.
6. PanovkoYa. G. Osnovyprikladnojteoriiuprugihkolebanij (Basics of applied theory of elastic vibrations), Moscow: Mashinostroenie, 1967, 318 p.
7. Tarasov V. N. Metodyoptimizacii v issledovaniikonstruktivnonelinejnyhzadachmekhanikiuprugihsistem (Optimization methods in a research of constructively nonlinear problems of mechanics of elastic systems), Syktyvkar: KNC UrO RAN, 2013, 238 p.
8. Ulam S. M. Nereshennyematematicheskiezadachi (A Collection of mathematical problems), New York: 1960, 150 p.
9. Faddeev L. D., Yakubovskii O. A. Lekciipokvantovojmekhanikedlyastudentov-matematikov (Lectures on Quantum Mechanics for Mathematics Students), Leningrad: Izd-voLeningradskogouniversiteta, 1980, 200 p.
For citation:Mikhailov A. V. The fluctuations of the ring supported with threads, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 13–28.
III. Pimenov R. R. The interpretation and generalizations of the Pappus’s theorems: involutions and perpendicularity
If we draw the arrows on the picture of projection we can see involutive transformation. Geometric picture now is a diagram of involutions and their compositions. It gives useful interpretation for theorems of projective geometry.We generalize these arrows to multidimensional spaces that connect geometry of spheres with projective space and non-euclidian geometries. We study perpendicularity also. We change the word incidence for the word perpendicularity in the Pappus’s theorem and get true and meaningful propositions.
Keywords: theory of numbers, Fermat’s little theorem, Dirichlet’s theorem.
References:
1. Bachmann F. Postroeniegeometriinaosnoveponyatiyasimmetrii (Aufbau der GeometrieausdemSpiegelungsbegriff), М.: Nauka, 1969, 380 p.
2. Pimenov R. R. ObobshcheniyateoremyDezarga: geometriyaperpendikulyarnogo (The generalization of the Desargues’s theorem and geometry of perpendicularity), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2016, №1 (21), pp. 28–43.
3. Pimenov R. R. ObobshcheniyateoremyDezarga: skrytyeprostranstva (The generalization of the Desargues’s theorem and hidden subspaces), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2016, №1 (21), pp. 44–57.
4. Pimenov R. R. Otobrazheniyasferyineevklidovygeometrii (Mapping the sphere and non-euclidian geometries), Mathematical Education, 1999, ser. 3, № 3, pp. 158–166.
5. Pimenov R. R. Ehsteticheskayageometriyailiteoriyasimmetrij (Aesthetic geometry or theory of symmetries). SPb: School league, 2014, 288 p.
6. Hartshorne R. Osnovyproektivnojgeometrii (Foundations of Projective Geometry). Lecture notes at Harvard University. W. A. Benjamin 1nc, New York, 1967.
7. Tabachnikov S. Skewers, Arnold Mathematical Journal, 2, 2016, pp. 171–193.
For citation:Pimenov R. R. 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.
IV. Makarov P. A. On the variational principles of the mechanics of conservative and non-conservative systems
On the basis of the Hamilton—Ostrogradsky principle, applied to the motion of conservative and non-conservative systems, homogeneous and inhomogeneous Euler—Lagrange equations are compiled. An example of a plane motion of a material point is considered. The influence of dissipative forces on the characteristics of motion is determined.
Keywords: Hamilton’s mechanical action, variational principles of motion, the Euler—Lagrange equation, straight and circuitous paths, energy dissipation.
References:
1. Veretennikov V. G., Sinitsin V. A. Metodperemennogodejstviya (Method of variable action), 2 ed, M.: FIZMATLIT, 2005, 272 p.
2. Veretennikov V. G., Sinitsin V. A. Teoreticheskayamekhanika (Theoretical mechanics (additions to the general sections)), M.: FIZMATLIT, 2006, 416 p.
3. Gantmacher F. R. Lekciipoanaliticheskojmekhanike (Lectures on analytical mechanics), 2 ed, M.: Science, 1966, 300 p.
4. Goldstein G. Klassicheskayamekhanika (Classical mechanics), M.: Science, 1975, 416 p.
5. Landau L. D., Lifshitz E. M. Teoreticheskayafizika (Theoretical physics: V.I, Mechanics), 5 ed, M.: FIZMATLIT, 2007, 224 p.
6. Sludsky F. A. Zametka o nachalenaimen’shegodejstviya (A note on the principle of least action), Varational principles of mechanics, M.: FIZMATGIZ, 1959, pp. 388–391.
For citation: Makarov P. A. On the variational principles of the mechanics of conservative and non-conservative systems, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 46–59.
V. Odyniec W. P. Zenon IvanovichBorewicz (1922–1995) (To the 95th anniversary)
The article is devoted to the biography of the famous algebraist professor ZenonIvanovichBorewicz (to pronounce Borevich), the dean of Mathematics and Mechanics faculty of the Leningrad state university in 1973–83 years, seen by Polish mathematicians, and also to Z.I. Borewicz contacts with Poland, with detailed comments of the author.
Keywords: Z.I. Borewicz, the Siege of Leningrad, homologous algebra, linear groups theory, society of «Polonia».
References:
1. Narkiewicz W., Wie¸s law W. ZenonBorewicz (1922–1995), Wiadomo´sciMatematyczne, 36, 2000, pp. 65–72.
2. Odyniec W. P. About mathematicians of Leningrad, Wiadomo´sciMatematyczne, 27, 1987, pp. 279–292.
3. Odyniec W. P. About mathematicians of Leningrad – (St. Petersburg) – and not only of them, Wiadomo´sciMatematyczne, 34, 1998, pp. 149–158.
4. Jakovlev A. V. ZenonIvanovichBorevich. Questions of the theory of representations of algebras and groups. 5, Zapiskinauchnyhseminarov POMI, t. 236, 1997, pp. 9–12.
For citation:Odyniec W. P. ZenonIvanovichBorevich (1922–1995) (To the 95th anniversary), Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 60–69.
VI. Lubyagina E. N., Timshina L. V. Experience in the organization of students’ educational and research activities in the study of second-order curves
In the article, we offer materials that can be used to organize students’ educational and research activities in studying second-order curves. We give examples of the use of the GeoGebra environment.
Keywords: research activity, second-order curves, GeoGebra.
References:
1. Akopyan A. V., Zaslavsky A. A. Geometricheskiesvojstvakrivyhvtorogoporyadka (Geometric properties of second-order curves), М ., 2007, 136 p.
2. Atanasyan L. S., Atanasyan V. A. Sbornikzadachpogeometrii (Collection of problems on geometry), Textbook for students of physical and mathematical sciences, I. M.: Enlightenment, 1973, 480 p.
3. Bezumova O. L., Ovchinnikova R. P., Troitskaya O. N., Troitsky A. G., Vorkunova L. V., Shabanova M. V., Shirokova T. S., Tomilova O. M. Obucheniegeometrii s ispol’zovaniemvozmozhnostejGeoGebra (Geometry training using GeoGebra capabilities), Arkhangelsk: Kira, 2011, 140 p.
4. Boltyanskii V. G. Ogibayushchaya (Envelope), Kvant, N. 3, 1987, pp. 2–7.
5. Vechtomov E. M. ,Lubyagina E. N. Geometricheskieosnovykomp’yuternojgrafiki (Geometric Foundations of Computer Graphics: A Training Manual), Kirov. Publishing house: «Rainbow-Press», 2015, 164 p.
6. Gurov A. E. Zamechatel’nyekrivyevokrugnas (Wonderful curves around us), M., 1989, 112 p.
7. Zabelina C. B. Formirovanieissledovatel’skojkompetentnostimagistrantovmatematicheskogoobrazovaniya (Formation of research competence of undergraduates of mathematical education (direction pedagogical education)). Dis. … cand. ped. sciences, M., 2015.
8. Kachalova L. P. Issledovatel’skayakompetenciyamagistrantov: strukturno-soderzhatel’nyjanaliz (Research competence of undergraduates: structurally-substantial analysis), Political journal of scientific publications «Discussion», 3 (55), 2015.
9. Ruinsky A. Inversnyepreobrazovaniyagiperboly (Inverted hyperbola transformations), Mathematical education, s. 3, 4 (2000), pp. 120–126.
10. Smirnov V. I. Kursvysshejmatematiki (Course of Higher Mathematics), t. 2, M.: «Science», 1974, 479 p.
11. Timshina L. V. Seminarskiezanyatiyapogeometrii v vuze (Seminars on Geometry in the University), Teaching Mathematics, Physics, Informatics in Universities and Schools: Content Problems, Technologies and Techniques: Proceedings of the V All-Russian Scientific Conference. Conf. Glazov: «Glazov printing house», 2015, p. 131–133.
12. Chebotareva E. V. Komp’yuternyjehksperiment s GeoGebra (Computer experiment with GeoGebra), Kazan: Kazan University, 2015, 61 p.
13. Shabanova M. V., Ovchinnikova R. P., Yastrebov A. V., Pavlova M. A., Tomilova A. E., Forkunova L. V., Udovenko L. N., Novoselova N. N., Fomina N. I., Artemieva M. V., Shirikova T. S., Bezumova O. L., Kotova S. N., Parsheva V. V., Patronova N. N., Belorukova M. V., Teplyakov V. V., Rogushina T. P., Tarkhov E. A., Troitskaya O. N., Chirkova L. N. Ehksperimental’nayamatematika v shkole. Issledovatel’skoeobuchenie (Experimental mathematics in the school. Research training. Monograph on research activities), M.: Publishing house Academy of Natural History, 2016, 300 p.
14. Shirikova T. S. Metodikaobucheniyauchashchihsyaosnovnojshkolydokazatel’stvuteorempriizucheniigeometrii s ispol’zovaniem Geo Gebra (Method of teaching students of the basic school the proof of theorems in the study of geometry using GeoGebra). Diss. … cand. ped. sciences. Arkhangelsk, 2014.
15. Jaglom I. M., Ashkinuz V. G. Ideiimetodyaffinnojiproektivnojgeometrii: CH. I (Ideas and methods of affine and projective geometry, I). M: 1962, 247 p.
For citation:Lubyagina E. N., Timshina L. V. Experience in the organization of students’ educational and research activities in the study of second-order curves, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 70–84.
VII. Yermolenko A. V., Osipov K. S. Parallel programming in contact problems with a free boundary
The method of generalized reaction requires a large number of iterations, on each of which a large number of calculations is carried out. To accelerate calculations, the article considers parallelizing a contact problem using the OpenMP technology in C ++.
Keywords: plate, method of generalized reaction, contact problem, parallel computing.
References:
1. Antonov A. S. Parallel’noeprogrammirovanie s ispol’zovaniemtexnologiiOpenMP (Parallel Programming Using OpenMP Technology), Moscow.: Publishing house of MSU, 2009, 77 p.
2. Yermolenko A. V., Gintner A. N. Vliyaniepoperechny’xsdvigovnaponizhenienapryazhennogosostoyaniyaplastiny (The effect of transverse shear on the lowering of the stressed state of the plate), Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2015, №1 (20), pp. 91–96.
3. Yermolenko A. V. TeoriyaploskixplastintipaKarmana–Timoshenko–Nagdiotnositel’noproizvol’nojbazovojploskosti (The Karman– Timoshenko–Naghdi theory of plane plates relative to arbitrary base surface), In the world of scientific discoveries, Krasnoyarsk: SIS, 2011, № 8.1 (20), pp. 336–347.
4. Mikhailovskii E. I., Yermolenko A. V., Mironov V. V., Tulubenskaya E. V. Utochnenny’enelinejny’euravneniya v neklassicheskixzadachaxmexanikiobolochek (Refined nonlinear equations in nonclassical problems of shell mechanics), Syktyvkar: Publishing house of the Syktyvkar university, 2009, 141 p.
5. Mikhailovskii E. I., Tarasov V. N. O sxodimostimetodaobobshhennojreakcii v kontaktny’xzadachax so svobodnojgranicej (On the convergence of the generalized reaction method in contact problems with a free boundary, Journal of Applied Mathematics and Mechanics, 1993, v. 57, №. 1, pp. 128–136.
For citation:Yermolenko A. V., Osipov K. S. Parallel programming in contact problems with a free boundary, Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 85–91.
VIII. Chuprakov D. V., Vedernikova A. V. About structure of finite cyclic semirings with idempotent commutative addition
The paper deals with finite idempotent cyclic semirings with commutative addition. Authors present a criterion for existence of finite idempotent cyclic semirings with commutative addition, associated with ideal of nonnegative integers. They derive estimates of the cardinality of FIC-semiring. The article offers algorithms for calculation of cardinality of FIC-semiring by basis of associated ideal of nonnegative integers.
Keywords:semiring, cyclic semiring, monogenoussemiring, idempotent, ideal, positive integer.
References:
1. Bestujev A.S. Konechnyeidempotentnyeciklicheskiepolukol’ca (Finite Idempotent Cyclic Semirings), MatekaticheskiyVestnikPedvuzov I UniversitetovVolgo-VyatskogoRegiona, 2011, n. 13, pp. 71–78.
2. Bestuzev A.S., Vechtomov E.M. Ciklicheskiepolukol’ca s kommutativnymslozheniem (Cyclic SemiringsWith Commutative Addition), Bulletin of Syktyvkar State University. Series 1: Mathematics. Mechanics. Informatics, vol. 1 (20), 2015, pp. 8–39.
3. Vedernikova A.V., Chuprakov D.V. O predstavleniikonechnyhidempotentnyhciklicheskihpolukoleckortezhamicelyh chisel (About Representation of Finite Idempotent Cyclic Semirings by Tuples of Integers), Mathematical Bulletin of Universities and Pedagogical Unyversities of Volgo-Vyatskiy Region, 2017, n. 19, pp. 70–76.
4. Vechtomov E.M. Vvedenie v polukol’ca (Introduction to Semirings), Kirov: VGPU, 2000, 44 p.
5. Vechtomov E.M., Lubyagina (Orlova) I. V. Ciklicheskiepolukol’ca s idempotentnymnekommutativnymslozheniem (Cyclic SemiringsWith Idempotent Noncommutative Addition), Fundamentalnaya I PrikladnayaMatematika, 2011/2012, vol. 17, n. 1, pp. 33–52.
6. Vechtomov E.M., Orlova I.V. Ciklicheskiepolukol’ca s neidempotentnymnekommutativnymslozheniem (Cyclic SemiringsWithNonidempotent Noncommutative Addition), Fundamentalnaya I PrikladnayaMatematika, 2015, vol. 20, n. 6, pp. 17–41.
7. Vechtomov E.M. Mul’tiplikativnociklicheskiepolukol’ca (Multiplicative Cyclic Semirings), Technologies of Productive Learning of Mathematics: Traditions And Innovations, Arzamas, 2016, pp. 130–140.
8. Vechtomov E.M., Orlova I.V. Idealyikongruehnciiciklicheskihpolukolec (Ideals and Congruences of Cyclic Semirings), Bulletin of Syktyvkar University, Series 1: Mathematics. Mechanics. Informatics, 2017, n. 1 (22), pp. 29–40.
9. Lubyagina I.V. O ciklicheskihpolukol’cah s nekommutativnymslozheniem (About Cyclic SemiringsWith Noncommutative Addition), Trudy MatematicheskogoChentraIm. N.I. lobachevskogo, Kazan, 2010, vol. 40, pp. 212–215.
10. Naudin P., Quitt`e C. Algebraicheskayaalgoritmika s uprazhneniyamiiresheniyami (AlgorithmiqueAlg`ebrique Avec ExercicesCorrig`es), M.: Mir, 1999, 720 p.
11. Chermnyh V.V., Nikolaeva O.V. Ob idealahpolukol’canatural’nyh chisel (Amout Ideals of Semiring of Posivive Integers), Mathematical Bulletin of Universities and Pedagogical Unyversities of Volgo-Vyatskiy Region, 2009, n. 11, pp. 118–121.
12. Bestugev A.S., Vechtomov E.M. Multiplicatively Cyclic Semirings, International Scientific Conference Named After Academician M. Kravchuk, Kiev: National Technical University of Ukraine, 2010, p. 39.
For citation:Chuprakov D. V., Vedernikova A. V. About structure of finite cyclic semirings with idempotent commutative addition, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2017, 2 (23), pp. 92–109.